Forcing theory for transverse trajectories of surface homeomorphisms
FFORCING THEORY FOR TRANSVERSE TRAJECTORIES OF SURFACEHOMEOMORPHISMS
P. LE CALVEZ AND F. A. TAL
Abstract.
This paper studies homeomorphisms of surfaces isotopic to the identity by means ofpurely topological methods and Brouwer theory. The main development is a novel theory of orbitforcing using maximal isotopies and transverse foliations. This allows us to derive new proofs forsome known results as well as some new applications, among which we note the following: we extendFranks and Handel’s classification of zero entropy maps of S for non-wandering homeomorphisms;we show that if f is a Hamiltonian homeomorphism of the annulus, then the rotation set of f iseither a singleton or it contains zero in the interior, proving a conjecture posed by Boyland; weshow that there exist compact convex sets of the plane that are not the rotation set of some torushomeomorphisms, proving a first case of the Franks-Misiurewicz Conjecture; we extend a boundeddeviation result relative to the rotation set to the general case of torus homeomorphisms. Introduction
Let us begin by recalling some facts about Sharkovski’s theorem, which can be seen as a typical exampleof an orbit forcing theory in dynamical systems. In this theorem, an explicit total order (cid:22) on the setof natural integers is given that satisfies the following: every continuous transformation f on [0 ,
1] thatcontains a periodic orbit of period m contains a periodic orbit of period n if n (cid:22) m . Much more canbe said. If f admits a periodic orbit of period different from a power of 2, one can construct a Markovpartition and codes orbits with the help of the associated finite subshift. In particular one can provethat the topological entropy of f is positive. There exists a forcing theory about periodic orbits forsurface homeomorphisms related to Nielsen-Thurston classification of surface homeomorphisms, withmany interesting dynamical applications (see for example [Bo] or [Mo] for survey articles). In caseof homeomorphisms isotopic to the identity, this theory deals with the braid types associated to theperiodic orbits. A more subtle theory (homotopic Brouwer theory) was introduced by M. Handel forsurface homeomorphisms and developed by J. Franks and Handel to become a very efficient tool intwo-dimensional dynamics.The goal of the article is to give a new orbit forcing theory for surface homeomorphisms that areisotopic to the identity, theory that will be expressed in terms of maximal isotopy , transverse foliations and transverse trajectories . Note first that the class of surface homeomorphisms isotopic to the identitycontains the time one maps of time dependent vector fields. Consequently, what is proved in this articlecan be applied to the dynamical study of a time dependent vector field on a surface, periodic in time.In what follows, a surface M is orientable and furnished with an orientation. If f is a homeomorphismof M isotopic to the identity, the choice of an isotopy I = ( f t ) t ∈ [0 , from the identity to M shouldnot be very important, as we are looking at the iterates of f . What looks like the trajectory of a point z , that means the path I ( z ) : t (cid:55)→ f t ( z ) seems useless. It appears that this is not the case: there are F. A. Tal was partially supported by CAPES, FAPESP and CNPq-Brasil. a r X i v : . [ m a t h . D S ] N ov P. LE CALVEZ AND F. A. TAL isotopies that are better than the other ones. This is clear if f is the time one map of a completetime independent vector field ξ . The isotopy ( f t ) [0 , defined by the restriction of the flow ( f t ) t ∈ R isclearly better than any other choice of an isotopy, in the sense that it will be useful while studying thedynamics of f . It is easy to see that in this last case, there is no fixed point of f in the complement ofthe singular set of the vector field whose trajectory is contractible relative to this same singular set. Inthis situation, the singular points correspond to the fixed points of I , which means the points whosetrajectory is constant.In general, let us say that an isotopy I = ( f t ) t ∈ [0 , , that joins the identity to a homeomorphism f , isa maximal isotopy if there is no fixed point of f whose trajectory is contractible relative to the fixedpoint set of I . A very recent result of F. B´eguin, S. Crovisier and F. Le Roux [BCL] asserts thatsuch an isotopy always exists if f is isotopic to the identity (a slightly weaker result was previouslyproved by O. Jaulent [J]). A fundamental result [Lec1] asserts that a maximal isotopy always admits atransverse foliation. This is a singular oriented foliation F whose singular set coincides with the fixedpoint set of I and such that every non trivial trajectory is homotopic (relative to the endpoints) to apath that is transverse to the foliation (which means that it locally crosses every leaf from the right tothe left). This path I F ( z ), the transverse trajectory, is uniquely defined up to a natural equivalencerelation (meaning that the induced path in the space of leaves is unique). In the case where f is thetime one map of a complete time independent vector field ξ , it is very easy to construct a transversefoliation by taking the integral curves of any vector field η that is transverse to ξ , and in that case thetrajectories I ( z ) are transverse. In a certain sense, maximal isotopies are isotopies that are as close aspossible to isotopies induced by flows.Maximal isotopies and transverse foliations are known to be efficient tools for the dynamical study ofsurface homeomorphisms (see [D1], [D2], [KT2], [Lec1], [Lec2][Ler], [Mm], [T] for example). Usuallythey are used in the following way. Properties of f are transposed “by duality” to properties of F ,then one studies the dynamics of the foliation and comes back to f . Roughly speaking, the leavesof the foliation are pushed along the dynamics. This property is cleverly used in the articles of P.D´avalos ([D1], [D2]). Our original goal was a boundedness displacement result (Theorem H of thisintroduction) which needed a formalization of the ideas of D´avalos. This was nothing but a forcingtheory for transverse trajectories. For every integer n (cid:62)
1, let us define by concatenation the paths I n ( z ) = (cid:81) (cid:54) k Let f be a homeomorphism of A = T × [0 , that is isotopic to the identity and ˇ f a liftto R × [0 , . Suppose that rot( ˇ f ) is a non trivial segment and that ρ is an endpoint of rot( ˇ f ) that isrational. Define M ρ = { µ ∈ M ( f ) , rot( µ ) = ρ } , X ρ = (cid:91) µ ∈M ρ supp( µ ) . Then every invariant measure supported on X ρ belongs to M ρ . We deduce immediately the following positive answer to a question of P. Boyland: Corollary B. Let f be a homeomorphism of A that is isotopic to the identity and preserves a probabilitymeasure µ with full support. Let us fix a lift ˇ f . Suppose that rot( ˇ f ) is a non trivial segment. Therotation number rot( µ ) cannot be an endpoint of rot( ˇ f ) if this endpoint is rational. Let us explain what happens for torus homeomorphisms. Here again M ( f ) is the set of invariantBorel probability measures µ of f , the set supp( µ ) the support of µ and the rotation vector rot µ )the integral (cid:82) T ϕ dµ , where ϕ : T → R is the map lifted by ˇ f − Id. The set of rotation vectors ofinvariant measures rot( ˇ f ) is a compact and convex subset of R . Nothing is known about the planesubsets that can be written as such a rotation set. The following result gives the first obstruction: Theorem C. Let f be a homeomorphism of T that is isotopic to the identity and ˇ f a lift of f to R .The frontier of rot( ˇ f ) does not contain a segment with irrational slope that contains a rational pointin its interior. It was previously conjectured by Franks and Misiurewicz in [FM] that a line segment L could not berealized as a rotation set of a torus homeomorphism in the following conditions: (i) L has irrationalslope and a rational point in its interior, (ii) L has rational slope but no rational points and (iii) L has P. LE CALVEZ AND F. A. TAL irrational slope and no rational points. While Theorem C implies the conjecture for case (i), A. ´Avilahas given a counter-example for case (iii).The second result is a boundedness result: Theorem D. Let f be a homeomorphism of T that is isotopic to the identity and ˇ f a lift of f to R .If rot( ˇ f ) has a non empty interior, then there exist a constant L such that for every z ∈ R and every n (cid:62) , one has d ( ˇ f n ( z ) − z, n rot( ˇ f )) (cid:54) L . Note that by definition of the rotation set one knows thatlim n → + ∞ n (cid:18) max z ∈ R d ( ˇ f n ( z ) − z, n rot( ˇ f )) (cid:19) = 0Theorem D clarifies the speed of convergence. It was already known for homeomorphisms in thespecial case of a polygon with rational vertices (see D´avalos [D2]) and for C (cid:15) diffeomorphisms (seeAddas-Zanata [AZ]). As already noted in [AZ], we can deduce an interesting result about maximizingmeasures , which means measure µ ∈ M ( f ) whose rotation vector belongs to the frontier of rot( ˇ f ).The rotation number of such a measure belongs to at least one supporting line of rot( ˇ f ). Such a lineadmits the equation ψ ( z ) = α ( ψ ) where ψ is a non trivial linear form on R and α ( ψ ) = max µ ∈M ( f ) ψ (rot( µ )) = max µ ∈M ( f ) (cid:90) T ψ ◦ ϕ dµ. Set M ψ = { µ ∈ M ( f ) , ψ (rot( µ )) = α ( ψ ) } , X ψ = (cid:91) µ ∈M ψ supp( µ ) . The following result, that can be easily deduced from Theorem 63 and Atkinson’s Lemma in ErgodicTheory (see [A]), tells us that the sets X ψ behave like the Mather sets of the Tonelli Lagrangiansystems. Proposition E. Let f be a homeomorphism of T that is isotopic to the identity and ˇ f a lift of f to R .Assume that rot( ˇ f ) has a non empty interior. Then, every measure µ supported on X ψ belongs to M ψ .Moreover, if z lifts a point of X ψ , then for every n (cid:62) , one has | ψ ( ˇ f n ( z )) − ψ ( z ) − nβ ( ψ ) | (cid:54) L (cid:107) ψ (cid:107) ,where L is the constant given by Theorem D. It admits as an immediate corollary the torus version of Boyland’s question: Corollary F. Let f be a homeomorphism of T that is isotopic to the identity, preserving a measure µ of full support, and ˇ f a lift of f to R . Assume that rot( ˇ f ) has a non empty interior. Then rot( µ ) belongs to the interior of rot( ˇ f ) . This result was known for C (cid:15) diffeomorphisms (see [AZ]).The next resut is due to Llibre and MacKay, see [LlM]. Its original proof uses Thurston-Nielsen theoryof surface homeomorphisms, more precisely the authors prove that there exists a finite invariant set X such that f | T \ X is isotopic to a pseudo-Anosov map. We will give here an alternative proof byexhibiting ( n, ε ) separated sets constructed with the help of transverse trajectories. Theorem G. Let f be a homeomorphism of T that is isotopic to the identity and ˇ f a lift of f to R .If rot( ˇ f ) has a non empty interior, then the topological entropy of f is positive. ORCING THEORY FOR TRANSVERSE TRAJECTORIES OF SURFACE HOMEOMORPHISMS 5 Our original goal, while writing this article, was to prove the following boundedness displacementresult: Theorem H. We suppose that M is a compact orientable surface furnished with a Riemannian struc-ture. We endow the universal covering space ˇ M with the lifted structure and denote by d the induceddistance. Let f be a homeomorphism of M isotopic to the identity and ˇ f a lift to ˇ M naturally definedby the isotopy. Assume that there exists an open topological disk U ⊂ M such that the fixed points setof ˇ f projects into U . Then;- either there exists K > such that d ( ˇ f n (ˇ z ) , ˇ z ) (cid:54) K , for all n (cid:62) and all bi-recurrent point ˇ z of ˇ f ;- or there exists a nontrivial covering automorphism T and q > such that, for all r/s ∈ ( − /q, /q ) ,the map ˇ f q ◦ T − p has a fixed point. In particular, f has non-contractible periodic points of arbitrarilylarge prime period. Theorem H has an interesting consequence for torus homeomorphisms. Say a homeomorphism f of T is Hamiltonian if it preserves a measure µ with full support and it has a lift ˇ f (called the Hamiltonianlift of f ) such that the rotation vector of µ is null. Corollary I. Let f be a Hamiltonian homeomorphism of T such that all its periodic points arecontractible, and such that it fixed point set is contained in a topological disk. Then there exists K > such that if ˇ f is the Hamiltonian lift of f , then for every z and every n (cid:62) , one has (cid:107) ˇ f n ( z ) − z (cid:107) (cid:54) K . The study of non-contractible periodic orbits for Hamiltonian maps of sympletic manifolds has beenreceiving increased attention (see for instance [GG]). A natural question in the area, posed by V.Ginzburg, is to determine if the existence of non-contractible periodic points is generic for smoothHamiltonians. A consequence of Corollary I is an affirmative answer for the case of the torus: Proposition J. Let Ham ∞ ( T ) be the set of Hamiltonian C ∞ diffeomorphisms of T endowed withthe Whitney C ∞ - topology. There exists a residual subset A of Ham ∞ ( T ) such that every f in A hasnon-contractible periodic points. Let us explain now the results related to the entropy. For example we can give a short proof of thefollowing improvement of a result due to Handel [H1]. Theorem K. Let f : S → S be an orientation preserving homeomorphism such that the complementof the fixed point set is not an annulus. If f is topologically transitive then the number of periodic pointsof period n for some iterate of f grows exponentially in n . Moreover, the entropy of f is positive. Another entropy result we obtain is related to the existence and continuous variation of rotationnumbers for homeomorphisms of the open annulus. A stronger version of this result for diffeomorphismswas already proved in an unpublished paper of Handel [H2]. Given a homeomorphism of T × R anda lift ˇ f to R , we say that a point z ∈ T × R has a rotation number rot( z ) if the ω -limit of its orbit isnot empty, and if for any compact set K ⊂ T × R and every increasing sequence of integers n k suchthat f n k ( z ) ∈ K and any ˇ z ∈ π − ( z ),lim k →∞ n k (cid:0) π ( ˇ f n k (ˇ z ) − π (ˇ z ) (cid:1) = rot( z ) , where π is the covering projection from R to T × R and π : R → R is the projection on the firstcoordinate. P. LE CALVEZ AND F. A. TAL Theorem L. Let f be a homeomorphism of the open annulus T × R isotopic to the identity, ˇ f alift of f to the universal covering and f sphere be the natural extension of f to the sphere obtained bycompactifying each end with a point. If the topological entropy of f sphere is zero, then each bi-recurrentpoint (meaning forward and backward recurrent) has a rotation number, and the function z (cid:55)→ rot( z ) is continuous on the set of bi-recurrent points. Let us finish with a last application. J. Franks and M. Handel recently gave a classification result forarea preserving diffeomorphisms of S with entropy 0 (see [FH]). Their proofs are purely topologicalbut the C assumption is needed to use a Thurston-Nielsen type classification result relative to the fixedpoint set (existence of a normal form) and the C ∞ assumption to use Yomdin results on arcs whoselength growth exponentially by iterates. We will give a new proof of the fundamental decompositionresult (Theorem 1.2 of [FH]) which is the main building block in their structure theorem. In fact wewill extend their result to the case of homeomorphisms and replace the area preserving assumption bythe fact that every point is non wandering. Theorem M. Let f : S → S be an orientation preserving homeomorphism such that Ω( f ) = S and h ( f ) = 0 . Then there exists a family of pairwise disjoint invariant open sets ( A α ) α ∈A whose union isdense such that: i) each A α is an open annulus; ii) the sets A α are the maximal fixed point free invariant open annuli; iii) the α -limit set of a point z (cid:54)∈ (cid:83) α ∈A A α is included in a single connected component of the fixedpoint set fix( f ) of f , and the same holds for the ω -limit set of z ; iv) let C be a connected component of the frontier of A α in S \ fix( f ) , then the connected componentsof fix( f ) that contain α ( z ) and ω ( z ) are independent of z ∈ C . Let us explain now the plan of the article. In the second section we will introduce the definitions ofmany mathematical objects, including precise definitions of rotation vectors and rotation sets. Thethird section will be devoted to the study of transverse paths to a surface foliation. We will introducethe notion of a pair of equivalent paths, of a recurrent transverse path and of F -transverse intersectionbetween two transverse paths. An important result, which will be very useful in the proofs of TheoremsK and M is Proposition 2 which asserts that a transverse recurrent path to a singular foliation on S that has no F -transverse self-intersection is equivalent to the natural lift of a transverse simple loop (i.e.an adapted version of Poincar´e-Bendixson theorem). We will recall the definition of maximal isotopies,transverse foliations and transverse trajectories in Section 4. We will state the fundamental result about F -transverse intersections of transverse trajectories (Proposition 20) and its immediate consequences.An important notion that will be introduced is the notion of linearly admissible transverse loop. Toany periodic orbit is naturally associated such a loop. A realization result (Proposition 26) will giveus sufficient conditions for a linearly admissible transverse loop to be associated to a periodic orbit.Section 5 will be devoted to the proofs of Theorem 29 (about exponential growth of periodic orbits)and Theorem 36 (about positiveness of the entropy). We will give the proofs of Theorem H, A and Kin Section 6 while Section 7 will be almost entirely devoted to the proof of Theorem M (we will proveTheorem L at the end of it).We will begin by stating a “local version” relative to a given maximal ORCING THEORY FOR TRANSVERSE TRAJECTORIES OF SURFACE HOMEOMORPHISMS 7 isotopy (Theorem 53). We will study torus homeomorphisms in Section 8 and will give there the proofsof Theorems C, D and G.We would like to thank Fr´ed´eric Le Roux for informing us of some important gaps in the original proofsof Theorem 29, and Theorem 36. We would also like to thank Andr´es Koropecki for his useful commentsand for discussions regarding Proposition J, and to Victor Ginzburg for presenting us the question onthe genericity of non-contractible periodic points for Hamiltonian diffeomorphisms. Finally, we wouldlike to thank the anonymous referee for the careful work and suggestions which greatly improved ourtext. 2. Notations We will endow R with its usual scalar product (cid:104) (cid:105) and its usual orientation. We will write (cid:107) (cid:107) for theassociated norm. For every point z ∈ R and every set X ⊂ R we write d ( z, X ) = inf z (cid:48) ∈ X (cid:107) z − z (cid:48) (cid:107) . Wedenote by π : ( x, y ) (cid:55)→ x and π : ( x, y ) (cid:55)→ y the two projections. If z = ( x, y ), we write z ⊥ = ( − y, x ).The r -dimensional torus R r / Z r will be denoted T r , the 2-dimensional sphere will be denoted S . Asubset X of a surface M is called an open disk if it is homeomorphic to D = { z ∈ R , (cid:107) z (cid:107) < } and aclosed disk if it is homeomorphic to D = { z ∈ R , (cid:107) z (cid:107) (cid:54) } . It is called an annulus if it homeomorphicto T × J , where J is a non trivial interval of R . In case where J = [0 , J = (0 , J = [0 , X is a closed annulus , an open annulus , a semi-closed annulus respectively.Given a homeomorphism f of a surface M and a point z ∈ M we define the α -limit set of z by (cid:84) n (cid:62) (cid:83) k (cid:62) n f − k ( z ) and we denote it α ( z ). We also define the ω -limit set of z by (cid:84) n (cid:62) (cid:83) k (cid:62) n f k ( z ) andwe denote it ω ( z ).2.1. Paths, lines, loops. A path on a surface M is a continuous map γ : J → M defined on aninterval J ⊂ R . In absence of ambiguity its image will also be called a path and denoted by γ . We willdenote γ − : − J → M the path defined by γ − ( t ) = γ ( − t ). If X and Y are two disjoint subsets of M ,we will say that a path γ : [ a, b ] → M joins X to Y if γ ( a ) ∈ X and γ ( b ) ∈ Y . A path γ : J → M is proper if J is open and the preimage of every compact subset of M is compact. A line is an injectiveand proper path λ : J → M , it inherits a natural orientation induced by the usual orientation of R .If M = R , the complement of λ has two connected components, R ( λ ) which is on the right of λ and L ( λ ) which is on its left. More generally, if M is a non connected surface with connected componentshomeomorphic to R , and if M (cid:48) is the connected component of M containing λ , the two connectedcomponents of M (cid:48) \ λ will similarly be denoted R ( λ ) and L ( λ ).Let us suppose that λ and λ are two disjoint lines of R . We will say that they are comparable iftheir right components are comparable for the inclusion. Note that λ and λ are not comparable ifand only if λ and ( λ ) − are comparable.Let us consider three lines λ , λ , λ in R . We will say that λ is above λ relative to λ (and λ is below λ relative to λ ) if:- the three lines are pairwise disjoint;- none of the lines separates the two others; P. LE CALVEZ AND F. A. TAL - if γ , γ are two disjoints paths that join z = λ ( t ), z = λ ( t ) to z (cid:48) ∈ λ , z (cid:48) = λ respectively,and that do not meet the three lines but at the ends, then t > t . λ λ λ z z z z Figure 1. Order of lines relative to λ . This notion does not depend on the orientation of λ and λ but depends of the orientation of λ (seeFigure 1) . If λ is fixed, note that we get in that way an anti-symmetric and transitive relation onevery set of pairwise disjoint lines that are disjoint from λ .A proper path γ of R induces a dual function δ on its complement, defined up to an additive constantas follows: for every z and z (cid:48) in R \ γ , the difference δ ( z (cid:48) ) − δ ( z ) is the algebraic intersection number γ ∧ γ (cid:48) where γ (cid:48) is any path from z to z (cid:48) . If γ is a line, there is a unique dual function δ γ that is equalto 0 on R ( γ ) and to 1 on L ( γ ).Consider a unit vector ρ ∈ R , (cid:107) ρ (cid:107) = 1. Say that a proper path γ : R → R is directed by ρ iflim t →±∞ (cid:107) γ ( t ) (cid:107) = + ∞ , lim t → + ∞ γ ( t ) / (cid:107) γ ( t ) (cid:107) = ρ, lim t →−∞ γ ( t ) / (cid:107) γ ( t ) (cid:107) = − ρ. Observe that if γ is directed by ρ , then γ − is directed by − ρ and that for every z ∈ R , the translatedpath γ + z : t (cid:55)→ γ ( t ) + z is directed by ρ . Among the connected components of R \ γ , two of them R ( γ ) and L ( γ ) are uniquely determined by the following: for every z ∈ R , one has z − sρ ⊥ ∈ R ( γ )and z + sρ ⊥ ∈ L ( γ ) if s is large enough. In the case where γ is a line, the definitions agree with theformer ones. Note that two disjoint lines directed by ρ are comparable.Instead of looking at paths defined on a real interval we can look at paths defined on an abstractinterval J , which means a one dimensional oriented manifold homeomorphic to a real interval. If γ : J → M and γ (cid:48) : J (cid:48) → M are two paths, if J has a right end b and J (cid:48) a left end a (cid:48) (in the naturalsense), and if γ ( b ) = γ (cid:48) ( a (cid:48) ), we can concatenate the two paths and define the path γγ (cid:48) defined on theinterval J (cid:48)(cid:48) = J (cid:116) J (cid:48) /b ∼ a (cid:48) coinciding with γ on J and γ (cid:48) on J (cid:48) . One can define in a same way theconcatenation (cid:81) l ∈ L γ l of paths indexed by a finite or infinite interval of Z .A path γ : R → M such that γ ( t + 1) = γ ( t ) for every t ∈ R lifts a continuous map Γ : T → M .We will say that Γ is a loop and γ its natural lift . If n (cid:62) 1, we denote Γ n the loop lifted by the path t (cid:55)→ γ ( nt ). Here again, if M is oriented and Γ homologous to zero, one can define a dual function δ In all figures in the text, we will represent the plane R as the open disk. The reason being that in many cases weare dealing with the universal covering space of an a hyperbolic surface. ORCING THEORY FOR TRANSVERSE TRAJECTORIES OF SURFACE HOMEOMORPHISMS 9 defined up to an additive constant on M \ Γ as follows: for every z and z (cid:48) in R \ Γ, the difference δ ( z (cid:48) ) − δ ( z ) is the algebraic intersection number Γ ∧ γ (cid:48) where γ (cid:48) is any path from z to z (cid:48) .2.2. Rotations vectors. Let us recall the notion of rotation vector and rotation set for a homeomor-phism of a closed manifold, introduced by Schwartzman [Sc] (see also Pollicott [Po]). Let M be anoriented closed connected manifold and I an identity isotopy on M , which means an isotopy ( f t ) t ∈ [0 , such that f is the identity. The trajectory of a point z ∈ M is the path I ( z ) : z (cid:55)→ f t ( z ). If ω is a closed 1-form on M , one can define the integral (cid:82) I ( z ) ω on every trajectory I ( z ). Write f = f and denote M ( f ) the set of invariant Borel probability measures. For every µ ∈ M ( f ), the integral (cid:82) M (cid:16)(cid:82) I ( z ) ω (cid:17) dµ ( z ) vanishes when ω is exact. One deduces that ω → (cid:82) M (cid:16)(cid:82) I ( z ) ω (cid:17) dµ ( z ) defines anatural linear form on the first cohomology group H ( M, R ), and by duality an element of the firsthomology group H ( M, R ), which is called the rotation vector of µ and denoted rot( µ ). The set M ( f ),endowed with the weak ∗ topology, being convex and compact and the map µ (cid:55)→ rot( µ ) being affine,one deduces that the set rot( I ) = { rot( µ ) , µ ∈ M ( f ) } is a convex compact subset of H ( M, R ). If M is a surface of genus greater than 1 and I (cid:48) is a different identity isotopy given by ( f (cid:48) t ) t ∈ [0 , suchthat f (cid:48) = f , then for all z ∈ M the trajectories I ( z ) and I (cid:48) ( z ) are homotopic with fixed endpoints.Therefore the rotation vectors (and the rotation set) are independent of the isotopy, depending onlyon f . If M is a torus, it depends on a given lift of f . Let us clarify this case (see Misiurewicz-Zieman[MZ]). Let f be a homeomorphism of T that is isotopic to the identity and (cid:101) f a lift of f to the universalcovering space R . The map (cid:101) f − Id is invariant by the integer translations z (cid:55)→ z + p , p ∈ Z , and liftsa continuous map ϕ : T → R . The rotation vector of a Borel probability measure invariant by f isthe integral (cid:82) T ϕ dµ . If µ is ergodic, then for µ -almost every point z , the Birkhoff means converge torot( µ ). If (cid:101) z ∈ R is a lift of z , one haslim n → + ∞ (cid:101) f n ( (cid:101) z ) − (cid:101) zn = lim n → + ∞ n n − (cid:88) k =0 ϕ ( f k ( z )) = rot( µ ) . We will say that z (or (cid:101) z ) has a rotation vector rot( µ ). The rotation set rot( (cid:101) f ) is a non empty compactconvex subset of R . It is easy to prove that every extremal point of rot( (cid:101) f ) is the rotation vectorof an ergodic measure. Indeed the set of Borel probability measures of rotation vector ρ ∈ rot( f ) isconvex and compact, moreover its extremal points are extremal in M ( f ) if ρ is extremal in rot( f ).Observe also that for every p ∈ Z and every q ∈ Z , the map (cid:101) f q + p is a lift of f q and one hasrot( (cid:101) f q + p ) = q rot( (cid:101) f ) + p .We will also be concerned with annulus homeomorphisms. Let f be a homeomorphism of A = T × [0 , (cid:101) f a lift of f to the universal covering space R × [0 , π ◦ f − π is invariant by the translation T : z (cid:55)→ z + (1 , 0) and lifts a continuous map ϕ : A → R .The rotation number rot( µ ) of a Borel probability measure invariant by f is the integral (cid:82) A ϕ dµ . If µ is ergodic, then for µ -almost every point z , the Birkhoff means converge to rot( µ ). If (cid:101) z ∈ R × [0 , 1] isa lift of z , one has lim n → + ∞ π ◦ (cid:101) f n ( (cid:101) z ) − π ( (cid:101) z ) n = lim n → + ∞ n n − (cid:88) k =0 ϕ ( f k ( z )) = rot( µ ) . Here again we will say that (cid:101) z (or z ) has a rotation number rot( µ ). The rotation set rot( (cid:101) f ) is anon empty compact real segment and every endpoint of rot( (cid:101) f ) is the rotation number of an ergodic measure. Here again, for every p ∈ Z and every q ∈ Z , the map (cid:101) f q ◦ T p is a lift of f q and one hasrot( (cid:101) f q ◦ T p ) = q rot( (cid:101) f ) + p .Note that if J is a real interval, one can also define the rotation number of an invariant probabilitymeasure of a homeomorphism of T × J isotopic to the identity, for a given lift to R × J , provided thesupport of the measure is compact.3. Transverse paths to surface foliations General definitions. Let us begin by introducing some notations that will be used throughoutthe whole text. A singular oriented foliation on an oriented surface M is an oriented topologicalfoliation F defined on an open set of M . We will call this set the domain of F and denote it dom( F ),its complement will be called the singular set (or set of singularities) and denoted sing( F ). If thesingular set is empty, we will say that F is non singular . A subset of M is saturated if it is the unionof singular points and leaves. A trivialization neighborhood is an open set W ⊂ dom( F ) endowed witha homeomorphism h : W → (0 , that sends the restricted foliation F| W onto the vertical foliation.If ˇ M is a covering space of M and ˇ π : ˇ M → M the covering projection, F can be naturally lifted toa singular foliation ˇ F of ˇ M such that dom( ˇ F ) = ˇ π − (dom( F )). If ˇ N is a covering space of dom( F ),then the restriction of F to dom( F ) can also be naturally lifted to a non singular foliation of ˇ N . Wewill denote (cid:103) dom( F )the universal covering space of dom( F ) and (cid:101) F the foliation lifted from F | dom( F ) .For every z ∈ dom( F ) we will write φ z for the leaf that contains z , φ + z for the positive half-leaf and φ − z for the negative one. One can define the α -limit and ω -limit sets of φ as follows: α ( φ ) = (cid:92) z ∈ φ φ − z , ω ( φ ) = (cid:92) z ∈ φ φ + z . Suppose that a point z ∈ φ has a trivialization neighborhood W such that each leaf of F contains nomore than one leaf of F| W . In that case every point of φ satisfies the same property. If furthermoreno closed leaf of F meets W , we will say that φ is wandering . Recall the following facts, in the casewhere M = R and F is non singular (see Haefliger-Reeb [HR]):- every leaf of F is a wandering line;- the space of leaves Σ, furnished with the quotient topology, inherits a structure of connected andsimply connected one-dimensional manifold;- Σ is Hausdorff if and only if F is trivial (which means that it is the image of the vertical foliationby a plane homeomorphism) or equivalently if all the leaves are comparable.A path γ : J → M is positively transverse to F if its image does not meet the singular set and if, forevery t ∈ J , there exists a (continuous) chart h : W → (0 , at γ ( t ) compatible with the orientationand sending the restricted foliation F W onto the vertical foliation oriented downward such that themap π ◦ h ◦ γ is increasing in a neighborhood of t . Let ˇ M be a covering space of M and ˇ π : ˇ M → M the covering projection. If γ : J → dom( F ) is positively transverse to F , every lift ˇ γ : J → ˇ M istransverse to the lifted foliation ˇ F . Moreover, every lift (cid:101) γ : J → (cid:103) dom( F ) to the universal coveringspace (cid:103) dom( F ) is transverse to the lifted non singular foliation (cid:101) F . in the whole text “transverse” will mean “positively transverse” ORCING THEORY FOR TRANSVERSE TRAJECTORIES OF SURFACE HOMEOMORPHISMS 11 Suppose first that M = R and that F is non singular. Say that two transverse paths γ : J → R and γ (cid:48) : J (cid:48) → R are equivalent for F or F -equivalent if they satisfy the three following equivalentconditions:- there exists an increasing homeomorphism h : J → J (cid:48) such that φ γ (cid:48) ( h ( t )) = φ γ ( t ) , for every t ∈ J ;- the paths γ and γ (cid:48) meet the same leaves;- the paths γ and γ (cid:48) project onto the same path of Σ.Moreover, if J = [ a, b ] and J (cid:48) = [ a (cid:48) , b (cid:48) ] are two segments, these conditions are equivalent to this lastone:- one has φ γ ( a ) = φ γ (cid:48) ( a (cid:48) ) and φ γ ( b ) = φ γ (cid:48) ( b (cid:48) ) .In that case, note that the leaves met by γ are the leaves φ such that R ( φ γ ( a ) ) ⊂ R ( φ ) ⊂ R ( φ γ ( b ) ) . Ifthe context is clear, we just say that the paths are equivalent and omit the dependence on F .If γ : J → R is a transverse path, then for every a < b in J , the set L ( φ γ ( a ) ) ∩ R ( φ γ ( b ) ) is a topologicalplane and γ | ( a,b ) a line of this plane. Let us say that γ has a leaf on its right if there exists a < b in J and a leaf φ in L ( φ γ ( a ) ) ∩ R ( φ γ ( b ) ) that lies in the right of γ | ( a,b ) . Similarly, one can define the notionof having a leaf on its left . φ φ γ ( a ) φ γ ( b ) γ φ L ( φ γ ( a ) ) ∩ R ( φ γ ( b ) ) Figure 2. γ : [ a, b ] → R has both a leaf on its right ( φ ) and a leaf on its left ( φ ). γ | ( a,b ) is also a line in L ( φ γ ( a ) ) ∩ R ( φ γ ( b ) ). All previous definitions can be naturally extended in case every connected component of M is a planeand F is not singular. Let us return to the general case. Two transverse paths γ : J → dom( F ) and γ (cid:48) : J (cid:48) → dom( F ) are equivalent for F or F -equivalent if they can be lifted to the universal coveringspace (cid:103) dom( F ) of dom( F ) as paths that are equivalent for the lifted foliation (cid:101) F . This implies that thereexists an increasing homeomorphism h : J → J (cid:48) such that, for every t ∈ J , one has φ γ (cid:48) ( h ( t )) = φ γ ( t ) .Nevertheless these two conditions are not equivalent. In Figure 3, such a homeomorphism can beconstructed but the two loops are not equivalent. Nonetheless, one can show that γ and γ (cid:48) areequivalent for F if, and only if, there exists a holonomic homotopy between γ and γ (cid:48) , that is, ifthere exists a continuous transformation H : J × [0 , → dom( F ) and an increasing homeomorphism h : J → J (cid:48) satisfying:- H ( t, 0) = γ ( t ) , H ( t, 1) = γ (cid:48) ( h ( t ));- for all t ∈ J and s , s ∈ [0 , φ H ( t,s ) = φ H ( t,s ) . γ γ p p p Figure 3. The paths γ and γ are not equivalent for F , even though they cross the same leafs. By definition, a transverse path has a leaf on its right if it can be lifted to (cid:103) dom( F ) as a path with aleaf of (cid:101) F on its right (in that case every lift has a leaf on its right) and has a leaf on its left if it canbe lifted as a path with a leaf on its left. Note that if γ and γ (cid:48) have no leaf on their right and γγ (cid:48) iswell defined, then γγ (cid:48) has no leaf on its right. Note also that if γ and γ (cid:48) are F -equivalent, and if γ hasa leaf on its right, then γ (cid:48) has a leaf on its right. We say that an F -equivalence class has a leaf on itsright (on its left) if some representative of the class has a leaf on its right (on its left).Similarly, a loop Γ : T → dom( F ) is called positively transverse to F if it is the case for its naturallift γ : R → dom( F ). It has a leaf on its right or its left if it is the case for γ . Two transverse loopsΓ : T → dom( F ) and Γ (cid:48) : T → dom( F ) are equivalent if there exists two lifts (cid:101) γ : R → (cid:103) dom( F ) and (cid:101) γ (cid:48) : R → (cid:103) dom( F ) of Γ and Γ (cid:48) respectively, a covering automorphism T and an orientation preservinghomeomorphism h : R → R , such that, for every t ∈ R , one has (cid:101) γ ( t + 1) = T ( (cid:101) γ ( t )) , (cid:101) γ (cid:48) ( t + 1) = T ( (cid:101) γ (cid:48) ( t )) , h ( t + 1) = h ( t ) + 1 , φ (cid:101) γ (cid:48) ( h ( t )) = φ (cid:101) γ ( t ) . Of course Γ n and Γ (cid:48) n are equivalent transverse loops, for every n (cid:62) 1, if it is the case for Γ and Γ (cid:48) . Atransverse loop Γ will be called prime if there is no transverse loop Γ (cid:48) and integer n (cid:62) (cid:48) n .If two transverse loops Γ and Γ (cid:48) are equivalent, there exists a holonomic homotopy between them andtherefore they are freely homotopic in dom( F ), but the converse does not need to hold, as Figure 4shows.A transverse path γ : R → M will be called F -positively recurrent if for every segment J ⊂ R andevery t ∈ R there exists a segment J (cid:48) ⊂ [ t, + ∞ ) such that γ | J (cid:48) is equivalent to γ | J . It will be called F -negatively recurrent if for every segment J ⊂ R and every t ∈ R there exists a segment J (cid:48) ⊂ ( −∞ , t ]such that γ | J (cid:48) is equivalent to γ | J . It is F -bi-recurrent if it is both F -positively and F -negativelyrecurrent. Note that, if γ : R → M and γ (cid:48) : R → M are F -equivalent and if γ is F -positively recurrent(or F -negatively recurrent), then so is γ (cid:48) . We say that an F -equivalence class is positively recurrent(negatively recurrent, bi-recurrent) if some representative of the class is F -positively recurrent (resp. F -negatively recurrent, F -bi-recurrent).We will very often use the following remarks. Suppose that Γ is a transverse loop homologous to zeroand δ a dual function. Then δ decreases along each leaf with a jump at every intersection point. Onededuces that every leaf met by Γ is wandering. In particular, Γ does not meet any set α ( φ ) or ω ( φ ), ORCING THEORY FOR TRANSVERSE TRAJECTORIES OF SURFACE HOMEOMORPHISMS 13 ΓΓ p Figure 4. The transversal loops Γ and Γ (cid:48) are not equivalent for F , even though they arefreely homotopic. which implies that for every leaf φ , there exist z − and z + on φ such that Γ does not meet neither φ − z − nor φ + z + . Writing n + and n − for the value taken by δ on φ − z − and φ + z + respectively, one deduces that n + − n − is the number of times that Γ intersect φ . Note that n + − n − is uniformly bounded. Indeed,the fact that every leaf that meets Γ is wandering implies that T can be covered by open intervalswhere Γ is injective and does not meet any leaf more than once. By compactness, T can be coveredby finitely many such intervals, which implies that there exists N such that Γ meets each leaf at most N times. We have similar results for a multi-loop Γ = (cid:80) (cid:54) i (cid:54) p Γ i homologous to zero. In case where M = R , we have similar results for a proper transverse path with finite valued dual function. In caseof an infinite valued dual function, everything is true but the existence of z − , z + , n − , n + and thefiniteness condition about intersection with a given leaf. In particular a transverse line λ meets everyleaf at most once (because the dual function takes only two values) and one can define the sets r ( λ )and l ( λ ), union of leaves included in R ( λ ) and L ( λ ) respectively. They do not depend on the choice of λ in the equivalence class. Note that if the diameter of the leaves of F are uniformly bounded, everypath equivalent to λ is still a line. We have similar results for directed proper paths. If γ is a properpath directed by a unit vector ρ , one can define the sets r ( γ ) and l ( γ ), union of leaves included in R ( γ )and L ( γ ) respectively. They do not depend on the choice of γ in the equivalence class. Moreover, ifthe leaves of F are uniformly bounded, every path equivalent to γ is still a path directed by ρ .3.2. F -transverse intersection for non singular plane foliations. We suppose here that M = R and that F is non singular.Let γ : J → R and γ : J → R be two transverse paths. The set X = { ( t , t ) ∈ J × J | φ γ ( t ) = φ γ ( t ) } , if not empty, is an interval that projects injectively on J and J as does its closure. Moreover, forevery ( t , t ) ∈ X \ X , the leaves φ γ ( t ) and φ γ ( t ) are not separated in Σ. To be more precise,suppose that J and J are real intervals and that φ γ ( t ) = φ γ ( t ) . Set J − = J ∩ ( −∞ , t ] and J − = J ∩ ( −∞ , t ]. Then either one of the paths γ | J − , γ | J − is equivalent to a subpath of the otherone, or there exist a < t and a < t such that:- γ | ( a ,t ] and γ | ( a ,t ] are equivalent; - φ γ ( a ) ⊂ L ( φ γ ( a ) ) , φ γ ( a ) ⊂ L ( φ γ ( a ) )- φ γ ( a ) and φ γ ( a ) are not separated in Σ.Observe that the second property (but not the two other ones) is still satisfied when a , a are replacedby smaller parameters. Note also that φ γ ( a ) is either above or below φ γ ( a ) relative to φ γ ( t ) andthat this property remains satisfied when a , a are replaced by smaller parameters and t by anyparameter in ( a , t ]. We have a similar situation on the possible right end of X .Let γ : J → R and γ : J → R be two transverse paths such that φ γ ( t ) = φ γ ( t ) = φ . We will saythat γ and γ intersect F -transversally and positively at φ (and γ and γ intersect F -transversallyand negatively at φ ) if there exist a , b in J satisfying a < t < b , and a , b in J satisfying a < t < b , such that:- φ γ ( a ) is below φ γ ( a ) relative to φ ;- φ γ ( b ) is above φ γ ( b ) relative to φ .See Figure 5.Note that, if γ intersects F -transversally γ , if γ (cid:48) is equivalent to γ and γ (cid:48) is equivalent to γ , then γ (cid:48) intersects F -transversally γ (cid:48) , and we say that the equivalence class of γ intersect transversally theequivalence class of γ .As none of the leaves φ , φ γ ( a ) , φ γ ( a ) separates the two others, one deduces that φ γ ( a ) ⊂ L ( φ γ ( a ) ) , φ γ ( a ) ⊂ L ( φ γ ( a ) )and similarly that φ γ ( b ) ⊂ R ( φ γ ( b ) ) , φ γ ( b ) ⊂ R ( φ γ ( b ) ) . φ γ ( a ) φ γ ( a ) φ γ ( b ) φ γ ( b ) γ γ φ Figure 5. F -transverse intersection. The tangency point is also a point of F -transverse intersection. As explained above, these properties remain true when a , a are replaced by smaller parameters, b , b by larger parameters and φ by any other leaf met by γ and γ . Note that γ and γ have at least oneintersection point and that one can find two transverse paths γ (cid:48) , γ (cid:48) equivalent to γ , γ respectively,such that γ (cid:48) and γ (cid:48) have a unique intersection point, located on φ , with a topologically transverseintersection. Note that, if γ and γ are two paths that meet the same leaf φ , then either they intersect ORCING THEORY FOR TRANSVERSE TRAJECTORIES OF SURFACE HOMEOMORPHISMS 15 F -transversally, or one can find two transverse paths γ (cid:48) , γ (cid:48) equivalent to γ , γ , respectively, with nointersection point.3.3. F -transverse intersection in the general case. Here again, the notion of F -transverse in-tersection can be naturally extended in case every connected component of M is a plane and F isnot singular. Let us return now to the general case of a singular foliation F on a surface M . Let γ : J → M and γ : J → M be two transverse paths that meet a common leaf φ = φ γ ( t ) = φ γ ( t ) .We will say that γ and γ intersect F -transversally at φ if there exist paths (cid:101) γ : J → (cid:103) dom( F ) and (cid:101) γ : J → (cid:103) dom( F ), lifting γ and γ , with a common leaf (cid:101) φ = φ (cid:101) γ ( t ) = φ (cid:101) γ ( t ) that lifts φ , andintersecting (cid:101) F -transversally at (cid:101) φ . If φ is closed the choices of (cid:101) γ and (cid:101) γ do not need to be unique, seeFigure 6. γ γ p p p Figure 6. Given a lift (cid:101) γ of γ , there are two different lifts of γ intersecting (cid:101) F -transversally (cid:101) γ . Here again, we can give a sign to the intersection. As explained in the last subsection, there exist t (cid:48) and t (cid:48) such that γ ( t (cid:48) ) = γ ( t (cid:48) ) and such that γ and γ intersect F -transversally at φ γ ( t (cid:48) ) = φ γ ( t (cid:48) ) .In this case we will say that γ and γ intersect F -transversally at γ ( t (cid:48) ) = γ ( t (cid:48) ). In the case where γ = γ we will talk of an F -transverse self-intersection. A transverse path γ has an F -transverseself-intersection if for every lift (cid:101) γ to the universal covering space of the domain, there exists a nontrivial covering automorphism T such that (cid:101) γ and T ( (cid:101) γ ) have a (cid:101) F -transverse intersection. We will oftenuse the following fact. Let γ : J → M and γ : J → M be two transverse paths that meet a commonleaf φ = φ γ ( t ) = φ γ ( t ) . If J (cid:48) , J (cid:48) are two sub-intervals of J , J that contain t , t respectively andif γ | J (cid:48) and γ | J (cid:48) intersect F -transversally at φ , then γ and γ intersect F -transversally at φ .Similarly, let Γ be a loop positively transverse to F and γ its natural lift. If γ intersects F -transversallya transverse path γ (cid:48) at a leaf φ , we will say that Γ and γ (cid:48) intersect F -transversally at φ . Moreover if γ (cid:48) is the natural lift of a transverse loop Γ (cid:48) we will say that Γ and Γ (cid:48) intersect F -transversally at φ .Here again we can talk of self-intersection.As a conclusion, note that if two transverse paths have an F -transverse intersection, they both have aleaf on their right and a leaf on their left. Some useful results. In this section, we will state different results that will be useful in therest of the article. Observe that the finiteness condition for the next proposition is satisfied if everyleaf of F is wandering, or when M has genus 0. Proposition 1. Let F be an oriented singular foliation on a surface and (Γ i ) (cid:54) i (cid:54) m a family of primetransverse loops that are not pairwise equivalent. We suppose that the leaves met by the loops Γ i arenever closed and that there exists an integer N such that no loop Γ i meets a leaf more than N times.Then, for every i ∈ { , . . . , m } , there exists a transverse loop Γ (cid:48) i equivalent to Γ i such that: i) Γ (cid:48) i and Γ (cid:48) j do not intersect if Γ i and Γ j have no F -transverse intersection; ii) Γ (cid:48) i is simple if Γ i has no F -transverse self-intersection.Proof. There is a natural partial order on dom( F ) defined as follows: write z (cid:54) z (cid:48) if φ z is not closedand z (cid:48) ∈ φ + z . One can suppose, without loss of generality, that the loops Γ i are included in the sameconnected component W of dom( F ). One can lift F| W to an oriented foliation (cid:101) F on the universalcovering space (cid:102) W of W . We will parameterize Γ i by a copy T i of T and consider the T i as disjointcircles. We will endow the set T ∗ = (cid:116) (cid:54) i (cid:54) m T i with the natural topology generated by the open setsof the T i . We get a continuous map Γ : T ∗ → W (a multi-loop) by setting Γ( t ) = Γ i ( t ) ( t ), where t ∈ T i ( t ) . Suppose that t (cid:54) = t (cid:48) and φ Γ( t ) = φ Γ( t (cid:48) ) . One can lift the loops Γ i ( t ) and Γ i ( t (cid:48) ) to lines (cid:101) γ i ( t ) : R → (cid:102) W and (cid:101) γ i ( t (cid:48) ) : R → (cid:102) W transverse to (cid:101) F such that (cid:101) φ (cid:101) γ i ( t ) ( (cid:101) t ) = (cid:101) φ (cid:101) γ i ( t (cid:48) ) ( (cid:101) t (cid:48) ) = (cid:101) φ , where (cid:101) t and (cid:101) t (cid:48) lift t and t (cid:48) respectively. The fact that the loops are prime and not equivalent implies that (cid:101) γ i ( t ) | [ (cid:101) t, + ∞ ) and (cid:101) γ i ( t (cid:48) ) | [ (cid:101) t (cid:48) , + ∞ ) are not equivalent and similarly that (cid:101) γ i ( t ) | ( −∞ , (cid:101) t ] and (cid:101) γ i ( t (cid:48) ) | ( −∞ , (cid:101) t (cid:48) ] are not equivalent.So, (cid:101) φ (cid:101) γ i ( t (cid:48) ) ( (cid:101) t (cid:48)(cid:48) ) is above or below (cid:101) φ (cid:101) γ i ( t ) ( (cid:101) t (cid:48)(cid:48) ) relative to (cid:101) φ if (cid:12)(cid:12)(cid:101) t (cid:48)(cid:48) (cid:12)(cid:12) is sufficiently large. Moreover the optiondoes not depend on the choice of the lifts. We will write t ≺ t (cid:48) in the case where (cid:101) φ (cid:101) γ i ( t (cid:48) ) ( (cid:101) t (cid:48)(cid:48) ) is above (cid:101) φ (cid:101) γ i ( t ) ( (cid:101) t (cid:48)(cid:48) ) and (cid:101) φ (cid:101) γ i ( t (cid:48) ) ( − (cid:101) t (cid:48)(cid:48) ) is above (cid:101) φ (cid:101) γ i ( t ) ( − (cid:101) t (cid:48)(cid:48) ) for (cid:101) t (cid:48)(cid:48) sufficiently large. Observe that one has t ≺ t (cid:48) or t (cid:48) ≺ t in the two following cases:- i ( t ) (cid:54) = i ( t (cid:48) ) and Γ i ( t ) and Γ i ( t (cid:48) ) have no F -transverse intersection;- i ( t ) = i ( t (cid:48) ) and Γ i ( t ) has no F -transverse self-intersection.We will say that t ∈ T ∗ is a good parameter of Γ, if for every t (cid:48) ∈ T ∗ , one has t ≺ t (cid:48) ⇒ Γ( t ) < Γ( t (cid:48) ) . To get the proposition it is sufficient to construct, for every i ∈ { , . . . , m } , a transverse loop Γ (cid:48) i equivalent to Γ i such that the induced multi-loop Γ (cid:48) has only good parameters. Let us define the order o ( t ) of t ∈ T ∗ to be the number of t (cid:48) ∈ T ∗ such that t ≺ t (cid:48) . Note that every parameter of order 0 isa good parameter. We will construct Γ (cid:48) by induction, supposing that every parameter of order (cid:54) r isgood and constructing Γ (cid:48) such that every parameter of order (cid:54) r + 1 is good. Note that for every s ,the set T (cid:54) s of parameters of order (cid:54) s is closed and the set T good of good parameters is open. Theset T bad = T (cid:54) r +1 \ T good is closed and disjoint from T (cid:54) r : it contains only parameters of order r + 1.Let us fix an open neighborhood O of T bad disjoint from T (cid:54) r . By hypothesis, for every t ∈ T bad , onecan find r + 1 points θ ( t ), . . . , θ r ( t ) in T ∗ such that t ≺ θ i ( t ) for every i ∈ { , . . . , r } and among theΓ( θ i ( t )) a smallest one Γ( θ ( t )) (for the order (cid:54) ). Each θ i ( t ) belongs to T (cid:54) r and therefore is disjointfrom O . Note that each function θ i can be chosen continuous in a neighborhood of a point t , whichimplies that t (cid:55)→ Γ( θ ( t )) is continuous on T bad . It is possible to make a perturbation of Γ supported ORCING THEORY FOR TRANSVERSE TRAJECTORIES OF SURFACE HOMEOMORPHISMS 17 on O by sliding continuously each point Γ( t ) on φ − Γ( t ) to obtain a transverse multi-loop Γ (cid:48) such thatΓ (cid:48) ( t ) < Γ( θ ( t )). Since the perturbation is a holonomic homotopy, Γ (cid:48) must be equivalent to Γ.Since θ i ( t ) ∈ T (cid:54) r for every i ∈ { , . . . , r } , we have Γ( θ i ( t )) = Γ (cid:48) ( θ i ( t )) and so Γ (cid:48) ( t ) < Γ (cid:48) ( θ ( t )). (cid:3) Let us continue with the following adapted version of Poincar´e-Bendixson Theorem. Proposition 2. Let F be an oriented singular foliation on S and γ : R → S an F -bi-recurrenttransverse path. The following properties are equivalent: i) γ has no F -transverse self-intersection; ii) there exists a transverse simple loop Γ (cid:48) such that γ is equivalent to the natural lift γ (cid:48) of Γ (cid:48) ; iii) the set U = (cid:83) t ∈ R φ γ ( t ) is an open annulus.Proof. To prove that ii) implies iii) , just note that a dual function of Γ (cid:48) takes only two consecutivevalues, which implies that every leaf of F meets Γ (cid:48) at most once.To prove that iii) implies i) it is sufficient to note that if (cid:83) t ∈ R φ γ ( t ) is an annulus, each connectedcomponent of its preimage in the universal covering space of dom( F ) is an open set, union of leaves,where the lifted foliation (cid:101) F is trivial. This implies that γ has no F -transverse self-intersection.It remains to prove that i) implies ii) . The path γ being F -bi-recurrent, one can find a < b suchthat φ γ ( a ) = φ γ ( b ) . Replacing γ by an equivalent transverse path, one can suppose that γ ( a ) = γ ( b ).Let Γ be the loop naturally defined by the closed path γ | [ a,b ] . As explained previously, every leafthat meets Γ is wandering and consequently, if t and t (cid:48) are sufficiently close, one has φ Γ( t ) (cid:54) = φ Γ( t (cid:48) ) .Moreover, because Γ is positively transverse to F , one cannot find an increasing sequence ( a n ) n (cid:62) anda decreasing sequence ( b n ) n (cid:62) , such that φ γ ( a n ) = φ γ ( b n ) . So, there exist a (cid:54) a (cid:48) < b (cid:48) (cid:54) b such that t (cid:55)→ φ γ ( t ) is injective on [ a (cid:48) , b (cid:48) ) and satisfies φ γ ( a (cid:48) ) = φ γ ( b (cid:48) ) . Replacing γ by an equivalent transversepath, one can suppose that γ ( a (cid:48) ) = γ ( b (cid:48) ). The set U = (cid:83) t ∈ [ a (cid:48) ,b (cid:48) ] φ γ ( t ) is an open annulus and the loopΓ (cid:48) naturally defined by the closed path γ | [ a (cid:48) ,b (cid:48) ] is a simple loop.Let us prove now that γ is equivalent to the natural lift γ (cid:48) of Γ (cid:48) . Being F -bi-recurrent it cannot beequivalent to a strict subpath of γ (cid:48) . So it is sufficient to prove that it is included in U . We will give aproof by contradiction. We denote the two connected components of the complement of U as X , X .Suppose that there exists t ∈ R such that γ ( t ) (cid:54)∈ U . The path γ being F -bi-recurrent and the sets X i saturated, there exists t (cid:48) ∈ R separated from t by [ a (cid:48) , b (cid:48) ] such that γ ( t (cid:48) ) is in the same component X i than γ ( t ). More precisely, one can find real numbers t < a (cid:48)(cid:48) (cid:54) a (cid:48) < b (cid:48) (cid:54) b (cid:48)(cid:48) < t and an integer k (cid:62) 1, uniquely determined such that- γ | [ a (cid:48)(cid:48) ,b (cid:48)(cid:48) ] is equivalent to γ | k [ a (cid:48) ,b (cid:48) ] ;- γ | ( t ,a (cid:48)(cid:48) ) and γ | ( b (cid:48)(cid:48) ,t ) are included in U but do not meet φ γ ( a (cid:48) ) ;- γ ( t ) and γ ( t ) do not belong to U .Moreover, if γ ( t ) does not belong to the same component X i than γ ( t ), one can find real numbers t (cid:54) t < t uniquely determined such that- γ ( t ) belongs to the same component X i than γ ( t ); - γ | [ t ,t ) does not meet this component,- γ | ( t ,t ) is included in U ;- γ ( t ) does not belong to U .Observe now that if γ ( t ) and γ ( t ) belong to the same component X i , then γ [ t ,b (cid:48)(cid:48) ] and γ [ a (cid:48)(cid:48) ,t ] intersect F -transversally at φ γ ( a (cid:48)(cid:48) ) = φ γ ( b (cid:48)(cid:48) ) . Suppose now that γ ( t ) and γ ( t ) do not belong to thesame component X i . Fix t ∈ ( t , t ). There exists t (cid:48) ∈ [ a (cid:48) , b (cid:48) ] such that φ γ ( t (cid:48) ) = φ γ ( t ) . Observe that γ | [ t ,t ] and γ | [ t ,t ] intersect F -transversally at φ γ ( t (cid:48) ) = φ γ ( t ) . (cid:3) Remark . Note that the proof above tells us that if γ is F -positively or F -negatively recurrent, thereexists a transverse simple loop Γ (cid:48) such that γ is equivalent to a subpath of the natural lift γ (cid:48) of Γ (cid:48) . φ γ ( a ) γ ( t ) γ ( t ) γ ( a ) γ ( b ) X X U φ γ ( a ) γ ( t ) γ ( t ) γ ( a ) γ ( b ) X X Uγ ( t ) γ ( t ) Figure 7. Proof of Proposition 2. The next result is a slight modification. Proposition 4. Let F be an oriented singular foliation on R with leaves of uniformly bounded diam-eter and γ be a transverse proper path. The following properties are equivalent: i) γ has no F -transverse self-intersection; ii) γ meets every leaf at most once; iii) γ is a line.Proof. The fact that ii) implies iii) is obvious, as is the fact that iii) implies i) . It remains to provethat i) implies ii) . Let us suppose that φ γ ( a ) = φ γ ( b ) , where a < b . We will prove that γ has atransverse self-intersection. Like in the proof of the previous proposition, replacing γ by an equivalenttransverse path, one can find a (cid:54) a (cid:48) < b (cid:48) (cid:54) b such that γ ( a (cid:48) ) = γ ( b (cid:48) ), such that U = (cid:83) t ∈ [ a (cid:48) ,b (cid:48) ] φ γ ( t ) is an open annulus and such that the loop Γ (cid:48) naturally defined by the closed path γ | [ a (cid:48) ,b (cid:48) ] is a simpleloop. Write X for the unbounded connected component of R \ Γ (cid:48) and X for the bounded one. Thepath γ being proper, one can find real numbers t < a (cid:48)(cid:48) (cid:54) a (cid:48) < b (cid:48) (cid:54) b (cid:48)(cid:48) < t and an integer k (cid:62) 1, uniquely determined such that- γ | [ a (cid:48)(cid:48) ,b (cid:48)(cid:48) ] is equivalent to γ | k [ a (cid:48) ,b (cid:48) ] ; ORCING THEORY FOR TRANSVERSE TRAJECTORIES OF SURFACE HOMEOMORPHISMS 19 - γ | ( t ,a (cid:48)(cid:48) ) and γ | ( b (cid:48)(cid:48) ,t ) are included in U but do not meet φ γ ( a (cid:48) ) ;- γ ( t ) and γ ( t ) do not belong to U .As seen in the proof of the previous proposition, if γ ( t ) and γ ( t ) belong to the same component X i ,then γ [ t ,b (cid:48)(cid:48) ] and γ [ a (cid:48)(cid:48) ,t ] intersect F -transversally at φ γ ( a (cid:48)(cid:48) ) = φ γ ( b (cid:48)(cid:48) ) . If γ ( t ) ∈ X and γ ( t ) ∈ X ,using the fact that γ is proper, one can find real numbers t (cid:54) t < t uniquely determined such that- γ ( t ) belongs to X ;- γ | [ t ,t ) does not meet X ,- γ | ( t ,t ) is included in U ;- γ ( t ) belongs to X .As seen in the proof of the previous proposition, γ | [ t ,t ] and γ | [ t ,t ] intersect F -transversally. Thecase where γ ( t ) ∈ X and γ ( t ) ∈ X can be treated analogously. (cid:3) Let us add another result describing paths with no F -transverse self-intersection: Proposition 5. Let F be an oriented singular foliation on R , γ a transverse proper path and δ adual function of γ . If γ (cid:48) is a transverse path that does not intersect F -transversally γ , then δ takes aconstant value on the union of the leaves met by γ (cid:48) but not by γ .Proof. Let us suppose that γ (cid:48) meets two leaves φ and φ , disjoint from γ and such that δ does nottake the same value on φ and on φ . One can suppose that γ (cid:48) joins φ to φ . Let W be the connectedcomponent of dom( F ) that contains γ . Write (cid:102) W for the universal covering space of W and (cid:101) F for thelifted foliation. Every lift of γ is a line. Fix a lift (cid:101) γ (cid:48) , it joins a leaf (cid:101) φ that lifts φ to a leaf (cid:101) φ thatlifts φ . By hypothesis, there exists a lift (cid:101) γ of γ such that the dual function δ (cid:101) γ do not take the samevalue on (cid:101) φ and (cid:101) φ . One can suppose that (cid:101) φ ⊂ r ( (cid:101) γ ) and (cid:101) φ ⊂ l ( (cid:101) γ ) for instance (recall that r ( (cid:101) γ ) isthe union of leaves included in the connected component of (cid:102) W \ (cid:101) γ on the right of (cid:101) γ and l ( (cid:101) γ ) the unionof leaves included in the other component). The foliation (cid:101) F being non singular, the sets r ( (cid:101) γ ) and l ( (cid:101) γ )are closed. Consequently, there exists a subpath (cid:101) γ (cid:48)(cid:48) of (cid:101) γ (cid:48) that joins a leaf of r ( (cid:101) γ ) to a leaf of l ( (cid:101) γ ) andthat is contained but the ends in the open set (cid:101) U , union of leaves met by (cid:101) γ . Observe now that (cid:101) γ and (cid:101) γ (cid:48)(cid:48) intersect (cid:101) F -transversally and positively. (cid:3) We deduce immediately Corollary 6. Let F be an oriented singular foliation on R , γ a transverse path that is either a lineor a proper path directed by a unit vector ρ and γ (cid:48) a transverse path. If γ and γ (cid:48) do not intersect F -transversally, then γ (cid:48) cannot meet both sets r ( γ ) and l ( γ ) . Given a transverse loop Γ with a F -transverse self-intersection and its natural lift γ , there exists someinteger K for which γ | [0 ,K ] also has an F -transverse self-intersection. Let us continue this section withan estimate of the minimal such K when Γ is homologous to zero. Proposition 7. Let F be an oriented singular foliation on M and Γ : T → M a transverse loophomologous to zero in M with an F -transverse self-intersection. If γ : R → M is the natural lift of Γ ,then γ | [0 , has an F -transverse self-intersection. Proof. Write (cid:103) dom( F ) for the universal covering space of dom( F ). If (cid:101) γ : J → (cid:103) dom( F ) is a path and T a covering automorphism, write T ( (cid:101) γ ) : J → (cid:103) dom( F ) for the path satisfying T ( (cid:101) γ )( t ) = T ( (cid:101) γ ( t )) forevery t ∈ J . Choose a lift (cid:101) γ of γ to (cid:103) dom( F ) and write T for the covering automorphism such that (cid:101) γ ( t + 1) = T ( (cid:101) γ )( t ), for every t ∈ R . Since γ has an F -transverse self-intersection and is periodic ofperiod 1, there exist a covering automorphism S and a < t < b , a < t < b , such that- (cid:101) γ | ( a ,b ) is equivalent to S ( (cid:101) γ ) | ( a ,b ) ;- (cid:101) γ | [ a ,b ] and S ( (cid:101) γ ) | [ a ,b ] have a (cid:101) F -transverse intersection at (cid:101) γ ( t ) = S ( (cid:101) γ )( t ),- both a , a belong to [0 , b (cid:54) a + 1 and b (cid:54) a + 1 , which implies that γ | [0 , has an F -transverse self-intersection. Assume for a contradiction that b > a + 1 (the case where b > a + 1 is treatedsimilarly). Then we can find a (cid:48) , a (cid:48) , b (cid:48) with a < a (cid:48) < t , a < a (cid:48) < t < b (cid:48) < b such that (cid:101) γ | [ a (cid:48) ,a (cid:48) +1] is equivalent to S ( (cid:101) γ ) | [ a (cid:48) ,b (cid:48) ] .Consider first the case where b (cid:48) = a (cid:48) + 1. In that situation there exists an increasing homeomorphism h : [ a (cid:48) , a (cid:48) + 1] → [ a (cid:48) , a (cid:48) + 1] , such that h ( t ) = t and φ (cid:101) γ ( t ) = φ S ( (cid:101) γ )( h ( t )) . This implies that T ( φ (cid:101) γ ( a (cid:48) ) ) = φ (cid:101) γ ( a (cid:48) +1) = φ S ( (cid:101) γ )( a (cid:48) +1) = ST S − φ (cid:101) γ ( a (cid:48) ) = ST S − φ (cid:101) γ ( a (cid:48) ) . In case ST S − = T , one can extend h to a homeomorphism of the real line that commutes withthe translation t (cid:55)→ t + 1 such that φ (cid:101) γ ( t ) = φ S ( (cid:101) γ )( h ( t )) , for every t ∈ R . If K is large enough, then[ − K, K ] contains [ a , b ] and h ([ − K, K ]) contains [ a , b ]. This contradicts the fact that (cid:101) γ | [ a ,b ] and S ( (cid:101) γ ) | [ a ,b ] have a (cid:101) F -transverse intersection at (cid:101) γ ( t ) = S ( (cid:101) γ )( t ). In case ST S − (cid:54) = T , the leaf φ (cid:101) γ ( a (cid:48) ) is invariant by the commutator T − ST S − and so projects into a closed leaf of F that is homologicalto zero in dom( F ), which means that it bounds a closed surface in this domain. This closed surface,being a subsurface of dom( F ), is naturally foliated by F , a non singular foliation, and one gets acontradiction by Poincar´e-Hopf formula. One also gets a contradiction since this closed leaf has a nonzero intersection number with the loop Γ.Now assume that b (cid:48) < a (cid:48) + 1. Let s ∈ ( b (cid:48) , a (cid:48) + 1) and consider φ γ ( s ) . As noted in the last paragraphof Subsection 3.1, since Γ is homologous to zero, it intersects every given leaf a finite number of times.Let n be the number of times it intersects φ γ ( s ) . It is equal to the number of times γ | [ a (cid:48) ,a (cid:48) +1) or γ | [ a (cid:48) ,a (cid:48) +1) intersect φ γ ( s ) . On the other hand, since γ | [ a (cid:48) ,b (cid:48) ) is equivalent to γ | [ a (cid:48) ,a (cid:48) +1) , it must alsointersect φ γ ( s ) exactly n times, and since s ∈ ( b (cid:48) , a (cid:48) + 1), γ | [ a (cid:48) ,a (cid:48) +1) needs to intersect φ γ ( s ) at least n + 1 times, a contradiction (see Figure 8).Finally, if b (cid:48) > a (cid:48) + 1, then γ | [ a (cid:48) ,a (cid:48) +1] is equivalent to γ | [ a (cid:48) ,b (cid:48) ] for some b (cid:48) < a (cid:48) + 1 and the samereasoning as above may be applied. (cid:3) We will finish this subsection with a result (Proposition 9) that will be useful later. Let us begin withthis simple lemma. Lemma 8. Let F be an oriented singular foliation on M and (cid:101) F the lifted foliation on the universalcovering space (cid:103) dom( F ) of dom( F ) . Let (cid:101) γ and (cid:101) γ (cid:48) be two lines of (cid:103) dom( F ) transverse to (cid:101) F , invariant ORCING THEORY FOR TRANSVERSE TRAJECTORIES OF SURFACE HOMEOMORPHISMS 21 e γ ( a ) e γ ( a + 1) S e γ ( a ) S e γ ( b ) S e γ ( a + 1) e φ γ ( s ) T ( e φ γ ( s ) ) T ( e φ γ ( s ) ) Figure 8. Contradiction from Proposition 7. (cid:101) γ | [ a (cid:48) ,a (cid:48) +1] and S (cid:101) γ | [ a (cid:48) ,a (cid:48) +1] must cross thesame finite number of lifts of φ γ ( s ) . by T and T (cid:48) respectively, where T and T (cid:48) are non trivial covering automorphisms. Up to a rightcomposition by a power of T and a left composition by a power of T (cid:48) there are finitely many coveringautomorphisms S such that (cid:101) γ and S ( (cid:101) γ (cid:48) ) intersect (cid:101) F -transversally.Proof. Suppose (cid:101) γ : R → (cid:103) dom( F ) and (cid:101) γ (cid:48) : R → (cid:103) dom( F ) parameterized such that (cid:101) γ ( t +1) = T ( (cid:101) γ ( t )) and (cid:101) γ (cid:48) ( t + 1) = T (cid:48) ( (cid:101) γ ( t )), for every t ∈ R . The group of covering automorphisms acts freely and properly.So there exists L < + ∞ automorphisms S such that (cid:101) γ | [0 , ∩ S ( (cid:101) γ (cid:48) | [0 , ) (cid:54) = ∅ . If (cid:101) γ and S ( (cid:101) γ (cid:48) ) intersect (cid:101) F -transversally, there exist t and t (cid:48) such that (cid:101) γ ( t ) = S ( (cid:101) γ (cid:48) )( t (cid:48) ). Write [ x ] for the integer part of a realnumber x . One has (cid:101) γ ( t − [ t ]) = T − [ t ] ST (cid:48) [ t (cid:48) ] ( (cid:101) γ )( t (cid:48) − [ t (cid:48) ]), which implies that T − [ s ] ST (cid:48) [ t ] is one of the L previous automorphisms. (cid:3) Let F be an oriented singular foliation on M and (cid:101) F the lifted foliation on the universal covering space (cid:103) dom( F ) of dom( F ). Let Γ be a loop on M transverse to F and (cid:101) γ a lift of Γ to (cid:103) dom( F ). Write T for the covering automorphism such that (cid:101) γ ( t + 1) = T ( (cid:101) γ ( t )) for every t ∈ R . If δ : J → (cid:103) dom( F ) isa transverse path equivalent to a subpath of (cid:101) γ we define its width (relative to (cid:101) γ ) to be the largestinteger l (possibly infinite) such that δ meets l translates of a leaf by a power of T . More precisely,width (cid:101) γ ( δ ) = ∞ if there exists a leaf φ such that δ meets infinitely many translates of φ by a power of T ,and width( δ ) = l < + ∞ if there exists a leaf φ such that δ meets every leaf T k ( φ ), 0 (cid:54) k < l , and if l +1does not satisfy this property. By Lemma 8, up to a left composition by a power of T there are finitelymany lifts S ( (cid:101) γ ) such that (cid:101) γ and S ( (cid:101) γ ) intersect (cid:101) F -transversally. This number is clearly independent ofthe chosen lift (cid:101) γ , we denote it self(Γ). Saying that Γ has a F -transverse self-intersection means thatself(Γ) (cid:54) = 0. If (cid:101) γ and S ( (cid:101) γ ) intersect (cid:101) F -transversally, one can consider the maximal subpath of S ( (cid:101) γ )that is equivalent to a subpath of (cid:101) γ . Note that its width (relative to (cid:101) γ ) is finite. Looking at all the S ( (cid:101) γ ) that intersect (cid:101) F -transversally (cid:101) γ and taking the supremum, one gets a finite number because the S ( (cid:101) γ ) are finite up to a composition by a power of T . This number is independent of the choice of (cid:101) γ ,we denote it width(Γ). One gets a total order (cid:22) (cid:101) γ on the set of leaves met by a lift (cid:101) γ , where (cid:101) φ (cid:22) (cid:101) γ (cid:101) φ (cid:48) ⇔ R ( (cid:101) φ ) ⊂ R ( (cid:101) φ (cid:48) ) . Note that if (cid:101) γ and S ( (cid:101) γ ) intersect transversally and (cid:101) φ is a leaf met by (cid:101) γ , then there exist at mostwidth(Γ) translates of (cid:101) φ by a power of T that meet S ( (cid:101) γ ). Moreover, there exists k ∈ Z such that f z S k ( e γ ) f γ f z f z f z f z f z f γ Figure 9. Cases described in Lemma 10 (left) and Lemma 11(right). every leaf (cid:101) φ (cid:48) met by (cid:101) γ and S ( (cid:101) γ ) satisfies T k ( (cid:101) φ ) ≺ (cid:101) γ (cid:101) φ (cid:48) ≺ (cid:101) γ T width(Γ)+1+ k ( (cid:101) φ ) and consequently that (cid:101) φ and T width(Γ)+1 ( (cid:101) φ ) are separated by T − k S ( (cid:101) γ ). Proposition 9. Let F be an oriented singular foliation on M , let Γ be a transverse loop with a F -self-transverse intersection and γ its natural lift. Write M (Γ) = self(Γ)width(Γ)(width(Γ) + 1) + 1 . Consider two points z and z (cid:48) disjoint from Γ and look at the set of homotopy classes, with fixed end-points, of paths starting at z and ending at z (cid:48) . There are at most M (Γ) classes which are representedboth by a path disjoint from Γ and by a (possibly different) transverse path equivalent to a subpath of γ .Proof. Fix a lift (cid:101) z of z in (cid:103) dom( F ) and denote (cid:101) X the set of lifts (cid:101) z (cid:48) of z (cid:48) such that there exists a pathfrom (cid:101) z to (cid:101) z (cid:48) disjoint from all lifts of γ and such that there exists a transverse path from (cid:101) z to (cid:101) z (cid:48) thatis (cid:101) F -equivalent to a subpath of at least one lift of γ . The proposition is equivalent to showing that (cid:101) X does not contain more than 2 M (Γ) points.By definition, for every (cid:101) z (cid:48) ∈ (cid:101) X , there exists a transverse path (cid:101) δ (cid:101) z (cid:48) from (cid:101) z to (cid:101) z (cid:48) , unique up to equivalence.Moreover the set (cid:101) X ∪ { (cid:101) z } is included in a connected component (cid:102) W of the complement of the unionof lifts of γ . There is no lift (cid:101) γ of γ that separates points of (cid:101) X : for each lift (cid:101) γ , the set (cid:101) X is includedin R ( (cid:101) γ ) or in L ( (cid:101) γ ). One can write (cid:101) X = (cid:101) X r ∪ (cid:101) X l , where (cid:101) z (cid:48) ∈ (cid:101) X r if there exists a lift (cid:101) γ of γ satisfying (cid:101) X ⊂ R ( (cid:101) γ ) such that (cid:101) δ (cid:101) z (cid:48) is a subpath of (cid:101) γ (up to equivalence). One define similarly (cid:101) X l replacing thecondition (cid:101) X ⊂ R ( (cid:101) γ ) by (cid:101) X ⊂ L ( (cid:101) γ ). We will prove that (cid:101) X r and (cid:101) X l do not contain more than M (Γ)points. The two situations being similar, we will study the first one. Lemma 10. If (cid:101) z (cid:48) and (cid:101) z (cid:48) are two different points in (cid:101) X , then φ (cid:101) z (cid:48) (cid:54) = φ (cid:101) z (cid:48) .Proof. See Figure 9 for the following construction. Suppose for example that (cid:101) z (cid:48) ∈ φ + (cid:101) z (cid:48) and denote by S the covering automorphism such that (cid:101) z (cid:48) = S ( (cid:101) z (cid:48) ). There exists a lift (cid:101) γ of γ such that (cid:101) δ (cid:101) z (cid:48) is a subpathof (cid:101) γ (up to equivalence). Note that there exists k ∈ Z such that (cid:101) z (cid:48) ∈ R ( S k ( (cid:101) γ )) and (cid:101) z (cid:48) ∈ L ( S k ( (cid:101) γ )).The lift S k ( (cid:101) γ ) separates (cid:101) z (cid:48) and (cid:101) z (cid:48) , we have a contradiction. (cid:3) Lemma 11. If (cid:101) z (cid:48) and (cid:101) z (cid:48) are two different points in (cid:101) X r , then up to equivalence, one of the paths (cid:101) δ (cid:101) z (cid:48) , (cid:101) δ (cid:101) z (cid:48) is a subpath of the other one. ORCING THEORY FOR TRANSVERSE TRAJECTORIES OF SURFACE HOMEOMORPHISMS 23 Proof. Each path (cid:101) δ (cid:101) z (cid:48) i , i ∈ { , } , joins φ (cid:101) z to φ (cid:101) z (cid:48) i . We claim that either L ( φ (cid:101) z (cid:48) ) ⊂ L ( φ (cid:101) z (cid:48) ) or L ( φ (cid:101) z (cid:48) ) ⊂ L ( φ (cid:101) z (cid:48) ). If this is not the case, one of the leaves φ (cid:101) z (cid:48) , φ (cid:101) z (cid:48) is above the other one relative to φ (cid:101) z . Supposethat it is φ (cid:101) z (cid:48) . By definition of (cid:101) X r , there exists a lift (cid:101) γ of γ satisfying (cid:101) X r ⊂ R ( (cid:101) γ ) such that (cid:101) δ (cid:101) z (cid:48) is asubpath of (cid:101) γ (up to equivalence). Note now that φ (cid:101) z (cid:48) ⊂ L ( (cid:101) γ ) which contradicts the fact that (cid:101) z (cid:48) ∈ R ( (cid:101) γ ).Since L ( φ (cid:101) z ) contains both L ( φ (cid:101) z (cid:48) ) and L ( φ (cid:101) z (cid:48) ), and since, by the previous lemma, φ (cid:101) z , φ (cid:101) z (cid:48) , and φ (cid:101) z (cid:48) areall distinct, either φ (cid:101) z (cid:48) separates φ (cid:101) z from φ (cid:101) z (cid:48) , or φ (cid:101) z (cid:48) separates φ (cid:101) z from φ (cid:101) z (cid:48) . In the first case, (cid:101) δ (cid:101) z (cid:48) isequivalent to a subpath of (cid:101) δ (cid:101) z (cid:48) , and in the second case (cid:101) δ (cid:101) z (cid:48) is equivalent to a subpath of (cid:101) δ (cid:101) z (cid:48) . (cid:3) We will suppose that (cid:101) X r has at least K points, and we will show that K (cid:54) M (Γ), which proves theproposition. Using Lemmas 10 and 11, one can find a family ( (cid:101) z (cid:48) i ) (cid:54) i (cid:54) K − of points of (cid:101) X r such that (cid:101) δ (cid:101) z (cid:48) i is a strict subpath of (cid:101) δ (cid:101) z j (up to equivalence) if i < j . One knows that there exists a lift (cid:101) γ of γ suchthat (cid:101) δ (cid:101) z (cid:48) M (Γ) is equivalent to a subpath of (cid:101) γ . One deduces that every leaf φ (cid:101) z (cid:48) i , 0 (cid:54) i (cid:54) K − 1, is met by (cid:101) γ and that φ (cid:101) z (cid:48) i ≺ (cid:101) γ φ (cid:101) z (cid:48) j if i < j . Write (cid:101) z (cid:48) i = T i ( (cid:101) z (cid:48) ) and note that T i belongs to stab( (cid:102) W ), the stabilizerof (cid:102) W in the group of covering automorphisms. φ e γ ( a ) f γ T i ( f γ ) f βT i ( φ e γ ( a ) ) φ e γ ( b ) φ e β ( a ) φ e β ( b ) T i ( φ e β ( a ) ) T i ( φ e β ( b ) ) T i ( φ ) Figure 10. Final case of Lemma 12. Lemma 12. The lifts (cid:101) γ and T i ( (cid:101) γ ) intersect transversally for every i ∈ { , . . . , K − } .Proof. Fix i ∈ { , . . . , K − } . One can find a transverse line (cid:101) β invariant by T i passing through (cid:101) z (cid:48) and (cid:101) z (cid:48) i . Write (cid:101) δ for the maximal subpath of (cid:101) γ that is equivalent to a subpath of (cid:101) β . If (cid:101) γ and (cid:101) β intersect (cid:101) F -transversally, then (cid:101) γ separates T − ki ( φ (cid:101) z (cid:48) ) and T ki ( φ (cid:101) z (cid:48) ) if k is large enough. So, it separates T − ki ( (cid:101) z (cid:48) ) and T ki ( (cid:101) z (cid:48) ). This contradicts the fact that T i ∈ stab( (cid:102) W ). If (cid:101) δ is unbounded or equivalently ifwidth β ( γ ) = + ∞ , then (cid:101) δ is forward or backward invariant by T i . Look at the first case. Since γ has a F -transverse self-intersection, there exists a covering automorphism S such that (cid:101) γ and S ( (cid:101) γ ) intersect (cid:101) F -transversally. For every integer n , the lines T n S ( (cid:101) γ ) and (cid:101) γ intersect (cid:101) F -transversally. One deducesthat if n is large enough, then T n S ( (cid:101) γ ) and (cid:101) δ intersect (cid:101) F -transversally and consequently T n S ( (cid:101) γ )and (cid:101) β intersect (cid:101) F -transversally. So, if k is large enough, T n S ( (cid:101) γ ) separates T − ki ( φ (cid:101) z (cid:48) ) and T ki ( φ (cid:101) z (cid:48) ).This again contradicts the fact that T i ∈ stab( (cid:102) W ). The case where (cid:101) δ is backward invariant can betreated analogously. It remains to study the case where δ is bounded and (cid:101) γ and (cid:101) β do not intersect (cid:101) F -transversally. See Figure 10 for the following construction. Write (cid:101) δ (cid:48) for the maximal subpath of (cid:101) β that is equivalentto (cid:101) δ . There exist a < b such that (cid:101) δ = (cid:101) γ | ( a,b ) and a (cid:48) < b (cid:48) such that (cid:101) δ (cid:48) = (cid:101) β | ( a (cid:48) ,b (cid:48) ) . The sets R ( (cid:101) γ ( a )) and R ( (cid:101) β ( a (cid:48) )) are disjoint, as are the sets L ( (cid:101) γ ( b )) and L ( (cid:101) β ( b (cid:48) )). There exists a leaf φ such that (cid:101) δ and (cid:101) δ (cid:48) meet φ and T i ( φ ). In particular T i ( φ ) is met by (cid:101) γ | [ a,b ] and T i ( (cid:101) γ ) | [ a,b ] . By assumption, (cid:101) γ and (cid:101) β donot intersect (cid:101) F -transversally. There is no loss of generality by supposing that φ (cid:101) β ( a (cid:48) ) is below φ (cid:101) γ ( a ) and φ (cid:101) β ( b (cid:48) ) below φ (cid:101) γ ( b ) relative to φ or T i ( φ ). The leaf T i ( φ (cid:101) β ( a (cid:48) ) ) is above T i ( φ (cid:101) γ ( a ) ) = φ T i ( (cid:101) γ )( a ) relativeto T i ( φ ). Moreover, since T i preserves (cid:101) β , one has that T i ( φ (cid:101) β ( a (cid:48) ) ) separates φ (cid:101) β ( a (cid:48) ) and T i ( φ ), andtherefore it is crossed by both (cid:101) δ and (cid:101) δ (cid:48) . This implies that there exists a transverse path that joins φ (cid:101) γ ( a ) to T i ( φ (cid:101) β ( a (cid:48) ) ). Consequently, φ (cid:101) γ ( a ) belongs to R ( T i ( φ (cid:101) β ( a (cid:48) ) )) and is above φ T i ( (cid:101) γ )( a ) relative to T i ( φ ).Similarly, there exists a transverse path that joins φ (cid:101) β ( b (cid:48) ) to φ T i ( (cid:101) γ )( b ) . Consequently, φ T i ( (cid:101) γ )( b ) belongsto L ( φ (cid:101) β ( b (cid:48) ) ) and is above φ (cid:101) γ ( b ) relative to T i ( φ ). We have proved that the paths (cid:101) γ [ a,b ] and T i ( (cid:101) γ ) [ a,b ] intersect (cid:101) F -transversally. (cid:3) By the definition of self(Γ) and Lemma 12, there exists covering automorphisms ( S l ) (cid:54) l< self(Γ) suchthat (cid:101) γ and S l ( (cid:101) γ ) intersect (cid:101) F -transversally, a family ( l i ) (cid:54) i (cid:54) K − in { , . . . , self(Γ) − } and families ofrelative integers ( n i ) (cid:54) i (cid:54) K − , ( m i ) (cid:54) i (cid:54) K − , such that T i = T n i S l i T m i . Note that, if 1 (cid:54) i, j (cid:54) K − i (cid:54) = j , then T i (cid:54) = T j , and the function that assigns for each i the triple ( n i , l i , m i ) is injective.Define, for 0 (cid:54) l < self(Γ) the set I l = { (cid:54) i (cid:54) K − , l i = l } and fix some l in { , ..., self(Γ) } . If i ∈ I l , each leaf T m i ( φ (cid:101) z (cid:48) ) = S − l T − n i ( φ (cid:101) z (cid:48) i ) is met both by (cid:101) γ and by S − l ( (cid:101) γ ), and since (cid:101) γ and S − l ( (cid:101) γ )also intersect (cid:101) F -transversally, we deduce that there are at most width(Γ) possible values for m i with i ∈ I l . Fix such a value m and consider the set I l,m = { (cid:54) i (cid:54) K − , l i = l, m i = m } . Note that, if i ∈ I l,m , then the leaf φ (cid:101) z (cid:48) i = T n i S l T m ( φ (cid:101) z (cid:48) ) is met by both (cid:101) γ and S l ( (cid:101) γ ). The fact that no line T k S (cid:101) γ , k ∈ Z separates the leaves φ (cid:101) z (cid:48) i , i ∈ I l,m implies, as noted just before Proposition 9, that there at mostwidth(Γ) + 1 such leaves, that means at most width(Γ) + 1 possible values of n i and elements in I l,m .One deduces that K (cid:54) self(Γ)width(Γ)(width(Γ) + 1) = M (Γ) , as desired. (cid:3) Transverse homology set. For any loop Γ on M , let us denote [Γ] ∈ H ( M, Z ) its singular homology class. The transversehomology set of F is the smallest set THS( F ) of H ( M, Z ), that is stable by addition and contains allclasses of loops positively transverse to F . The following result will also be useful: Proposition 13. Let F be a singular oriented foliation on T and (cid:101) F its lift to R . If one can findfinitely many classes κ i ∈ THS( F ) , (cid:54) i (cid:54) r , that linearly generate the whole homology of the torusand satisfy (cid:80) (cid:54) i (cid:54) r κ i = 0 , then the diameters of the leaves of (cid:101) F are uniformly bounded.Proof. Decomposing each class κ i and taking out all the loops homologous to zero, one can suppose(changing r is necessary) that for every i ∈ { , . . . , r } , there exists a transverse loop Γ i such that[Γ i ] = κ i . The fact that the κ i linearly generate the whole homology of the torus implies that themulti-loop Γ = (cid:80) (cid:54) i (cid:54) p Γ i is connected (as a set) and that the connected components of its complementare simply connected. Moreover, these components are lifted in uniformly bounded simply connecteddomains of R , let us say by a constant K . The multi-loop Γ being homologous to zero induces a dualfunction δ on its complement. It has been explained before that δ decreases on each leaf of F and is ORCING THEORY FOR TRANSVERSE TRAJECTORIES OF SURFACE HOMEOMORPHISMS 25 bounded. Consequently, there exists an integer N such that every leaf meets at most N components.If one lifts it to R , one find a path of diameter bounded by N K . (cid:3) Maximal isotopies, transverse foliations, admissible paths Singular isotopies, maximal isotopies. Let us begin by introducing mathematical objects related to isotopies. Let f be a homeomorphismof an oriented surface M . An identity isotopy of f is a path joining the identity to f in the space ofhomeomorphisms of M , furnished with the C topology (defined by the uniform convergence of mapsand their inverse on every compact set). We will write I for the set of identity isotopies of f and willsay that f is isotopic to the identity if this set is not empty. If I = ( f t ) t ∈ [0 , ∈ I is such an isotopy,we can define the trajectory of a point z ∈ M , which is the path I ( z ) : t (cid:55)→ f t ( z ). More generally,for every n (cid:62) I n ( z ) = (cid:81) (cid:54) k For every I ∈ I her there exists I (cid:48) ∈ I her , maximal in I her , satisfying I (cid:22) I (cid:48) . Such anisotopy I (cid:48) is maximal in I sing and so there is no point z ∈ fix( f ) ∩ dom( I (cid:48) ) such that I (cid:48) ( z ) is homotopicto zero in dom( I (cid:48) )Note that if ˇ M is a covering space of M and ˇ π : ˇ M → M the covering projection, then for everysingular isotopies I , I (cid:48) satisfying I (cid:22) I (cid:48) , the respective lifts ˇ I , ˇ I (cid:48) satisfy ˇ I (cid:22) ˇ I (cid:48) . Note also that asingular isotopy I is maximal if and only if its lift ˇ I is maximal.Let us explain the reason why hereditary singular isotopies are important. It is related to the followingproblem. If I is a singular isotopy, does there exists a global isotopy I (cid:48) ∈ I such that fix( I (cid:48) ) =sing( I ) ∪ fix( I ) and I (cid:48) | M \ fix( I (cid:48) ) equivalent to I ? Such an isotopy I (cid:48) always exists in the case wherefix( f ) is totally disconnected. Indeed, in that case, I naturally extends to an isotopy on M that fixesthe ends of dom( I ) corresponding to points of M . The problem is much more difficult in the case wherefix( f ) is not totally disconnected. The fact that I is a hereditary singular isotopy is necessary becausethe restriction of a global isotopy to the complement of its fixed point set is obviously hereditary. Itappears that this condition is sufficient. This is the purpose of a recent work by B´eguin-Crovisier-LeRoux [BCL]. Following [BCL], Jaulent’s theorem about existence of maximal isotopies can be statedin the following much more natural form: for every I ∈ I , there exists I (cid:48) ∈ I such that: i) fix( I ) ⊂ fix( I (cid:48) ); ii) I (cid:48) is homotopic to I relative to fix( I ); iii) there is no point z ∈ fix( f ) \ fix( I (cid:48) ) whose trajectory I (cid:48) ( z ) is homotopic to zero in M \ fix( I (cid:48) ).The last condition can be stated in the following equivalent form:- if (cid:101) I (cid:48) = ( (cid:101) f (cid:48) t ) t ∈ [0 , is the identity isotopy that lifts I (cid:48) | M \ fix( I (cid:48) ) to the universal covering space of M \ fix( I (cid:48) ), then (cid:101) f (cid:48) is fixed point free.The typical example of an isotopy I ∈ I verifying iii) is the restricted family I = ( f t ) t ∈ [0 , of atopological flow ( f t ) t ∈ R on M . Indeed, one can lift the flow ( f t | M \ fix( I ) ) t ∈ R as a flow ( (cid:101) f t ) t ∈ R on theuniversal covering space of M \ fix( I ). This flow has no fixed point and consequently no periodic point.So (cid:101) f is fixed point free, which exactly means that the condition iii) is fulfilled. In particular, therestriction of f to M \ fix( I ) is a hereditary maximal isotopy. To construct a maximal singular isotopythat is not hereditary, let us consider the flow ( f t ) t ∈ R on R defined as follows in polar coordinates f t ( r, θ ) = ( r, θ + 2 πth ( r ))where h ( r ) = (cid:40) r (1 − r ) , if r ∈ [0 , ,r − (1 − r − ) , if r ∈ [1 , + ∞ ) . ORCING THEORY FOR TRANSVERSE TRAJECTORIES OF SURFACE HOMEOMORPHISMS 27 and set f = f . The fixed point set of the flow is the union of the origin and the unit circle S and so,the restriction of the isotopy ( f t ) t ∈ [0 , to the complement of this fixed point set is a maximal hereditarysingular isotopy. The isotopy I (cid:48) = ( f (cid:48) t ) t ∈ [0 , , whose domain is the complement of { (0 , } ∪ S , thatcoincides with ( f t ) t ∈ [0 , on the set of points such that 0 < r < r > f (cid:48) t ( r, θ ) = (cid:40) ( r, θ + 4 πtr ) , if t ∈ [0 , / , ( r, θ + 4 π ( t − / h ( r )) , if t ∈ [1 / , . is not a hereditary singular isotopy. There is no singular isotopy I (cid:48)(cid:48) of f whose domain is the comple-ment of { (0 , , (1 , } such that I (cid:48)(cid:48) (cid:22) I (cid:48) for the following reason. If (cid:101) I (cid:48)(cid:48) is the lift of I (cid:48)(cid:48) to the universalcovering of dom( I (cid:48)(cid:48) ), and if z is a lift of (0 , − (cid:101) z n are lifts of (0 , − − /n ) such that (cid:101) z n convergesto (cid:101) z , then the trajectory of (cid:101) I (cid:48)(cid:48) ( (cid:101) z ) would be a closed loop, but the endpoints of the trajectories of (cid:101) I (cid:48)(cid:48) ( (cid:101) z n )do not converge to (cid:101) z , since the trajectories of I (cid:48) ( z n ) and I (cid:48)(cid:48) ( z n ) are homotopic in dom( I ).Since the proof of [BCL] is not published yet, we will use the formalism of singular isotopies in thearticle.4.2. Transverse foliations. Let f be an orientation preserving plane homeomorphism. By definition, a Brouwer line of f is atopological line λ such that f ( L ( λ )) ⊂ L ( λ ) (or equivalently a line λ such that f ( λ ) ⊂ L ( λ ) and f − ( λ ) ⊂ R ( λ )). The classical Brouwer Plane Translation Theorem asserts that R can be coveredby Brouwer lines in case f is fixed point free (see [Br]). Let us recall now the equivariant foliatedversion of this theorem (see [Lec2]). Suppose that f is a homeomorphism isotopic to the identity onan oriented surface M . Let I be a maximal singular isotopy and write (cid:101) I = ( (cid:101) f t ) t ∈ [0 , for the liftedidentity defined on the universal covering space (cid:103) dom( I ) of dom( I ). Recall that (cid:101) f = (cid:101) f is fixed pointfree. Suppose first that dom( I ) is connected. In that case, (cid:103) dom( I ) is a plane and we have [Lec2]: Theorem 15. There exists a non singular topological oriented foliation (cid:101) F on (cid:103) dom( I ) , invariant bythe covering automorphisms, whose leaves are Brouwer lines of (cid:101) f . Consequently, for every point (cid:101) z ∈ (cid:103) dom( I ), one has (cid:101) f ( (cid:101) z ) ∈ L ( φ (cid:101) z ) , (cid:101) z ∈ R ( φ (cid:101) f ( (cid:101) z ) ) . This implies that there exists a path (cid:101) γ positively transverse to (cid:101) F that joins (cid:101) z to (cid:101) f ( (cid:101) z ). As noted insection 3.1, this path is uniquely defined up to (cid:101) F -equivalence, provided the endpoints remain the same.The leaves of the lifted foliation (cid:101) F met by (cid:101) γ are the leaves φ such that R ( φ (cid:101) z ) ⊂ R ( φ ) ⊂ R ( φ (cid:101) f ( (cid:101) z ) ).In particular, every leaf met by (cid:101) γ is met by (cid:101) I ( (cid:101) z ). Of course, (cid:101) F lifts a singular foliation F such thatdom( F ) = dom( I ). We immediately get the following result, still true in case dom( I ) is not connected: Corollary 16. There exists a singular topological oriented foliation F satisfying dom( F ) = dom( I ) such that for every z ∈ dom( I ) the trajectory I ( z ) is homotopic, relative to the endpoints, to a path γ positively transverse to F and this path is uniquely defined up to equivalence. We will say that a foliation F satisfying the conclusion of Corollary 16 is transverse to I . Observe thatif ˇ M is a covering space of M and ˇ π : ˇ M → M the covering projection, a foliation F transverse to amaximal singular isotopy I lifts to a foliation ˇ F transverse to the lifted isotopy ˇ I .We will write I F ( z ) for the class of paths that are positively transverse to F , that join z to f ( z ) andthat homotopic in dom( I ) to I ( z ), relative to the endpoints. We will also use the notation I F ( z ) forevery path in this class and called it the transverse trajectory of z . Similarly, for every n (cid:62) 1, one candefine I n F ( z ) = (cid:81) (cid:54) k Fix z ∈ dom( I ) , n (cid:62) , and parameterize I n F ( z ) by [0 , . For every < a < b < , thereexists a neighborhood V of z such that, for every z (cid:48) ∈ V , the path I n F ( z ) | [ a,b ] is equivalent to a subpathof I n F ( z (cid:48) ) . Moreover, there exists a neighborhood W of z such that, for every z (cid:48) and z (cid:48)(cid:48) in W , the path I n F ( z (cid:48) ) is equivalent to a subpath of I n +2 F ( f − ( z (cid:48)(cid:48) )) Proof. Keep the notations introduced above. Fix a lift (cid:101) z ∈ (cid:94) dom( I ) of z and denote by φ and φ (cid:48) theleaves of (cid:101) F containing (cid:101) I n (cid:101) F ( (cid:101) z )( a ) and (cid:101) I n (cid:101) F ( (cid:101) z )( b ) respectively. One has R ( φ (cid:101) z ) ⊂ R ( φ ) ⊂ R ( φ ) ⊂ R ( φ (cid:48) ) ⊂ R ( φ (cid:48) ) ⊂ R ( φ (cid:101) f n ( (cid:101) z ) ) . If V ⊂ dom( I ) is a topological disk, small neighborhood of z , the lift (cid:101) V that contains (cid:101) z satisfies (cid:101) V ⊂ R ( φ ) , (cid:101) f n ( (cid:101) V ) ⊂ L ( φ (cid:48) ) . Consequently, for every z (cid:48) ∈ V , the path I n F ( z ) | [ a,b ] is equivalent to a subpath of I n F ( z (cid:48) ).Let us prove the second assertion. One can find a leaf φ of the lifted foliation such that R ( φ (cid:101) f − ( (cid:101) z ) ) ⊂ R ( φ ) ⊂ R ( φ ) ⊂ R ( φ (cid:101) z )and a leaf φ (cid:48) such that R ( φ (cid:101) f n ( (cid:101) z ) ) ⊂ R ( φ (cid:48) ) ⊂ R ( φ (cid:48) ) ⊂ R ( φ (cid:101) f n +1 ( (cid:101) z ) ) . If W ⊂ dom( I ) is a topological disk, small neighborhood of z , the lift (cid:102) W that contains (cid:101) z satisfies (cid:101) f − ( (cid:102) W ) ⊂ R ( φ ) , (cid:102) W ⊂ L ( φ ) , (cid:101) f n ( (cid:102) W ) ⊂ R ( φ (cid:48) ) , (cid:101) f n +1 ( (cid:102) W ) ⊂ L ( φ (cid:48) ) . Consequently, for every z (cid:48) and z (cid:48)(cid:48) in W , the path I n F ( z (cid:48) ) is equivalent to a subpath of I n +2 F ( f − ( z (cid:48)(cid:48) )). (cid:3) Say that z ∈ M is positively recurrent if z ∈ ω ( z ), which means that there is a subsequence of thesequence ( f n ( z )) n (cid:62) that converges to z . Say that z ∈ M is negatively recurrent if z ∈ α ( z ), whichmeans that there is a subsequence of the sequence ( f − n ( z )) n (cid:62) that converges to z . Say that z ∈ M is bi-recurrent if it is positively and negatively recurrent. An immediate consequence of the previouslemma is the fact that if z ∈ dom( I ) is positively recurrent, negatively recurrent or bi-recurrent, then I Z F ( z ) is F -positively recurrent, F -negatively recurrent or F -bi-recurrent respectively. ORCING THEORY FOR TRANSVERSE TRAJECTORIES OF SURFACE HOMEOMORPHISMS 29 δ φ φ φ γ ( a ) φ γ ( b ) V a V b γ δ λz z f ( z ) f ( φ ) z z β α α Figure 11. Lemma 18, construction of V a and V b . Lemma 18. Suppose that γ : [ a, b ] → dom( I ) is a transverse path that has a leaf on its right and aleaf on its left. Then, there exists a compact set K ⊂ dom( I ) such that for every n > and for everytransverse trajectory I n F ( z ) that contains a subpath equivalent to γ , there exists k ∈ { , . . . , n − } suchthat f k ( z ) belongs to K .Proof. Lifting our path to the universal covering space of the domain, it is sufficient to prove the resultin the case where dom( I ) is a plane.Figure 11 illustrates the following construction. Suppose that φ is on the right of γ and φ on itsleft and write W for the connected component of the complement of φ ∪ φ that contains γ . Since φ and φ are Brouwer lines, every orbit that goes from R ( φ γ ( a ) ) to L ( φ γ ( b ) ) is contained in W . Let δ be a simple path that joins a point z of φ to a point z of φ , that is contained in W but theendpoints and that does not meet neither R ( φ γ ( a ) ) nor L ( φ γ ( b ) ). Write V b for the connected componentof W \ δ that contains L ( φ γ ( b ) ). We will extend δ as a line α β δβ α as follows. If W is containedin R ( φ ), set α = f ( φ − z ) and choose for β a simple path that joins f ( z ) to z and is containedin R ( f ( φ )) ∩ L ( φ ) but the endpoints. If W is contained in L ( φ ), set α = ( φ + z ) − and β = { z } .Similarly, if W is contained in R ( φ ), choose for β a simple path that joins z to f ( z ) and iscontained in L ( φ ) ∩ R ( f ( φ )) but the endpoints and set α = f ( φ + z ). Otherwise, if W is containedin L ( φ ), set β = { z } and α = ( φ − z ) − . Note that λ = α β δβ α is a line.The image of β δβ by f − is compact and the images of α and α by f − are disjoint from W . So,one can find a simple path δ (cid:48) that joins a point z (cid:48) of φ to a point z (cid:48) of φ , that is contained in W butthe endpoints, that does not meet V b and such that the connected component V a of W \ δ that doesnot contain V b (and that meets R ( φ γ ( a ) )), does not intersect f − ( λ ). This implies that f ( V a ) and V b are separated by λ and satisfy f ( V a ) ∩ V b = ∅ . So, every orbit that goes from R ( φ γ ( a ) ) to L ( φ γ ( b ) ) hasto meet both sets V a and V b but is not included in the union of these sets. It must meet the compactset K = W \ ( V a ∪ V b ). (cid:3) Admissible paths. Until the end of the whole section, we suppose given a homeomorphism f isotopic to the identity onan oriented surface M and a maximal singular isotopy I . We write (cid:101) I = ( (cid:101) f t ) t ∈ [0 , for the lifted identitydefined on the universal covering space (cid:103) dom( I ) of dom( I ) and set (cid:101) f = (cid:101) f for the lift of f | dom( I ) induced by the isotopy. We suppose that F is a foliation transverse to I and write (cid:101) F for the lifted foliation on (cid:103) dom( I ).We will say that a path γ : [ a, b ] → dom( I ), positively transverse to F , is admissible of order n if itis equivalent to a path I n F ( z ), z ∈ dom( I ), in the sense defined in subsection 3.1. It means that if (cid:101) γ : [ a, b ] → (cid:103) dom( I ) is a lift of γ , there exists a point (cid:101) z ∈ (cid:103) dom( I ) such that (cid:101) z ∈ φ (cid:101) γ ( a ) and (cid:101) f n ( (cid:101) z ) ∈ φ (cid:101) γ ( b ) ,or equivalently, that (cid:101) f n ( φ (cid:101) γ ( a ) ) ∩ φ (cid:101) γ ( b ) (cid:54) = ∅ . We will say that γ is admissible of order (cid:54) n if it is a subpath of an admissible path of order n . If (cid:101) γ : [ a, b ] → (cid:103) dom( I ) is a lift of γ , this means that (cid:101) f n ( R ( φ (cid:101) γ ( a ) )) ∩ L ( φ (cid:101) γ ( b ) ) (cid:54) = ∅ . More generally, we will say that a transverse path γ : J → dom( I ) defined on an interval is admissible if for every segment [ a, b ] ⊂ J , there exists n (cid:62) γ | [ a,b ] is admissible of order (cid:54) n . If (cid:101) γ : J → (cid:103) dom( I ) is a lift of γ , this means that for every a < b in J , there exists n (cid:62) (cid:101) f n ( R ( φ (cid:101) γ ( a ) )) ∩ L ( φ (cid:101) γ ( b ) ) (cid:54) = ∅ . Similarly, we will say that a transverse loop Γ is admissible if its natural lift is admissible. If thecontext is clear, we will say that a path is of order n (order (cid:54) n ) if it is admissible of order n (resp.admissible of order (cid:54) n ).Let us finish this subsection with a useful result which says that except in some particular trivial cases,there is no difference between being of order (cid:54) n and being of order n (and so of being of order (cid:54) n and being of order m for every m (cid:62) n ). Proposition 19. Let γ : [ a, b ] → dom( I ) be a transverse path of order (cid:54) n but not of order n , then γ has no leaf on its right and no leaf on its left.Proof. Lifting the path to the universal covering space of the domain, it is sufficient to prove theresult in case where dom( I ) is a plane. By hypothesis, one has: f n ( φ γ ( a ) ) ∩ φ γ ( b ) = ∅ , f n ( R ( φ γ ( a ) )) ∩ L ( φ γ ( b ) ) (cid:54) = ∅ . This implies that f n ( L ( φ γ ( a ) )) ⊂ L ( φ γ ( b ) ) and f − n ( R ( φ γ ( b ) )) ⊂ R ( φ γ ( a ) ). Suppose that there existsa leaf φ in L ( φ γ ( a ) )) ∩ R ( φ γ ( b ) ) that does not meet γ . Recall that φ is a Brouwer line. One of thesets R ( φ ) or L ( φ ) is included in L ( φ γ ( a ) )) ∩ R ( φ γ ( b ) ). It cannot be R ( φ ), because f − n ( R ( φ )) wouldbe contained both in R ( φ ) and in R ( φ γ ( a ) ); it cannot be L ( φ ), because f n ( L ( φ )) would be containedboth in L ( φ ) and in L ( φ γ ( b ) ). We have a contradiction. (cid:3) The fundamental proposition. The next proposition is a new result about maximal isotopies and transverse foliations. It gives us anoperation that permits to construct admissible paths from a pair of admissible paths and its proof isvery simple. Nevertheless, this fundamental result will have many interesting consequences. ORCING THEORY FOR TRANSVERSE TRAJECTORIES OF SURFACE HOMEOMORPHISMS 31 Proposition 20. Suppose that γ : [ a , b ] → M and γ : [ a , b ] → M are transverse paths thatintersect F -transversally at γ ( t ) = γ ( t ) . If γ is admissible of order n and γ is admissible oforder n , then γ | [ a ,t ] γ | [ t ,b ] and γ | [ a ,t ] γ | [ t ,b ] are admissible of order n + n . Furthermore,either one of these paths is admissible of order min( n , n ) or both paths are admissible of order max( n , n ) .Proof. By lifting to the universal covering space of the domain, it is sufficient to prove the result inthe case where M is a plane and F is non singular.By Proposition 19, each path γ , γ , γ | [ a ,t ] γ | [ t ,b ] and γ | [ a ,t ] γ | [ t ,b ] , having a leaf on its rightor on its left, will be admissible of order m if it is admissible of order (cid:54) m . Note first that for everyintegers k , k in Z , one has f k ( R ( φ γ ( a ) )) ∩ f k ( R ( φ γ ( a ) )) = f k ( L ( φ γ ( b ) )) ∩ f k ( L ( φ γ ( b ) )) = ∅ . For every i ∈ { , } define the sets X i = f n i ( R ( φ γ i ( a i ) )) ∪ L ( φ γ i ( b i ) ) , Y i = f − n i ( L ( φ γ i ( b i ) )) ∪ R ( φ γ i ( a i ) ) , which are connected according to the admissibility hypothesis.If γ | [ a ,t ] γ | [ t ,b ] is not admissible of order n , then X ∩ L ( φ γ ( b ) ) = ∅ and so X separates R ( φ γ ( a ) )and L ( φ γ ( b ) ). This implies that none of the sets X ∩ X and X ∩ Y is empty. The first propertyimplies that f n ( R ( φ γ i ( a ) )) ∩ L ( φ γ ( b ) ) (cid:54) = ∅ , which means that γ | [ a ,t ] γ | [ t ,b ] is admissible oforder n . The second one implies that f − n ( L ( φ γ ( b ) )) ∩ f n ( R ( φ γ ( a ) )) (cid:54) = ∅ , which means that γ | [ a ,t ] γ | [ t ,b ] is admissible of order n + n .If γ | [ a ,t ] γ | [ t ,b ] is not admissible of order n , then Y ∩ R ( φ γ ( b ) ) = ∅ and so Y separates R ( φ γ ( a ) )and L ( φ γ ( b ) ). This implies that none of the sets Y ∩ Y and Y ∩ X is empty. The first propertyimplies that γ | [ a ,t ] γ | [ t ,b ] is admissible of order n . The second one implies that γ | [ a ,t ] γ | [ t ,b ] is admissible of order n + n .In conclusion, γ | [ a ,t ] γ | [ t ,b ] is admissible of order n + n . Moreover, if it is not admissible of ordermin( n , n ) then γ | [ a ,t ] γ | [ t ,b ] is admissible of order max( n , n ). The paths γ and γ playing thesame role, we get the proposition. (cid:3) φ γ ( a ) φ γ ( a ) φ γ ( b ) φ γ ( b ) γ γ f n ( φ γ ( a ) ) f n ( φ γ ( a ) ) φ γ ( a ) φ γ ( b ) γ f n ( φ γ ( a ) ) X f − n ( φ γ ( b ) ) Y Figure 12. Fundamental lemma, the case where γ | [ a ,t ] γ | [ t ,b ] is not admissible of order n . One deduces immediately the following: Corollary 21. Let γ i : [ a i , b i ] → M , (cid:54) i (cid:54) r , be a family of r (cid:62) transverse paths. We supposethat for every i ∈ { , . . . , r } there exist s i ∈ [ a i , b i ] and t i ∈ [ a i , b i ] , such that: i) γ i | [ s i ,b i ] and γ i +1 | [ a i +1 ,t i +1 ] intersect F -transversally at γ i ( t i ) = γ i +1 ( s i +1 ) if i < r ; ii) one has s = a < t < b , a r < s r < t r = b r and a i < s i < t i < b i if < i < r ; iii) γ i is admissible of order n i .Then (cid:81) (cid:54) i (cid:54) r γ i | [ s i ,t i ] is admissible of order (cid:80) (cid:54) i (cid:54) r n i .Proof. Here again, it is sufficient to prove the result when M = R and F is not singular. One mustprove by induction on q ∈ { , . . . , r } that (cid:89) (cid:54) i Let γ i : [ a i , b i ] → M , (cid:54) i (cid:54) r , be a family of r (cid:62) transverse paths. We supposethat for every i ∈ { , . . . , r } there exist s i ∈ [ a i , b i ] and t i ∈ [ a i , b i ] , such that: i) γ i and γ i +1 intersect F -transversally and positively at γ i ( t i ) = γ i +1 ( s i +1 ) if i < r ; ii) one has s = a < t < b , a r < s r < t r = b r and a i < s i < t i < b i if < i < r ; iii) γ i is admissible of order n i .Then (cid:81) (cid:54) i (cid:54) r γ i | [ s i ,t i ] is admissible of order (cid:80) (cid:54) i (cid:54) r n i .Proof. Here again, it is sufficient to prove the result when M = R and F is not singular. Here again,one must prove by induction on q ∈ { , . . . , r } that (cid:89) (cid:54) i Suppose that γ : [ a, b ] → M is a transverse path admissible of order n and that γ intersects itself F -transversally at γ ( s ) = γ ( t ) where s < t . Then γ | [ a,s ] γ | [ t,b ] is admissible of order n and γ | [ a,s ] (cid:0) γ | [ s,t ] (cid:1) q γ | [ t,b ] is admissible of order qn for every q (cid:62) .Proof. See Figure 13 illustrating the construction below. Applying Corollary 22 to the family γ i = γ, s i = s if 1 < i (cid:54) q , t i = t if 1 (cid:54) i < q, one knows that γ | [ a,t ] (cid:0) γ | [ s,t ] (cid:1) q − γ | [ s,b ] = γ | [ a,s ] (cid:0) γ | [ s,t ] (cid:1) q γ | [ t,b ] is admissible of order qn for every q (cid:62) 2. Moreover the induction argument and the last sentence ofProposition 20 tell us either that γ | [ a,s ] γ | [ t,b ] is admissible of order n , or that γ | [ a,s ] (cid:0) γ | [ s,t ] (cid:1) q γ | [ t,b ] isadmissible of order n for every q (cid:62) 1. To get the proposition, one must prove that the last case isimpossible.We do not lose any generality by supposing that dom( I ) is connected. Fix a lift (cid:101) γ of γ and denote T the covering automorphism such that (cid:101) γ ( t ) = T ( (cid:101) γ ( s )). The quotient space (cid:100) dom( I ) = (cid:103) dom( I ) /T is anannulus and one gets an identity isotopy (cid:98) I = ( (cid:98) f t ) ∈ [0 , on (cid:100) dom( I ) by projection, as a homeomorphism (cid:98) f = (cid:98) f and a transverse foliation (cid:98) F . The path (cid:101) γ projects onto a transverse path (cid:98) γ . The path (cid:101) γ (cid:48) = (cid:81) k ∈ Z T k ( (cid:101) γ | [ s,t ] ) is a line that lifts a loop (cid:98) Γ (cid:48) of (cid:100) dom( I ) transverse to (cid:98) F . The union of leaves thatmeet (cid:101) γ (cid:48) is a plane (cid:101) U that lifts an annulus (cid:98) U of (cid:100) dom( I ). The fact that γ intersects itself F -transversallyat γ ( t ) = γ ( s ) means that (cid:101) γ and T ( (cid:101) γ ) intersect (cid:101) F -transversally at (cid:101) γ ( t ) = T ( (cid:101) γ ( s )). One deduces thefollowing:- the paths (cid:101) γ [ a,s ] and (cid:101) γ [ t,b ] are not contained in (cid:101) U ;- if a (cid:48) ∈ [ a, s ) is the largest value such that (cid:101) γ ( a (cid:48) ) (cid:54)∈ (cid:101) U and b (cid:48) ∈ ( t, b ] the smallest value such that (cid:101) γ ( b (cid:48) ) (cid:54)∈ (cid:101) U , then (cid:101) γ ( a (cid:48) ) and (cid:101) γ ( b (cid:48) ) are in the same connected component of (cid:103) dom( I ) \ (cid:101) γ (cid:48) .The fact that γ | [ a,s ] (cid:0) γ | [ s,t ] (cid:1) q γ | [ t,b ] is admissible of order n implies that (cid:101) γ | [ a,s ] (cid:89) (cid:54) k Proof of Proposition 23. Corollary 24. Let γ : [ a, b ] → M be a transverse path admissible of order n . Then, there exists atransverse path of order n , γ (cid:48) : [ a, b ] → M such that γ (cid:48) has no F -transverse self-intersections, and φ γ (cid:48) ( a ) = φ γ ( a ) , φ γ (cid:48) ( b ) = φ γ ( b ) .Proof. Note first that there exists a transverse path γ (cid:48) : [ a, b ] → M equivalent to γ with finitelymany self-intersections (not necessarily F -transverse). Indeed, choose for every z on γ , a trivializationneighborhood W z . Divide the interval in n intervals J i = [ a i , b i ] of equal length and set γ i = γ J i , sothat γ = (cid:81) (cid:54) i (cid:54) n γ i . If n is large enough, then for every i , the union of γ i and all paths γ j that meet γ i is contained in a set W z . Let us begin by perturbing each γ i to find an equivalent path γ (cid:48) i , such that γ (cid:48) i ( b i ) = γ (cid:48) i +1 ( a i ), if i < n , and such that the γ (cid:48) i ( b i ) are all distinct. One can also suppose that thatfor every i , the union of γ (cid:48) i and all γ (cid:48) j that meet γ (cid:48) i is contained in a set W z . Suppose that for every i < i and every j (cid:54) = i , the paths γ (cid:48) i and γ (cid:48) j have finitely many points of intersection. One can perturbin an equivalent way each γ (cid:48) j on ( a j , b j ), j > i , such that it intersects γ i finitely many often, without ORCING THEORY FOR TRANSVERSE TRAJECTORIES OF SURFACE HOMEOMORPHISMS 35 changing the intersection points with γ i if i < i and such that condition concerning the trivializationneighborhoods is still satisfied. One knows that for every i (cid:54) i and every j (cid:54) = i , the new paths γ (cid:48) i and γ (cid:48) j have finitely many points of intersection.Let G be the collection of all transverse paths that are admissible of order n whose initial leaf is φ γ ( a ) and whose final leaf is φ γ ( b ) . Let γ (cid:48) : [ a, b ] → M be a path in G that is minimal with regards to thenumber of self-intersections. Then γ (cid:48) has no F -transverse self-intersections. Indeed, if γ (cid:48) had an F -transverse self-intersection at γ (cid:48) ( t ) = γ (cid:48) ( s ) where s < t , by the Proposition 23 the path γ (cid:48) | [ a,s ] γ (cid:48) | [ t,b ] would also be also contained in G and it would have a strictly smaller number of self-intersections. (cid:3) Realizability of transverse loops. Let Γ be a transverse loop associated to a periodic point z of period q . Recall that it means that Γ is equivalent to a transverse loop Γ (cid:48) whose natural lift γ (cid:48) is equivalent to the whole transverse trajectory of z . In particular, if γ is the natural lift of Γ, thereexists t ∈ ( − , 0] such that φ γ ( t ) = φ z and such that for every n (cid:62) γ [ t,t + n ] is equivalent to I nq F ( z ).So, the loop satisfies the following:( P q ) : for every n (cid:62) , γ | [0 ,n − is admissible of order (cid:54) nq. The following question is natural: Let Γ be a transverse loop that satisfies ( P q ) . Is Γ associated to a periodic orbit of period q ? We will see that in many situations, it is the case. In such situations, f will have infinitely manyperiodic orbits. More precisely, for every rational number r/s ∈ (0 , /q ] written in an irreducible way,the loop Γ r will be associated to a periodic orbit of period s . In fact the weaker following propertywill be sufficient:( Q q ) : there exist two sequences ( r k ) k (cid:62) and ( s k ) k (cid:62) of natural integers satisfyinglim k → + ∞ r k = lim k → + ∞ s k = + ∞ , lim sup k → + ∞ r k /s k (cid:62) /q such that γ | [0 ,r k ] is admissible of order (cid:54) s k .We will say that a transverse loop Γ is linearly admissible of order q if it satisfies ( Q q ) (note that everyequivalent loop will satisfy the same condition).Let us define now the natural covering associated to Γ (or to its natural lift γ ) and introduce someuseful notations. Fix a lift (cid:101) γ of γ and denote T the covering automorphism such that (cid:101) γ ( t +1) = T ( (cid:101) γ ( t ))for every t ∈ R . The path (cid:101) γ is a line and the union of leaves that it crosses is a topological plane (cid:101) U . Moreover it projects onto the natural lift of a loop (cid:98) Γ in the quotient space (cid:100) dom( I ) = (cid:103) dom( I ) /T .One gets an identity isotopy (cid:98) I = ( (cid:98) f t ) ∈ [0 , on (cid:100) dom( I ) by projection, as a homeomorphism (cid:98) f = (cid:98) f anda transverse foliation (cid:98) F . The loop (cid:98) Γ is transverse to (cid:98) F and the union of leaves that it crosses is atopological annulus (cid:98) U .Before stating the realization result, let us recall the following lemma (for example, see [Lec2], Theorem9.1, for a proof that uses maximal isotopies and transverse foliations). A loop in an annulus will becalled essential if it is not homotopic to zero. Lemma 25. Let J be a real interval, f a homeomorphism of T × J isotopic to the identity and (cid:101) f alift of f to R × J . We suppose that:- every essential simple loop Γ ⊂ T × J meets its image by f ;- there exist two probability measures µ and µ with compact support, invariant by f , such that theirrotation numbers (for (cid:101) f ) satisfy rot( µ ) < rot( µ ) .Then, for every r/s ∈ [rot( µ ) , rot( µ )] written in an irreducible way, there exists a point z ∈ R × J such that (cid:101) f s ( z ) = z + ( r, . Let us state now the principal result of this subsection. Proposition 26. Let Γ be a linearly admissible transverse loop of order q that satisfies one of the threefollowing conditions. Then for every rational number r/s ∈ (0 , /q ] written in an irreducible way, Γ r is associated to a periodic orbit of period s . i) The loop Γ has a leaf on its left and a leaf on its right, and the annulus (cid:98) U does not contain a simpleloop homotopic to (cid:98) Γ disjoint from its image by (cid:98) f . ii) There exists both an admissible transverse path that intersects Γ F -transversally and positively,and an admissible transverse path that intersects Γ F -transversally and negatively. iii) The loop Γ has an F -transverse self-intersection.Proof . The condition iii) is stronger than ii) because Γ intersects itself F -transversally positivelyand negatively. The condition ii) tells us that there is an admissible transverse path that intersects (cid:98) Γ (cid:98) F -transversally and positively, and an admissible transverse path that intersects (cid:98) Γ (cid:98) F -transversallyand negatively. But this implies that i) is satisfied because there exists orbits that cross (cid:98) U in bothways. It remains to prove the result under the assumption i) .We do not lose any generality by supposing that dom( I ) is connected, which means that (cid:103) dom( I ) is aplane and (cid:100) dom( I ) an annulus. By assumption, we know that there exists a leaf on the left of (cid:101) γ and aleaf on its right. One can compactify (cid:100) dom( I ) with a point S at the end on the right of (cid:98) Γ and a point N at the end on the left of (cid:98) Γ. We will denote by (cid:100) dom( I ) sph this compactification and still write (cid:98) f forthe extension that fixes the added points. The ω -limit set ω ( (cid:98) φ ) in (cid:100) dom( I ) sph of a leaf (cid:98) φ ⊂ (cid:98) U does notdepend on (cid:98) φ . Lemma 27. The set ω ( (cid:98) φ ) is reduced to S .Proof. If not, ω ( (cid:98) φ ) is either a closed leaf that bounds (cid:98) U or the union of S and of leaves homoclinic to S (which means that the two limit sets are reduced to S ). In the first case, the closed leaf that bounds (cid:98) U is homotopic to (cid:98) Γ and disjoint from its image by (cid:98) f . A simple loop included in (cid:98) U sufficiently closewill satisfy the same properties. This contradicts the assumptions of the proposition.Let us study now the second case. Choose a point (cid:98) z ∗ ∈ ω ( (cid:98) φ ) \ { S } and denote by (cid:98) φ ∗ the leaf thatcontains (cid:98) z ∗ . One can suppose that (cid:98) Γ is on the right of (cid:98) φ ∗ . This is independent of the choice of (cid:98) z ∗ andin that case, the leaf (cid:98) φ is on the right of (cid:98) φ ∗ . Let us present two arguments to deal with this situation. ORCING THEORY FOR TRANSVERSE TRAJECTORIES OF SURFACE HOMEOMORPHISMS 37 c Γ c φ ∗ c z ∗ c φ ˆ z ∗∗ z z c β f z ∗ n f z ∗ n +1 f β n f β n +1 f z n f z n +1 f λ n f α n f φ ∗ n f φ ∗ n +1 f ζ n f ζ n +1 f φT − ( f φ ) Figure 14. Construction of (cid:101) γ n in Lemma 27. The first argument is the following: As (cid:100) dom( I ) has only finitely many ends and (cid:98) F is transversal to (cid:98) I , one could adapt the proof of Lemma 3.3 of [Ler] to show that, given any point (cid:98) y in (cid:100) dom( I ), thereexists a neighborhood V (cid:98) y of (cid:98) y such that, if (cid:98) F (cid:48) is an oriented foliation of (cid:100) dom( I ) that is equal to (cid:98) F in the complement of V (cid:98) y , then (cid:98) F (cid:48) is also transversal to (cid:98) I . Let V (cid:48) be a neighborhood of (cid:98) z ∗ as givenby this result, and we further assume that there exists a homeomorphism h : V (cid:48) → [ − , × [ − , (cid:98) F ∩ V (cid:48) onto horizontal line segments oriented from right to left, and such that h ( (cid:98) z ∗ ) = (0 , h − ([ − , × (0 , ⊂ (cid:98) U , and since (cid:98) z ∗ ∈ ω ( (cid:98) φ ), there exist t < t such that (cid:98) φ ( t ) , (cid:98) φ ( t ) both belong to V (cid:48) , and such that (cid:98) φ ( s ) is disjoint from V (cid:48) if t < s < t . This implies thatthere exists x , x in (0 , 1) such that h ( (cid:98) φ ( t )) = ( − , x ) and h ( (cid:98) φ ( t )) = (1 , x ). One can find thenan oriented foliation (cid:98) F (cid:48) of (cid:92) dom( I ) that agrees with (cid:98) F in the complement of h − ([ − , × (0 , σ is the line segment in h ( V (cid:48) ) connecting ( − , x ) to (1 , x ), then h − ( σ ) is containedin a single leaf of the foliation. In particular, the leaf (cid:98) φ (cid:48) of (cid:98) F (cid:48) that contains h − ( σ ) is a closed leafcontained in (cid:98) U . Since (cid:98) F (cid:48) is also transversal to the isotopy (cid:98) I , one obtains again a contradiction as inthe first case.Since Lemma 3.3 of [Ler] is not stated in the form we used above, let us present a complete argumentfor the second case: We have the following result; for every neighborhood V of S , there exists aneighborhood W of ω ( (cid:98) φ ) in (cid:100) dom( I ) sph such that (cid:98) f ( W \ V ) ∩ (cid:98) U = ∅ . Let us consider a simple path (cid:98) β joining a point (cid:98) z ∗∗ ∈ (cid:98) U to (cid:98) z ∗ positively transverse to (cid:98) F , included (but the end (cid:98) z ∗ ) in (cid:98) U and sufficientlysmall that its image by (cid:98) f will be included in the connected component of (cid:100) dom( I ) \ (cid:98) φ ∗ that is on the leftof (cid:98) φ ∗ . The leaf (cid:98) φ meets (cid:98) β in a“monotone” sequence ( (cid:98) z n ) n (cid:62) , where lim n → + ∞ (cid:98) z n = (cid:98) z ∗ . More precisely,for every real parameterization of (cid:98) φ , one has (cid:98) z n = (cid:98) φ ( t n ), where t n +1 > t n , and lim n → + ∞ t n = + ∞ .Moreover, (cid:98) z n +1 is “closer” to (cid:98) z ∗ than (cid:98) z n on (cid:98) β . We will prove that if n is large enough, the simpleloop (cid:98) Γ n obtained by concatenating the segment (cid:98) α n ⊂ (cid:98) φ joining (cid:98) z n to (cid:98) z n +1 and the subpath (cid:98) ξ n of (cid:98) β − joining (cid:98) z n +1 to (cid:98) z n is disjoint from its image by (cid:98) f , see Figure 14 for the following construction. Wewill begin by extending (cid:98) β in a simple proper path (with the same name) contained in (cid:100) dom( I ) \ ω ( (cid:98) φ )“joining” the end N to (cid:98) z ∗ . One can find a neighborhood W (cid:48) of ω ( (cid:98) φ ) in (cid:100) dom( I ) sph that intersects (cid:98) β only between (cid:98) z and (cid:98) z ∗ . If n is large enough, (cid:98) α n will be contained in W (cid:48) and so will intersect (cid:98) β onlyat the points (cid:98) z n and (cid:98) z n +1 . We will suppose n large enough to satisfy this property. Fix a lift (cid:101) z of (cid:98) z ,write (cid:101) β for the lift of (cid:98) β that contains (cid:101) z , write (cid:101) z ∗ for its end and (cid:101) φ ∗ for the lift of (cid:98) φ ∗ that contains (cid:101) z ∗ . For every n (cid:62) (cid:101) z ∗ n = T − n ( (cid:101) z ∗ ) , (cid:101) β n = T − n ( (cid:101) β ) , (cid:101) φ ∗ n = T − n ( (cid:101) φ ∗ ) . Write (cid:101) z n for the lift of (cid:98) z n that lies on (cid:101) β n , write (cid:101) ζ n for the segment of (cid:101) β n that joins (cid:101) z n to (cid:101) z ∗ n and (cid:101) α n for the lift of (cid:98) α n that joins (cid:101) z n to (cid:101) z n +1 . Choose a parameterization (cid:98) φ ∗ : R → (cid:100) dom( I ) of (cid:98) φ ∗ sending 0onto (cid:101) z ∗ and lift it to parameterize the leaves (cid:101) φ ∗ n . We will prove that the line (cid:101) λ n = (cid:101) φ ∗ n | ( −∞ , (cid:101) ζ − n (cid:101) α n (cid:101) ζ n +1 (cid:101) φ ∗ n +1 | [0 , + ∞ ] is a Brouwer line if n is large enough. Observe first that one has L ( (cid:101) φ ∗ n ) ∪ L ( (cid:101) φ ∗ n +1 ) ⊂ L ( (cid:101) λ n ) ⊂ L (cid:16) (cid:101) φ ∗ n | ( −∞ , (cid:101) β − n (cid:17) ∩ L (cid:16) (cid:101) β n +1 (cid:101) φ ∗ n +1 | [0 , + ∞ ) (cid:17) , then note that if K is large enough one has (cid:101) f − (cid:16) (cid:101) φ ∗ n | ( −∞ , − K ] (cid:17) ⊂ R (cid:16) (cid:101) φ ∗ n | ( −∞ , (cid:101) β − n (cid:17) , (cid:101) f − (cid:16) (cid:101) φ ∗ n +1 | [ K, + ∞ ] (cid:17) ⊂ R (cid:16) (cid:101) β n +1 (cid:101) φ ∗ n +1 | [0 , + ∞ ) (cid:17) . Let V be a neighborhood of S such that (cid:98) f ( V ) ∩ (cid:16) (cid:98) φ ∗ ([ − K, K ]) ∪ (cid:98) β (cid:17) = ∅ and W a neighborhood of ω ( (cid:98) φ ) such that f ( W \ V ) ∩ (cid:98) U = ∅ . If n is large enough, then (cid:98) Γ n is included in W . Let us prove that (cid:101) λ n is a Brouwer line of (cid:101) f and then that (cid:98) Γ n is disjoint from its image by (cid:98) f . The leaves (cid:101) φ ∗ n and (cid:101) φ ∗ n +1 being Brouwer lines of (cid:101) f , one has (cid:101) f ( (cid:101) φ ∗ n | ( −∞ , ) ⊂ L ( (cid:101) φ ∗ n ) ⊂ L ( (cid:101) λ n ) , (cid:101) f ( (cid:101) φ ∗ n +1 | [0 , + ∞ ) ) ⊂ L ( (cid:101) φ ∗ n +1 ) ⊂ L ( (cid:101) λ n ) . By hypothesis on (cid:98) β , one knows that (cid:101) f ( (cid:101) ζ n ) ⊂ L ( (cid:101) φ ∗ n ) ⊂ L ( (cid:101) λ n ) , (cid:101) f ( (cid:101) ζ n +1 ) ⊂ L ( (cid:101) φ ∗ n +1 ) ⊂ L ( (cid:101) λ n ) . The path (cid:101) α n being included in a leaf of (cid:101) F and each leaf being a Brouwer line of (cid:101) f , one knows that (cid:101) f ( (cid:101) α n ) ∩ (cid:101) α n = ∅ . To prove that (cid:101) λ n is a Brouwer line, it remains to prove that (cid:101) f ( (cid:101) α n ) does not meet any of the paths (cid:101) φ ∗ n | ( −∞ , , (cid:101) φ ∗ n +1 | [0 , + ∞ ) , (cid:101) ζ n , (cid:101) ζ n +1 . By hypothesis, one knows that (cid:98) f ( (cid:98) α n ) does not meet neither (cid:98) φ ∗ ([ − K, K ]), nor (cid:98) ζ . Moreover, one knowsthat (cid:101) α n does not meet neither (cid:101) f − ( (cid:101) φ ∗ n | ( −∞ ,K ] ), nor (cid:101) f − ( (cid:101) φ ∗ n +1 | [ K, + ∞ ) ). So, we are done.To prove that (cid:98) Γ n is disjoint from its image by (cid:98) f , one must prove that (cid:98) Γ n is lifted to a path that isdisjoint from its image by (cid:101) f . This path will be included in the union of the images by the iterates of T of the path (cid:101) ζ − n (cid:101) α n (cid:101) ζ n +1 . So it is sufficient to prove that the union of these translates is disjoint fromits image by (cid:101) f . Observe now that every path T k ( (cid:101) ζ − n (cid:101) α n (cid:101) ζ n +1 ), k ∈ Z , is disjoint from L ( (cid:101) λ n ), whichimplies that it is disjoint from (cid:101) f ( (cid:101) ζ − n (cid:101) α n (cid:101) ζ n +1 ). (cid:3) Lemma 28. There is no simple loop included in (cid:100) dom( I ) homotopic to (cid:98) Γ that is disjoint from its imageby (cid:98) f .Proof. Suppose that there exists a simple loop (cid:98) Γ included in (cid:100) dom( I ) that is homotopic to (cid:98) Γ anddisjoint from its image by (cid:98) f . One can suppose for instance that (cid:98) f ( (cid:98) Γ ) is included in the component of (cid:100) dom( I ) sph \ (cid:98) Γ that contains N , and orient (cid:98) Γ in such a way that this component, denoted by L ( (cid:98) Γ ), ison the left of (cid:98) Γ . The loop (cid:98) Γ meets finitely many leaves of (cid:98) F homoclinic to S that are on the frontierof (cid:98) U . We denote them (cid:98) φ i , 1 (cid:54) i (cid:54) p . Let us prove first that (cid:98) Γ is on the left side of each (cid:98) φ i . Indeed, if (cid:98) Γ is on the right side of (cid:98) φ i , writing (cid:101) Γ for the lift of (cid:98) Γ and (cid:101) φ i for a lift of (cid:98) φ i , one finds a non empty ORCING THEORY FOR TRANSVERSE TRAJECTORIES OF SURFACE HOMEOMORPHISMS 39 c Γ c φ c φ c Γ c Γ S N Figure 15. Construction of (cid:98) Γ in Lemma 28. compact subset L ( (cid:101) Γ )) ∩ L ( (cid:101) φ i ) of (cid:103) dom( I ) that is forward invariant by (cid:101) f . But such a set does not existbecause (cid:101) f is fixed point free.Each loop (cid:98) φ i ∪ { S } bounds a Jordan domain (cid:98) L i of (cid:100) dom( I ) sph that contains N . By a classical result ofKer´ekj´art´o [Ke], one knows that the connected component of L ( (cid:98) Γ ) ∩ ( (cid:84) (cid:54) i (cid:54) p (cid:98) L i ) that contains N is aJordan domain whose boundary (cid:98) Γ is a simple loop homotopic to (cid:98) Γ in (cid:100) dom( I ), disjoint from its imageby (cid:98) f , and included in (cid:98) U ∪ L (Γ) (see Figure 15). By intersecting (cid:98) Γ with the leaves of (cid:98) F homoclinicto N that are on the frontier of (cid:98) U , one constructs similarly a simple loop (cid:98) Γ included in (cid:100) dom( I ) ∩ (cid:98) U that is homotopic to (cid:98) Γ in (cid:99) M and disjoint from its image by (cid:98) f . It remains to approximate (cid:98) Γ by asimple loop included in (cid:98) U and we get a contradiction since we are assuming condition i) in Proposition26. (cid:3) End of the proof of Proposition 26 . One must prove that for every rational number r/s ∈ (0 , /q ]written in an irreducible way, there exists a point (cid:101) z ∈ (cid:103) dom( I ) such that (cid:101) f s ( (cid:101) z ) = T r ( (cid:101) z ). Indeed, theorbit of (cid:101) z should be contained in (cid:101) U , the point (cid:101) z will project in dom( I ) onto a periodic point z of f ofperiod s , finally the loop Γ r will be associated to z .Write (cid:101) φ for the leaf containing (cid:101) γ (0) and (cid:98) φ for its projection in (cid:100) dom( I ). Using the analogous ofLemma 27 for α -limit sets, one can suppose that (cid:98) φ is a line. Let us fix a leaf (cid:98) φ N homoclinic to N anda leaf (cid:98) φ S homoclinic to S , which exists since we are assuming that the loop Γ has a leaf on its rightand a leaf on its left. Each of them is disjoint from all its images by the (non trivial) iterates of (cid:98) f .By a result of B´eguin, Crovisier, Le Roux (see [Ler], Proposition 2.3.3) one knows that there exists acompactification (cid:100) dom( I ) ann obtained by blowing up the two ends N and S replaced by circles (cid:98) Σ N and (cid:98) Σ S such that (cid:98) f extends to a homeomorphism (cid:98) f ann that admits fixed points on each added circle witha rotation number equal to zero for the lift that extends (cid:101) f . Moreover, one can suppose that each set ω ( (cid:98) φ N ) and ω ( (cid:98) φ S ) is reduced to a unique point on (cid:98) Σ N and (cid:98) Σ S respectively. One can join a point of (cid:98) φ N to a point of (cid:98) φ S by a segment disjoint from (cid:98) φ . Consequently, one can construct a line (cid:98) λ in (cid:100) dom( I ),disjoint from (cid:98) φ , that admits a limit on each added circle. Write (cid:103) dom( I ) ann = (cid:103) dom( I ) (cid:116) (cid:101) Σ N (cid:116) (cid:101) Σ S for theuniversal covering space of (cid:100) dom( I ) ann and keep the notation T for the natural covering automorphism.Write (cid:101) λ for the lift of (cid:98) λ located between (cid:101) φ and T ( (cid:101) φ ). One can construct a continuous real function (cid:101) g on (cid:103) dom( I ) ann that satisfies (cid:101) g ( T ( (cid:101) z )) = (cid:101) g ( z ) + 1 and vanishes on (cid:101) λ . The function (cid:101) h = (cid:101) g ◦ (cid:101) f − (cid:101) g is invariant by T and lifts a continuous function (cid:98) h : (cid:100) dom( I ) ann → R . If µ is a Borel probability measure invariant by (cid:98) f , the quantity (cid:82) (cid:100) dom( I ) ann h dµ is the rotation number of the measure µ for thelift (cid:101) f ann . Let us consider now the real function (cid:101) g on (cid:103) dom( I ) ann , that coincides with (cid:101) g on (cid:101) Σ N ∪ (cid:101) Σ S ,that satisfies (cid:101) g ( T ( (cid:101) z )) = (cid:101) g ( (cid:101) z ) + 1 and that vanishes on (cid:101) φ and at every point located between (cid:101) φ and T ( (cid:101) φ ). Note that (cid:101) g − (cid:101) g is uniformly bounded by a certain number K and invariant by T . Theproperty ( Q q ) satisfied by Γ tells us that for every k (cid:62) 0, one can find a point (cid:101) z k ∈ R ( (cid:101) φ ) such that (cid:101) f s k ( (cid:101) z k ) ∈ L ( T r k ( (cid:101) φ )). Write (cid:98) z k for its projection in (cid:100) dom( I ). Observe that (cid:101) g ( f s k ( (cid:101) z k )) − (cid:101) g ( (cid:101) z k ) (cid:62) r k . By taking a subsequence, one can suppose thatlim k → + ∞ s k ( (cid:101) g ( f s k ( (cid:101) z k )) − (cid:101) g ( (cid:101) z k )) = ρ ∈ [ 1 q , + ∞ ]and so that (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) s k s k − (cid:88) i =0 (cid:98) h ( (cid:98) f i ( (cid:98) z k )) − ρ (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) = (cid:12)(cid:12)(cid:12)(cid:12) s k ( (cid:101) g ( f s k ( (cid:101) z k )) − (cid:101) g ( (cid:101) z k )) − ρ (cid:12)(cid:12)(cid:12)(cid:12) (cid:54) (cid:12)(cid:12)(cid:12)(cid:12) s k ( (cid:101) g ( f s k ( (cid:101) z k )) − (cid:101) g ( (cid:101) z k )) − ρ (cid:12)(cid:12)(cid:12)(cid:12) + 2 Ks k . Write δ (cid:98) z for the Dirac measure at a point (cid:98) z ∈ (cid:100) dom( I ) ann and choose a measure µ that is the limit ofa subsequence of (cid:16) s k (cid:80) s k − i =0 δ (cid:98) f i ann ( (cid:98) z k ) (cid:17) k (cid:62) for the weak ∗ topology. One knows that µ is an invariantmeasure of rotation number ρ . As the rotation number induced on the boundary circles are equal to0, one deduces that the rotation set rot( (cid:101) f ann ) contains [0 , ρ ]. The intersection property supposed in i) implies by Lemma 25 that for every rational number r/s ∈ (0 , /q ] written in an irreducible way,there exists a point (cid:101) z ∈ (cid:103) dom( I ) ann such that (cid:101) f s ann ( (cid:101) z ) = T r ( (cid:101) z ). But this point does not belong to theboundary circles because the induced rotation numbers are equal to 0. So its belongs to (cid:103) dom( I ). (cid:3) Exponential growth of periodic points and entropy In this section we give a sufficient condition for the exponential growth of periodic points of a surfacehomeomorphism. This condition will imply that the topological entropy is positive in the compactcase. We will make use of these criteria later.We assume here, as in the previous section, that f is a homeomorphism isotopic to the identity onan oriented surface M and that I = ( f t ) t ∈ [0 , is a maximal hereditary singular isotopy, which impliesthat f = f | dom( I ) . We write (cid:101) I = ( (cid:101) f t ) t ∈ [0 , for the lifted identity defined on the universal coveringspace (cid:103) dom( I ) of dom( I ) and set (cid:101) f = (cid:101) f for the lift of f | dom( I ) induced by the isotopy. We supposethat F is a foliation transverse to I and write (cid:101) F for the lifted foliation on (cid:103) dom( I ).5.1. Exponential growth of periodic points. The main result of this section is Theorem 29. Let γ , γ : R → M be two admissible positively recurrent transverse paths (possiblyequal) with an F -transverse intersection. Then the number of periodic points of period n for someiterate of f grows exponentially in n . Theorem 29 is a direct consequence of Lemma 30 and Proposition 31. ORCING THEORY FOR TRANSVERSE TRAJECTORIES OF SURFACE HOMEOMORPHISMS 41 Lemma 30. Let γ , γ be two admissible F -positively recurrent transverse paths (possibly equal) withan F -transverse intersection, and let I and I be two real segments. Then there exists a linearly admis-sible transverse loop Γ with an F -transverse self-intersection, such that γ | I and γ | I are equivalentto subpaths of the natural lift of Γ .Proof. As explained at the end of subsection 3.3, we can find a , b , t , a , b , t such that I ⊂ [ a , b ], I ⊂ [ a , b ] and such that γ | [ a ,b ] intersects F -transversally γ | [ a ,b ] at γ ( t ) = γ ( t ). Since γ is F -positively recurrent, we can find b < a (cid:48) < b (cid:48) < a (cid:48)(cid:48) < b (cid:48)(cid:48) such that γ | [ a ,b ] , γ | [ a (cid:48) ,b (cid:48) ] and γ | [ a (cid:48)(cid:48) ,b (cid:48)(cid:48) ] are equivalent. In particular, there exists a (cid:48) < t (cid:48) < b (cid:48) < a (cid:48)(cid:48) < t (cid:48)(cid:48) < b (cid:48)(cid:48) such that- γ | [ a ,t ] , γ | [ a (cid:48) ,t (cid:48) ] and γ | [ a (cid:48)(cid:48) ,t (cid:48)(cid:48) ] are equivalent;- γ | [ t ,b ] , γ | [ t (cid:48) ,b (cid:48) ] and γ | [ t (cid:48)(cid:48) ,b (cid:48)(cid:48) ] are equivalent.Moreover, replacing γ by an equivalent path, one can suppose that γ ( t ) = γ ( t (cid:48) ) = γ ( t (cid:48)(cid:48) ). Since γ is F -positively recurrent, we can also find b < a (cid:48) < t (cid:48) < b (cid:48) < a (cid:48)(cid:48) < t (cid:48)(cid:48) < b (cid:48)(cid:48) and replace γ by an equivalent path such that a similar statement holds with the necessary changes.Note that this implies that γ is F -transverse to γ at both γ ( t (cid:48)(cid:48) ) = γ ( t ) and γ ( t ) = γ ( t (cid:48)(cid:48) ).Suppose that γ | [ a ,b (cid:48)(cid:48) ] and γ | [ a ,b (cid:48)(cid:48) ] are admissible of order (cid:54) q and apply Corollary 21 to the families( γ i ) (cid:54) i (cid:54) n , ( s i ) (cid:54) i (cid:54) n , ( t i ) (cid:54) i (cid:54) n where γ j +1 = γ | [ a ,b (cid:48)(cid:48) ] , γ j = γ | [ a ,b (cid:48)(cid:48) ] and s j +1 = t if j > , s j = t , t j +1 = t (cid:48)(cid:48) , t j = t (cid:48)(cid:48) if j < n. One deduces that for every n (cid:62) γ | [ a ,t ] (cid:0) γ | [ t ,t (cid:48)(cid:48) ] γ | [ t ,t (cid:48)(cid:48) ] (cid:1) n γ | [ t (cid:48)(cid:48) ,b ] is admissible of order 2 nq and consequently that (cid:0) γ | [ t ,t (cid:48)(cid:48) ] γ | [ t ,t (cid:48)(cid:48) ] (cid:1) n is admissible of order (cid:54) nq . So,the closed path γ (cid:48) = γ | [ t ,t (cid:48)(cid:48) ] γ | [ t ,t (cid:48)(cid:48) ] defines a loop that is linearly admissible: it satisfies the condition( Q q ) stated in the previous section. Furthermore, since both γ | [ a (cid:48) ,b (cid:48) ] and γ | [ a (cid:48) ,b (cid:48) ] are subpaths of γ (cid:48) , the induced loop has an F -transverse self-intersection. (cid:3) Proposition 31. If there exists a linearly admissible transverse loop Γ with an F -transverse self-intersection, then the number of periodic points of period n for some iterate of f grows exponentiallyin n .Proof. The proof of Proposition 31 will last until the end of this subsection. Suppose that Γ satisfiesthe condition ( Q q ) and denote γ its natural lift. By assumption, there exist s < t such that γ hasan F -transverse self-intersection at γ ( s ) = γ ( t ). So, one can apply the realization result (Proposition26) and deduce that Γ is associated to a fixed point of f q . Modifying Γ in its equivalence class ifnecessary, one can suppose that for every r (cid:62) 1, the path γ | [0 ,r ] is admissible of order rq . Adding the same positive integer to both s and t , one can find a positive integer K such that γ | [0 ,K ] has an F -transverse self-intersection at γ ( s ) = γ ( t ) and one knows that γ | [0 ,mK ] is admissible of order mKq for every m (cid:62) 1. To get our proposition, one needs a preliminary result. Set γ = γ | [ s,t ] , γ = γ [ t,K + s ] . Lemma 32. For every sequence ( ε i ) i ∈ N ∈ { , } N , every n (cid:62) , and every m (cid:62) the path γ | [0 ,s ] (cid:89) (cid:54) i The final case to consider is if ε n = 2. We must prove that γ | [0 ,s ] (cid:89) (cid:54) i Relative position of the leafs of the endpoints of (cid:101) γ and (cid:101) γ , when both start at (cid:101) z ∗ in Lemma 33. Lemma 33. Let e = ( ε i ) i ∈ N ∈ { , } N be a periodic word of period q which is not periodic of period 1.Then the loop Γ e , defined by the closed path (cid:81) (cid:54) i There exists a constant L > such that, given a palindromic word e of length n , thereare at most Ln different palindromic words e (cid:48) of length n such that Γ (cid:48) e and Γ (cid:48) e (cid:48) are equivalent.Proof. Let (cid:101) γ (cid:48) and (cid:101) γ (cid:48) be two respective lifts of γ (cid:48) and γ (cid:48) to (cid:103) dom( I ). The group of covering auto-morphisms acts freely and properly. So there exists a constant L (cid:48) such that there are at most L (cid:48) automorphisms S such that (cid:101) γ (cid:48) ∩ S ( (cid:101) γ (cid:48) ) (cid:54) = ∅ , at most L (cid:48) automorphisms S such that (cid:101) γ (cid:48) ∩ S ( (cid:101) γ (cid:48) ) (cid:54) = ∅ andat most L (cid:48) automorphisms S such that (cid:101) γ (cid:48) ∩ S ( (cid:101) γ (cid:48) ) (cid:54) = ∅ . Of course, L (cid:48) is independent of the choicesof (cid:101) γ (cid:48) and (cid:101) γ (cid:48) . We deduce that for every palindromic word e of length 2 n , there are at most 8 L (cid:48) n automorphisms S such that (cid:101) γ (cid:48) e ∩ S ( (cid:101) γ (cid:48) e ) (cid:54) = ∅ . This implies that there are at most 8 L (cid:48) n automorphisms S such that (cid:101) γ (cid:48) e and S ( (cid:101) γ (cid:48) e ) have a (cid:101) F -transverse intersection.Suppose that e and e (cid:48) are two palindromic words of length 2 n such that Γ (cid:48) e and Γ (cid:48) e (cid:48) are equivalent.There exists a covering automorphism S e (cid:48) such that (cid:101) γ (cid:48)∞ e (cid:48) is equivalent to S e (cid:48) ( (cid:101) γ (cid:48)∞ e ) and such that S e (cid:48) ◦ T e ◦ S − e (cid:48) = T e (cid:48) . Composing S e (cid:48) on the left by a power of T e (cid:48) if necessary, one can suppose that (cid:101) γ (cid:48) e (cid:48) is equivalent to a subpath of S e (cid:48) ( (cid:101) γ (cid:48) e ). By Lemma 34, one deduces that (cid:101) γ (cid:48) e and S e (cid:48) ( (cid:101) γ (cid:48) e ) intersect (cid:101) F -transversally. It remains to prove that S e (cid:48) (cid:54) = S e (cid:48)(cid:48) if e (cid:48) (cid:54) = e (cid:48)(cid:48) . But if this the case, then (cid:101) Γ (cid:48) e (cid:48) and (cid:101) Γ (cid:48) e (cid:48)(cid:48) are equivalent, which is impossible because this two paths intersect (cid:101) F -transversally at (cid:101) z . (cid:3) Since there exists 2 n different palindromic words of length 2 n , one concludes by Lemmas 33 and 35that f nKq has at least n Ln distinct fixed points, proving Proposition 31.5.2. Topological entropy. By the previous result, it is natural to believe that the topological entropyis positive, in case M is compact. The next result asserts that this is the case: Theorem 36. Let M be a compact surface, γ , γ : R → M be two admissible F -positively recurrenttransverse paths (possibly equal) with an F -transverse intersection. Then the topological entropy of f is positive.Remark . As we will see in the proof, Theorem 36 will be stated even in case where M is not compactby proving that its Alexandrov extension has positive entropy. More precisely, write dom( I ) alex for theAlexandrov compactification of dom( I ) if it is not compact, and f alex for the extension of f | dom( I ) thatfixes the point at infinity (otherwise set dom( I ) alex = dom( I ) and f alex = f | dom( I ) in what follows).Of course, f alex is a factor of f and so h ( f ) (cid:62) h ( f alex ) if M is compact.Theorem 36 will be the direct consequence of Lemma 30 and the following result: Proposition 38. Let Γ be a transverse loop with an F -transverse self-intersection, and γ its naturallift. Assume that there exists integers K, r such that γ | [0 ,K ] has an F -transverse self-intersection, andsuch that γ | [0 ,mK ] is admissible of order mr for every m (cid:62) . Then the topological entropy of f alex isat least equal to log 2 / (4 r ) . Before proving the proposition, we will need the following lemma: Lemma 39. There exists a covering ( V z ) z ∈ dom( I ) of dom( I ) satisfying the following properties: i) V z is an open disk that contains z ; ii) for every z , z in dom( I ) , for every integer p (cid:62) and for every z ∈ V z ∩ f − p ( V z ) there existsa transverse path joining z to z equivalent to a subpath of I p +2 F ( f − ( z )) , that is homotopic, withendpoints fixed, to the path α I p ( z ) α − , where α is a path in V z that joins z to z and α is a pathin V z that joins z to f p ( z ) ; iii) in the previous assertion, if p = 1 , the homotopy class of the path that joins z to z does notdepend on z ∈ V z ∩ f − ( V z ) . Proof. One can construct an increasing sequence ( K i ) i (cid:62) of compact sets of dom( I ) that cover dom( I )and such K i +1 is a neighborhood of K i ∪ f ( K i ) ∪ f − ( K i ), a distance on (cid:103) dom( I ), denoted by d , that isinvariant under the action of the group of covering transformations, and an equivariant family of leaves( φ ∗ (cid:101) z ) (cid:101) z ∈ (cid:103) dom( I ) , where φ ∗ (cid:101) z separates (cid:101) z and (cid:101) f ( (cid:101) z ) and consequently is met by (cid:101) I (cid:101) F ( (cid:101) z ) (equivariant means that φ ∗ T ( (cid:101) z ) = T ( φ ∗ (cid:101) z ) for every covering transformation T ). Then one can construct an equivariant family ofrelatively compact open sets ( W (cid:101) z ) (cid:101) z ∈ (cid:103) dom( I ) , where W (cid:101) z contains (cid:101) z , projects onto an open disk of dom( I )and satisfies (cid:101) f − ( W (cid:101) z ) ⊂ R ( φ ∗ (cid:101) f − ( z ) ) , (cid:101) f ( W (cid:101) z ) ⊂ L ( φ ∗ (cid:101) z ) . Note that, given (cid:101) z, φ ∗ (cid:101) z and W (cid:101) z as above, if (cid:101) z (cid:48) is sufficiently close to (cid:101) z , then one could choose φ ∗ (cid:101) z (cid:48) tobe equal to φ ∗ (cid:101) z and also W (cid:101) z (cid:48) = W (cid:101) z . Therefore we may assume that the family ( W (cid:101) z ) (cid:101) z ∈ (cid:103) dom( I ) is locallyfinite.Set K i = ∅ if i (cid:54) 0. One knows that f ( K i \ K i − ) ⊂ int( K i +1 \ K i − ). By a compactness argu-ment, there exists a positive and decreasing sequence ( η i ) i (cid:62) such that for every (cid:101) z ∈ π − ( K i \ K i − ),the open ball B ( (cid:101) z, η i ) is included in W (cid:101) z ∩ π − (int( K i +1 \ K i − )) and its image (cid:101) f ( B ( (cid:101) z, η i )) included in π − (int( K i +1 \ K i − )), the ball B ( (cid:101) z, η i ) being defined with respect to d . Then, one considers a decreas-ing and positive sequence ( η (cid:48) i ) i (cid:62) satisfying η (cid:48) i < η i +4 / (cid:101) z ∈ π − ( K i \ K i − ),one has (cid:101) f ( B ( (cid:101) z, η (cid:48) i )) ⊂ B ( (cid:101) f ( (cid:101) z ) , η i +1 / . Finally, one constructs an equivariant family of open sets ( V (cid:101) z ) (cid:101) z ∈ (cid:103) dom( I ) , where V (cid:101) z contains (cid:101) z , is includedin B ( (cid:101) z, η (cid:48) i ) if (cid:101) z ∈ π − ( K i \ K i − ) and projects onto an open disk of dom( I ). Note that V (cid:101) z ⊂ W (cid:101) z .By projection on dom( I ), one gets a family ( V z ) z ∈ dom( I ) satisfying i) . To prove that it satisfies ii) , onemust prove that if there exists a point (cid:101) z ∈ V (cid:101) z such that (cid:101) f p ( (cid:101) z ) ∈ V (cid:101) z , then there exists a transversepath from (cid:101) z to (cid:101) z that is equivalent to a subpath of (cid:101) I p +2 (cid:101) F ( (cid:101) f − ( (cid:101) z )). As V (cid:101) z i ⊂ W (cid:101) z i , i ∈ { , } , by theproperties of the chosen family ( W (cid:101) y ) (cid:101) y ∈ (cid:103) dom( I ) , R ( φ (cid:101) f − ( (cid:101) z ) ) ⊂ R ( φ ∗ (cid:101) f − ( (cid:101) z ) ) ⊂ R ( φ (cid:101) z ) ⊂ R ( φ ∗ (cid:101) z ) ⊂ R ( φ (cid:101) f ( (cid:101) z ) )and R ( φ (cid:101) f p − ( (cid:101) z ) ) ⊂ R ( φ ∗ (cid:101) f − ( (cid:101) z ) ) ⊂ R ( φ (cid:101) z ) ⊂ R ( φ ∗ (cid:101) z ) ⊂ R ( φ (cid:101) f p +1 ( (cid:101) z ) ) . One deduces that (cid:101) I p +2 (cid:101) F ( (cid:101) f − ( (cid:101) z )) meets φ (cid:101) z and φ (cid:101) z . It remains to prove that R ( φ (cid:101) z ) ⊂ R ( φ (cid:101) z ) to ensurethat φ (cid:101) z is met before φ (cid:101) z and to prove the existence of a transverse path from (cid:101) z to (cid:101) z . The casewhere p (cid:62) R ( φ (cid:101) z ) ⊂ R ( φ (cid:101) f ( (cid:101) z )) ) ⊂ R ( φ (cid:101) f p − ( (cid:101) z ) ) ⊂ R ( φ (cid:101) z ) . To prove the result in the case where p = 1 it is sufficient to prove that V (cid:101) z is included in W (cid:101) f ( (cid:101) z ) becauseevery point in W (cid:101) f ( (cid:101) z ) belongs to L ( φ ∗ (cid:101) z ). Moreover one will get iii) because W (cid:101) f ( (cid:101) z ) is disjoint from itsimages by the non trivial covering transformations. Set η j = η and η (cid:48) j = η (cid:48) if j (cid:54) 0, and let i be suchthat z ∈ K i \ K i − . One knows that (cid:101) z ∈ π − (int( K i +1 \ K i − )), so (cid:101) f ( (cid:101) z ) ∈ π − (int( K i +2 \ K i − )) and (cid:101) z ∈ π − (int( K i +3 \ K i − )). One also knows that (cid:101) f ( (cid:101) z ) ∈ π − (int( K i +1 \ K i − )). Using the fact that η (cid:48) i − < η i +1 / 4, that d ( (cid:101) f ( (cid:101) z ) , (cid:101) f ( (cid:101) z )) < η i +1 / 2, and that d ( (cid:101) f ( (cid:101) z ) , (cid:101) z ) < η (cid:48) i − , one deduces that V (cid:101) z ⊂ B ( (cid:101) z , η (cid:48) i − ) ⊂ B ( (cid:101) f ( (cid:101) z ) , η i +1 ) ⊂ W (cid:101) f ( (cid:101) z ) . (cid:3) ORCING THEORY FOR TRANSVERSE TRAJECTORIES OF SURFACE HOMEOMORPHISMS 47 Proof of Proposition 38. We keep the notations of Proposition 31. Since, by Lemma 34 applied to n = 1, the paths ( γ (cid:48) ) and ( γ (cid:48) ) intersect F -transversally, one deduces that γ (cid:48) has a leaf on its rightand a leaf on its left, which implies that γ (cid:48) satisfies the same property. One proves similarly that γ (cid:48) hasa leaf on its right and a leaf on its left. By Lemma 18, there exists a compact set K such that for every n (cid:62) z ∈ dom( I ) \ (cid:83) (cid:54) k The entropy of f alex relative to the covering V p is at least equal to log 2 / (4 r ) − log M (Γ (cid:48) ) / p .Proof. As seen in the proof of Theorem 29, to every palindromic word e = ( ε i ) (cid:54) i< n of length 2 n is associated a fixed point z e of f nr and an associate F -transverse loop defined by (cid:81) (cid:54) j< n γ (cid:48) ε i .Moreover, by Lemma 35, there exists L > 0, independent of n , such that there are at least 2 n /Ln different equivalent classes among the associated loops. We will prove that every open set of thecovering (cid:87) (cid:54) k< nr f − k ( V p ) contains at most Ln M (Γ (cid:48) ) nr/p points z e . We deduce that every finitesub-covering of (cid:87) (cid:54) k< nr f − k ( V p ) has at least 2 n /Ln M (Γ (cid:48) ) nr/p open sets and so h ( f alex , V p ) (cid:62) lim n → + ∞ nr log(2 n /Ln M (Γ (cid:48) ) nr/p ) = log 2 / (4 r ) − log M (Γ (cid:48) ) / p. Let us consider an element W = (cid:84) (cid:54) j< nr f − j ( V j ) of V p . We suppose that it contains at least onepoint z e . Denote J ∞ the set of j ∈ { , . . . , nr − } such that there exists j (cid:48) ∈ { , . . . , nr } satisfying | j − j (cid:48) | < p and V j (cid:48) = V ∞ ,p and denote J < ∞ the complement of J ∞ . Note that f j ( z e ) ∈ V ∞ if j ∈ J ∞ .By Lemma 18, one knows that the orbit of z e cannot be contained in V ∞ . So J < ∞ (cid:54) = ∅ .Let us begin with the case where 0 ∈ J < ∞ and write J < ∞ = { j , . . . , j l ∗ } , where j = 0 < j . . . < j l ∗ ,and add j l ∗ +1 = 4 nr . Every open set V j l can be written V j l = V z jl . For every l , choose a path δ l in V z jl from z j l to f j l ( z e ). By Lemma 39, there exists a F -transverse path β l from z j l to z j l +1 equivalentto a subpath of I j l +1 − j l +2 F ( f j l − ( z e )) and homotopic to δ l I j l +1 − j j ( f j l ( z e )) δ − l +1 . In the case where j l +1 − j l = 1, the homotopy class of β i (with endpoints z i l and z i l +1 fixed) is uniquely determined.Let us explain now why there are at most M (Γ (cid:48) ) possible homotopy classes if j l +1 − j l (cid:62) 2. Note firstthat j l +1 − j l (cid:62) p in that case, and that all points f j ( z e ), j l − (cid:54) j (cid:54) j l +1 + 1 belong to V ∞ . Thisimplies that neither γ (cid:48) nor γ (cid:48) are equivalent to subpaths of I j l +1 − j l +2 F ( f j l − ( z e )). Note that the latteris equivalent to a subpath of I nr F ( z e ), which is equivalent to (cid:81) (cid:54) j< n γ (cid:48) ε i . We remark that, if σ is anytransverse path that is equivalent to a subpath of (cid:81) (cid:54) j< n γ (cid:48) ε i , but that does not contain a subpaththat is equivalent to either γ (cid:48) or γ (cid:48) , then σ must be equivalent to a subpath of one the six possiblefollowing paths: γ (cid:48) , γ (cid:48) , γ (cid:48) γ (cid:48) , γ (cid:48) γ (cid:48) , γ (cid:48) γ (cid:48) , γ (cid:48) γ (cid:48) and therefore σ (and thus I j l +1 − j l +2 F ( f j l − ( z e ))) must equivalent to a subpath of γ (cid:48) . Furthermore β i = δ l I j l +1 − j j ( f j l ( z e )) δ − l +1 is disjoint from Γ (cid:48) by definition of V ∞ because it is the case for I j l +1 − j j ( f j l ( z e )) and for the disks V z jl and V z jl +1 . One can apply Proposition 9 and obtain that β i must belong to oneof the at most M (Γ (cid:48) ) different homotopy classes of paths connecting z i l and z i l +1 , as claimed before.The path (cid:81) (cid:54) l (cid:54) l ∗ β l is a closed path based at z . Noting that there exist at most 4 nr/ p = 2 nr/p integers l such that j l +1 − j l (cid:54) = 1, we deduce that there exist at most M (Γ (cid:48) ) nr/p homotopy classes(with fixed base point) possible. The loop defined by (cid:81) (cid:54) l (cid:54) l ∗ β l is freely homotopic to Γ (cid:48) e . So thereexist at most M (Γ (cid:48) ) nr/p free homotopy classes defined by the loops Γ (cid:48) e such that z e ∈ W . By Lemma35, to prove that there is at most Ln M (Γ (cid:48) ) nr/p points z e in W , it is sufficient to prove the followingstronger result: there exist at most M (Γ (cid:48) ) nr/p classes defined by the loops Γ (cid:48) e that are equivalent astransverse paths and such that z e ∈ W . Suppose that z e and z e (cid:48) belong to W and that the paths β l ,0 (cid:54) l (cid:54) l ∗ , constructed with z e and z e (cid:48) are all homotopic. Fix a lift V (cid:101) z of V z and note (cid:101) z e and (cid:101) z e (cid:48) the respective lifts of z e and z e (cid:48) that belongs to V (cid:101) z . The whole (cid:101) F -transverse trajectories of (cid:101) z e and (cid:101) z e (cid:48) are invariant by the same non trivial covering automorphism T , which is naturally defined by thelift (cid:81) (cid:54) l (cid:54) l ∗ (cid:101) β l of (cid:81) (cid:54) l (cid:54) l ∗ β l . Moreover, (cid:101) β is (cid:101) F -equivalent to a subpath of (cid:101) I (cid:101) F ( (cid:101) z e ) and to a subpathof (cid:101) I (cid:101) F ( (cid:101) z e (cid:48) ). So these two lines meet a common leaf (cid:101) φ . One deduces that they meet T k ( (cid:101) φ ), for every k ∈ Z and consequently that they meet the same leaves. So there are equivalent.The case where 0 ∈ J ∞ can be reduced to the case where 0 ∈ J < ∞ after a cyclic permutation on { , . . . , nr − } because the points z e are all fixed by f nr . (cid:3) As a direct application of Proposition 38 we can obtain the following result, which is connected to thestudy of the minimal entropy of pure braids in S . There are sharper results with a larger lower boundfor the entropy (see [So]), but they use very different techniques. Theorem 41. Let f be an orientation preserving homeomorphism on S and I a maximal hereditarysingular isotopy. Assume that there exists z ∈ dom( I ) ∩ fix( f ) such that the loop naturally defined bythe trajectory I ( z ) is not homotopic in dom( I ) to a multiple of a simple loop. Then the entropy of f is at least equal to log(2) / (4) .Proof. Let F be a foliation transverse to the isotopy. By hypothesis, the transverse loop Γ associatedto z is not a multiple of a simple loop, so by Proposition 2, it has an F -transverse self-intersection.If γ is the natural lift of Γ, then for all integers K, γ | [0 ,K ] is admissible of order K . Furthermore,by Proposition 7, γ | [0 , has an F -transverse self-intersection. The theorem then follows directly fromProposition 38. (cid:3) Associated subshifts. Let us give a natural application of Corollary 21, Corollary 22 and theresults of this section. We keep the assumptions and notations given at the beginning of the section.Consider a transverse path γ : [ a, b ] → dom( F ) with finitely many double points, none of themcorresponding to an end of the path and no triple points (by a slight modification of the argumentgiven in the proof of Corollary 24 one can show that every transverse path is equivalent to such a path).There exists real numbers a < t < . . . < t r < b and a fixed point free involution σ on { , . . . , r } such that γ ( t i ) = γ ( t σ ( i ) ), for every i ∈ { , . . . , r } and such that γ is injective on the complement of { t , . . . , t r } . Set t = a and t r +1 = b and define for every i ∈ { , . . . , r } the path γ i = γ | [ t i ,t i +1 ] .Consider the incidence matrix P ∈ M r +1 ( Z ) (indexed by { , . . . , r } ) such that P i,j = 1 if j = i + 1 or j = σ ( i + 1) and 0 otherwise (in particular if i = 2 r ). Note that the first column and the last row onlycontain 0. For every P -admissible word ( i s ) (cid:54) s (cid:54) s , which means that P i s ,i s +1 = 1 if s < s , the path ORCING THEORY FOR TRANSVERSE TRAJECTORIES OF SURFACE HOMEOMORPHISMS 49 (cid:81) (cid:54) s (cid:54) s γ i s is transverse to F . Note that every transverse path γ : [ a (cid:48) , b (cid:48) ] → dom( F ) whose image iscontained in the image of γ is a subpath of such a path (cid:81) (cid:54) s (cid:54) s γ i s .Suppose now that γ is admissible of order n . Can we decide when a path (cid:81) (cid:54) s (cid:54) s γ i s is admissible andwhat is its order? More precisely, do there exist other incidence matrices P (cid:48) smaller than P (whichmeans that P (cid:48) i,j = 0 if P i,j = 0) such that (cid:81) (cid:54) s (cid:54) s γ i s is admissible if ( i s ) (cid:54) s (cid:54) s is P (cid:48) -admissible?Corollaries 21 and 22 imply that the following three matrices satisfy this property:- the matrix P strong , where P strong i,j = 1 if and only if j = i + 1, or if j = σ ( i + 1) and γ i γ i +1 and γ j − γ j have an F -transverse intersection at γ ( t i +1 ) = γ ( t j );- the matrix P left , where P left i,j = 1 if and only if j = i + 1, or if j = σ ( i + 1) and γ has an F -transversepositive self-intersection at γ ( t i +1 ) = γ ( t j );- the matrix P right , where P right i,j = 1 if and only if j = i + 1, or if j = σ ( i + 1) and γ has an F -transverse negative self-intersection at γ ( t i +1 ) = γ ( t j ).More precisely, if P (cid:48) is one of the three previous matrices, then for every P (cid:48) -admissible word ( i s ) (cid:54) s (cid:54) s ,the path (cid:81) (cid:54) s (cid:54) s γ i s is admissible of order kn , where k is the number of s < s such that i s +1 = σ ( i s + 1). As explained in Proposition 23, its order can be less. One can adapt the proof of Theorem36 to give a lower bound to the topological entropy of f alex . For example it is at least equal to 1 /n times the logarithm of the spectral radius of P (cid:48) if the paths γ i have a leaf on their right and a leafon their left, otherwise one has to replace these paths by finite admissible words. One can adapt theproof of Theorem 29 to show that for every P (cid:48) -admissible word ( i s ) (cid:54) s (cid:54) s such that i = i s , the loopnaturally defined by (cid:81) (cid:54) s Example 1 - Leafs of the foliation are represented as dashed lines, while trans-verse paths are solid. For the first example, see Figure 17, the admissibility matrices are P strong1 = , P left1 = , P right1 = . The matrix P strong1 does not tell us anything, the only admissible paths are subpaths of γ = γ . . . γ .The only interesting informations got from P left1 and P right1 respectively are the facts that the loopsnaturally defined by γ γ and γ γ are linearly admissible of order 2. Nevertheless the first loop hasno leaf on its left while the second one has no leaf on its right. So, one cannot apply Proposition 26and deduce that the loops are equivalent to transverse loops associated to periodic points of period2. Note also that the spectral radius of P left1 and P right1 are equal to 1. In this example, one cannotdeduce neither the positivity of entropy, nor the existence of periodic orbits. γ γ γ γ γ Figure 18. Example 2 - Leafs of the foliation are represented as dashed lines, while trans-verse paths are solid. For the second example, see Figure 18, the admissibility matrices are P strong2 = ,P left2 = , P right2 = . The matrix P strong2 does not tell us anything. The matrix P right2 is nilpotent and the only admissiblepaths are γ γ , γ γ γ γ and γ , all of them admissible of order 1 by Proposition 23. The matrix P left2 is much more interesting: its spectral radius, the real root of the polynomial X − X − 1, is largerthan 1. The loop naturally defined by γ is linearly admissible of order 1 but has no leaf on its left: one ORCING THEORY FOR TRANSVERSE TRAJECTORIES OF SURFACE HOMEOMORPHISMS 51 cannot deduce that it is equivalent to a transverse loop associated to a fixed point. If p (cid:62) 1, the loopnaturally defined by γ γ p γ is linearly admissible of order p and has leaf on its right and a leaf on itsleft. More precisely, it has a transverse self-intersection, so one can apply Proposition 26 and deducethat it is equivalent to a transverse loop associated to a periodic point of period p . In particular, theloop defined by γ γ γ is equivalent to a transverse loop associated to a fixed point: one can applyTheorem 41 and deduce that the entropy of f is at least log 2 / γ γ γ γ γ Figure 19. Example 3 - Leafs of the foliation are represented as dashed lines, while trans-verse paths are solid. In third example, see figure 19, the trajectory is the same as in the first example but the foliation isdifferent. The admissibility matrices are P strong3 = ,P left3 = , P right3 = . The matrices P right3 and P left3 are the same as in the first example. Nevertheless, one can say more.Indeed the loops defined by γ γ and γ γ , which are linearly admissible of order 2, now have a leaf ontheir left and a leaf on their right. They intersect F -transversally and negatively the paths γ γ and γ γ respectively but they do not interest F -transversally and positively a path drawn on γ . So, onecannot apply the second item of Proposition 26. However, by the first item of the same proposition, ifthey are not equivalent to transverse loops associated to periodic points of period 2, they are homotopicin the domain to a simple loop that does not meet its image by f . In particular, if Ω( f ) = S , theymust be equivalent to transverse loops associated to periodic points of period 2. The matrix P strong3 is much more interesting: its spectral radius is equal to √ 2. Every path defined by a word of length n in the alphabet { γ γ , γ γ } is admissible of order 2 n and intersect γ transversally. The proofs of Theorem 36 and Theorem 29 tell us that the topological entropy of f is at least equal to log 2 / 2, andthat the number of fixed point of f n in the domain is at least equal to e n if Ω( f ) = S . γ γ γ γ γ γ γ γ γ Figure 20. Example 4 - Leafs of the foliation are represented as dashed lines, while trans-verse paths are solid. In the fourth example, see Figure 20, the foliation is the same as in the first example but the trajectoryis different. In particular, there are three points of self-intersection of γ , and all are F -transverse. Theadmissibility matrices are: P strong1 = ,P left1 = , P right1 = . By inspection of P left1 one assures the existence of a single admissible loop γ γ , while inspection of P right1 we see that both loops γ and γ γ γ γ are admissible and that the entropy of f must bepositive. 6. First applications In this section we give two applications for homeomorphisms of compact oriented surfaces. The firstone is a new proof of Handel’s result on transitive homeomorphisms of the sphere. The second appli-cation provides sufficient conditions for the existence of non-contractible periodic orbits, and has as aconsequence a positive answer to a problem posed by P. Boyland for the annulus. ORCING THEORY FOR TRANSVERSE TRAJECTORIES OF SURFACE HOMEOMORPHISMS 53 Transitive maps of surfaces of genus . In [H1], Handel prove that a transitive orientationpreserving homeomorphism f of S with at least three fixed points, but finitely many, has infinitelymany periodic orbits: more precisely the number of periodic points of period n for some iterate of f grows exponentially in n . We will improve this result as follows, with Theorem K of the introduction: Theorem 42. Let f : S → S be an orientation preserving homeomorphism such that the complementof the fixed point set is not an annulus. If f is topologically transitive then the number of periodic pointsof period n for some iterate of f grows exponentially in n . Moreover, the entropy of f is positive.Proof. Recall that, in our case, the transitivity implies the existence of a point z whose ω -limit and α -limit sets are the whole sphere. One knows that every connected component of S \ fix( f ) is invariant(seeBrown-Kister [BK]). Since f has a dense orbit, this complement must be connected. Moreover it cannotbe a disk because f has a dense orbit. Indeed the Brouwer Plane Translation Theorem implies thatevery fixed point free orientation preserving homeomorphism of the plane has only wandering points.One deduces that the fixed point set has at least three connected components. Choose three fixedpoints in different connected components and an isotopy I (cid:48) from identity to f that fixes these threefixed points (this is always possible). The restriction of I (cid:48) to the complement of these three points isa hereditary singular isotopy. Using Theorem 14 one can find a maximal hereditary singular isotopy I larger than I (cid:48) . Let F be a foliation transverse to this isotopy. It has the same domain as I , and thisdomain is not an annulus because I is larger than I (cid:48) . The fact that ω ( z ) = α ( z ) = S implies that I Z F ( z )is an admissible F -bi-recurrent transverse path that contains as a subpath (up to equivalence) everyadmissible segment and consequently that crosses all leaves of F . Since dom( I ) is not a topologicalannulus, this implies that I Z F ( z ) has an F -transverse self-intersection by Proposition 2. The resultfollows from Theorems 29 and 36. (cid:3) Existence of non-contractible periodic orbits. Let f be a homeomorphism isotopic to iden-tity on an oriented connected surface M and I (cid:48) an identity isotopy of f . A periodic point z ∈ M of period q is said to have a contractible orbit if I (cid:48) q ( z ) naturally defines a homotopically trivial loop,otherwise it is said to be non-contractible . In this subsection we examine some conditions that ensurethe existence of non-contractible periodic orbits of arbitrarily high period. Through this subsectionwe assume that ˇ M is the universal covering space of M and write ˇ π : ˇ M → M for the coveringprojection. Write ˇ I (cid:48) for the lifted identity isotopy and ˇ f for the associated lift of f . One can finda maximal hereditary singular isotopy I larger than I (cid:48) . It can be lifted to an identity isotopy ˇ I onˇdom( I ) = ˇ π − (dom( I )). This isotopy is a maximal singular isotopy of ˇ f larger than ˇ I (cid:48) . Let F be afoliation transverse to I , its lift to ˇdom( I ), denoted by ˇ F is transverse to ˇ I .The main technical result is the following proposition: Proposition 43. Suppose that there exist an admissible ˇ F -bi-recurrent path ˇ γ for ˇ f , a leaf ˇ φ of ˇ F and three distinct covering automorphisms T i , (cid:54) i (cid:54) , such that ˇ γ crosses each T i ( ˇ φ ) . Then thereexists q > and a non trivial covering automorphism T = T i ◦ T − j such that for all r/s ∈ (0 , /q ] , themaps ˇ f s ◦ T − r and ˇ f s ◦ T r have fixed points. In particular, f has non-contractible periodic points ofarbitrarily large prime period.Proof. By assumptions, there are non trivial covering automorphisms. So M is not simply connectedand ˇ M is a topological plane. For every loop ˇΓ in ˇ M , we will denote δ ˇΓ the dual function that vanisheson the unbounded connected component of ˇ M \ ˇΓ. It is usually called the winding number of ˇΓ. Sub-lemma 44. If ˇΓ is a loop positively transverse to ˇ F , the set of singular points ˇ z of ˇ F such that δ ˇΓ (ˇ z ) (cid:54) = 0 is a non empty compact subset Σ ˇΓ of ˇ M . Furthermore Σ ˇΓ = Σ ˇΓ (cid:48) if ˇΓ and ˇΓ (cid:48) are equivalenttransverse loops.Proof. The fact that Σ ˇΓ = Σ ˇΓ (cid:48) if ˇΓ and ˇΓ (cid:48) are equivalent transverse loops is obvious as is the fact thatΣ ˇΓ compact. To prove that this set is not empty, let us consider a leaf ˇ φ that meets ˇΓ. As recalled inthe first section, at least one the two following assertions is true:- the set α ( ˇ φ ) is a non empty compact set and δ ˇΓ takes a constant positive value on it;- the set ω ( ˇ φ ) is a non empty compact set and δ ˇΓ takes a constant negative value on it.Suppose for instance that we are in the first situation. If α ( ˇ φ ) contains a singular point, we are done.If not, it is a closed leaf disjoint from ˇΓ. More precisely, α ( ˇ φ ) is contained in a bounded connectedcomponent of ˇ M \ ˇΓ where δ ˇΓ takes a constant positive value. This component contains the boundedcomponent of the complement of α ( ˇ φ ) and so contains a singular point. (cid:3) Let us prove first that there exists an admissible loop ˇΓ that crosses each T i ( ˇ φ ). Suppose first that ˇ γ has a ˇ F -transverse self-intersection and choose a , b , t , a , b , t be such that ˇ γ | [ a ,b ] intersects ˇ F -transversally ˇ γ | [ a ,b ] at ˇ γ ( t ) = ˇ γ ( t ) and such that ˇ γ | [ a ,b ] crosses each T i ( ˇ φ ). The constructiondone in the proof of Lemma 30 gives us such a loop ˇΓ. Suppose now that ˇ γ has no ˇ F -transverseself-intersection. By Proposition 2, one knows that ˇ γ is equivalent to the natural lift of a simple loopˇΓ and this loop satisfies the desired property.Let us prove now that one can find at least two distinct loops among the T − i (ˇΓ) that have a ˇ F -transverse intersection. If not, by Proposition 1, one can find for every i ∈ { , , } a transverse loopˇΓ (cid:48) i equivalent to T − i (ˇΓ) such that the ˇΓ (cid:48) i are pairwise disjoint. The three functions δ ˇΓ (cid:48) i are decreasingon the leaf ˇ φ . For each i , either δ ˇΓ (cid:48) i is not null in α ( ˇ φ ) or δ ˇΓ (cid:48) i is not null in ω ( ˇ φ ), and therefore eitherthere exists two different indices i and j such that for all points in α ( ˇ φ ) , δ ˇΓ (cid:48) i (cid:54) = 0 and δ ˇΓ (cid:48) j (cid:54) = 0, or thereexists two different indices i and j such that for all points in ω ( ˇ φ ) , δ ˇΓ (cid:48) i (cid:54) = 0 and δ ˇΓ (cid:48) j (cid:54) = 0. In any case,there exists a point ˇ z ∈ ˇ φ and two different indices i and j such that δ ˇΓ (cid:48) i (ˇ z ) (cid:54) = 0 and δ ˇΓ (cid:48) j (ˇ z ) (cid:54) = 0. Thefact that there exists a point where the two dual functions do not vanish tells us that one of the loops,let us say ˇΓ (cid:48) i , is included in a bounded connected component of the complement of the other one ˇΓ (cid:48) j ,and that ˇΓ (cid:48) j is included in the unbounded connected component of the complement of ˇΓ (cid:48) i . One deducesthat Σ ˇΓ (cid:48) i ⊂ Σ ˇΓ (cid:48) j . Setting T = T j ◦ ( T i ) − , one gets the inclusion T (Σ ˇΓ ) ⊂ Σ ˇΓ , where Σ ˇΓ is a non emptycompact set. We have found a contradiction because T is a non trivial covering automorphism.We have proved that there exist i (cid:54) = j such that T − i (ˇΓ) and T − j (ˇΓ) intersect ˇ F -transversally. Thisimplies that ˇΓ and T (ˇΓ) intersect ˇ F -transversally, where T = T j ◦ ( T i ) − . Write ˇ γ for the naturallift of ˇΓ and choose an integer L sufficiently large, so that ˇ γ | [0 ,L ] has a ˇ F -transverse intersection with T (ˇ γ ) | [0 ,L ] at ˇ γ ( t ) = T (ˇ γ )( s ), with s < t . The loop ˇΓ being admissible, there exists q > γ | [ − L, L ] is admissible of order q . It follows from Corollaries 22 and Proposition19 that, for any n > n − (cid:89) i =0 T i (cid:0) ˇ γ | [ s − L,t + L ] (cid:1) , n − (cid:89) i =0 T − i (cid:0) ˇ γ | [ t − L,s + l ] (cid:1) are admissible of order nq , and both have ˇ F -transverse self-intersections. Therefore the pathsˇ γ | [ s − L,t + L ] and ˇ γ | [ t − L,s + l ] project onto closed paths of M and the two loops naturally defined have ORCING THEORY FOR TRANSVERSE TRAJECTORIES OF SURFACE HOMEOMORPHISMS 55 F -transverse self-intersection and are linearly admissible. So, one can deduce Proposition 43 fromProposition 26. (cid:3) Let us state a first application of Proposition 43. In [T] conditions are given for a homeomorphism f , isotopic to the identity, of a compact surface M to have only contractible periodic points. There itis shown, using Nielsen-Thurston theory, that for such f , under a suitable condition on the size of itsfixed point set, there exists an uniform bound on the diameter of the orbits of periodic points. Thenext theorem improves the main result of that note, by extending the uniform bound on the diameterof orbits from ˇ f periodic points to ˇ f recurrent points. Note that the hypothesis that the fixed pointset of ˇ f project in a disk cannot be removed. There exists an example of a C ∞ diffeomorphism f of T preserving the Lebesgue measure and ergodic such that every periodic orbit of f is contractible,and such that almost all points in the lift have orbits unbounded in every direction (see [KT1]). Thefollowing is Theorem H of the introduction: Theorem 45. We suppose that M is compact and furnished with a Riemannian structure. We endowthe universal covering space ˇ M with the lifted structure and denote by d the induced distance. Let f be a homeomorphism of M isotopic to the identity and ˇ f a lift to ˇ M naturally defined by the isotopy.Assume that there exists an open topological disk U ⊂ M such that the fixed point set of ˇ f projects into U . Then;- either there exists K > such that d ( ˇ f n (ˇ z ) , ˇ z ) (cid:54) K , for all n (cid:62) and all bi-recurrent point ˇ z of ˇ f ;- or there exists a nontrivial covering automorphism T and q > such that, for all r/s ∈ ( − /q, /q ) ,the map ˇ f s ◦ T − r has a fixed point. In particular, f has non-contractible periodic points of arbitrarilylarge prime period.Proof. Let I be a maximal hereditary singular isotopy larger than the given isotopy and F a foliationtransverse to I . Denote ˇ M the universal covering space of M and ˇ π : ˇ M → M the covering projection.Write ˇ I (cid:48) for the lifted identity isotopy on ˇdom( I ) = ˇ π − (dom( I )) and ˇ F for the lifted foliation. Thetheorem follows directly from the next lemma and Proposition 43. Lemma 46. There exists K > such that, for all ˇ z in ˇ M and all n (cid:62) , if d ( ˇ f n (ˇ z ) , ˇ z ) (cid:62) K , thenthere exists a leaf ˇ φ and three distinct covering automorphisms T i , (cid:54) i (cid:54) , such that I n ˇ F (ˇ z ) crosseseach T i ( ˇ φ ) .Proof. One can find a neighborhood V ⊂ U of sing( I ) such that for every point ˇ z ∈ π − ( V ), thepoints ˇ z and ˇ f (ˇ z ) belong to the same connected component of π − ( U ). For reasons explained in theproof of Lemma 18, one knows that for every z ∈ M \ V , there exists a small open disk O z ⊂ dom( F )containing z such that I F ( f − ( z (cid:48) )) crosses φ z if z (cid:48) ∈ O z . By compactness of M \ V , one can coverthis set by a finite family ( O z i ) (cid:54) i (cid:54) r . One constructs easily a partition ( X z i ) (cid:54) i (cid:54) r of M \ V such that X z i ⊂ O z i . We have a unique partition ( ˇ X α ) α ∈A of ˇ M such that, either ˇ X α is contained in a connectedcomponent of π − ( U ) and projects onto V , or there exists i ∈ { , . . . , r } such that ˇ X α is contained ina connected component of π − ( O z i ) and projects onto X z i . Write α (ˇ z ) = α if ˇ z ∈ ˇ X α . Let us define K = max ˇ z ∈ ˇ M d ( ˇ f (ˇ z ) , ˇ z )) , K = max α ∈A diam( ˇ X α ) . Fix ˇ z ∈ ˇ M , n (cid:62) n < n < . . . < n s in the following inductive way: n = 0 , n j +1 = 1 + sup { k ∈ { n j , . . . , n − } | α ( ˇ f k (ˇ z )) = α ( ˇ f n j (ˇ z )) } , n s = n. Note that d ( ˇ f n j (ˇ z ) , ˇ f n j +1 (ˇ z )) (cid:54) K + K , if j < s . Note also that, if ˇ X α ( ˇ f nj (ˇ z )) projects on V , then f n j +1 − ( z ) also belongs to V and by the choice of V both ˇ f n j +1 − (ˇ z ) and ˇ f n j +1 (ˇ z ) belong to thesame connected component of π − ( U ). As α ( ˇ f n j (ˇ z )) (cid:54) = α ( ˇ f n j +1 (ˇ z )), one gets that ˇ X α ( ˇ f nj +1 (ˇ z )) do notproject on V . Fix K > (6 r + 1)( K + K ). If d ( ˇ f n (ˇ z ) , ˇ z ) (cid:62) K , then s (cid:62) r + 1 and there exist at least3 r sets ˇ X α ( ˇ f nj (ˇ z )) , 0 < j < s , that do not project on V . This implies that there exist three points f n jl ( π ( z )) that belong to the same X z i , and therefore one finds that there exist a point ˇ z i ∈ π − ( z i )and two distinct nontrivial covering automorphisms T , T such that the orbit of ˇ z intersects the threedistinct connected components of π − ( O z i ) that contain ˇ z i , T (ˇ z i ) and T (ˇ z i ), respectively. By thechoice of O z i , this implies that I n ˇ F (ˇ z ) intersect φ ˇ z i , T ( φ ˇ z i ) and T ( φ ˇ z i ). (cid:3) Proposition 43 is also fundamental in solving the following conjecture posed by P. Boyland (see, forinstance, [AT] where the conjecture is shown to be true generically for sufficiently smooth diffeomor-phisms): Let f be a homeomorphism of the closed annulus preserving a probability measure µ withfull support, and let ˇ f be a lift of f to the universal covering space of the annulus. If the rotation setof f is a non trivial segment and the rotation number of µ is null, is it true that rot( µ ) belongs to theinterior of the rotation set?We recall first Atkinson’s Lemma on ergodic theory, that will be very useful in this paper (see [A]). Proposition 47. Let ( X, B , µ ) be a probability space, and let T : X → X be an ergodic automorphism.If ϕ : X → R is an integrable map such that (cid:82) ϕ dµ = 0 , then for every B ∈ B and every ε > , onehas µ (cid:32)(cid:40) x ∈ B, ∃ n (cid:62) , T n ( x ) ∈ B and (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) n − (cid:88) k =0 ϕ ( T k ( x )) (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) < ε (cid:41)(cid:33) = µ ( B ) . We have the following, Theorem A of the introduction: Theorem 48. Let f be a homeomorphism of A = T × [0 , that is isotopic to the identity and ˇ f alift to R × [0 , . Suppose that rot( f ) is a non trivial segment and that one of its endpoint ρ is rational.Define M ρ = { µ ∈ M ( f ) , rot( µ ) = ρ } , X ρ = (cid:91) µ ∈M ρ supp( µ ) . Then every invariant measure supported on X ρ belongs to M ρ .Proof. Replacing f by a power f q and ˇ f by a lift ˇ f q ◦ T p , one can assume that ρ = 0 and rot( f ) = [0 , a ],where a > 0. The fact that 0 is extremal implies that for every µ ∈ M , each ergodic measure µ (cid:48) that appears in the ergodic decomposition of µ also belongs to M . Atkinson’s Lemma, with T = f and ϕ the map lifted by ˇ ϕ : z (cid:55)→ π ( ˇ f (ˇ z ) − ˇ z ), tells us that µ (cid:48) -almost every point of A is lifted toa recurrent point of ˇ f . The union of the supports of such ergodic measures being dense in supp( µ ),one deduces that the recurrent set of ˇ f is dense in π − ( X ). Writing f = ( f , f ), one can extend f to a homeomorphism of T × R such that f ( x, y ) = ( f ( x, , y ) if y (cid:62) f ( x, y ) = ( f ( x, , y ) if y (cid:54) f the lift that extends the initial lift. Let I (cid:48) be an identity isotopy of f thatis lifted to an identity isotopy ˇ I (cid:48) of ˇ f . Let I be a maximal hereditary singular isotopy larger than I (cid:48) and F a foliation transverse to I . Consider the lift ˇ I of I and the lifted foliation ˇ F . If there exists aninvariant measure supported on X whose rotation number is positive, there exists a recurrent point z of rotation number strictly larger than 0. Let us fix a lift ˇ z . As ˇ z is not fixed by ˇ f , it belongs to thedomain of ˇ I and the path I Z ˇ F (ˇ z ) meets infinitely many translates of φ ˇ z . But ˇ z can be approximated by ORCING THEORY FOR TRANSVERSE TRAJECTORIES OF SURFACE HOMEOMORPHISMS 57 a recurrent point ˇ z (cid:48) of ˇ f because we have seen that the recurrent set of ˇ f was dense in π − ( X ). So wecan suppose that I Z ˇ F (ˇ z (cid:48) ) meets at least three translates of φ ˇ z . The result now follows from Proposition43. Indeed, one finds some power n of T such that for any pair of integers r, s with s > | r/s | is sufficiently small, there exists a fixed point ˇ z r,s of ˇ f r ◦ T − ns . The points ˇ z r,s project toperiodic points z r,s in A such that the rotation number of z r,s is r/ns . In particular both 1 /sn and − /sn belong to the rotation set of ˇ f if s is sufficiently large, in contradiction with the fact that 0 isan end of rot( f ). (cid:3) We deduce immediately the positive answer to Boyland’s question, Corollary B of the introduction: Corollary 49. Let f be a homeomorphism of A that is isotopic to the identity and preserves a prob-ability measure µ with full support. Let us fix a lift ˇ f . Suppose that rot( f ) is a non trivial segment.The rotation number rot( µ ) cannot be an endpoint of rot( f ) if this endpoint is rational.Proof. If rot( f ) is an endpoint of rot( f ) and this endpoint is a rational ρ , by Theorem 48 we get that,if X ρ = supp( µ ) is the whole annulus A , then every invariant measure supported on A has rotationvector ρ , which implies that rot( f ) = { ρ } . (cid:3) Entropy zero conservative homeomorphisms of the sphere We will prove in this section the improvement of Franks-Handel’s result about area preserving dif-feomorphisms of S with entropy zero, stated in the introduction as Theorem M. Let us begin byintroducing an important notion due to Franks and Handel: let f : S → S be an orientation preserv-ing homeomorphism, a point z is free disk recurrent if there exist an integer n > D containing z and f n ( z ) such that f ( D ) ∩ D = ∅ . We will also need the notion of heteroclinicpoint , which means that its α -limit and ω -limit sets are included in connected subsets of fix( f ).Let us state first some easy but useful facts. By definition if f is a homeomorphism of a topologicalspace X , a subset Y is free if f ( Y ) ∩ Y = ∅ . Proposition 50. One has the following results: i) the set of free disk recurrent points is an invariant open set fdrec( f ) ; ii) it contains every positively or negatively recurrent point outside fix( f ) ; iii) every point in S \ fdrec( f ) is heteroclinic; iv) every periodic connected component of fdrec( f ) is fixed.Proof. If D is a free disk that contains z and f n ( z ), it contains z (cid:48) and f n ( z (cid:48) ) if z (cid:48) is close to z . Moreover f k ( D ) is a free disk that contains f k ( z ) and f k + n ( z ), for every k ∈ Z . So i) is true.For every z ∈ S \ fix( f ), one can choose a free disk D that contains z . If z is positively recurrent, thereexists n > f n ( z ) ∈ D . If z is negatively recurrent, there exist n > f − n ( z ) ∈ D ,which implies that f n ( D ) is a free disk that contains z and f n ( z ). In both cases, z belongs to fdrec( f ),which means that ii) is true.It is sufficient to prove iii) for the ω -limit set, the proof for the α -limit set being similar. Let us provefirst that ω ( z ) ⊂ fix( f ) if z (cid:54)∈ fdrec( f ). Indeed, if z (cid:48) ∈ ω ( z ) \ fix( f ), one can choose a free disk D containing z (cid:48) and two integers n (cid:48) > n such that f n ( z ) and f n (cid:48) ( z ) belong to D . It implies that f − n ( D ) is a free disk that contains z and f n (cid:48) − n ( z ). This contradicts the fact that z (cid:54)∈ fdrec( f ). To prove that ω ( z ) is included in a connected component of fix( f ), it is sufficient to prove that it is contained ina connected component of O , for every neighborhood O of fix( f ). If O is such a neighborhood,thereexists a neighborhood O (cid:48) ⊂ O of fix( f ) such that for every z ∈ O (cid:48) ∩ f − ( O (cid:48) ), the points z and f ( z )belong to the same connected component of O . There exists N such that f n ( z ) ∈ O (cid:48) for every n (cid:62) N .This implies that the f n ( z ), n (cid:62) N , belong to the same connected component of O .It remains to prove iv) . If W is a connected component of fdrec( f ) of period q > 1, it is not aconnected component of S \ fix( f ) (see Brown-Kister [BK]) and so one can find a path α in S \ fix( f )joining a point z ∈ W to a point z (cid:48) (cid:54)∈ W . Taking a subpath if necessary, one can suppose that γ isincluded in W but the endpoint z (cid:48) (which is in the frontier of W and not fixed). Let us choose a path β in W joining z to f q ( z ). It is a classical fact that there exists a simple path γ joining z (cid:48) to f q ( z (cid:48) )whose image is included in the image of α − βf q ( α ). The point z (cid:48) is not periodic because it is neitherin fix( f ) nor in fdrec( f ) and so the points z (cid:48) , f ( z (cid:48) ), f q ( z (cid:48) ) and f q +1 ( z (cid:48) ) are distinct (recall that q > γ ⊂ W and W is free, the path γ is free and so one can find a free disk thatcontains it, which contradicts the fact that z (cid:48) is not in fdrec( f ). (cid:3) Suppose now that the set of positively recurrent points is dense. It is equivalent to say that Ω( f ) = S and in that case the set of positively recurrent points is a dense G δ set, as is the set of bi-recurrentpoints (these conditions are satisfied in the particular case of an area preserving homeomorphism).Note that, in this case, every connected component of fdrec( f ) is periodic and so is fixed. Write( W α ) α ∈A f for the family of connected components of fdrec( f ) and define A α to be the interior in S \ fix( f ) of the closure of W α . Note that A α = S \ (cid:91) α (cid:48) ∈A f \{ α } A α (cid:48) ∪ fix( f )because the recurrent points are dense in S and contained in fdrec( f ) if not fixed.We will prove the following result, which implies Theorem M of the introduction, and that extendsTheorem 1.2 of [FH]. Theorem 51. Let f : S → S be an orientation preserving homeomorphism such that Ω( f ) = S and h ( f ) = 0 . Then one has the following results: i) each A α is an open annulus; ii) the sets A α are the maximal fixed point free invariant open annuli; iii) every point that is not in a A α is heteroclinic; iii) let C be a connected component of the frontier of A α in S \ fix( f ) , then the connected componentsof fix( f ) that contain α ( z ) and ω ( z ) are independent of z ∈ C . We will begin by stating a local version of this result, which means a version relative to a given maximalhereditary singular isotopy I . We denote (cid:101) I the lifted identity isotopy to the universal covering space (cid:103) dom( I ) of dom( I ) and (cid:101) f the induced lift of f | dom( I ) . Say that a point z ∈ dom( I ) is free disk recurrentrelative to I or I free disk recurrent if there exists an integer n > D ⊂ dom( I ) containing z and f n ( z ), such that each lift to (cid:103) dom( I ) is disjoint from its image by (cid:101) f (wewill say that D is I -free). Let us state the local version of Proposition 50. ORCING THEORY FOR TRANSVERSE TRAJECTORIES OF SURFACE HOMEOMORPHISMS 59 Proposition 52. One has the following results: i) the set of I -free disk recurrent points is an invariant open set fdrec( I ) ; ii) it contains every positively or negatively recurrent point in dom( I ) ; iii) every point in S \ fdrec( I ) is heteroclinic and its α -limit and ω -limit sets are included in connectedsubsets of sing( I ) ; iv) every periodic connected component of fdrec( I ) is fixed and lifted to fixed subsets of ˇ f .Proof. Replacing free disks by I -free disks, one proves the three first assertions exactly like in theglobal situation. Similarly, one can prove that every periodic connected component of fdrec( I ) is fixed.Writing π : (cid:103) dom( I ) → dom( I ) for the universal covering projection, it remains to prove that theconnected components of π − ( W ) are fixed by (cid:101) f , if W is a fixed connected component of fdrec( I ).If they are not fixed, they are not connected components of (cid:103) dom( I ), which means that W is not aconnected component of dom( I ). So one can find a simple path α joining a point z ∈ W to a point z (cid:48) ∈ ∂W ∩ dom( I ) and included in W but the endpoint z (cid:48) , and then construct a simple path γ joining z (cid:48) to f ( z (cid:48) ) included in W but the two endpoints. It will lift to a (cid:101) f - free simple path and so one canfind a I -free disk that contains γ . This contradicts the fact that z (cid:48) is not in fdrec( I ). (cid:3) Suppose now that Ω( f ) = S . Write ( W β ) β ∈B I for the family of connected components of fdrec( I ) anddefine A β to be the interior in dom( I ) of the closure of W β . One knows that the sets W β , A β are fixedand lifted to fixed subsets of (cid:101) f . Here again, one has A β = dom( I ) \ (cid:91) β (cid:48) ∈B I \{ β } A β (cid:48) . The local version of Theorem 51 is the following: Theorem 53. Let f : S → S be an orientation preserving homeomorphism such that Ω( f ) = S and h ( f ) = 0 , and I a hereditary singular maximal isotopy. Then one has the following results: i) each A β is an open annulus; ii) the sets A β are the maximal invariant open annuli contained in dom( I ) ; iii) every point that is not in a A β is heteroclinic and its α -limit and ω -limit sets are included inconnected subsets of sing( I ) ; iv) let C be connected component of the frontier of A β in dom( I ) , then the connected components of sing( I ) that contain α ( z ) and ω ( z ) are independent of z ∈ C . Let us explain first why the local theorem implies the global one. If A is a topological annulus, an openset will be called essential if it contains an essential loop and inessential otherwise. A closed set willbe called inessential if there exists a connected component of its complement that is a neighborhoodof the two ends in the case where A is open, that meets the two boundary circle in the case where A isclosed, and that is a neighborhood of the unique end and meets the boundary circle in the remainingcase. Otherwise, we will say that this set is essential . Proof of Theorem 51, first part, proof of assertions (i), (ii), and (iii) . Let us explain first why everyfixed point free invariant open annulus A is contained in an A α , α ∈ A f . It is sufficient to prove that fdrec( f ) ∩ A is connected. Indeed fdrec( f ) ∩ A will be contained in a W α , α ∈ A f , and consequently A will be contained in A α . Let W be a connected component of fdrec( f ) ∩ A . Applying Proposition 50to the end compactification of A , one knows that W is fixed. If it is inessential, one gets an invariantopen disk D ⊂ A by filling W , which means adding the inessential components of its complement.By Brouwer’s plane translation Theorem, since the restriction of f to D has non wandering points,there must exist a fixed point in this disk, which is impossible. So, every connected component offdrec( f ) ∩ A is essential. Suppose now that fdrec( f ) ∩ A has at least two connected components. Thecomplement in A of the union of two such components has a unique compact connected component. Itis located “between” these components. This last set is invariant (by uniqueness) and contains pointsthat are not free disk recurrent. But one knows that the α -limit and ω -limit sets of such points containfixed points and A is fixed point free. We have a contradiction.Let us prove now that each A α , α ∈ A f , is an annulus. It is sufficient to prove that it is contained in afixed point free invariant annulus. Let us consider a sequence ( z i ) i (cid:62) dense in fix( f ), sequence whichwill be finite if there are finitely many fixed points. Let us fix A α . Let I be a maximal hereditarysingular isotopy whose singular set contains z , z , z . The set W α is connected and included infdrec( I ) so it is contained in a connected component W β , β ∈ B I . One deduces that A α ⊂ A β . If A β is fixed point free, we stop the process. If not, we consider the first z k that belongs to A β andconsider a maximal hereditary singular isotopy I of f | A β whose singular set contains z k . Similarly,there exists β ∈ B I such that A α ⊂ A β . If A β is fixed point free, we stop the process. If not, weconsider the first z k that belongs to A β and we continue. If the process stops, the last annulus willbe fixed point free. If the process does not stop, A α is contained in the interior of (cid:84) i (cid:62) A β i . Theconnected component W of the interior of (cid:84) i (cid:62) A β i that contains A α is invariant. Moreover, it is fixedpoint free because it is open and because the sequence ( z i ) i (cid:62) is dense in fix( f ) and away from W . Letus prove that for i large enough, A β i +1 is essential in A β i and that W is an annulus. Let us supposethat A β i +1 is inessential in A β i for infinitely many i . Consider a simple loop Γ in W . It bounds adisk (uniquely determined) included in A β i , every time A β i +1 is inessential in A β i , which implies thatit bounds a disk included in (cid:84) i (cid:62) A β i , and so included in W . This means that W is a disk, whichcontradicts the fact it is fixed point free. Suppose now that A β i +1 is essential in A β i for every i (cid:62) i .By the same reasoning, if Γ ⊂ W is a simple loop such that Γ is inessential in the A β i , i (cid:62) i , then Γbounds a disk in W . This implies that W is an open annulus, that is essential in the A β i , i (cid:62) i .The assertion iii) is obvious because every free disk recurrent point is contained in a W α and so in an A α . We will postpone the proof of iv) to the end of this section because we need a little bit more thanwhat is stated in the local theorem. (cid:3) Before proving Theorem 53, we will state a result relative to a couple ( I, F ), where F is a foliationtransverse to I . By Theorem 36 and the density of the set of recurrent points, one knows that twotransverse trajectories never intersect F -transversally. In particular, there is no transverse trajectorywith F -transverse self-intersection and by Proposition 2 every whole transverse trajectory of an F -bi-recurrent point is equivalent to the natural lift of a transverse simple loop Γ. We denote by G I, F the setof such loops (well defined up to equivalence) and rec( f ) Γ the set of bi-recurrent points whose wholetransverse trajectory is equivalent to the natural lift of Γ. Consider a point z ∈ dom( I ). For any givensegment of I Z F ( z ) there exists a neighborhood of z such that this segment is equivalent to a subpathof I Z F ( z (cid:48) ) if z (cid:48) belongs to this neighborhood. Suppose now that this segment meets a leaf more thanonce. The transverse simple loop Γ associate to z (cid:48) does not depend on z (cid:48) , if z (cid:48) is chosen bi-recurrent ORCING THEORY FOR TRANSVERSE TRAJECTORIES OF SURFACE HOMEOMORPHISMS 61 (remind that the set of bi-recurrent points is dense). Summarizing, we have stated that any segment of I Z F ( z ) is equivalent to a subpath of the natural lift of a transverse simple loop, but this loop is uniquelydefined (up to equivalence) if this segment meets a leaf more than once. Consequently, if I Z F ( z ) meetsa leaf more than once, it is equivalent to a subpath of the natural lift of a uniquely defined transversesimple loop. One deduces that the set of points whose whole transverse trajectory meets a leaf morethan once, admits a partition (cid:70) Γ ∈G I, F W Γ in disjoint invariant open sets, where z ∈ W Γ if I Z F ( z ) meetsa leaf at least twice and is a subpath of the natural lift of Γ. Define A Γ = int( W Γ ). Note that A Γ = int(rec( f ) Γ ) = dom( I ) \ (cid:91) Γ (cid:48) ∈G I, F \{ Γ } A Γ (cid:48) . Recall that U Γ is the union of leaves that meet Γ. Proposition 54. One has the following results: i) the set A Γ is an essential open annulus of U Γ ; ii) every point in dom( I ) \ (cid:83) Γ ∈G I,F A Γ is heteroclinic and its α -limit and ω -limit sets are included inconnected subsets of sing( I ) ; iii) let C be a connected component of the frontier of A Γ in dom( I ) , then the connected componentsof sing( I ) that contain α ( z ) and ω ( z ) are independent of z ∈ C . Proof of Proposition 54. This subsection is devoted entirely to the proof of Proposition 54.The assertion ii) is an immediate consequence of the following: if z (cid:48) ∈ dom( I ) belongs to the α -limitor ω -limit set of z ∈ dom( I ), then the whole transverse trajectory of z meets infinitely often the leaf φ z (cid:48) and so z belongs to (cid:83) Γ ∈G I, F W Γ .Let us prove i) . One can always suppose that dom( I ) is connected, otherwise one must replace dom( I )by its connected component that contains Γ in what follows. Fix a lift (cid:101) γ of Γ in (cid:103) dom( I ), write T forthe covering automorphism such that (cid:101) γ ( t + 1) = T ( (cid:101) γ ( t )), write (cid:100) dom( I ) = (cid:103) dom( I ) /T for the annularcovering space associated to Γ. Denote by (cid:98) π : (cid:100) dom( I ) → dom( I ) the covering projection, by (cid:98) I theinduced identity isotopy, by (cid:98) f the induced lift of f , by (cid:98) F the induced foliation. The line (cid:101) γ projectsonto the natural lift of a loop (cid:98) Γ. The union of leaves that meet (cid:98) Γ, denoted by U (cid:98) Γ , is the annularcomponent of (cid:98) π − ( U Γ ). We note that there cannot be an essential simple closed curve (cid:98) Γ (cid:48) contained in U (cid:98) Γ whose image by (cid:98) f is disjoint from itself, otherwise the region bounded by (cid:98) π ( (cid:98) Γ (cid:48) ) and (cid:98) π ( (cid:98) f ( (cid:98) Γ (cid:48) )) wouldbe wandering for any f . In particular, by the same reasoning as in Lemma 27, one gets that the α and ω limit of any given leaf of (cid:98) F that is contained in U (cid:98) Γ must be different ends of (cid:100) dom( I ), and every leafof (cid:98) F that does not intersect U (cid:98) Γ disconnects (cid:100) dom( I ). One gets a sphere (cid:100) dom( I ) sph by adding the end N of (cid:100) dom( I ) at the left of Γ and the end S at the right. The complement of U (cid:98) Γ has two connectedcomponents l ( (cid:98) Γ) ∪ { N } and r ( (cid:98) Γ) ∪ { S } . Note that (cid:98) Γ is the unique simple loop (up to equivalence)that is transverse to (cid:98) F . Like in dom( I ), transverse trajectories do not intersect (cid:98) F -transversally. Theset of points that lift a bi-recurrent point of f is dense. If the trajectory of such a point z meets a leafat least twice, then (cid:98) I Z (cid:98) F ( z ) is the natural lift of (cid:98) Γ. Denote rec( f ) (cid:98) Γ the set of such points. Otherwise (cid:98) I Z (cid:98) F ( z ) meets either l ( (cid:98) Γ) or r ( (cid:98) Γ), the two situations being excluded, because (cid:98) I Z (cid:98) F ( z ) does not intersect (cid:98) Γ (cid:98) F -transversally. Denote rec( f ) N and rec( f ) S the set of points z that lift a bi-recurrent point of f and such that (cid:98) I Z (cid:98) F ( z ) meets l ( (cid:98) Γ) and r ( (cid:98) Γ) respectively. Note that the intersection of the completetransverse trajectory of z ∈ rec( f ) N and U (cid:98) Γ , when not empty is equivalent to (cid:98) Γ | J where J is an open interval of T , and a similar statement holds if z ∈ rec( f ) S . In particular there exists n (cid:62) (cid:98) I N (cid:98) F ( (cid:98) f n ( z )) ⊂ l ( (cid:98) Γ) and (cid:98) I − N (cid:98) F ( (cid:98) f − n ( z )) ⊂ l ( (cid:98) Γ). Write W (cid:98) Γ for the set of points such that (cid:98) I Z (cid:98) F ( z ) meets a leafat least twice, write W N for the set of points z ∈ (cid:100) dom( I ) such that (cid:98) I Z (cid:98) F ( z ) meets l ( (cid:98) Γ), write W S forthe set of points such that (cid:98) I Z (cid:98) F ( z ) meets r ( (cid:98) Γ). We get three disjoint invariant open sets, that containrec( f ) (cid:98) Γ , rec( f ) N , rec( f ) S respectively and whose union is dense. Note that the α -limit and ω -limitsets of a point z (cid:54)∈ W (cid:98) Γ are reduced to one of the ends. These ends are both equal to N if z ∈ rec( f ) N and both equal to S if z ∈ rec( f ) S . We will see later that they are both equal to N if z ∈ W N andboth equal to S if z ∈ W S . Note also that W N = (cid:91) k ∈ Z f − k ( l (Γ)) , W S = (cid:91) k ∈ Z f − k ( r (Γ)) . Indeed, every leaf φ that is not in U (cid:98) Γ bounds a disk disjoint from U (cid:98) Γ . So, if (cid:98) I (cid:98) F ( z ) meets φ and φ ⊂ l (Γ),then one of the point z or (cid:98) f ( z ) is in l (Γ), and if φ ⊂ r (Γ), then one of the point z or (cid:98) f ( z ) is in r (Γ).Observe that W (cid:98) Γ projects homeomorphically on W Γ and that A (cid:98) Γ = int( W (cid:98) Γ ) = (cid:100) dom( I ) sph \ W N ∪ W S projects homeomorphically on A Γ . We want to prove that A (cid:98) Γ is an annulus. Lemma 55. There exists a leaf φ S in U (cid:98) Γ that does not meet W N .Proof. Recall that the intersection of the whole transverse trajectory of z ∈ rec( f ) N and U (cid:98) Γ , whennot empty is equivalent to (cid:98) Γ | J where J is an open interval of T . Consider the set J of such intervals.The fact that there are no transverse intersection tells us that these intervals do not overlap: if twointervals intersect, one of them contains the other one. One deduces that there exists t ∈ T that doesnot belong to any J . Indeed, by a compactness argument, if T can be covered by the intervals of J ,there exists r (cid:62) r such intervals but not less. By connectedness, atleast two of the intervals intersect and one can lower the number r . Set φ S = φ (cid:98) Γ( t ) . The set rec( f ) N being dense in W N , the leaf φ S does not meet the whole transverse trajectories of points in W N . Inparticular, it does not meet W N . (cid:3) Lemma 56. The set O S of points whose whole transverse trajectory meets φ S is a connected essentialopen set.Proof. Fix a lift (cid:101) φ of φ S in (cid:103) dom( I ). The set (cid:101) O of points whose trajectory meets (cid:101) φ is equal to (cid:83) k ∈ Z (cid:101) f − k (cid:16) L ( (cid:101) φ ) ∩ R ( (cid:101) f ( (cid:101) φ )) (cid:17) , it its connected and simply connected. So its projection O S is con-nected. Every lift of a point in rec( f ) (cid:98) Γ belongs to all the translates T k ( (cid:101) O ), k ∈ Z . So the union of thetranslates is connected, which means that O S is essential. (cid:3) Lemma 57. The set W N does not contains S and for every z ∈ W N , one has α ( z ) = ω ( z ) = { N } .Proof. The set W N is connected because it can be written W N = (cid:91) k ∈ Z f − k ( l (Γ) ∪ { N } ) . It does not contain S because it is connected and does not intersect the essential open set O S . Moreover,one knows that the α -limit and ω -limit sets of points in W N are reduced to one of the ends. Theyboth are equal to N , because S (cid:54)∈ W N . (cid:3) ORCING THEORY FOR TRANSVERSE TRAJECTORIES OF SURFACE HOMEOMORPHISMS 63 Similarly, there exists a leaf φ N in U (cid:98) Γ that does not meet W S and the set O N of points whose wholetransverse trajectory meets φ N is a connected essential open set. Moreover, N (cid:54)∈ W S and for every z ∈ W S , one has α ( z ) = ω ( z ) = { S } . Consequently W N and W S do not intersect. Two points in O S ∩ O N are not separated neither by W N nor by W S , because O S and O N are connected and disjointfrom W N and W S respectively. So they are not separated by W S ∪ W N because W S ∩ W N = ∅ . Onededuces that O S ∩ O N is contained in a connected component O of the complement of W S ∪ W N , whichis nothing but A (cid:98) Γ . So we have W (cid:98) Γ ⊂ O S ∩ O N ⊂ O ⊂ A (cid:98) Γ ⊂ W (cid:98) Γ . We deduce that the sets appearing in the inclusions have the same closure and that A (cid:98) Γ is connected be-cause O ⊂ A (cid:98) Γ ⊂ O . To conclude that A (cid:98) Γ is an essential annulus, it is sufficient to use the connectednessof W N and W S , they are the two connected components of the complement of A (cid:98) Γ .It remains to prove iii) . Note first that every leaf of F is met by a transverse simple loop and so iswandering. It implies that the α -limit and ω -limit sets of a leaf are included in two different connectedcomponents of sing( I ). Let us fix Γ ∈ G I, F . The complement of A Γ has two connected components.One of them contains all singularities at the left of Γ and all leaves in l (Γ), denote it by L ( A Γ ). Onedefines similarly R ( A Γ ). Write Ξ for the union of intervals J ∈ J defined in the proof of Lemma 55.A point t ∈ T belongs to Ξ if and only if there exists z ∈ rec( f ) ∩ L ( A Γ ) whose whole transversetrajectory meets φ Γ( t ) or equivalently, if there exists z ∈ L ( A Γ ) whose whole transverse trajectorymeets φ Γ( t ) . Note that if C is a connected component of ( ∂A Γ \ sing( I )) ∩ L ( A Γ ), then the set J C = { t ∈ T , C ∩ φ Γ( t ) (cid:54) = ∅} is an interval contained in Ξ. Denote by ( t − , t + ) the connected component of Ξ that contains thisinterval. The assertion iii) is an immediate consequence of the following: Lemma 58. The interval J C is equal to ( t − , t + ) . Moreover, for every z ∈ C , the connected componentsof sing( I ) that contain α ( z ) and ω ( z ) coincide with the connected components of sing( I ) that contain ω ( φ Γ( t − ) ) and ω ( φ Γ( t + ) ) respectively.Proof. Fix z ∈ C . Every point f k ( z ) belongs to a leaf φ Γ( t k ) , where t k ∈ T . By definition of Ξ, oneknows that the whole transverse trajectory of z never meets a leaf φ Γ( t ) , t (cid:54)∈ Ξ, and so t ∈ ( t − , t + ) if φ Γ( t ) meets this trajectory. In particular, the sequence ( t k ) k ∈ Z is an increasing sequence in ( t − , t + ).We set t (cid:48)− = lim k →−∞ t k and t (cid:48) + = lim k → + ∞ t k . We write F (cid:48) + for the connected component of sing( I )that contains ω ( φ Γ( t (cid:48) + ) ). We will prove first that the connected component of sing( I ) that contains ω ( z ) is F (cid:48) + and then that t (cid:48) + = t + . We can do the same for the α -limit set. One knows that ω ( z )is contained in L ( A Γ ) ∩ sing( I ). So, there exists a sequence ( z (cid:48) k ) k (cid:62) such that z (cid:48) k ∈ φ − ( z k ) for every k (cid:62) 0, that “converges to F (cid:48) + ” in the following sense: every neighborhood of F (cid:48) + contains z (cid:48) k for k sufficiently large. Let us prove now that every neighborhood of F (cid:48) + contains the segment γ k of φ Γ( t k ) between z (cid:48) k and z k , for k sufficiently large. If not, there exists a subsequence of ( γ k ) k (cid:62) that convergesfor the Hausdorff topology to a set that contains a point z (cid:54)∈ sing( I ). This point belongs to l (Γ) andthe leaf φ z is met by a loop Γ (cid:48) ∈ G I, F . For convenience choose the loop passing through z , so thatwe know that z k belongs to L (Γ (cid:48) ), for infinitely many k . One deduces that the connected componentof sing( I ) that contains ω ( z ) belongs to L (Γ (cid:48) ). But this implies that it also belongs to L ( A Γ (cid:48) ). Thisconnected component being included in the open disk A Γ (cid:48) ∪ L ( A Γ (cid:48) ), every point z k belongs to this diskfor k large enough. This contradicts the fact that z ∈ ∂A Γ , because A Γ (cid:48) ∪ L ( A Γ (cid:48) ) is in the interior of L ( A Γ ). It remains to prove that t (cid:48) + = t + . If t (cid:48) + < t + , then φ Γ( t (cid:48) + ) is met by a loop Γ (cid:48) ∈ G I, F such that A Γ (cid:48) ⊂ L ( A Γ ) and we prove similarly that for k large enough z k belongs to the open disk A Γ (cid:48) ∪ L ( A Γ (cid:48) )getting the same contradiction. (cid:3) Proofs of Theorems 53 and 51. Proof of Theorem 53. Note that if Γ and Γ (cid:48) are two distinct elements of G I, F , then Γ is not freelyhomotopic to Γ (cid:48) in dom( I ). Indeed, there exists a leaf φ ∈ U Γ \ U Γ (cid:48) . The two sets α ( φ ) and ω ( φ ) areseparated by Γ but not by Γ (cid:48) which implies that these two loops are not freely homotopic. Let us explainnow why the families (rec( f ) Γ ) Γ ∈G I, F and ( A Γ ) Γ ∈G I, F are independent of F (up to reindexation), theydepend only on I . In particular, if F (cid:48) is another foliation transverse to I , then every Γ ∈ G I, F is freelyhomotopic to a unique Γ (cid:48) ∈ G I, F (cid:48) and one has rec( f ) Γ = rec( f ) Γ (cid:48) . Let z be a recurrent point and D ⊂ dom( I ) an open disk containing z . For every couple of points ( z (cid:48) , z (cid:48)(cid:48) ) in D , choose a path γ z (cid:48) ,z (cid:48)(cid:48) in D joining z (cid:48) to z (cid:48)(cid:48) . Let ( n k ) k (cid:62) be an increasing sequence of integers such that lim k → + ∞ f n k ( z ) = z .For k large enough, the path I n k ( z ) γ f nk ( z ) ,z defines a loop whose homotopy class is independent ofthe choices of D and γ f nk ( z ) ,z . If z belongs to rec( f ) Γ , this class is a multiple of the class of Γ. Thismeans that the family of classes of loops Γ ∈ G I, F does not depend on F . It implies that the familyof sets rec( f ) Γ does not depend on F either. We will denote ( A κ ) κ ∈K I and (rec( f ) κ ) κ ∈K I our familiesindexed by homotopy classes.The fact that every invariant annulus contained in dom( I ) is contained in an A β , β ∈ B I , can beproven exactly like in the global case. So, to prove Theorem 53, particularly the fact that every A β isan annulus, it is sufficient to prove that it is equal to an A κ , κ ∈ K I . Note that an A κ is an invariantannulus contained in dom( I ) and so is contained in an A β . If we prove that every I -free disk recurrentpoint is contained in an A κ , we will deduce that each A β is a union of A κ , which implies that it is equalto one A κ because it is connected. We will prove in fact that for every I -free disk recurrent point z ,there exists a transverse foliation F such that z belongs to a W Γ , Γ ∈ G I, F . Let us give the reason. Inthe construction of transverse foliations we have the following: if X is a finite set included in an I -freedisk D , one can construct a transverse foliation such that X is included in a leaf (see Proposition 59at the next subsection). Consequently, if D contains two points z and f n ( z ), n > 0, one can constructa transverse foliation such that z and f n ( z ) are on the same leaf, which implies that z belongs to a W Γ . (cid:3) Proof of Theorem 51, second part, proof of assertion iv) . Fix α ∈ A f . The assertion iv) is obviouslytrue if the complement of A α is the union of two fixed points. Let us prove it in case exactly one theconnected components of the complement of A α is a fixed point z . By assertion iii) there exists atleast one connected component X (cid:54) = { z } of fix( f ) that meets the frontier of A α . If { z } and X are the only connected components of fix( f ), the result is also obviously true. If not, choose a thirdcomponent X , then choose z ∈ X ∩ ∂ ( A α ) and z ∈ X and finally a maximal hereditary singularisotopy I whose singular set contains z , z and z . We will prove that the connected component A β , β ∈ B I , that contains A α is reduced to A α . This will imply iv) . Suppose that A β is not reduced to A α . In that case it contains other A α , α ∈ A f , and the union of such sets is dense in A β and containall the recurrent points. The two ends of A β are adjacent to A α because z ∈ sing( I ). It implies that A α is the unique A α that is essential in A β . So, if A α is included in A β and α (cid:54) = α , the union of A α and of the connected component of its complement that are included in A β is an invariant open ORCING THEORY FOR TRANSVERSE TRAJECTORIES OF SURFACE HOMEOMORPHISMS 65 disk D α ⊂ A β disjoint from A α . Let us consider a foliation F transverse to I and the loop Γ ∈ G I, F such that A Γ = A β . We will work in the annular covering space, where A (cid:98) Γ is homeomorphic to A Γ and will write D (cid:98) α ⊂ A (cid:98) Γ for the disk corresponding to D α and A (cid:98) α for the annulus corresponding to A α .The fact that { z } is a connected component of S \ A α and that the singular set of I contains z andtwo other points in different components of the fixed point set of f , implies that one of the sets r ( (cid:98) Γ)or l ( (cid:98) Γ) is empty and the other one is not. We will assume for instance that r ( (cid:98) Γ) = ∅ and l ( (cid:98) Γ) (cid:54) = ∅ .We have seen in the proof of Proposition 26 that there exists a compactification (cid:100) dom( I ) ann obtainedby blowing up the end N at the left of (cid:98) Γ by a circle (cid:98) Σ N such that (cid:98) f extends to a homeomorphism (cid:98) f ann that admits fixed points on the added circle with a rotation number equal to zero for the lift (cid:101) f ann that extends (cid:101) f . Note now that every recurrent point of (cid:98) f that belongs to a D (cid:98) α has a rotation number(for the lift (cid:101) f ) and that this number is a positive integer because D (cid:98) α is fixed and included in A (cid:98) Γ . So,every periodic orbit whose rotation number is not an integer belongs to A (cid:98) α .There are different ways to get a contradiction. Let us begin by the following one. The closure of A (cid:98) β in (cid:100) dom( I ) ann is an invariant essential closed set that contains A (cid:98) α and meets Σ N . In particularit contains fixed points of rotation number 0 on Σ N . Denote by K the complement of A (cid:98) α in theclosure of A (cid:98) β in (cid:100) dom( I ) ann . It contains the fixed points located on Σ N and all the D (cid:98) α , which meansthat it contains fixed points of positive rotation number. It is an essential compact set, because A (cid:98) α is an essential annulus which is a neighborhood of the end of (cid:100) dom( I ) ann . All points in K being nonwandering, one can apply a result of S. Matsumoto [Mm] saying that K contains a periodic orbit ofperiod q and rotation number p/q for every p/q ∈ (0 , A (cid:98) α . We have a contradiction.Let us give another explanation. We will need the following intersection property: every essentialsimple loop in (cid:100) dom( I ) ann meets its image by (cid:98) f ann . The reason is very simple. Perturbing our loop,it is sufficient to prove that every essential simple loop in (cid:100) dom( I ) meets its image by (cid:98) f . Such a loopmeets A (cid:98) β because the two ends of (cid:100) dom( I ) are adjacent to A (cid:98) β and so contains a non wandering point(every point of A (cid:98) β is non wandering). This implies that the loop meets its image by (cid:98) f .Using the fact that the entropy of f is zero, one can consider the family of annuli ( A α (cid:48) ) α (cid:48) ∈A ( f ) , anddenote by A (cid:98) α (cid:48) the annulus of A (cid:98) β that corresponds to an annulus A α (cid:48) contained in A β . Every periodicpoint z of period 3 and rotation number 1 / / A (cid:98) α (cid:48) and this annulus is (cid:98) f -invariant. It must be essential in A (cid:98) β , otherwise the rotation number of z should be a multiple of 1 / (cid:98) f -invariant, its (cid:98) f -period cannot be 2. It is included in A (cid:98) α , otherwiseit would be included in a non essential A (cid:98) α . Being given such an essential annulus, note that the set ofperiodic points of period 3 and rotation 1 / / (cid:98) Σ N ) is compact. Indeed, the rotation number induced on the added circle is 0. One deduces that thereare finitely many annuli A (cid:98) α (cid:48) that contains periodic points of period 3 and rotation number 1 / / A (cid:98) α (cid:48) that contains periodic pointsof period 3 and rotation number 1 / / 3. If one adds the connected component of (cid:100) dom( I ) ann \ A (cid:98) α (cid:48) containing (cid:98) Σ N to A (cid:98) α (cid:48) , one gets an invariant semi-open annulus A that contains (cid:98) Σ N and all disks D (cid:98) α .The restriction (cid:98) f ann | A satisfies the intersection property stated in Lemma 25 because A is essential in (cid:100) dom( I ) ann . The annulus A contains a fixed point of rotation number 0 and a fixed point of positiverotation number, so, by Lemma 25, it contains at least one periodic orbit of period 3 and rotation number 1 / / 3. These two orbits must beincluded in A (cid:98) α (cid:48) by definition of this set. But (cid:98) f | A (cid:98) α (cid:48) satisfies the intersection property because A (cid:98) α (cid:48) isessential in (cid:100) dom( I ) ann . So A (cid:98) α (cid:48) contains a periodic point of period 2 and rotation number 1 / 2, whichis impossible.In the case where none of the connected components of the complement of A α is a fixed point, onecan crush one of these components to a point and used what has been done in the new sphere. (cid:3) Let us add some comments on the boundary of the annuli A α .Let f : S → S be an orientation preserving homeomorphism such that Ω( f ) = S and h ( f ) = 0.Suppose moreover than the fixed point set is totally disconnected. Every annulus A α , α ∈ A f , admitsaccessible fixed points on its boundary. More precisely, if X is a connected component of S \ A α ,there exists a simple path γ joining a point z ∈ A α to a point z (cid:48) ∈ sing( f ) ∩ X and contained in A α but the end z (cid:48) . Indeed, one can always suppose that the other connected component of S \ A α isreduced to a point z and that f has least three fixed points (otherwise the result is obvious). Whathas been done in the previous proof tells us that there exists a maximal hereditary singular isotopy I ,a transverse foliation F and Γ ∈ G I, F such that A α = A Γ . There exists a leaf φ ⊂ U Γ that is not metby any transverse trajectory that intersects X . This leaf (or the inverse of the leaf) joins z to a fixedpoint z ∈ X and is contained in A Γ .Let f : S → S be an orientation preserving homeomorphism such that Ω( f ) = S and h ( f ) = 0.Let I be a maximal hereditary singular isotopy and F a transverse foliation. Every annulus A Γ ,Γ ∈ G I, F , that meets φ is such that the connected components of Fix( I ) that contains α ( φ ) and ω ( φ )are separated by A Γ . One deduces immediately that a point z ∈ dom( I ) belongs to the frontier of atmost two annuli A Γ , Γ ∈ G I, F . Of course this means that a point z ∈ dom( I ) belongs to the frontierof at most two annuli A β , β ∈ B I , but it also implies that a point z (cid:54)∈ fix( f ) belongs to the frontierof at most two annuli A α , α ∈ A f . Indeed, suppose that z (cid:54)∈ fix( f ) belongs to the frontier of A α i ,0 (cid:54) i (cid:54) 2. If X i is the connected component of S \ A α i that does not contain z , then the three sets X i are disjoint. Choose a fixed point z i in each X i (such a fixed point exists because X i ∪ A α i is aninvariant disk and A α i has no fixed points). Choose a maximal hereditary singular isotopy I that fixesthe z i and denote A β i , β i ∈ B I , the annulus that contains A α i . Note that the three annuli A β i aredistinct and that z belongs to their frontier. We have a contradiction. (cid:3) Transverse foliation and free disks. We conclude this section by justifying a point used abovein the proof of Theorem 53. Proposition 59. Let f : M → M be a homeomorphism isotopic to the identity on a surface M and I a maximal singular isotopy. Let X be a finite set contained in an I -free disk. Then, there exists atransverse foliation F such that X is contained in a leaf of F .Proof. The proof can be deduced immediately from the construction of transverse foliations, that werecall now (see [Lec2]). A brick decomposition D = ( V, E, B ) on a surface is given by a one dimensionalstratified set, the skeleton Σ( D ), with a zero-dimensional submanifold V such that any vertex v ∈ V is locally the extremity of exactly three edges e ∈ E . A brick b ∈ B is the closure of a connectedcomponent of the complement of Σ( D ). Say that a brick decomposition D = ( V, E, B ) on dom( I ) is I -free, if every brick is I -free, or equivalently, if its lifts to a brick decomposition (cid:101) D = ( (cid:101) V , (cid:101) E, (cid:101) B ) on the ORCING THEORY FOR TRANSVERSE TRAJECTORIES OF SURFACE HOMEOMORPHISMS 67 universal covering (cid:103) dom( I ), whose bricks are (cid:101) f -free, where (cid:101) f is the lift associated to (cid:101) I . Say that D isminimal if there is no I -free brick decomposition whose skeleton is strictly included in the skeleton of D . Such a decomposition always exists.Write G for the group of automorphisms of the universal covering space. Using the classical Franks’lemma on free disk chains [F1], one constructs a natural order (cid:54) on (cid:101) B that satisfies the following:- it is G -invariant;- if (cid:101) f ( (cid:101) β ) meets β (cid:48) , then β (cid:48) (cid:54) β ;- two adjacent bricks are comparable.One can define an orientation on Σ( (cid:101) D ) (inducing an orientation on Σ( D )) such that the brick on theleft of an edge (cid:101) e ∈ (cid:101) E is smaller than the brick on the right. Moreover, every vertex (cid:101) v ∈ (cid:101) V is the endingpoint of at least one oriented edge and the starting point of at least one oriented edge. In other wordsthere is no sink and no source on the oriented skeleton. We have three possibilities for the bricks of (cid:101) B :- it can be a closed disk with a sink and a source on the boundary (seen from inside);- it can be homeomorphic to [0 , + ∞ [ × R with a sink on the boundary and a source at infinity;- it can be homeomorphic to [0 , + ∞ [ × R with a source on the boundary and a sink at infinity;- it can be homeomorphic to [0 , × R with a sink and a source at infinity (in this case it can projectonto a closed annulus).Let us state now the fundamental result, easy to prove in the case where G is abelian and much moredifficult in the case it is not (see Proposition 3.2 of [Lec2]): one can cover Σ( (cid:101) D ) with a G -invariantfamily of Brouwer lines of (cid:101) f , such that two lines never intersect transversally in the following sense:if λ and λ (cid:48) are two lines in this family, either they do not intersect, or one of the sets R ( λ ), R ( λ (cid:48) )contains the other one.Such family of lines inherits a natural order (cid:54) , where λ (cid:54) λ (cid:48) ⇔ R ( λ ) ⊂ R ( λ (cid:48) ) . One can “complete” this family to get a larger family, with the same properties, that possesses thetopological properties of a lamination (in particular every line admits a compact and totally orderedneighborhood). Then one can arbitrarily foliate each brick b ∈ B such that, when lifted to a foliationon a brick (cid:101) b ∈ (cid:101) B , every leaves goes from the source to the sink. We obtain then, in a natural way, adecomposition of (cid:103) dom( I ) by a G -invariant family of Brouwer lines that do not intersect transversally,and that possesses the topological structure of a plane foliation (it is a non Hausdorff one dimensionalmanifold).It remains to blow up each vertex, by a desingularization process (see [Lec1]) to obtained a G -invariantfoliation by Brouwer lines. (cid:3) Zero entropy annulus homeomorphisms. In this subsection we prove some results for a gen-eral open annulus homeomorphism whose extension to the ends compactification has zero topologicalentropy. A stronger version of the first result for diffeomorphisms was already proved in an unpublishedpaper of Handel [H2]. As noted in the introduction, given a homeomorphism of an open annulus T × R and a lift ˇ f to R ,denote by π : R → T × R the covering projection, and by π : R → R the projection in the firstcoordinate. For any point z ∈ T × R such that its ω -limit set is not empty, we say that z has arotation number rot( z ) if, for any compact set K ⊂ T × R , any increasing sequence of integers n k suchthat f n k ( z ) ∈ K and any ˇ z ∈ π − ( z ),lim k →∞ n k (cid:0) π ( ˇ f n k (ˇ z ) − π (ˇ z ) (cid:1) = rot( z ) . In general it is not expected that every point will have a rotation number, but if we assume that f haszero entropy this must be the case, at least for recurrent points, as shown by the following theorem,which is a restatement of Theorem L Theorem 60. Let f be a homeomorphism of T × R isotopic to the identity, ˇ f a lift of f to the universalcovering space, and let f sphere be the natural extension of f to the sphere obtained by compactifyingeach end with a point. If the topological entropy of f sphere is zero, then each bi-recurrent point has arotation number rot( z ) . Moreover, the function z (cid:55)→ rot( z ) is continuous on the set of bi-recurrentpoints.Proof. For every compact set K of T × R define the set rot ˇ f,K ( z ) ⊂ R ∪ {−∞ , ∞} as following: ρ belongs to rot (cid:101) f,K ( z ) if there exists an increasing sequence of integers n k such that f n k ( z ) ∈ K andsuch that for any ˇ z ∈ π − ( z ), one haslim k →∞ n k (cid:0) π ( ˇ f n k (ˇ z ) − π (ˇ z ) (cid:1) = ρ. Writing T : ( x, y ) (cid:55)→ ( x + 1 , y ) for the fundamental covering automorphism, one immediately getsrot ˇ f ◦ T p ,K ( z ) = rot ˇ f,K ( z )+ p for every p ∈ Z . One can prove quite easily that rot ˇ f q ,K q ( z ) = q rot ˇ f,K ( z ),for every q (cid:62) 1, where K q = (cid:83) (cid:54) k 1. The set R of integers r such that there exists a subsequence of ( f nq ( z )) n (cid:62) that convergesto f r ( z ) or equivalently a subsequence of ( f nq − r ( z )) n (cid:62) that converges to z , is non empty because z is positively recurrent. Note that R is stable by addition. Indeed if r and r (cid:48) belong to R , one canapproximate f r + r (cid:48) ( z ) = f r ( f r (cid:48) ( z )) by a point f r ( f n (cid:48) q ( z )) = f r + n (cid:48) q ( z ) as close as we want, and thenapproximate f r + n (cid:48) q ( z ) = f n (cid:48) q ( f r ( z )) by a point f n (cid:48) q ( f nq ( z )) = f ( n + n (cid:48) ) q ( z ) as close as we want. Onededuces that qr belongs to R if it is the case for r , which implies that z is a positively recurrent pointof f q . Similarly, every bi-recurrent point z of f is a bi-recurrent point of f q , for every q (cid:62) p, q ) of integers relatively prime ( q (cid:62) I ∗ p,q of f q that is lifted to an identity isotopy of ˇ f q ◦ T − p , then a maximal hereditary singular isotopy I p,q such that I (cid:48) p,q (cid:22) I p,q , and finally a singular foliation F p,q transverse to I p,q . The singular points of I p,q are periodic points of period q and rotation number p/q . Let z be a bi-recurrent point of f thatis not a singular point of I p,q . By Theorem 36 and Proposition 2, the whole trajectory ( I p,q ) Z F p,q ( z ) isthe natural lift of a simple loop Γ p,q ( z ) (uniquely defined up to equivalence). In particular, one hasΓ p,q ( z ) = Γ p,q ( f q ( z )). Write U Γ p,q ( z ) for the open annulus, union of leaves met by Γ p,q ( z ). Every leafcontaining a point of O ( f q , z ) is met by ( I p,q ) Z F p,q ( z ). It implies that O ( f q , z ) ⊂ U Γ p,q ( z ) ∪ sing( I p,q ).Note also that the function z (cid:55)→ Γ p,q ( z ) is locally constant on the set of bi-recurrent points. Indeedif z is a bi-recurrent point and γ : [0 , → dom( I p,q ) a non simple transverse subpath of the natural ORCING THEORY FOR TRANSVERSE TRAJECTORIES OF SURFACE HOMEOMORPHISMS 69 lift of Γ p,q ( z ), then γ is a subpath of the whole trajectory of z (cid:48) , if z (cid:48) is a bi-recurrent point sufficientlyclose to z , which implies that Γ p,q ( z (cid:48) ) = Γ p,q ( z ).One can lift the isotopy I p,q to a singular maximal isotopy ˇ I p,q of ˇ f q ◦ T − p and the foliation F p,q to afoliation ˇ F p,q transverse to ˇ I p,q . Fix a lift ˇ z ∈ R of z . In the case where Γ p,q ( z ) is not essential, then( ˇ I p,q ) Z ˇ F p,q (ˇ z ) is the natural lift of a transverse simple loop Γ p,q (ˇ z ) that lifts Γ p,q ( z ). The (cid:101) f q ◦ T − p -orbit ofˇ z stays in the annulus U Γ p,q (ˇ z ) , union of leaves met by Γ p,q (ˇ z ). In the case where Γ p,q ( z ) is essential andthe upper end of T × R is on the left of Γ p,q ( z ), then ( ˇ I p,q ) Z ˇ F p,q (ˇ z ) is the natural lift of a transverse line γ p,q (ˇ z ) that lifts Γ p,q ( z ). The (cid:101) f q ◦ T − p -orbit of ˇ z stays in the strip U γ p,q (ˇ z ) , union of leaves met by γ p,q (ˇ z ).Fix a parameterization γ p,q (ˇ z ) : R → dom( ˇ I p,q ) such that γ p,q (ˇ z )( t + 1) = T ( γ p,q (ˇ z )( t )). For everyˇ z (cid:48) ∈ U γ p,q (ˇ z ) there exist a unique real number, denoted by π γ p,q (ˇ z ) (ˇ z (cid:48) ) such that ˇ z (cid:48) and γ p,q (ˇ z )( t ) are onthe same leaf. One gets a map π γ p,q (ˇ z ) : U γ p,q (ˇ z ) → R , such that the sequence ( π γ p,q (ˇ z ) ( ˇ f q ◦ T − p ) k (ˇ z )) k ∈ Z is increasing. In the case where Γ p,q ( z ) is essential and the upper end of T × R is on the right ofΓ p,q ( z ), one proves by the same argument that the sequence ( π γ p,q (ˇ z ) ( ˇ f q ◦ T − p ) k (ˇ z )) k ∈ Z is decreasing.Let z be a periodic point that is not a singular point of I p,q . If Γ p,q ( z ) is not essential, then the rotationnumber of z (defined for the lift (cid:101) f q ◦ T − p of f q ) is equal to zero, which implies that rot( z ) = p/q . IfΓ p,q ( z ) is essential and if the upper end of T × R is on the left of Γ p,q ( z ), the rotation number of z (defined for the lift (cid:101) f q ◦ T − p of f q ) is positive, which implies that rot( z ) > p/q . If Γ p,q ( z ) is essentialand if the upper end of T × R is on the right of Γ p,q ( z ), then rot( z ) < p/q .Let us prove now that if z is bi-recurrent, the periodic points that belong to the closure of O ( f, z )have the same rotation number. Otherwise, one can find a couple ( p, q ) of integers relatively prime( q (cid:62) z , z in the closure of O ( f q , z ) such that rot( z ) < p/q < rot( z ).One deduces that Γ p,q ( z ) = Γ p,q ( z ) = Γ p,q ( z ), which is impossible because the upper end of T × R is on the right of Γ p,q ( z ) and on the left of Γ p,q ( z ).Let K be a compact subset of T × R and z a bi-recurrent point. Let us suppose first that the closureof O ( f, z ) has no periodic points. For every couple ( p, q ) of integers relatively prime ( q (cid:62) O ( f, z ) ∩ K q is a compact subset of the annulus U Γ p,q ( z ) . In the case where Γ p,q ( z ) is not essential, onededuces that rot (cid:101) f q ◦ T − p ,K q ( z ) is reduced to { } and so rot (cid:101) f,K ( z ) is reduced to { p/q } . In the case whereΓ p,q ( z ) is essential and the upper end of T × R is on the left of Γ p,q ( z ), there exits a real number M suchthat for every point (cid:101) z (cid:48) that lifts a point of ( O ( f, z ) ∩ K q ) ∪{ z } , one has | π (ˇ z (cid:48) ) − π γ p,q (ˇ z ) (ˇ z (cid:48) ) | (cid:54) M . Usingthis property and the fact that the sequence ( π γ p,q (ˇ z ) ( ˇ f q ◦ T − p ) k (ˇ z )) k ∈ Z is increasing, one deduces thatrot (cid:101) f ◦ T − p ,K q ( z ) ⊂ [0 , + ∞ ] and consequently that rot (cid:101) f,K ( z ) ⊂ [ p/q, + ∞ ]. Similarly, in the case whereΓ p,q ( z ) is essential and the upper end of T × R is on the right of Γ p,q ( z ), one gets rot (cid:101) f,K ( z ) ⊂ [ −∞ , p/q ].One immediately concludes that rot (cid:101) f,K ( z ) is reduced to a number in R ∪ {−∞ , ∞} if not empty. Ofcourse, this number is independent of K , we denote it rot( z ). Suppose now that the closure of O ( f, z )contains periodic points. As said before, they have the same rotation number p /q . The argumentabove is still valid if p/q (cid:54) = p /q and permit us to concluded that rot (cid:101) f,K ( z ) is reduced to a numberin R ∪ {−∞ , ∞} independent of K . Of course the number is nothing but p /q . Note that in bothsituations rot( z ) is uniquely defined by the following property:- if p/q < rot( z ), then Γ p,q ( z ) is essential and the upper end of T × R is on the right of Γ( z ),- if p/q > rot( z ), then Γ p,q ( z ) is essential and the upper end of T × R is on the left of Γ( z ). Using the fact that each function z (cid:55)→ Γ p,q ( z ) is locally constant on the set of bi-recurrent points, onededuces immediately that the function z (cid:55)→ rot( z ) is continuous on the set of bi-recurrent points.It remains to prove that rot( z ) is finite. Of course one can suppose that the closure of O ( f, z ) does notcontain a periodic orbit (otherwise as said before rot( z ) is rational). By assumption, z is not periodic,so let us choose a free disk D containing z . There exists an integer s > f s ( z ) ∈ D andan integer r ∈ Z such that ˇ f s (ˇ z ) ∈ T r ( ˇ D ), if ˇ D ⊂ R is a lift of D and ˇ z is the lift of z containedin ˇ D . Let us consider the singular isotopy I k, , where k > r/s . As explained above the two points z and f s ( z ) belong to the free disk D . So, by Proposition 59 one can choose the foliation F k, suchthat z and f s ( z ) belong to the same leaf φ . This implies that Γ k, ( z ) is essential and the upper endof T × R is on the right of Γ k, ( z ). Consequently, one deduces that rot( z ) (cid:54) k . One proves similarlythat rot( z ) (cid:62) k (cid:48) if k (cid:48) is an integer smaller than r/s . (cid:3) An interesting consequence is: Proposition 61. Let f be an orientation preserving homeomorphism of S . Suppose that there existsa bi-recurrent point z such that the closure of its orbit contains periodic points of minimal period q < q , where q does not divide q . Then the entropy of f is positive.Proof. We suppose that the closure of the f -orbit O ( f, z ) of z contains a periodic point z of period q and a periodic point z of period q . One can choose z and z in O ( f q , z ). Writing r for the remainderof the Euclidean division of q by q , one knows that f q ( z ) = f r ( z ). Since q is not a multiple of q ,it is larger than 2 and f q must have at least three distinct fixed points. Choose a maximal hereditaryidentity isotopy I of f q whose singular set contains z , f r ( z ) and at least a third fixed point of f q ,then consider a singular foliation F transverse to I . Since f has zero topological entropy, f q has zerotopological entropy, and the path I Z F ( z ) is equivalent to the natural lift of a simple transverse loopΓ. Using the fact that there exist at least there singular points, one can find two singular points z (cid:48) and z (cid:48)(cid:48) of I that are separated by Γ and such that { z (cid:48) , z (cid:48)(cid:48) } (cid:54) = { z , f r ( z ) } . The isotopy I defines anatural lift of f q | S \{ z (cid:48) ,z (cid:48)(cid:48) } and for this lift, the rotation number of every point of the f q -orbit of z is a non vanishing number, while the rotation number of every singular point different from z (cid:48) and z (cid:48)(cid:48) is zero. As seen in the previous proposition, the points z and f r ( z ) are bi-recurrent points of f q .In the case where z (cid:54)∈ { z (cid:48) , z (cid:48)(cid:48) } , the closure of O ( f q , z ) contains two periodic points z and z withdifferent rotation numbers. In the case where f r ( z ) (cid:54)∈ { z (cid:48) , z (cid:48)(cid:48) } , the closure of O ( f q , f r ( z )) containstwo periodic points f r ( z ) = f q ( z ) and f r ( z ) with different rotation numbers. We have seen in theproof of the previous proposition that in both cases, the entropy of f is positive.. (cid:3) Applications to torus homeomorphims In this section an element of Z will be called an integer and an element of Q a rational. If K is aconvex compact subset of R , a supporting line is an affine line that meets K but does not separatetwo points of K , a vertex is a point that belongs to infinitely many supporting lines.Let us begin by stating the main results of this section, that are nothing but Theorems C, D and Gfrom the introduction. ORCING THEORY FOR TRANSVERSE TRAJECTORIES OF SURFACE HOMEOMORPHISMS 71 Theorem 62. Let f be a homeomorphism of T that is isotopic to the identity and ˇ f a lift of f to R .The frontier of rot( ˇ f ) does not contain a segment with irrational slope that contains a rational pointin its interior. Theorem 63. Let f be a homeomorphism of T that is isotopic to the identity and ˇ f a lift of f to R . If rot( ˇ f ) has a non empty interior, then there exists L (cid:62) such that for every z ∈ R and every n (cid:62) , one has d ( ˇ f n ( z ) − z, n rot( ˇ f )) (cid:54) L . Theorem 64. Let f be a homeomorphism of T that is isotopic to the identity and ˇ f a lift of f to R . If rot( ˇ f ) has a non empty interior, then the topological entropy of f is positive. Recall that Theorem 64 has been known for a long time and is due to Llibre and MacKay, see [LlM]and that Theorem 63 was known for homeomorphisms in the special case of a polygon with rationalvertices, see Davalos [D2], and for C (cid:15) diffeomorphisms, see Addas-Zanata [AZ].Let us state first some consequences of these results. Let f be a homeomorphism of T that is isotopicto the identity and ˇ f a lift of f to R . We suppose that rot( ˇ f ) has non empty interior. For every nontrivial linear form ψ on R , define α ( ψ ) = max { ψ (rot( µ )) , µ ∈ M ( f ) } . The affine line of equation ψ ( z ) = α ( ψ ) is a supporting line of rot( ˇ f ). Set M ψ = { µ ∈ M ( f ) , ψ (rot( µ )) = α ( ψ ) } , X ψ = (cid:91) µ ∈M ψ supp( µ ) . As already noted in [AZ], we can deduce from Theorem 63 and Proposition 47 (Atkinson’s Lemma)the following result, Proposition E of the introduction. Proposition 65. Every measure µ supported on X ψ belongs to M ψ . Moreover, if z lifts a point of X ψ , then for every n (cid:62) , one has | ψ ( ˇ f n ( z )) − ψ ( z ) − nα ( ψ ) | (cid:54) L (cid:107) ψ (cid:107) , where L is the constant givenby Theorem 63.Proof. We will prove the second statement, it obviously implies the first one. Note first that the ergodiccomponents of a measure µ ∈ M ψ also belong to M ψ . Furthermore, the set of points A (cid:48) having alift z satisfying that | ψ ( ˇ f n ( z )) − ψ ( z ) − nα ( ψ ) | (cid:54) L (cid:107) ψ (cid:107) for every n (cid:62) µ ∈ M ψ there exists a set A ⊂ A (cid:48) of full measure.As seen before, since µ ∈ M ψ , the function lifted by ψ ◦ ˇ f − ψ − α ( ψ ) has null mean, and we can applyAtkinson’s lemma to obtain that there exists a set A of full measure such that, for every point z liftinga point of A , there exists a subsequence ( n l ) l ∈ N such thatlim l → + ∞ ψ ( ˇ f n l ( z )) − ψ ( z ) − n l α ( ψ ) = 0 . By Theorem 63, one knows that for every z ∈ R and every n (cid:62) 1, one has ψ ( ˇ f n ( z )) − ψ ( z ) − nα ( ψ ) (cid:54) L (cid:107) ψ (cid:107) . It remains to prove that ψ ( ˇ f n ( z )) − ψ ( z ) − nα ( ψ ) (cid:62) − L (cid:107) ψ (cid:107) if z lifts a point of A . If n l isgreater than n one can write ψ ( ˇ f n ( z )) − ψ ( z ) − nα ( ψ )= (cid:0) ψ ( ˇ f n l ( z )) − ψ ( z ) − n l α ( ψ ) (cid:1) − (cid:0) ψ ( ˇ f n l ( z )) − ψ ( ˇ f n ( z )) − ( n l − n ) α ( ψ ) (cid:1) (cid:62) ψ ( ˇ f n l ( z )) − ψ ( z ) − n l α ( ψ ) − L (cid:107) ψ (cid:107) . Letting l tend to + ∞ , one gets our inequality. (cid:3) Let us state two corollaries. The first one, Corollary F of the introduction, as already noted in [AZ],follows immediately from the previous proposition. Corollary 66. Let f be a homeomorphism of T that is isotopic to the identity, preserving a measure µ of full support, and ˇ f a lift of f to R . Assume that rot( ˇ f ) has a non empty interior. Then rot( µ ) belongs to the interior of rot( ˇ f ) . P. Boyland had conjectured that, for a given f and ˇ f in the hypotheses of Corollary 66, if rot( µ ) wasan integer then it belonged to the interior of rot( ˇ f ). The previous result shows that the conjecture istrue, and that the hypothesis on the rationality of the rotation vector of µ is superfluous.The second corollary shows that, for points in the lift of the support of measures with rotation vectorin a vertex, the displacement from the corresponding rigid rotation is uniformly bounded. Corollary 67. Let ρ be a vertex of rot( ˇ f ) , and set M ρ = { µ ∈ M ( f ) , rot( µ ) = ρ } , X ρ = (cid:91) µ ∈M ρ supp( µ ) . There exists a constant L ρ such that if z lifts a point of X ρ , then for every n (cid:62) , one has d ( ˇ f n ( z )) − z − nρ ) (cid:54) L ρ .Proof. One can find two forms ψ and ψ (cid:48) , linearly independent such that ρ belongs to the supportinglines defined by these forms. Note that X ρ = X ψ ∩ X ψ (cid:48) and apply Proposition 65 . (cid:3) We remark that the conclusion from Corollary 67 does not hold if instead of requiring that ρ is a vertexof rot( ˇ f ) we assume that ρ is an extremal point of rot( ˇ f ), see Boyland-de Carvalho-Hall [BCH].Write ∂ (cid:0) rot( ˇ f ) (cid:1) for the frontier of rot( ˇ f ). Let us define now M ∂ = (cid:8) µ ∈ M ( f ) , rot( µ ) ∈ ∂ (cid:0) rot( ˇ f ) (cid:1)(cid:9) , X ∂ = (cid:91) µ ∈M ∂ supp( µ ) = (cid:91) ψ (cid:54) =0 X ψ . Similarly, we have: Proposition 68. Every ergodic measure µ supported on X ∂ belongs to M ∂ . Moreover, if z lifts apoint of X ∂ , then for every n (cid:62) , one has d (cid:0) ˇ f n ( z ) − z, n ∂ (cid:0) rot( ˇ f ) (cid:1)(cid:1) (cid:54) L , where L is the constantgiven by Theorem 63.Proof. Here again, it is sufficient to prove the second statement. To do so, let us choose a non triviallinear form ψ and let us prove that for every n (cid:62) 1, and for every point z lifting a point of X ψ , onehas d (cid:0) ˇ f n ( z ) − z, n ∂ (cid:0) rot( ˇ f ) (cid:1)(cid:1) (cid:54) L. The fact that, by Proposition 65, | ψ ( ˇ f n ( z )) − ψ ( z ) − nβ ( ψ ) | (cid:54) L (cid:107) ψ (cid:107) ORCING THEORY FOR TRANSVERSE TRAJECTORIES OF SURFACE HOMEOMORPHISMS 73 implies that d ( ˇ f n ( z ) − z, ∆) (cid:54) L where ∆ is the affine line of equation ψ ( z ) = nα ( z ). So, if f n ( z ) − z does not belong to n rot( ˇ f ), one has d (cid:0) ˇ f n ( z ) − z, n ∂ (cid:0) rot( ˇ f ) (cid:1)(cid:1) = d ( ˇ f n ( z ) − z, n rot( ˇ f )) (cid:54) L, and if ˇ f n ( z ) − z belongs to n rot( ˇ f ), one has d (cid:0) ˇ f n ( z ) − z, n ∂ (cid:0) rot( ˇ f ) (cid:1)(cid:1) (cid:54) d ( ˇ f n ( z ) − z, ∆) (cid:54) L. (cid:3) Another application is a classification result about Hamiltonian homeomorphisms . In our setting, aHamiltonian homeomorphism is a torus homeomorphism preserving a probability measure µ which hasa lift ˇ f (the Hamiltonian lift ) such that that rot( µ ) = (0 , . An illustrative example is given by thetime one map of a time dependent Hamiltonian flow, 1 periodic in time, and its natural lift.We will need the following result, which can be found in [KT2]: Proposition 69. Let f be a homeomorphism of T isotopic to the identity and ˇ f a lift of f . If (0 , is a vertex of rot( ˇ f ) then, for any measure µ ∈ M ( f ) such that rot( µ ) = (0 , , almost every point liftsto a recurrent point of ˇ f . We have: Theorem 70. Let f be a Hamiltonian homeomorphism of T such that its fixed point set is containedin a topological disk, and let ˇ f be its Hamiltonian lift. Then one of the following three conditions holds:- The set rot( ˇ f ) does not have empty interior: in that case the origin lies in its interior.- The set rot( ˇ f ) is a non trivial segment: in that case rot( ˇ f ) generates a line with rational slope, theorigin is not an end of rot( ˇ f ) , furthermore, there exists an invariant essential open annulus in T .- The set rot( ˇ f ) is reduced to the origin: in that case, there exists K > such that, for every z ∈ R and every k ∈ Z , one has (cid:107) ˇ f k ( z ) − z (cid:107) (cid:54) K. Proof. Suppose first that rot( ˇ f ) is reduced to the origin. The origin being a vertex, one knows byProposition 69 that the recurrent set of ˇ f is dense in R . So the assertion comes from Theorem 45.Suppose now that rot( ˇ f ) is a non trivial segment. If the origin was an end of rot( ˇ f ) its would be avertex and we would have a contradiction, still from from Proposition 69 and Theorem 45. The factthat rot( ˇ f ) generates a line with rational slope is a consequence of Theorem 62. The existence of anessential open annulus which is left invariant by the dynamics whenever rot( ˇ f ) is a non trivial segmentthat generates a line with rational slope is the main result of [GKT].The case where rot( ˇ f ) has non empty interior is nothing but Corollary 66. (cid:3) Here again, as in Theorem 45, the requirement that the fixed point set is contained in a topologicaldisk cannot be removed. As a consequence, we obtain the following boundedness result for areapreserving homeomorphisms of the torus with restriction on its rotational behaviour, Corollary I ofthe introduction: Corollary 71. Let f be a Hamiltonian homeomorphism of T such that all its periodic points arecontractible, and such that its fixed point set is contained in a topological disk. Then there exists K > such that if ˇ f is the Hamiltonian lift of f , then for every z ∈ R and every k ∈ Z , one has (cid:107) ˇ f k ( z ) − z (cid:107) (cid:54) K .Proof. By Theorem 70, f must belong to one of the three described possibilities. If f is a homeo-morphism of T such that all its periodic points are contractible, then by the main result of [F2] therotation set of any lift of f must have empty interior (see also Remark 74 later in the paper), and sothe first possibility in Theorem 70 is excluded. Furthermore, it was shown in [F3] that, if g is an areapreserving homeomorphism with lift ˇ g and the rotation set of ˇ g is a line segment, then for every pointin rot(ˇ g ) with bi-rational coordinates there exists a periodic point for f with the same rotation vector.Since f has no periodic points that are not contractible, the second possibility is also excluded. (cid:3) As a consequence we obtain the Proposition J: Proposition 72. Let Ham ∞ ( T ) be the set of Hamiltonian C ∞ diffeomorphisms of T endowed withthe Whitney C ∞ - topology. There exists a residual subset A of Ham ∞ ( T ) such that f has non-contractible periodic points if f ∈ A .Proof. We will prove that f has non contractible periodic points if the following properties are satisfied: • if f q ( z ) = z , then 1 is not an eigenvalue of Df q ( z ); • if z is an elliptic periodic point of period q (which means that the eigenvalues of Df q ( z ) areon the unit circle), then z is Moser stable (which means that z is surrounded by f q -invariantcurves arbitrarily close to z ); • if z , z (cid:48) are hyperbolic periodic points of period q , q (cid:48) respectively (which means that theeigenvalues of Df q ( z ) and Df q (cid:48) ( z (cid:48) ) are real), then the stable and unstable manifolds of z and z (cid:48) are either disjoint or they intersect transversally.The first property implies that the fixed point set of f is finite and so included in a topological disk.By Corollary 71, to get our result it remains to prove that there is no K > f is theHamiltonian lift of f , then for every z ∈ R and every k ∈ Z , one has (cid:107) ˇ f k ( z ) − z (cid:107) (cid:54) K . If such K exists,choose a bounded open set W containing the fundamental domain [0 , . The set (cid:83) k ∈ Z ˇ f k ( W ) is aninvariant bounded open set. One finds an invariant bounded open disk V containing [0 , by lookingat the complement of the unbounded component of the complement of W . Let us show first that ∂V has no periodic points. Since V is bounded, we may take a sufficiently large integer L such that, ifˆ T = R / ( L Z ) is the torus that finitely covers T , ˆ f is the induced homeomorphism and ˆ π : R → ˆ T is the projection, then ˆ π ( V ) is contained in a topological disk. The diffeomorphism ˆ f satisfies thefollowing properties: • if ˆ f q ( z ) = z , then 1 is not an eigenvalue of D ˆ f q ( z ); • every elliptic periodic point of ˆ f is Moser stable; • the stable and unstable manifolds of hyperbolic periodic points of ˆ f are either disjoint or theyintersect transversally.By a theorem of J. Mather (see [Mt]), one knows that the prime-end rotation number of ˆ π ( V ) isirrational. The main result from [KLN] shows that the frontier ∂ ˆ π ( V ) has no periodic point becausethe prime-end rotation number of ˆ π ( V ) is irrational. This implies that ∂V has no periodic points.In fact it is not necessary to use [KLN]. Indeed, working directly with V and ˇ f , the boundednesscondition implies that the stable and unstable manifolds of every hyperbolic periodic point z are ORCING THEORY FOR TRANSVERSE TRAJECTORIES OF SURFACE HOMEOMORPHISMS 75 bounded. Mather’s arguments implies that, under our generic conditions, the branches of z have allthe same closure. By a result of Pixton ([Pi]), every stable branch of z intersect every unstable branchand one can find surrounding curves arbitrarily close to z contained in the union of the stable andunstable manifolds. By Mather’s argument again, one knows that such a point z cannot be containedin ∂V . Moreover there is no elliptic periodic point on ∂V .The fact that V contains [0 , implies that (cid:83) p ∈ Z ( ∂V + p ) is connected. Moreover, the interior of (cid:83) p ∈ Z ( ∂V + p ) is empty. This set projects onto a compact subset of T whose interior is empty,which is totally essential (the connected components of its complement are open disks) and whichdoes not contain periodic points of f . This contradicts a result of A. Koropecki (see [Ko]) that statesthe following: if K is an invariant closed connected subset of a homeomorphism defined on a closedorientable surface and having no wandering points, and if K has no periodic point, then either M isa torus and K coincides with M , or K is a decreasing sequence of compact annuli. (cid:3) Before proving our three theorems, let us state some introductory results. In what follows (Proposition73 and Proposition 75) f is a homeomorphism of T that is isotopic to the identity and ˇ f a lift of f to R . We consider an identity isotopy I (cid:48) of f that is lifted to an identity isotopy ˇ I (cid:48) of ˇ f . We considera maximal hereditary singular isotopy I larger than I (cid:48) and its lift ˇ I to R . We consider a foliation F transverse to I an its lift ˇ F to R . Proposition 73. If (0 , belongs to the interior of rot( ˇ f ) or to the interior of a segment with irrationalslope included in ∂ (cid:0) rot( ˇ f ) (cid:1) , then the leaves of ˇ F are uniformly bounded.Proof. Suppose first that (0 , 0) belongs to the interior of rot( ˇ f ). One can find finitely many extremalpoints ρ i of rot( ˇ f ), 1 (cid:54) i (cid:54) r , that linearly generate the plane and positive numbers t i , 1 (cid:54) i (cid:54) r , suchthat: (cid:88) (cid:54) i (cid:54) r t i = 1 , (cid:88) (cid:54) i (cid:54) r t i ρ i = (0 , . Each ρ i is the rotation number of an ergodic measure µ i ∈ M ( f ). Applying Poincar´e RecurrenceTheorem and Birkhoff Ergodic Theorem, one can find a positively recurrent point z i of f having ρ i as a rotation number. Fix a lift ˇ z i of z i and a small neighborhood ˇ W i of ˇ z i that trivializes ˇ F .One can find a subsequence ( ˇ f n l ( z i )) l (cid:62) of ˇ f n ( z i ) n (cid:62) and a sequence ( p i,l ) l (cid:62) of integers such thatˇ f n l (ˇ z i ) ∈ ˇ W i + p i,l and such that lim l → + ∞ p i,l /n l = ρ i . One deduces that the transverse homologicalspace THS( F ) contains p i,l . If l is large enough, the p i,l generate the plane and (0 , 0) is contained inthe interior of the polygonal defined by these points. By Proposition 13, we deduce that the leaves ofˇ F are uniformly bounded.Suppose now that (0 , 0) belongs to the interior of a segment with irrational slope included in ∂ (cid:0) rot( ˇ f ) (cid:1) .If this segment [ ρ , ρ ] is chosen maximal, then ρ and ρ are extremal points of rot( ˇ f ) and respectivelyequal to the rotation number of ergodic measures µ and µ in M ( f ). Let W i ⊂ T be a trivializing boxof F such that µ i ( W i ) (cid:54) = 0 and ˇ W i ⊂ R a lift of W i . The first return map Φ i : W i → W i , z (cid:55)→ f τ i ( z )(where τ i : W i → N ) is defined µ i -almost everywhere on W i as the displacement function ξ i : W i → Z ,where ˇ f τ i ( z ) (ˇ z ) ∈ ˇ W i + ξ i ( z ), if ˇ z is the lift of z that belongs to ˇ W i . Let ψ : R → R be a non triviallinear form that vanishes on our segment. Using Birkhoff Ergodic Theorem, one knows that µ i -almostevery point z has a rotation number ρ i , and solim n → + ∞ (cid:80) n − k =0 ξ i (Φ ki ( z )) (cid:80) n − k =0 τ i (Φ ki ( z )) = ρ i . By Kac’s theorem, one knows that τ i is µ i -integrable and satisfies (cid:82) W i τ i dµ i = µ i ( (cid:83) k ∈ Z f k ( W i )) ∈ (0 , ξ i /τ i is bounded, which implies that ξ i is µ i -integrable. Consequently, onehas lim n → + ∞ (cid:80) n − k =0 ξ i (Φ ki ( z )) (cid:80) n − k =0 τ i (Φ ki ( z )) = (cid:82) W i ξ i dµ i (cid:82) W i τ i dµ i , which implies that (cid:90) W i ξ i dµ i = (cid:18)(cid:90) W i τ i dµ i (cid:19) ρ i (cid:54) = 0and (cid:90) W i ψ ◦ ξ i dµ i = ψ (cid:18)(cid:90) W i ξ i dµ i (cid:19) = 0 . Note that ψ ◦ ξ i ( z ) (cid:54) = 0 if ξ i ( z ) (cid:54) = 0, because ξ i ( z ) is an integer and the kernel of ψ is generated by asegment with irrational slope. We deduce that there exists z , z (cid:48) in W such that ψ ◦ ξ ( z ) < < ψ ◦ ξ ( z (cid:48) ) . Consequently, one can find z (cid:48)(cid:48) ∈ W , z (cid:48)(cid:48) ∈ W and integers n , n such that (0 , 0) is in the interior ofthe quadrilateral determined by ξ ( z ) , ξ ( z (cid:48) ) , (cid:80) n − k =0 ξ (Φ k ( z (cid:48)(cid:48) )) (cid:80) n − k =0 τ (Φ k ( z (cid:48)(cid:48) )) , (cid:80) n − k =0 ξ (Φ k ( z )) (cid:80) n − k =0 τ (Φ k ( z (cid:48)(cid:48) )) , because the last two points may be chosen arbitrarily close to ρ and ρ . The set THS( F ) containingthe integers ξ ( z ) , ξ ( z (cid:48) ) , n − (cid:88) k =0 ξ (Φ k ( z (cid:48)(cid:48) )) , n − (cid:88) k =0 ξ (Φ k ( z )) , one can apply Proposition 13 to conclude that the leaves of ˇ F are uniformly bounded. (cid:3) Remark . As a corollary, one deduces that F is singular and that ˇ f is not fixed point free. Applyingthis to ˇ f q − p , for every rational p/q ∈ int(rot( ˇ f )), one deduces that there exists a point z ∈ R suchthat ˇ f q ( z ) = z + q . This result was already well known, due to Franks [F2]. Proposition 75. We suppose that the leaves of ˇ F are uniformly bounded. If there exists an admissibletransverse path ˇ γ : [ a, b ] → dom( ˇ F ) of order q and an integer p ∈ Z such that ˇ γ and ˇ γ + p intersect ˇ F -transversally at φ ˇ γ ( t ) = φ (ˇ γ + p )( s ) , where s < t , then p/q belongs to rot( ˇ f ) .Proof. By Corollary 22 one deduces that for every k (cid:62) γ | [ a,t ] (cid:32) (cid:89)
1, one has (cid:107) ˇ f kq (ˇ z k ) − ˇ z k − ( k − p (cid:107) (cid:54) K . Denote z k theprojection of ˇ z k in T . Choose a measure µ that is the limit of a subsequence of (cid:16) kq (cid:80) kq − i =0 δ f i ( z k ) (cid:17) k (cid:62) for the weak ∗ topology. It is an invariant measure of f of rotation number p/q for ˇ f . (cid:3) Let us state the following improved version of Atkinson’s Lemma: ORCING THEORY FOR TRANSVERSE TRAJECTORIES OF SURFACE HOMEOMORPHISMS 77 Proposition 76. Let ( X, B , µ ) be a probability space and T : X → X an ergodic automorphism. If ϕ : X → R is an integrable map such that (cid:82) ϕ dµ = 0 , then for every B ∈ B and every ε > , one has µ (cid:32)(cid:40) x ∈ B, ∃ n (cid:62) , T n ( x ) ∈ B and (cid:54) n − (cid:88) k =0 ϕ ( T k ( x )) < ε (cid:41)(cid:33) = µ ( B ) . Proof. Let us consider B ∈ B and set A = B \ (cid:40) x ∈ B, ∃ n (cid:62) , T n ( x ) ∈ B and (cid:54) n − (cid:88) k =0 ϕ ( T k ( x )) < ε (cid:41) } . Atkinson’s result directly implies that there exists a set A (cid:48) ⊂ A with µ ( A (cid:48) ) = µ ( A ) such that, for every x ∈ A (cid:48) , there exists a subsequence ( n l ) l ∈ N such that T n l ( x ) ∈ A and lim l →∞ (cid:80) n l − k =0 ϕ ( T k ( x )) = 0.Assume, for a contradiction, that µ ( A ) > 0. There exists some x ∈ A (cid:48) and n > y = T n ( x ) ∈ A and a = (cid:80) n − k =0 ϕ ( T k ( x )) ∈ ( − ε, ε ), and since x ∈ A we know that a < 0. Since x ∈ A (cid:48) there exists some n > n such that T n ( x ) ∈ A and a < (cid:80) n − k =0 ϕ ( T k ( x )) < ε + a . Thisimplies that T n − n ( y ) ∈ A and that 0 < (cid:80) n l − n − k =0 ϕ ( T k ( y )) < ε , which is a contradiction since y ∈ A proving the claim. (cid:3) Proof of Theorem 62. We will give a proof by contradiction. Replacing f by f q and ˇ f by ˇ f q − p , where q ∈ N and p ∈ Z , we can suppose that the frontier of rot( ˇ f ) contains a segment [ ρ , ρ ] with irrationalslope, that (0 , 0) is in its interior and that ρ and ρ are extremal points of rot( ˇ f ). We can supposemoreover than for every ρ ∈ rot( ˇ f ), one has (cid:104) ρ ⊥ , ρ (cid:105) (cid:54) (cid:54) (cid:104) ρ ⊥ , ρ (cid:105) . We consider two ergodic measures µ and µ in M ( f ) whose rotation vectors are ρ and ρ respectively. We know that there exists apoint z ∈ R such that rot( z ) = ρ and that projects onto a bi-recurrent point of f . By Proposition76, we have a stronger result: Lemma 77. There exists a point z , projecting to a bi-recurrent point and with rot( z ) = ρ , suchthat for every ε ∈ {− , } one can find a sequence ( p l , q l ) l (cid:62) in Z × N satisfying: lim l → + ∞ q l = + ∞ , lim l → + ∞ ˇ f q l ( z ) − z − p l = 0 , lim l → + ∞ (cid:104) ρ ⊥ , p l (cid:105) = 0 , ε (cid:104) ρ ⊥ , p l (cid:105) > and a sequence ( p (cid:48) l , q (cid:48) l ) l (cid:62) in Z × N satisfying: lim l → + ∞ q (cid:48) l = + ∞ , lim l → + ∞ ˇ f − q (cid:48) l ( z ) − z − p (cid:48) l = 0 , lim l → + ∞ (cid:104) ρ ⊥ , p (cid:48) l (cid:105) = 0 , ε (cid:104) ρ ⊥ , p (cid:48) l (cid:105) > . Proof. Let W be a small disk such that µ ( W ) (cid:54) = 0 and ˇ W a lift of W . Define the maps τ and ξ like in Proposition 73. The measure µ being ergodic, one knows that (cid:82) W τ dµ = 1 andthat (cid:82) W ξ dµ = ρ . Let us define on W the first return map T : z (cid:55)→ f τ ( z ) ( z ) and the function ϕ : z (cid:55)→ ε (cid:104) ρ ⊥ , ξ ( z ) (cid:105) .For each integer i (cid:62) 1, let ( B i,j ) (cid:54) j (cid:54) k i be a covering of W by open sets with diameter smaller than1 /i , and define C i,j = (cid:40) x ∈ B i,j ∩ W , ∃ n (cid:62) , T n ( x ) ∈ B i,j ∩ W and 0 (cid:54) n − (cid:88) k =0 ϕ ( T k ( x )) < /i (cid:41) Set C i = (cid:83) k i j =1 C i,j and C = (cid:84) i (cid:62) C i . By Proposition 76, one knows that µ ( C i ) = µ ( C ) = µ ( W ),and if C (cid:48) is the subset of the bi-recurrent points of C with rotation vector ρ , then µ ( C (cid:48) ) = µ ) ( C ).. If z belongs to C (cid:48) , one can find an increasing integer sequence ( m l ) l (cid:62) such thatlim l → + ∞ T m l ( z ) = z , lim l → + ∞ m l − (cid:88) k =0 ε (cid:104) ρ ⊥ , ξ ( T k ( z ) (cid:105) = 0 , m l − (cid:88) k =0 ε (cid:104) ρ ⊥ , ξ ( T k ( z ) (cid:105) (cid:62) . Setting p l = (cid:80) m l − k =0 ξ ( T k ( z )) and q l = (cid:80) m l − k =0 τ ( T k ( z )), one gets the first assertion of the lemma,with a large inequality instead of a strict one. Noting that lim l → + ∞ (cid:107) p l (cid:107) = + ∞ and that the linegenerated by ρ has irrational slope, one deduces that the inequality is strict. The second assertioncan be proved analogously. (cid:3) We note that, by Proposition 73, the leaves of ˇ F are uniformly bounded. Let us choose z as in theprevious lemma. The fact that rot( z ) = ρ tells us that the whole trajectory ˇ I Z ( z ) is a proper pathdirected by ρ . The fact that the leaves of ˇ F are uniformly bounded and that every leaf met by ˇ I Z ˇ F ( z )is also met by ˇ I Z ( z ) implies that ˇ I Z ˇ F ( z ) is a transverse proper path directed by ρ . We parameterizeˇ I Z ˇ F ( z ) in such a way that ˇ I Z ˇ F ( z ) | [ l,l +1] = ˇ I ˇ F ( ˇ f l ( z )). We consider sequences ( p l , q l ) l (cid:62) and ( p (cid:48) l , q (cid:48) l ) l (cid:62) given by the previous lemma (the sign ε has no importance at the beginning). Lemma 78. For every closed segment [ a, b ] ⊂ R and every positive real numbers L , ε , there exists p ∈ Z and a segment [ a (cid:48) , b (cid:48) ] ⊂ R satisfying a (cid:48) − b > L such that |(cid:104) ρ ⊥ , p (cid:105)| < ε and such that the paths ˇ I Z ˇ F ( z ) | [ a,b ] and ( ˇ I Z ˇ F ( z ) + p ) | [ a (cid:48) ,b (cid:48) ] are equivalent. One has a similar result replacing the inequality a (cid:48) − b > L by a − b (cid:48) > L .Proof. Let us choose integers q and q (cid:48) such that [ a, b ] ⊂ ( q, q (cid:48) ). As z projects to a bi-recurrent point,one can find l , using Lemma 77, sufficiently large, such that q l > q (cid:48) − q + L and such that ˇ f q l ( ˇ f q ( z )) − p l is so close to ˇ f q ( z ) that we can affirm that ˇ I Z ˇ F ( z ) | [ a,b ] is equivalent to a path ( ˇ I Z ˇ F ( z ) − p l ) | [ a (cid:48) ,b (cid:48) ] , where[ a (cid:48) , b (cid:48) ] ⊂ ( q + q l , q (cid:48) + q l ). Note that |(cid:104) ρ ⊥ , p l (cid:105)| < ε if l is sufficiently large. The version with the inequality a − b (cid:48) > L can be proven similarly by using the sequences ( p (cid:48) l ) l (cid:62) and ( q (cid:48) l ) l (cid:62) . (cid:3) Lemma 79. There is no p ∈ Z \ { } such that ˇ I Z ˇ F ( z ) and ˇ I Z ˇ F ( z ) + p intersect ˇ F -transversally.Proof. Write ˇ I Z ˇ F ( z ) = γ for convenience. Suppose that γ and γ + p intersect ˇ F -transversally at φ = φ γ ( t ) = φ ( γ + p )( s ) . The leaves being uniformly bounded, one knows that φ γ ( t ) (cid:54) = φ γ ( t ) + p andso t (cid:54) = s . Replacing p with − p if necessary, one can suppose that s < t . By Proposition 75, there exists q (cid:62) p/q ∈ rot( ˇ f ). Consequently, one has (cid:104) ρ ⊥ , p (cid:105) (cid:54) 0. By assumption, the segment [0 , ρ ]has irrational slope and p (cid:54) = 0, so one deduces that (cid:104) ρ ⊥ , p (cid:105) < N such that γ | [ − N,N ] and ( γ + p ) | [ − N,N ] intersect ˇ F -transversally at φ . By Lemma 78, we can find some p (cid:48) ∈ Z such that |(cid:104) ρ ⊥ , p (cid:48) (cid:105)| is sufficiently small asto get (cid:104) ρ ⊥ , p + p (cid:48) (cid:105) < 0, and such that there exists some a (cid:48) , b (cid:48) with N < a (cid:48) < b (cid:48) , where ( γ + p (cid:48) ) | [ a (cid:48) ,b (cid:48) ] isequivalent to γ | [ − N,N ] . This implies that ( γ + p + p (cid:48) ) | [ a (cid:48) ,b (cid:48) ] is equivalent to ( γ + p ) | [ − N,N ] , and so γ and γ + p + p (cid:48) intersect ˇ F -transversally at φ γ ( t ) = φ ( γ + p + p (cid:48) )( s (cid:48) ) where s (cid:48) > t . So, one knows that γ and γ − p − p (cid:48) intersect ˇ F -transversally at φ γ ( s (cid:48) ) = φ ( γ − p − p (cid:48) )( t ) . We deduce as before, by Proposition75, that for some q (cid:48) > − p − p (cid:48) q (cid:48) ∈ rot( ˇ f ), a contradiction since (cid:104) ρ ⊥ , − p − p (cid:48) (cid:105) < (cid:3) Lemma 80. The path ˇ I Z ˇ F ( z ) is a line ORCING THEORY FOR TRANSVERSE TRAJECTORIES OF SURFACE HOMEOMORPHISMS 79 Proof. Here again, write ˇ I Z ˇ F ( z ) = γ . If γ is a not a line, by Proposition 4 one knows that thereexist two segments [ a , b ] and [ a , b ] such that γ | [ a ,b ] and γ | [ a ,b ] intersect ˇ F -transversally. ByLemma 78, one deduces that there exist p ∈ Z \ { } and a segment [ a (cid:48) , b (cid:48) ] such that γ | [ a ,b ] and( γ + p ) | [ a (cid:48) ,b (cid:48) ] intersect ˇ F -transversally. This contradicts Lemma 79. (cid:3) Similarly, we can find a point z of rotation number ρ that projects onto a recurrent point of f andsuch that ˇ I Z ˇ F ( z ) is a line directed by ρ that does not meet transversally its integer translated. Lemma 81. The line ˇ I Z ˇ F ( z ) intersects ˇ F -transversally one of the translates of ˇ I Z ˇ F ( z ) .Proof. Let us prove by contradiction that γ = ˇ I Z ˇ F ( z ) intersects ˇ F -transversally one of the translatesof γ = ˇ I Z ˇ F ( z ). If not, let us denote by U the union of leaves that are met by γ . Its complement canbe written R ( U ) (cid:116) L ( U ) where R ( U ) = R ( γ ) \ U is the union of r ( γ ) and of the set of singularitiesat the right of γ and L ( U ) = L ( γ ) \ U is the union of l ( γ ) and of the set of singularities at theleft of γ . If γ and γ do not intersect ˇ F -transversally, then by Corollary 6, one knows that either γ ∩ R ( U ) = ∅ or γ ∩ L ( U ) = ∅ . As γ is directed by ρ (cid:107) ρ (cid:107) and γ is directed by the opposite vector ρ (cid:107) ρ (cid:107) = − ρ (cid:107) ρ (cid:107) , one knows that if γ ∩ R ( U ) = ∅ , then R ( γ ) ∩ R ( U ) = ∅ , and if γ ∩ L ( U ) = ∅ , then L ( γ ) ∩ L ( U ) = ∅ . Consequently, γ and γ cannot meet a common leaf. Indeed if φ is such a leaf,one knows by Proposition 4 that it is met once by γ and γ . So, the α -limit set of φ is contained in L ( U ) ∩ L ( γ ) and the ω -limit set is included in R ( U ) ∩ R ( γ ), which is impossible.So, if the conclusion of our lemma is not true, there exists a partition Z = A − (cid:116) A + , where p ∈ A − ⇔ ( r ( γ ) ∪ U ) + p ⊂ l ( γ ) ,p ∈ A + ⇔ ( l ( γ ) ∪ U ) + p ⊂ r ( γ ) . Also, by Lemma 79 and the fact that γ is directed by ρ , one knows that l ( γ + p ) ⊂ l ( γ ) if 0 (cid:54) (cid:104) ρ ⊥ , p (cid:105) and one deduces this partition is a cut of the order on Z defined as follows p (cid:22) p (cid:48) ⇔ (cid:104) ρ ⊥ , p (cid:105) (cid:54) (cid:104) ρ ⊥ , p (cid:48) (cid:105) . Let us fix a leaf φ that intersects γ . By Lemma 78, one knows that there exists p (cid:54) = (0 , 0) suchthat φ intersects γ + p . One deduces that A − + p = A − and A + + p = A + , which of course isimpossible. (cid:3) End of the proof of Theorem 62. Replacing γ by a translate if necessary, we can always suppose that γ and γ intersect ˇ F -transversally at γ ( t ) = γ ( t ) and we define γ = γ | [ −∞ ,t ] γ | [ t , + ∞ ] whichis an admissible transverse proper path. There exist two segments [ a , b ] and [ a , b ] containing t and t respectively in their interior, such that γ | [ a ,b ] and γ | [ a ,b ] intersect transversally at γ ( t ) = γ ( t ). Using Lemma 78, one can find p and p in Z distinct, and segments [ a (cid:48) , b (cid:48) ] ⊂ ( −∞ , t ) and[ a (cid:48) , b (cid:48) ] ⊂ ( t , + ∞ ) such that ( γ + p ) | [ a (cid:48) ,b (cid:48) ] is equivalent to γ | [ a ,b ] and ( γ + p ) | [ a (cid:48) ,b (cid:48) ] is equivalentto γ | [ a ,b ] . We deduce that there exists t (cid:48) ∈ ( a (cid:48) , b (cid:48) ) and t (cid:48) ∈ ( a (cid:48) + t − t , b (cid:48) + t − t ) such that γ + p and γ + p intersect ˇ F -transversally at φ = φ ( γ + p )( t (cid:48) ) = φ ( γ + p )( t (cid:48) ) . So γ and γ + p − p intersect ˇ F -transversally at φ − p = φ γ ( t (cid:48) ) = φ ( γ + p − p )( t (cid:48) ) . Observing that t (cid:48) < t (cid:48) , one deduces, byProposition 75, that there exists q (cid:62) p − p ) /q ∈ rot( ˇ f ) and thus (cid:104) ρ ⊥ , p − p (cid:105) < 0. ButLemma 77 tells us that p and p can be chosen so that (cid:104) ρ ⊥ , p (cid:105) < (cid:104) ρ ⊥ , p (cid:105) > 0. We have founda contradiction. (cid:3) Proof of Theorem 63. In the proof, we will use the sup norm (cid:107) (cid:107) ∞ where (cid:107) ( x , x ) (cid:107) ∞ = max( | x | , | x | )which will be more convenient that the Euclidean norm and will write d ∞ ( z, X ) = inf z (cid:48) ∈ X (cid:107) z − z (cid:48) (cid:107) ∞ .Replacing f by f q and ˇ f by ˇ f q − p , where q ∈ N and p ∈ Z , we can suppose that (0 , 0) is in the interiorof rot( ˇ f ). Here again, we consider an identity isotopy I (cid:48) of f that is lifted to an identity isotopy ˇ I (cid:48) ofˇ f . We consider a maximal hereditary singular isotopy I larger than I (cid:48) and its lift ˇ I to R . We considera foliation F transverse to I an its lift ˇ F to R . One knows by Proposition 73 that the leaves of ˇ F areuniformly bounded. In the remainder of the proof we will usually work in the universal covering spaceof T , with paths transversal to the lifted foliation ˇ F . The theorem is an immediate consequence ofthe following, where the direction D ( γ ) of a path γ : [ a, b ] → R is defined as D ( γ ) = γ ( b ) − γ ( a ): Proposition 82. There exists a constant L such that for every transverse admissible path γ of order n , one has d ∞ ( D ( γ ) , n rot( f )) (cid:54) L . We will begin by proving: Lemma 83. There exist a transverse admissible path γ ∗ : [0 , → R , a real number K ∗ and aninteger p ∗ ∈ Z such that:- every transverse path γ whose diameter is larger than K ∗ intersects ˇ F -transversally an integer trans-late of γ ∗| [1 , ;- γ ∗| [2 , and γ ∗| [0 , + p ∗ intersect ˇ F -transversally.Proof. Let us choose N large enough such (1 /N, 0) and (0 , /N ) belong to the interior of rot( ˇ f ). Aspreviously noted in Remark 74, there exists a point z satisfying ˇ f N ( z ) = z + (1 , 0) and a point z satisfying ˇ f N ( z ) = z + (0 , I Z ˇ F ( z ) and ˇ I Z ˇ F ( z ) are parameterized, suchthat ˇ I Z ˇ F ( z )( t + 1) = ˇ I Z ˇ F ( z )( t ) + (1 , 0) and ˇ I Z ˇ F ( z )( t + 1) = ˇ I Z ˇ F ( z )( t ) + (0 , Sub-lemma 84. There exists a real number K such that if γ is a transverse path that does notintersect ˇ F -transversally ˇ I Z ˇ F ( z ) , then either π ( γ ( t )) > − K or π ( γ ( t )) < K and if it does notintersect ˇ F -transversally ˇ I Z ˇ F ( z ) , then either π ( γ ( t )) > − K or π ( γ ( t )) < K .Proof. There exists K > F is bounded by K and there exists K (cid:48) > I Z ˇ F ( z ) ⊂ R × ( − K (cid:48) , K (cid:48) ). Setting K = K + K (cid:48) , note that every leaf that intersects R × ( −∞ , − K ] belongs to r ( ˇ I Z ˇ F ( z )) and every leaf that intersects R × [ K, + ∞ ) belongs to l ( ˇ I Z ˇ F ( z )).It remains to apply Corollary 6. We have a similar argument for ˇ I Z ˇ F ( z ). (cid:3) Setting K ∗ = 2 K + 1, one deduces immediately: Corollary 85. If γ is a transverse path and if the diameter of π ◦ γ is larger than K ∗ , there exists p ∈ Z such that γ intersects ˇ F -transversally ˇ I Z ˇ F ( z ) + p and if the diameter of π ◦ γ is larger than K ∗ , there exists p ∈ Z such that γ intersects ˇ F -transversally ˇ I Z ˇ F ( z ) + p . In particular γ = ˇ I Z ˇ F ( z ) intersects γ = ˇ I Z ˇ F ( z ) ˇ F -transversally at a point γ ( t ) = γ ( t ). One canfind an integer r > γ | [ t − r,t + r ] and γ | [ t − r,t + r ] intersect ˇ F -transversally at γ ( t ) = γ ( t ). Let γ ∗ : [0 , → R be a path such that ORCING THEORY FOR TRANSVERSE TRAJECTORIES OF SURFACE HOMEOMORPHISMS 81 - γ ∗ [0 , is a reparameterization of γ | [ t − (4 r +2) ,t − (2 r +2)] ;- γ ∗ [1 , is a reparameterization of γ | [ t − (2 r +2) ,t ] γ | [ t ,t +(2 r +2)] ;- γ ∗ [2 , is a reparameterization of γ | [ t +(2 r +2) ,t +(4 r +2)] .Let us prove that γ ∗ satisfies the proposition. Observe first that γ ∗ is admissible of order (8 r + 4) N byCorollary 21 and that the paths γ ∗ | [0 , and γ ∗ | [1 , are admissible of order (6 r + 2) N . Note first that γ ∗| [2 , and γ ∗| [0 , + (3 r + 2 , r + 2) intersect ˇ F -transversally. One can set p ∗ = (3 r + 2 , r + 2). Let γ be a transverse path such that the diameter of π ◦ γ is larger than K ∗ . By Sub-lemma 84 one knowsthat there exists p ∈ R such that γ intersects ˇ F -transversally γ + p . This means that there existtwo real segments J and J such that- γ | J intersects ˇ F -transversally γ | J + p ;- γ | int J and γ | int J + p are equivalent.If the length of J is smaller than 2 r + 1, then J is included in an interval [ t − (2 r + 2) + l , t + l ].This implies that γ intersects ˇ F -transversally γ ∗ | [1 , + p + ( l , r +1, then J contains an interval [ t − r + l , t + r + l ]. This implies that γ intersects ˇ F -transversally γ | [ t − r,t + r ] + p + ( l , 0) and so intersects ˇ F -transversally γ ∗ | [1 , + p + ( l , − r ). We get the sameconclusion for a transverse path such that the diameter of π ◦ γ is larger than K ∗ . (cid:3) Proof of the Proposition 82. We denote by K ∗∗ the diameter of γ ∗ and by K ∗∗∗ the diameter ofrot( f ). Let γ : [ a, b ] → R be a transverse path such that (cid:107) D ( γ ) (cid:107) > K ∗ . One can find c , d in ( a, b )with c < d such that (cid:107) D ( γ | [ a,c ] ) (cid:107) = (cid:107) D ( γ | [ d,b ] ) (cid:107) = K ∗ . Note that, if c is chosen to be minimal withthis property, and d is chosen to be maximal, then the diameter of both γ | [ a,c ] and γ | [ d,b ] are at most2 K ∗ . There exist p and p (cid:48) in Z such that- γ | [ a,c ] and γ ∗ | [1 , + p intersect ˇ F -transversally at γ ( t ) = γ ∗ ( s ) + p ;- γ | [ d,b ] and γ ∗ | [1 , + p (cid:48) intersect ˇ F -transversally at γ ( t (cid:48) ) = γ ∗ ( s (cid:48) ) + p (cid:48) .If γ is admissible of order n , then the path γ (cid:48) = ( γ ∗ | [0 ,s ] + p ) γ | [ t,t (cid:48) ] ( γ ∗ | [ s (cid:48) , + p (cid:48) )is admissible of order n + (12 r + 4) N by Corollary 21 and one has (cid:107) D ( γ (cid:48) ) − D ( γ ) (cid:107) (cid:54) K ∗ + 2 K ∗∗ . Recall that γ ∗ [2 , intersects ˇ F -transversally γ ∗ [0 , + p ∗ . One deduces that ( γ ∗ | [ s (cid:48) , + p (cid:48) ) intersects ˇ F -transversally ( γ ∗ | [0 ,s ] + p ) + p (cid:48)(cid:48) , where p (cid:48)(cid:48) = p (cid:48) − p + p ∗ and so that γ (cid:48) intersects ˇ F -transversally γ (cid:48) + p (cid:48)(cid:48) .Proposition 75 tells us that p (cid:48)(cid:48) / ( n + (12 r + 4) N ) belongs to rot( ˇ f ), which implies d ( p (cid:48)(cid:48) , n rot( ˇ f )) (cid:54) (12 r + 4) N K ∗∗∗ . Observe now that (cid:107) p (cid:48)(cid:48) − D ( γ (cid:48) ) (cid:107) (cid:54) K ∗∗ and so (cid:107) p (cid:48)(cid:48) − D ( γ ) (cid:107) (cid:54) K ∗ + 4 K ∗∗ . So, one gets d ( D ( γ ) , n rot( ˇ f )) (cid:54) K ∗ + 4 K ∗∗ + (12 r + 4) N K ∗∗∗ . (cid:3) Proof of Theorem 64 . Here again, using the fact that for every q (cid:62) p ∈ Z , one hasrot( ˇ f q + p ) = q rot( ˇ f ) + p , it is easy to see that it is sufficient to prove the result in the case where(0 , 0) belongs to the interior of rot( ˇ f ). Here again, we consider an identity isotopy I (cid:48) of f that is liftedto an identity isotopy ˇ I (cid:48) of ˇ f . We consider a maximal hereditary singular isotopy I larger than I (cid:48) andits lift ˇ I to R . We consider a foliation F transverse to I an its lift ˇ F to R . We know that the leavesof ˇ F are uniformly bounded. We can immediately deduce the theorem from what has been done inthe previous proof and Theorem 36. Indeed, we know that there are two transverse loops associatedto periodic points that have a transverse intersection. We will give a proof that does not use Theorem36 by exhibiting separated sets.Let us begin with the following lemma: Lemma 86. There exists a constant K such that for every point z ∈ dom( ˇ F ) and any z (cid:48) for which φ z (cid:48) intersects I ˇ F ( z ) , one has d ( z, z (cid:48) ) (cid:54) K .Proof. There exists K (cid:48) > F is bounded by K (cid:48) . Moreover, theset (cid:83) t ∈ [0 , ,z ∈ [0 , I ( z ), being compact, is included in [ − K (cid:48)(cid:48) , K (cid:48)(cid:48) + 1] ,for some K (cid:48)(cid:48) > 0. The leavesthat ˇ I ˇ F ( z ) intersects, are also intersected by ˇ I ( z ) (see the beginning of Section 3). One deduces that K = K (cid:48) + K (cid:48)(cid:48) satisfies the conclusion of the lemma. (cid:3) We consider the paths γ = ˇ I ˇ F ( z ) and γ = ˇ I Z ˇ F ( z ) defined in the proof of Theorem 63. We keep thesame notations and set z ∗ = γ ( t ) = γ ( t ). Let us define K (cid:48)(cid:48)(cid:48) = max (cid:0) diam( γ | [ t ,t + r ] ) , diam( γ | [ t ,t + r ] ) (cid:1) and choose an integer m (cid:62) mr (cid:62) K (cid:48)(cid:48)(cid:48) + 2 K + K + 1. Set γ (cid:48) = γ | [ t ,t + mr ] , γ (cid:48) = γ | [ t ,t + mr ] . Fix n and for every e = ( ε , . . . , ε n ) ∈ { , } n define γ (cid:48) e = (cid:89) (cid:54) i (cid:54) n ( γ (cid:48) ε i + p i − ) , where the sequence ( p i ) (cid:54) i (cid:54) n satisfies k = 0 and is defined inductively by the relation: p i +1 = (cid:40) p i + ( mr, 0) if ε i = 0 ,p i + (0 , mr ) if ε i = 1 . The path γ (cid:48) ω is admissible of order l = nmrN . More precisely, there exists a point z e ∈ φ z ∗ such thatˇ f l ( z e ) ∈ φ z ∗ + k n , and such that γ (cid:48) e = ˇ I l ˇ F ( z e ). Lemma 87. If e and e (cid:48) are two different sequences in { , } n , there exists j ∈ { , . . . , l − } such that (cid:107) ˇ f j ( z e ) − ˇ f j ( z e (cid:48) ) (cid:107) (cid:62) .Proof. Consider the integer i ∗ such that ε i ∗ (cid:54) = ε (cid:48) i ∗ and ε i = ε (cid:48) i if i < i ∗ . The leaf φ z ∗ + p i ∗ is intersectedby γ (cid:48) e but not by γ (cid:48) e (cid:48) . More precisely d ( φ z ∗ + p i ∗ , γ (cid:48) e (cid:48) ) (cid:62) mr − K (cid:48) − K . Using Lemma 86 , onededuces that there exists j ∈ { , . . . , l } such that d ( ˇ f j ( z ω ) , φ z ∗ + p i ∗ ) (cid:54) K . Moreover, one knows that d ( ˇ f j ( z e (cid:48) ) , γ (cid:48) e (cid:48) ) (cid:54) K because γ (cid:48) e intersects φ ˇ f j ( z e (cid:48) ) . One deduces that (cid:107) ˇ f j ( z e ) − ˇ f j ( z e (cid:48) ) (cid:107) (cid:62) mr − K (cid:48) − K − K (cid:62) . ORCING THEORY FOR TRANSVERSE TRAJECTORIES OF SURFACE HOMEOMORPHISMS 83 (cid:3) To finish the proof of the proposition, let us define on T the distance d ( Z, Z (cid:48) ) = inf π ( z )= Z, π ( z (cid:48) )= Z (cid:48) (cid:107) z − z (cid:48) (cid:107) , where π : R → T ,z (cid:55)→ z + Z is the projection. Note that for every Z ∈ T , one has π − ( B ( Z, / (cid:71) π ( z )= Z B ( z, / π | B ( z, / is an isometry from B ( z, / 4) onto B ( Z, / ε ∈ (0 , / 4) such that for every z , z (cid:48) in R , one has (cid:107) z − z (cid:48) (cid:107) < ε ⇒ (cid:107) ˇ f ( z ) − ˇ f ( z (cid:48) ) (cid:107) < / . One deduces that two points Z and Z (cid:48) such that d ( f j ( Z ) , f j ( Z )) < ε , for every j ∈ { , . . . l − } arelifted by points z , z (cid:48) such that (cid:107) ˇ f j ( z ) − ˇ f j ( z ) (cid:107) < ε , for every j ∈ { , . . . l − } .Consequently, the points z e project on a ( nmrN, ε )-separated set of cardinality 2 n . One deduces that h ( f ) > log 2 /mrN . (cid:3) Bibliography [A] G. Atkinson: Recurrence of co-cycles and random walks, J. London Math. Soc. (2) , (1976), 486–488.[AZ] S. Addas–Zanata: Uniform bounds for diffeomorphisms of the torus and a conjecture by P. Boyland, e-printarXiv:1403.7533 (2014), to appear in Jour. Lon. Math. Soc. [AT] S. Addas-Zanata , F. 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q (cid:62) α (cid:48) does not meet neither T q ( φ (cid:101) γ ( b (cid:48) ) | ( −∞ , − K ] ) nor T q ( φ (cid:101) γ ( b (cid:48) ) | [ K, + ∞ ) ). One deduces that there exists q such that for every q (cid:62) q , α (cid:48) does not meet T q ( φ (cid:101) γ ( b (cid:48) ) ). This implies that if q (cid:62) q , then T q ( φ (cid:101) γ ( b (cid:48) ) )does not meet α (cid:48)(cid:48) and so is included in R ( α (cid:48)(cid:48) ). In particular it cannot intersect neither T ( (cid:101) f n ( φ − (cid:101) z )) nor (cid:101) f n ( φ + (cid:101) z ). Consequently, this implies that (cid:101) f n ( φ (cid:101) γ ( a (cid:48) ) ) does not meet T q ( (cid:101) φ (cid:101) γ ( b (cid:48) ) ), if q (cid:62) q . (cid:3) ˜ γ φ ˜ γ ( a ) T ( φ ˜ γ ( a ) ) φ ˜ γ ( b ) T q ( φ ˜ γ ( b ) ) f n ( φ ˜ γ ( a ) ) f n ( T ( φ ˜ γ ( a ) )) α Figure 13.