A uniqueness theorem for transitive Anosov flows obtained by gluing hyperbolic plugs
aa r X i v : . [ m a t h . D S ] M a y A uniqueness theorem for transitive Anosov flowsobtained by gluing hyperbolic plugs
Fran¸cois B´eguin and Bin Yu ∗ May 23, 2019
Abstract
In a previous paper with C. Bonatti ([5]), we have defined a general procedure to buildnew examples of Anosov flows in dimension 3. The procedure consists in gluing togethersome building blocks, called hyperbolic plugs , along their boundary in order to obtain a closed3-manifold endowed with a complete flow. The main theorem of [5] states that (under somemild hypotheses) it is possible to choose the gluing maps so the resulting flow is Anosov. Theaim of the present paper is to show a uniqueness result for Anosov flows obtained by sucha procedure. Roughly speaking, we show that the orbital equivalence class of these Anosovflows is insensitive to the precise choice of the gluing maps used in the construction. Theproof relies on a coding procedure which we find interesting for its own sake, and follows astrategy that was introduced by T. Barbot in a particular case.
In a previous paper written with C. Bonatti ([5]), we have proved a result allowing to constructtransitive Anosov flows in dimension 3 by “gluing hyperbolic plugs along their boundaries”. Thepurpose of the present paper is to study Anosov flows obtained by such a construction. We focusour attention on the diffeomorphisms that are used to glue together the boundaries of the hyper-bolic plugs. We aim to understand what is the impact of the choice of these diffeomorphisms onthe dynamics of the resulting Anosov flows. We will see that two gluing diffeomorphisms thatare “strongly isotopic” yield some Anosov flows that are orbitally equivalent. In other words,in [5], we have proved the existence of Anosov flows constructed by a certain gluing procedure,and the goal of the present paper is to prove a uniqueness result for these Anosov flows.In order to state some precise questions and results, we need to introduce some terminology.A hyperbolic plug is a pair (
U, X ) where U is a (not necessarily connected) compact three-dimensional manifold with boundary, and X is a vector field on U , transverse to ∂U , and suchthat the maximal invariant set Λ X := T t ∈ R X t ( U ) is a saddle hyperbolic set for the flow ( X t ).Given such a hyperbolic plug ( U, X ), we decompose ∂U as the disjoint union of an entranceboundary ∂ in U (the union of the connected component of ∂U where the vector field X is pointinginwards U ) and an exit boundary ∂ out U (the union of the connected component of ∂U wherethe vector field X is pointing outwards U ). The stable lamination W s (Λ X ) of the maximalinvariant set Λ X intersects transversally the entrance boundary ∂ in U and is disjoint from theexit boundary ∂ out U . Hence, L sX := W s (Λ X ) ∩ ∂U a one-dimensional lamination embedded inthe surface ∂ in U . Similarly, L uX := W u (Λ X ) ∩ ∂U a one-dimensional lamination embedded inthe surface ∂ out U . We call L sX and L uX the entrance lamination and the exit lamination of thehyperbolic plug ( U, X ). It can be proved that these laminations are quite simple: ∗ This work was partially carried during some stay of Fran¸cois B´eguin in Shanghai and Bin Yu in Paris. Wethank our universities for the financial support for these visits. Yu was partially supported by National NaturalScience Foundation of China (NSFC 11871374).
U, X ) and agluing diffeomorphism ψ : ∂ out U → ∂ in U . For such ( U, X ) and ψ , the quotient space M := U/ψ is a closed three-manifold, and the incomplete flow ( X t ) on U induces a complete flow ( Y t ) on M . The purpose of the paper [5] was to describe some sufficient conditions on U , X and ψ for( Y t ) to be an Anosov flow. We will now explain these conditions.We say that a one-dimensional lamination L is filling a surface S if every connected compo-nent C of S \ L is “a strip whose width tends to 0 at both ends”: more precisely, C is simplyconnected, the accessible boundary of C consists of two distinct non-compact leaves ℓ − , ℓ + of L , and these two leaves ℓ − , ℓ + are asymptotic to each other at both ends. We say that twolaminations L , L embedded in the same surface S are strongly transverse if they are transverseto each other, and moreover every connected component C of S \ ( L ∪ L ) is a topological discwhose boundary ∂C consists of exactly four arcs α , α , α ′ , α ′ where α , α ′ are arcs of leaves ofthe lamination L and α , α ′ are arcs of leaves of the lamination L . We say that a hyperbolicplug ( U, X ) has filling laminations if the entrance lamination L sX is filling the surface ∂ in U andthe exit lamination L uX is filling the surface ∂ out U . Given a hyperbolic plug ( U, X ), we saythat a gluing diffeomorphism ψ : ∂ out U → ∂ in U is strongly transverse if the laminations L sX and ψ ∗ L uX (both embedded in the surface ∂ in U ) are strongly transverse. If ( U, X ) and ( U, X )are two hyperbolic plugs with the same underlying manifold U , and ψ , ψ : ∂ out U → ∂ in U are two gluing diffeomorphisms, we say that the triples ( U, X , ψ ) and ( U, X , ψ ) are stronglyisotopic if one can find a continuous one-parameter family { ( U, X t , ψ t ) } t ∈ [1 , so that ( U, X t ) isa hyperbolic plug and ψ t : ∂ out U → ∂ in U is a strongly transverse gluing map for every t . Themain technical result of [5] can be stated as follows: Theorem 1.1.
Let ( U, X ) be a hyperbolic plug with filling laminations such that the maximalinvariant set of ( U, X ) contains neither attractors nor repellers, and let ψ : ∂ out U → ∂ in U be a strongly transverse gluing diffeomorphism. Then there exist a hyperbolic plug ( U, X ) withfilling laminations and a strongly transverse gluing diffeomorphism ψ : ∂ out U → ∂ in U such that ( U, X , ψ ) and ( U, X, ψ ) are strongly isotopic, and such that the vector field Y induced by X onthe closed manifold M := U/ψ is Anosov.
The idea of building transitive Anosov flows by gluing hyperbolic plugs goes back to [7] whereBonatti and R. Langevin consider a very simple hyperbolic plug (
U, X ) whose maximal invariantset is a single isolated periodic orbit and are able to find an explicit gluing diffeomorphism ψ : ∂ out U → ∂ in U so that the vector field Y induced by X on the closed manifold M := U/ψ generates a transitive Anosov flow. This example was later generalized by T. Barbot who defineda infinite family of transitive Anosov flows which he calls
BL-flows . These examples are obtainedby considering the same very simple hyperbolic plug (
U, X ) as Bonatti and Langevin, but moregeneral gluing diffeomorphisms.Theorem 1.1 naturally raises the following question (see [5, Question 1.7]): in the statementof Theorem 1.1, is the Anosov vector field Y well-defined up to orbitally equivalence ? (recall thattwo Anosov flows are said to be orbitally equivalent if there exists a homeomorphism betweentheir phase space mapping the oriented orbits of the first flow to the oriented orbitsthe secondone). One of the main purpose of the present paper is to provide a positive answer to thisquestion. More precisely, we will prove the following:2 heorem 1.2. Let ( U, X , ψ ) and ( U, X , ψ ) be two hyperbolic plugs endowed with stronglytransverse gluing diffeomorphisms. Let Y and Y be the vector fields induced by X and X onthe closed manifolds M := U/ψ and M := U/ψ . Suppose that:0. the manifolds U , M and M are orientable;1. both Y and Y are transitive Anosov flows;2. the triples ( U, X , ψ ) and ( U, X , ψ ) are strongly isotopic.Then the flows ( Y t ) and ( Y t ) are orbitally equivalent. It should be noted that, in the statement of Theorem 1.2, we do not require that the hy-perbolic plugs (
U, X ) and ( U, X ) have filling laminations. So Theorem 1.2 concerns a class ofAnosov flows which is larger than the class of Anosov flows provided by Theorem 1.1. For exam-ple, Bonatti-Langevin’s classical example and its generalizations by Barbot (BL-flows) satisfythe hypotheses of Theorem 1.2.Figure 1: Two examples of strongly transverse gluing diffeomorphisms. On the left-hand side,the laminations are filling. The right-hand side corresponds to Bonatti-Langevin’s example. Remark . A possible application of Theorem 1.2 is to get some finiteness results. Supposewe are given a hyperbolic plug (
U, X ) and a diffeomorphism ψ : ∂ out U → ∂ in U . Considerthe partition of the isotopy class of ψ into strong isotopy classes. Although we did not writedown a complete proof, it seems to us that this partition is finite. In view of Theorem 1.2, thismeans the following: up to orbital equivalence, there are only finitely many transitive Anosovflows that are built using the hyperbolic plug ( U, X ) and a gluing map ψ : ∂ out U → ∂ in U isotopic to ψ . A further consequence should be that, if we consider some given hyperbolicplugs ( U , X ) , . . . , ( U n , X n ) so that U , . . . , U n are hyperbolic manifolds, and if we consider amanifold M , then, up to orbital equivalence, there should only finitely many transitive Anosovflows on M that are obtained by gluing ( U , X ) , . . . , ( U n , X n ).An analog of Theorem 1.2 was proved by Barbot in the much more restrictive context ofBL-flows (see [2, second assertion of Theorem B]). Barbot’s result can actually be consideredas a particular case of Theorem 1.2: it corresponds to the case where the maximal invariantset of the hyperbolic plug ( U i , X i ) is a single isolated periodic orbit for i = 1 ,
2. Our proofof Theorem 1.2 roughly follows the same strategy as those of Barbot’s result, but is far moreintricate and requires some important new ingredients since we manipulate general hyperbolicplugs.The proof is based on a coding procedure that we will describe now. Consider a hyperbolicplug (
U, X ) and a strongly transverse gluing diffeomorphism ψ : ∂ out U → ∂ in U . Let Y be the3ector field induced by X on the closed manifold M := U/ψ , and assume that the flow ( Y t ) isa transitive Anosov flow. The projection in M of ∂U is a closed surface tranverse to the orbitsof the Anosov flow ( Y t ); we denote this surface by S . The projection in M of the entrancelamination of the plug ( U, X ) is a lamination in the surface S ; we denote it by L s . Consider theuniversal cover f M of the manifold M , and the lifts ( e Y t ) , e S, e L s of ( Y t ) , S, L s . We will consider the(countable) alphabet A whose letters are the connected components of e S \ e L s , and the symbolicspace Σ whose elements are bi-infinite words on the alphabet A . We will construct a codingmap χ from (a dense subset of) the surface e S to the symbolic space Σ, commuting with thenatural actions of the fundamental group of M , and conjugating the Poincar´e first return mapof the flow ( e Y t ) on the surface e S to the shift map on the symbolic space Σ. If Λ denotes theprojection in M of the maximal invariant set of the plug ( U, X ), and e Λ denotes the lift of Λin f M , then the map χ is defined at every point of e S which is neither in the stable nor in theunstable lamination of e Λ. This means that the dynamics of the flow ( Y t ) can be decomposedinto two parts: on the one hand, the orbits that converge towards to the maximal invariant setΛ in the past or in the future, on the other hand, the dynamics that is well-described by thecoding map χ . Remark . Besides being the cornerstone of the proof of Theorem 1.2, this coding procedureis interesting in its own sake. Indeed, it allows to understand the behavior of the recurrentorbits of the Anosov flow ( Y t ) that intersect the surface S ( i.e. which do not correspond torecurrent orbits of the incomplete flow ( X t )). In a forthcoming paper [6], we will use this codingprocedure to describe the free homotopy classes of theses orbits, and build new examples oftransitive Anosov flows.Let us now explain how this coding procedure yields a proof of Theorem 1.2. For i = 1 , i and a coding map χ i with value in Σ i . The strong isotopy between( U, X , ψ ) and ( U, X , ψ ) implies that there is a natural map between the symbolic space Σ and Σ . Together with the coding maps, this yields a conjugacy between the Poincar´e returnmaps of the flows ( e Y t ) , ( e Y t ) on the surfaces e S , e S . Unfortunately, this conjugacy is not well-defined on the whole surfaces e S , e S . So we need to extend it. In order to do that, we introducesome (partial) pre-orders on the leaf spaces of the lifts of the stable/unstable foliations of theAnosov flows ( Y t ) , ( Y t ), and prove that the conjugacy preserves these pre-orders. This is quitedelicate since the coding maps χ , χ do not behave very well with respect to these pre-orders.Once the extension has been achieved, we obtain a homeomorphism between the orbits spacesof the flows ( e Y t ) and ( e Y t ) that is equivariant with respect to the actions of the fundamentalgroups of the manifolds M and M . Using a classical result, this implies that the Anosov flows( Y t ) and ( Y t ) are orbitally equivalent. In this section, we will consider a transitive Anosov flow obtained by gluing hyperbolic plugs.Our goal is to define a coding procedure for the orbits of this Anosov flow. Actually, this codingprocedure will only describe the behavior of the orbits which do not remain in int( U ) forever. We consider a hyperbolic plug (
U, X ). Recall that this means that U is a (not necessarilyconnected ) compact 3-dimensional manifold with boundary, and X is a vector field on U , Hence, a finite collection of hyperbolic plugs can always be considered as a single, non connected, hyperbolicplug. ∂U , such that the maximal invariant setΛ X := \ t ∈ R X t ( U )is a saddle hyperbolic set for the flow of X . We decompose the boundary of U as ∂U := ∂ in U ⊔ ∂ out U where ∂ in U (resp. ∂ out U ) is the union of the connected component of ∂U where X is pointinginwards (resp. outwards) U . The stable manifold theorem implies that W sX (Λ X ) and W uX (Λ X )are two-dimensional laminations transverse to ∂U . Moreover, W sX (Λ X ) is obviously disjointfrom ∂ out U and W uX (Λ X ) is obviously disjoint from ∂ in U . As a consequence, L sX := W sX (Λ X ) ∩ ∂U = W sX (Λ X ) ∩ ∂ in U and L uX := W uX (Λ X ) ∩ ∂U = W uX (Λ X ) ∩ ∂ out U are one-dimensional laminations embedded in the surfaces ∂ in U and ∂ out U respectively. Notethat L sX can be described as the set of points in ∂ in U whose forward ( X t )-orbit remains in U forever, i.e. does not intersect ∂ out U . Similarly, L uX is the set of points in ∂ out U whose backward( X t )-orbit remains in U forever, i.e. does not intersect ∂ in U . These characterizations of L sX and L uX allow to define a map θ X : ∂ in U \ L sX −→ ∂ out U \ L uX where θ X ( x ) is the (unique) point of intersection the ( X t )-orbit of x with the surface ∂ out U .Clearly, θ X is a homeomorphism between ∂ in U \ L sX and ∂ out U \ L uX . We call θ X the crossingmap of the plug ( U, X ).In order to create a closed manifold equipped with a transitive Anosov flow, we consider adiffeomorphism ψ : ∂ out U → ∂ in U. The quotient space M := U/ψ. is a closed three-dimensional topological manifold. We denote by π : U → M the natural pro-jection map. The topological manifold M can equipped with a differential structure (compatiblewith the differential structure of U ) so that the vector field Y := π ∗ X is well-defined (and as smooth as X ). We make the following hypotheses:0. the manifolds U and M are orientable;1. the flow ( Y t ) is a transitive Anosov flow on the manifold M ;2. the diffeomorphism ψ is a strongly transverse gluing diffeomorphism.Recall that 2 means that the laminations L sX and ψ ∗ ( L uX ) are transverse in the surface ∂ in U and moreover that every connected component C of ∂ in U \ ( L sX ∪ ψ ∗ ( L uX )) is a topological discwhose boundary ∂C consists of exactly four arcs α s , α s ′ , α u , α u ′ where α s , α s ′ are arcs of leavesof L sX and α u , α u ′ are arcs of leaves ψ ∗ ( L uX )). We denote S := π ( ∂ in U ) = π ( ∂ out U ) Λ := π (Λ X ) L s := π ∗ ( L sX ) L u := π ∗ ( L uX ) .
5y construction, S is a closed surface, embedded in the manifold M , transverse to the vectorfield Y . The set Λ is the union of the orbits of ( Y t ) that do not intersect the surface S . It is aninvariant saddle hyperbolic set for the Anosov flow ( Y t ). Our assumptions imply that L s and L u are two strongly transverse one-dimensional laminations in the surface S . The lamination L s (resp. L u ) can be described as the sets of points in S whose forward (resp. backward) ( Y t )-orbitdoes not intersect S . Similarly, L u is a strict subset of W u (Λ) ∩ S . The homeomorphism θ X induces a homeomorphism θ = ( π | ∂ out U ) ◦ θ X ◦ ( π | ∂ in U ) − : S \ L s −→ S \ L u . Note that θ is nothing but the Poincar´e first return map of the orbits of the Anosov flow ( Y t )on the surface S .Since ( Y t ) is an Anosov flow, it comes with a stable foliation F s and an unstable foliation F u . These are two-dimensional foliations, transverse to each other, and transverse to the surface S . Hence, they induce two transverse one-dimensional foliations F s := F s ∩ S and F u := F u ∩ S on the surface S . Clearly, L s and L u are sub-laminations ( i.e. union of. leaves) of the foliations F s and F u respectively.In order to code the orbits of the Anosov flow ( Y t ), we cannot work directly in the manifold M , we need to unfold the leaves of the laminations L s and L u by lifting them to theuniversalcover of M . We denote this universal cover by p : f M −→ M , and we denote by e S, e Λ , f W s (Λ) , f W u (Λ) , e L s , e L u , e F s , e F u , e F s , e F u , the complete lifts of the surface S , the hyperbolic set Λ, the laminations W s (Λ) , W u (Λ) , L s , L u and the foliations F s , F u , F s , F u . We insist that e S is the complete lift of S , that is e S := p − ( S ).In particular, e S has infinitely many connected components. By construction, e F s and e F u aretwo transverse one-dimensional foliations on the surface e S , and e L s and e L u are sub-laminationsof e F s and e F u respectively. We also lift the vector field Y to a vector field e Y on M . Of course, e Y is transverse to the surface e S , so we can consider the Poincar´e return map˜ θ : e S \ e L s → e S \ e L u of the orbits of ( e Y t ) on the surface e S . Obviously, ˜ θ is a lift of the map θ . e S \ e L s The purpose of the present subsection is to collect some informations about the connectedcomponents of e S \ e L s and the action of the Poincar´e map e θ on these connected components.These informations will be used in Subsection 2.3. Let us start by the topology of the surface e S . Proposition 2.1.
Every connected component of e S is a properly embedded topological plane.Proof. The surface S is transverse to the Anosov flow ( Y t ). Hence, S is a collection of incom-pressible tori in M (see e.g. [8, Corollary 2.2]). The proposition follows.This allows us to describe the topology of the leaves of the foliations e F s and e F u : Proposition 2.2.
Every leaf of the foliations e F s and e F u is a properly embedded topological line.A leaf of e F s and a leaf of e F u intersect no more than one point. roof. The first assertion follows immediately from Proposition 2.1: it is a classical consequenceof Poincar´e-Hopf and Poincar´e-Bendixon theorems that the leaves of a foliation of a plane areproperly embedded topological lines.The second assertion is again a consequence Proposition 2.1, together with the transversalityof the foliations e F s and e F u . To prove it, we argue by contradiction: consider a leaf ℓ s of e F s , aleaf ℓ u of e F u , and assume that ℓ s and ℓ u intersect at more than one point. Then one can find twoarcs α s ⊂ ℓ s and α u ⊂ ℓ u , which share the same endpoints and have disjoint interiors. The union α s ∪ α s is a simple closed curve in e S . Since every connected component of e S is a topologicalplane, α s ∪ α s bounds a topological disc C ⊂ e S . Consider two copies of C , and glue them along α s in order to obtain a new topological disc D . The boundary of D is the union of two copiesof α u , hence is piecewise smooth. The foliation e F s provides a one-dimensional foliation on D ,which is topologically tranverse to boundary ∂D . This contradicts Poincar´e-Hopf Theorem.The next three propositions below concern the action of the Poincar´e map ˜ θ on the foliations e F s , e F u and the laminations e L s , e L u . We recall that e L s and e L u are sub-laminations ( i.e. union ofleaves) of the foliations e F s and e F u respectively. Proposition 2.3.
The Poincar´e map ˜ θ : e S − e L s → e S − e L u preserves the foliations e F s and e F u .Remark . Propostion 2.3 states that the foliation ( e F s ) | e S − e L s is mapped by ˜ θ to the foliation( e F s ) | e S − e L u . The leaves of ( e F s ) | e S − e L s are full leaves of the foliation e F s . On the contrary, a leaf ofthe foliation ( e F s ) | e S − e L u is never a full leaf of e F s (because every leaf of e F s is “cut into infinitelymany pieces” by the transverse lamination e L u ). As a consequence, ˜ θ maps leaves of e F s to piecesof leaves of e F s . Similarly, ˜ θ maps pieces of leaves of e F u to full leaves of e F u . Proof of Proposition 2.3.
Recall that e F s is defined as the intersection of the foliation e F s withthe transverse surface e S . The foliation e F s is leafwise invariant under the flow ( e Y t ). As aconsequence, e F s = e F s ∩ e S is invariant under the Poincar´e return map of ( e Y t ) on e S . Proposition 2.5.
For every n ≥ , S np =0 ˜ θ − p ( e L s ) is a closed sub-lamination of the foliation e F s .Proof. The foliation e F s is invariant under the Poincar´e map ˜ θ : e S − e L s → e S − e L u . Since e L s is aunion of leaves of e F s , it follows that ˜ θ − ( e L s ) is a union of leaves of e F s . Moreover, since e L s is aclosed subset of e S , its pre-image ˜ θ − ( e L s ) must be a closed subset of e S − e L s (remember that ˜ θ iswell-defined on e S − e L s ). Therefore S p =0 ˜ θ − p ( e L s ) is a closed subset of e S . So S p =0 ˜ θ − p ( e L s ) is aclosed union of leaves of e F s , i.e. a closed sub-lamination of e F s . Repeating the same arguments,one proves by induction that S np =0 ˜ θ − p ( e L s ) is a closed sub-lamination of e F s for every n ≥ Proposition 2.6. ∞ [ p =0 ˜ θ − p ( e L s ) = f W s (Λ) ∩ e S Proof.
By definition, W s (Λ) ∩ S is the set of all points x ∈ S so that the forward orbit of x converges towards the set Λ, which is disjoint from S . As a consequence, for every point x ∈ W s (Λ) ∩ S , the forward orbit of x intersects the surface S only finitely many times, say p ( x ) times. We have observed that L s is the set of all points y ∈ S so that the forward orbit of y does not intersect S and converges towards the set Λ (see Subsection 2.1). It follows that, forevery x ∈ W s (Λ) ∩ S , the last intersection point θ p ( x ) of the forward orbit of x with S is in L s .This proves the inclusion W s (Λ) ∩ S ⊂ S ∞ p =0 θ − p ( L s ). The converse inclusion is straightforward.Hence S ∞ p =0 θ − p ( L s ) = W s (Λ) ∩ S . The equality S ∞ p =0 ˜ θ − p ( e L s ) = f W s (Λ) ∩ e S follows by liftingto the universal cover. 7f course, f W s (Λ) ∩ e S and f W u (Λ) ∩ e S are union of leaves of the foliations e F s and e F u respectively. But these sets are not closed. More precisely: Proposition 2.7.
Both f W s (Λ) ∩ e S and e S − f W s (Λ) are dense in e S .Proof. Recall that ( Y t ) is a transitive Anosov flow on M . Hence every leaf of the weak stablefoliation F s is dense in M . Since both W s (Λ) and M \ W s (Λ) are non-empty union of leavesof the foliation F s , and since the leaves of F s are transversal to the surface S , it follows thatboth W s (Λ) ∩ S and S \ W s (Λ) are dense in S . Lifting to the universal cover, we obtain that g W s (Λ) ∩ e S and e S − g W s (Λ) are dense in e S .Of course, the analogs of Propositions 2.5, 2.6 and 2.7 for e L u and W u ( e Λ) hold (˜ θ − p should bereplaced by ˜ θ p in Propositions 2.5 and 2.6). We will now describe the topology of the connectedcomponents of e S \ e L s . We first introduce some vocabulary. Definition 2.8.
We call proper stable strip every topological open disc D of e S whose boundaryis the union of two leaves of the foliation e F s .If D is a proper stable strip, one can easily construct a homeomorphism h from the closureof D to R × [ − , Definition 2.9.
We say that a proper stable strip D is trivially bifoliated if there exists ahomeomorphism h from the closure of D to R × [ − ,
1] mapping the foliations e F s and e F u to thehorizontal and vertical foliations on R × [ − , proper unstable strips and trivially bifoliated proper unstable strips can be de-fined similarly. The proposition below gives a fairly precise description of the positions of theconnected components of e S − e L s with respect to the foliations e F s and e F u . Proposition 2.10.
Every connected component of e S − e L s is a trivially bifoliated proper stablestrip bounded by two leaves of the lamination e L s .Proof. Let D be a connected component of e S − e L s . Denote by P the connected component of e S containing D . Since P is a topological plane (Proposition 2.1), and since each leaf of e L s is8 properly embedded topological line (Proposition 2.2) which separates P into two connectedcomponents, it follows that D is a topological disc. The boundary of D is a union of leaves of e L s (which we call the boundary leaves of D ). We denote by D the closure of D . Claim 1. Let ℓ u be a leaf of the foliation e F u intersecting D , and α u be a connected componentof ℓ u ∩ D . Then α u is an arc joining two different boundary leaves of D . Let R be a connected component of D \ e L u , so that α u is included in the closure R of R (actually R is unique, but we will not use this fact). Observe that R is a connected componentof e S − ( e L s ∪ e L u ). Our assumptions (namely, the strong transversality of the gluing map ψ )imply that R is a relatively compact topological disc, whose boundary ∂R is made of four arcs α s − , α s + , α u − , α u + , where α s − and α s − are disjoint and lie in some leaves of e L s , and where α u − and α u + are disjoint and lie in some leaves of e L u . Loosely speaking, R is a rectangle with two sides α s − , α s + in e L s and two sides α u − , α u + in e L u . Proposition 2.2 implies that ℓ u intersects α s − and α s + at no more than one point. Since ℓ u is a proper line, and R is a compact set, it follows that α u must be an arc going from α s − to α s + . Using again Proposition 2.2, it also follows that α s − to α s + cannot be in the same leaf of e F s . The claim is proved. Claim 2. D has exactly two boundary leaves. In order to prove this claim, we endow the foliation e F u with an orientation (this is possible since e F u is a foliation on a collection of topological planes). For every x ∈ D , we denote by ℓ u ( x )the leaf of the foliation e F u passing through x , and denote by α u ( x ) the connected componentof ℓ ux ∩ D containing x . Note that ℓ u ( x ) and α u ( x ) are oriented by the orientation of e F u . ByClaim 1, α u ( x ) is an arc whose endpoints lie on two boundary leaves ℓ s − ( x ) and ℓ s + ( x ) of D .By transversality of the foliations e F u and e F s , the maps x ℓ s − ( x ) and x ℓ s + ( x ) are locallyconstant. Since D is connected, these maps are constant. In other words, one can find twoboundary leaves ℓ s − and ℓ s + of D , so that α u ( x ) is an arc from ℓ s − to ℓ s + for every x ∈ D . Itfollows that ℓ s − and ℓ s + are the only accessible boundary leaves of D : otherwise, one can consideranother boundary leaf ℓ s , take a point x ∈ ℓ s , and get a contradiction since one end of α ux is on ℓ s . As a further consequence, the accessible boundary of D is closed (recall that ℓ s − and ℓ s + areproperly embedded lines), and therefore coincides with the boundary of D . We finally concludethat ℓ s − and ℓ s + are the only boundary leaves of D and Claim 2 is proved.Claim 1 and 2 already imply that D is a proper stable strip bounded by two leaves ℓ s − , ℓ s + of e L s . We are left to prove that D a trivially bifoliated. Recall that e S is a topological plane(Proposition 2.1), and that ℓ s − , ℓ s + are properly embedded topological lines (Proposition 2.2).By easy planar topology, it folllows that there exists a homeomorphism h from D to R × [ − , ℓ s − and ℓ s + to R × {− } and R × { } respectively. Claim 1 implies that h ∗ ( e F uD is afoliation of R × [ − ,
1] by arcs going from R × {− } and R × { } . One can easily construct a self-homeomoprhism h ′ of R × [ − ,
1] mapping this foliation on the vertical foliation of R × [ − , h by h ′ ◦ h , we will assume that h maps e F uD on the vertical foliation of R × [ − , ℓ s of the foliation e F s included in D . According to Proposition 2.2, ℓ s intersects each leaf of e F u at no more than one point. Hence h ( ℓ s ) intersects each vertical segmentin R × [ − ,
1] at no more than one point. Let E be the set of t ∈ R so that h ( ℓ s ) intersects thevertical segment { t } × [ − , ℓ s is a proper topological line tranvsersal to e F u , the set E t must be open and closed in R . Therefore h ( ℓ s ) intersects every vertical segment in R × [ − , h ∗ ( e F sD ) are graphs over the first coordinatein R × [ − , h so that h ∗ ( e F sD ) is the horizontalfoliation of R × [ − , D is a trivially bifoliated proper stable strip.Of course, the unstable analog of Proposition 2.10 holds true: every connected componentof e S − e L u is a trivially bifoliated proper unstable strip bounded by two leaves of the lamination9 L u . On the other hand, ˜ θ maps connected components of e S − e L s to connected component of e S − e L u . So, we obtain: Corollary 2.11. If D is a connected component of e S − e L s , then ˜ θ ( D ) is a trivially bifoliatedproper unstable strip, disjoint from e L u , bounded by two leaves of the lamination e L u . The following proposition describes the action of ˜ θ on the connected components of e S − e L s . Proposition 2.12.
Let D be a connected component of e S − e L , and D ′ be any trivially bifoliatedproper stable strip. Assume that D ∩ ˜ θ − ( D ′ ) is non-empty. Then D ∩ ˜ θ − ( D ′ ) is a triviallybifoliated proper stable sub-strip of D .Proof. We call trivially bifoliated rectangle every topological open disc R ⊂ e S such that thereexists a homeomorphism from the closure of R to [ − , mapping the restrictions of e F s and e F u to the horizontal and vertical foliations of [ − , . In particular, the boundary of such atrivially bifoliated rectangle is made of two stable sides, and two unstable sides.According to Corollary 2.11, ˜ θ ( D ) is a trivially bifoliated proper unstable strip, disjoint from e L u , bounded by two leaves of e L u . By assumption, D ′ is a trivially bifoliated proper stable strip.It easily follows that ˜ θ ( D ) ∩ D ′ is a trivially bifoliated rectangle, disjoint from e L u , whose unstablesides are in e L u (see Figure 3). Observe that the interiors of two stable sides of ˜ θ ( D ) ∩ D ′ arefull leaves of e F s | e S − e L u . Hence:( ⋆ ) θ ( D ) ∩ D ′ is a connected subset of θ ( D ), and the boundary of θ ( D ) ∩ D ′ in θ ( D ) is madeof two disjoint leaves of e F s | e S − e L u .Now recall that ˜ θ − is a homeomorphism from e S − e L u to e S − e L s , mapping leaves of e F s | e S − e L u tofull leaves of e F s (see Proposition 2.3 and Remark 2.4). Also observe that D ∩ ˜ θ − ( D ′ ) is a subsetof D . As a consequence, Properties ( ⋆ ) implies:( ⋆ ′ ) D ∩ ˜ θ − ( D ′ ) is a connected subset of D , and the boundary of D ∩ ˜ θ − ( D ′ ) is made of twodisjoint leaves of e F s .Since D is a trivially foliated proper stable strip D , Property ( ⋆ ′ ) clearly implies that D ∩ ˜ θ − ( D ′ )is a trivially bifoliated proper stable sub-strip of D . See Figure 3. In this section, we will use the connected components of e S \ e L s to describe the itinerary of theorbits the flow ( e Y t ) that do not belong to f W s (Λ) ∪ f W u (Λ). We consider the alphabet A := { connected components of e S \ e L s } , and the symbolic spacesΣ s = n D s = ( D p ) p ≥ such that D p ∈ A and e θ ( D p ) ∩ D p +1 = ∅ for every p o , Σ u = n D u = ( D p ) p< such that D p ∈ A and e θ ( D p ) ∩ D p +1 = ∅ for every p o , Σ = n D = ( D p ) p ∈ Z such that D p ∈ A and e θ ( D p ) ∩ D p +1 = ∅ for every p o . In order to define the coding maps, we need to introduce some leaf spaces. We will denoteby f s the leaf space of the foliation e F s (equipped with the quotient topology). We will denote f s, ∞ the subset of f s made of the leaves that are not in f W s (Λ). Similarly, we denote by f u e F u , and by f u, ∞ the subset fo f u made of the leaves that are not in f W u (Λ).Finally we denote by e S ∞ the set of points e S that are neither in f W s (Λ) nor in f W u (Λ). f s, ∞ = { leaves of e F s that are not in f W s (Λ) } ,f u, ∞ = { leaves of e F u that are not in f W u (Λ) } , e S ∞ = e S − ( f W s (Λ) ∪ f W u (Λ)) . By Proposition 2.6, if ℓ s ∈ f s, ∞ , then ˜ θ p ( ℓ s ) is included in a connected component of e S − e L s for every p ≥
0. Similarly, if ℓ u ∈ f u, ∞ , then ˜ θ p ( ℓ u ) is included in a connected component of e S − e L u for every p ≤
0. Since ˜ θ − maps homeomorphically e S − e L u to e S − e L s , we deduce that:if ℓ u ∈ f u, ∞ , then ˜ θ p ( ℓ u ) is included in a connected component of e S − e L s for every p <
0. As afurther consequence, if x is a point of e S ∞ , then ˜ θ p ( x ) is in a connected component of e S − e L s forevery p ∈ Z . This shows that the following coding maps are well-defined: χ s : f s, ∞ −→ Σ s ℓ s D s = ( D p ) p ≥ where ˜ θ p ( ℓ s ) ⊂ D p for every p ≥ χ u : f u, ∞ −→ Σ u ℓ u D u = ( D p ) p< where ˜ θ p ( ℓ u ) ⊂ D p for every p < χ : e S ∞ −→ Σ x D = ( D p ) p ∈ Z where ˜ θ p ( x ) ∈ D p for every p ∈ Z The following proposition is an important step ingredient of the proof of Theorem 1.2.
Proposition 2.13.
The maps χ s , χ u and χ are bijective. Lemma 2.14.
1. For every D s = ( D p ) p ≥ ∈ Σ s , the set T p ≥ ˜ θ − p ( D p ) is a stable leaf ℓ s ∈ f s, ∞ .2. For every D u = ( D p ) p< ∈ Σ u , the set T p< ˜ θ − p ( D p ) is an unstable leaf ℓ u ∈ f u, ∞ . . For every D = ( D p ) p ∈ Z ∈ Σ , the set T p ∈ Z ˜ θ − p ( D p ) is a single point x ∈ e S ∞ .Remark . Lemma 2.14 is completely false if we replace the connected components of e S \ e L s by the connected components of S \ L s (and ˜ θ by θ ). For example, if ( D p ) p ≥ is a sequenceof connected components of S \ L s , then T p ≥ θ − p ( D p ), if not empty, will be the union ofuncountably many leaves of the foliation F s . This is the reason why, we need to work in theuniversal cover of M . Proof.
Let us prove the first item. Consider a sequence D s = ( D p ) p ≥ ∈ Σ s . By Proposition 2.10, D is a trivially bifoliated proper stable strip. Proposition 2.12 and a straightfoward inductionimply that for every n ∈ N , the set T np =0 e θ − p ( D p ) is a sub-strip of D . So ( T np =0 e θ − p ( D p )) n ≥ is a decreasing sequence of sub-strips of the trivially bifoliated proper stable strip D . It easilyfollows that T p ≥ e θ − p ( D p ) is a sub-strip of D . In particular, T p ≥ e θ − p ( D p ) is a connected unionof leaves of e F s . On the other hand, since D , D , . . . are connected components of e S − e L s , the set T ≥ e θ − p ( D p ) is disjoint from S p ≥ e θ − p ( e L s ) = f W s (Λ) ∩ e S (see Proposition 2.6). But f W s (Λ) ∩ e S is dense in e S (Proposition 2.7). It follows that T p ≥ e θ − p ( D p ) must be a single leaf of e F s . Thiscompletes the proof of item 1.Item 2 follows from exactly the same arguments as item 1. In order to prove the last item, weconsider a sequence D = ( D p ) p ∈ Z in Σ. According to the items 1 and 2, T p ≥ e θ − p ( D p ) is a leaf ℓ s of the foliation e F s and T p< e θ − p ( D p ) is a leaf ℓ u of the foliation e F u . Since D = ( D p ) p ∈ Z is in Σ,the intersection D ∩ ˜ θ ( D − ) is not empty. Since D is a trivially bifoliated proper stable strip(Proposition 2.10) and ˜ θ ( D − ) is a trivially bifoliated proper unstable strip (Corollary 2.11),every leaf of e F s in D must intersects every leaf of e F u in ˜ θ ( D − ) at exactly one point. Inparticular, T p ∈ Z e θ − p ( D p ) = ℓ s ∩ ℓ u is made of exactly one point x . Since the leaves ℓ s and ℓ u are disjoint from f W s (Λ) and f W u (Λ) respectively, the point x must be in e S ∞ . This completesthe proof. Proof of Proposition 2.13.
Lemma 2.14 allows to define some inverse maps for χ s , χ u and χ .Therefore, χ s , χ u and χ are bijective.Deck transformation preserve the surface e S , the foliations e F s , e F u , and the laminations W s ( e Λ) , W u ( e Λ). This induces some natural actions of π ( M ) on the set e S ∞ , on the leaf spaces f s, ∞ , f u, ∞ , on the alphabet A , and therefore on the symbolic spaces Σ , Σ s , Σ u . From the defi-nition of the coding maps, one easily checks that: Proposition 2.16.
The coding maps χ , χ s and χ u commute with the actions of the fundamentalgroup of M on e S ∞ f s , f u , Σ Σ s and Σ u . The definition of the coding maps also implies that:
Proposition 2.17.
The coding map χ (resp. χ s and χ u ) conjugates the action of Poincar´e firstreturn map e θ on e S ∞ (resp. f s and f u ) to the left shift on the symbolic space Σ (resp. Σ s and Σ u ). Given an integer n ≥ D , . . . , D n of e S − e L s , we define thecylinder [ D . . . D n ] s := { ( D p ) p ≥ ∈ Σ s such that D p = D p for 0 ≤ p ≤ n } . Similarly, given n < D n , . . . , D − of e S − e L s , we define thecylinder [ D n . . . D − ] u := { ( D p ) p< ∈ Σ u such that D p = D p for n ≤ p ≤ − } . The following proposition will be used in the next subsection.12 roposition 2.18.
1. for n ≥ and D , . . . , D n ∈ A , the set ( χ s ) − ([ D D . . . D n ] s ) = T ≤ p ≤ n ˜ θ − p ( D p ) is eitherempty or a sub-strip of the trivially foliated proper stable strip D bounded by two leavesof e θ − n ( e L s ) ;2. for n < and D n , . . . , D − ∈ A , the set ( χ u ) − ([ D n D n +1 . . . D − ]) = T − n ≤ p ≤− ˜ θ − p +1 ( D p ) is a sub-strip of the trivially foliated proper unstable strip ˜ θ ( D − ) bounded by two leavesof e θ K − ( f L u ) .Proof. This follows from the arguments of the proof of Lemma 2.14.
We will now describe a partial pre-order on the leaf space f s . The preservation of this partialpre-order will be a fundamental ingredient of our proof of Theorem 1.2 in Section 3.Let us start by choosing some orientations. First of all, we choose an orientation of thehyperbolic plug U . The orientation of U , together with the vector field X , provide an orientationof ∂U : if ω is a 3-form defining the orientation on U , then the 2-form i X U defines the orientationon ∂U . The orientation of U induces an orientation of the manifold M = U/ψ (we have assumedthat the manifold M is orientable, which is equivalent to assuming that the gluing map ψ preserves the orientation of ∂U ), and the orientation of ∂U induces an orientation of the surface S = π ( ∂ in U ) = π ( ∂ out U ). The orientation of M and S induce some orientations on f M and e S .Now, since every connected component of e S is a topological plane, the foliation e F s is orientable.We fix an orientation of e F s . This automatically induces an orientation of the foliation e F u asfollows: the orientation of e F u is chosen so that, if Z s and Z u are vector fields tangent to e F s and e F u respectively, and pointing in the direction of the orientation of the leaves, then the framefield ( Z s , Z u ) is positively oriented with respect to the orientation of e S . Remarks .
1. By construction, the orientations of the manifold f M and the surface e S arerelated as follows: if ω is a 3-form defining the orientation on f M , then the 2-form i e Y f M defines the orientation on e S . As a consequence, the Poincar´e return map ˜ θ of the orbitsof e Y on e S preserves the orientation of e S .2. Consequently, for any connected component D of e S − e L s , if the Poincar´e map ˜ θ | D preserves(resp. reverses) the orientation of the foliation e F s , then it also preserves (resp. reverses)the orientation of the foliation e F u .Let ℓ be a leaf of the foliation e F s , contained in a connected component e S ℓ of e S . Recall that e S ℓ is a topological plane, and ℓ is a properly embedded line in e S ℓ . As a consequence, e S ℓ \ ℓ hastwo connected components. Definition 2.20.
We denote by L ( ℓ ) and R ( ℓ ) the two connected component of e S \ ℓ so thatthe oriented leaves of e F u crossing ℓ go from L ( ℓ ) towards R ( ℓ ). The points of L ( ℓ ) are said tobe on the left of ℓ ; the points of R ( ℓ ) are said to be on the right of ℓ Now we can define a pre-order on the leaf space f s . Definition 2.21 (Pre-order on f s ) . Given two leaves ℓ = ℓ ′ of the foliation e F s , we write ℓ ≺ ℓ ′ if there exists an arc of a leaf of e F u with endpoints a ∈ ℓ and a ′ ∈ ℓ ′ , so that the orientation of e F u goes from a towards a ′ . Proposition 2.22. ≺ is a pre-order on f s : the relations ℓ ≺ ℓ ′ and ℓ ′ ≺ ℓ are incompatible roof. The relation ℓ ≺ ℓ ′ implies that the leaf ℓ ′ is on the right of ℓ , that is ℓ ′ ⊂ R ( ℓ ). Similarly,the relation ℓ ≺ ℓ ′ implies ℓ ′ ⊂ L ( ℓ ). The proposition follows since L ( ℓ ) ∩ R ( ℓ ) = ∅ .The proposition below is very easy to prove, but fundamental: Proposition 2.23. ≺ is a local total order on f s : for every leaf ℓ of e F s , there exists a neigh-bourhood V of ℓ in f s so that any two different leaves ℓ, ℓ ′ ∈ V are comparable ( i.e. satisfyeither ℓ ≺ ℓ ′ or ℓ ′ ≺ ℓ ).Proof. Consider a leaf ℓ of e F s and a leaf ℓ u of e F u so that ℓ u ∩ ℓ = ∅ . By transversality of thefoliations e F s and e F u , there exists a neighbourhood V of ℓ in f s so that ℓ u crosses every leaf in V . As a consequence, any two different leaves ℓ, ℓ ′ ∈ V are comparable for the pre-order ≺ .The proposition below shows that the pre-order ≺ is “compatible” with the connected com-ponents decomposition of e S − e L s . Proposition 2.24.
Given two different elements
D, D ′ of A , the following are equivalent:1. there exists some leaves ℓ , ℓ ′ ∈ f s so that ℓ ⊂ D , ℓ ′ ⊂ D ′ and ℓ ≺ ℓ ′ ;2. every leaves ℓ, ℓ ′ ∈ f s so that ℓ ⊂ D and ℓ ′ ⊂ D ′ satisfy ℓ ≺ ℓ ′ .Proof. Assume that 1 is satisfied. Since ℓ ≺ ℓ ′ , there must be a leaf ℓ u of the foliation e F u intersecting both ℓ and ℓ ′ . Proposition 2.10 implies that α := ℓ u ∩ D and α ′ := ℓ u ∩ D ′ are twodisjoint arcs in the leaf ℓ u . Consider some leaves ℓ, ℓ ′ of e F s contained in D and D ′ respectively.Again Proposition 2.10 implies that ℓ intersects ℓ u at some point a ℓ ∈ α and ℓ ′ intersects ℓ u atsome point a ℓ ′ ∈ α ′ . Since ℓ (cid:22) ℓ ′ the orientation of ℓ u goes from α towards α ′ , hence from a ℓ towards a ℓ ′ . This shows that ℓ ≺ ℓ ′ . Definition 2.25 (Pre-order on A ) . Given two different elements
D, D ′ of A , we write D ≺ D ′ if there exists some leaves ℓ , ℓ ′ ∈ f s so that ℓ ⊂ D , ℓ ′ ⊂ D ′ and ℓ ≺ ℓ ′ . Definition 2.26 (Pre-order on Σ s ) . The partial pre-order ≺ on A induces a lexicographic partialpre-order on Σ s ⊂ A N that will also be denoted by ≺ : for D = ( D p ) p ≥ and D ′ = ( D ′ p ) p ≥ inΣ s , we write D ≺ D ′ if and only if there exists p ≥ D p = D ′ p for p ∈ { , . . . , p − } , D p ≺ D ′ p .We have defined a pre-order on the leaf space f s (Definition 2.21) and a pre-order on thesymbolic space Σ s (Definition 2.26). It is natural to wonder whether the coding map χ s : f s, ∞ → Σ s is compatible with these pre-orders or not. For pedagogical reason, we first considerthe simple situation where the two-dimensional foliation F u is orientable: Proposition 2.27.
Assume that the unstable foliation F u is orientable. Then the coding map χ s : f s, ∞ → Σ s preserves the pre-orders, i.e. for ℓ, ℓ ′ ∈ f s, ∞ , ℓ ≺ ℓ ′ if and only if χ s ( ℓ ) ≺ χ s ( ℓ ′ ) .Proof. Since the two-dimension foliation F u is orientable, its lift e F u is also orientable. Recallthat the vector field e Y is tangent to the leaves of the foliation e F u . So, the orientatibility of thetwo-dimensional foliation e F u implies that the return map ˜ θ of the orbits of the vector field e Y on the surface e S preserves the orientation of the one-dimensional foliation e F u = e F u ∩ e S .Consider two leaves ℓ, ℓ ′ ∈ f s, ∞ so that ℓ ≺ ℓ ′ . Let χ s ( ℓ ) = ( D p ) p ≥ and χ s ( ℓ ) = ( D ′ p ) p ≥ .Recall that this means that ℓ = \ p ≥ e θ − p ( D p ) and ℓ ′ = \ p ≥ e θ − p ( D ′ p ) . p = min { p ≥ , D p = D ′ p } and the set b D := p − \ p =0 e θ − p ( D p ) . Both the leaves ℓ and ℓ ′ are included in b D , and, according to Proposition 2.18, b D is a triviallybifoliated proper stable strip. So we can consider an arc α u of a leaf ℓ u of the foliation e F u ,so that α u is included in the trivially bifoliated proper stable strip b D , so that the ends a, a ′ of α u are on ℓ and ℓ ′ respectively. Since ℓ ≺ ℓ ′ , the orientation of e F u goes from a towards a ′ . Now observe that b D is a connected component of e S − S p − p =0 e θ p ( e L s ). As a consequence, themap e θ p is well-defined on e D . In particular, we can consider β u := e θ p ( α u ). Observe that β u is an arc of a leaf of the foliation e F u . Its ends b := e θ p ( a ) and b ′ := e θ p ( a ′ ) are respectivelyin e θ p ( ℓ ) ⊂ D p and e θ p ( ℓ ′ ) ⊂ D ′ p . Since the return map e θ p preserves the orientation of thefoliation e F u , the orientation of e F u goes from b towards b ′ . It follows that e θ p ( ℓ ) ≺ e θ p ( ℓ ′ ) andtherefore D p ≺ D ′ p . As a further consequence, χ s ( ℓ ) = ( D , D , . . . , D p − , D p , . . . ) ≺ ( D , D , . . . , D p − , D ′ p , . . . ) = χ s ( ℓ ′ ) . This completes the proof of the implication ℓ ≺ ℓ ′ ⇒ χ s ( ℓ ) ≺ χ s ( ℓ ′ ). The converse implicationfollows from the very same arguments in reversed order.In general, the relationships between the order on the leaf space f s and the symbolic spaceΣ s is more complicated: Proposition 2.28.
Let ℓ, ℓ ′ be two different elements of f s, ∞ . Let ( D p ) p ≥ := χ s ( ℓ ) and ( D ′ p ) p ≥ := χ s ( ℓ ′ ) . Let p be the smallest interger p so that D p = D ′ p .1. If the map e θ p | T p − p =0 e θ − p ( D p ) preserves the orientation of the foliation e F u , then ( ℓ ≺ ℓ ′ ) ⇐⇒ ( D p ≺ D ′ p ) ⇐⇒ ( χ s ( ℓ ) ≺ χ s ( ℓ ′ )) .
2. If the map e θ p | T p − p =0 e θ − p ( D p ) reverses the orientation of the foliation e F u , then ( ℓ ≺ ℓ ′ ) ⇐⇒ ( D ′ p ≺ D p ) ⇐⇒ ( χ s ( ℓ ′ ) ≺ χ s ( ℓ )) . Proof.
The arguments are exactly the same as in the proof of Proposition 2.27.
We will now prove Theorem 1.2 with the help of the coding procedure implemented in section 2.
We begin by explaining why it is enough to prove Theorem 1.2 in the particular case where thevector fields X and X coincide.Let ( U, X , ψ ) and ( U, X , ψ ) be two triple satisfying the hypotheses of Theorem 1.2. Inparticular ( U, X , ψ ) and ( U, X , ψ ) are strongly isotopic. This means that there exists acontinuous one-parameter family { ( U, X t , ψ t ) } t ∈ [1 , so that ( U, X t ) is a hyperbolic plug and ψ t : ∂ out U → ∂ in U is a strongly transverse gluing map for every t . By standard hyperbolictheory, hyperbolic plugs are structurally stable. Hence, this means that we can find a continuous15amily ( h t ) t ∈ [1 , of self-homeomorphisms of U so that h = Id and so that h t induces an orbitalequivalence between X and X t . For t ∈ [1 , b ψ t := ( h t | ∂ in U ) − ◦ ψ t ◦ ( h t | ∂ out U )and observe that b ψ = ψ . For sake of clarity, let X := X . Then,– the triples ( U, X, b ψ ) and ( U, X, b ψ ) are strongly isotopic: the strong isotopy is given bythe continuous path { ( U, X, b ψ t ) } t ∈ [1 , ;– for t ∈ [1 , X on the manifold c M t := U/ b ψ t is orbitallyequivalent to the flow induced by the vector field X t on the manifold M t := U/ψ t : theorbital equivalence is induced by the homeomorphism h t .This shows that the hypotheses and the conclusion of Theorem 1.2 are satisfied for the triple( U, X , ψ ) and ( U, X , ψ ) if and only if they are satisfied for the triples ( U, X, b ψ ) and ( U, X, b ψ ).This allows us to replace the vector fields X and X by a single vector field X in the proof ofTheorem 1.2. From now on, we consider a hyperbolic plug (
U, X ) endowed with two strongly transverse gluingdiffeomorphisms ψ , ψ : ∂ out U → ∂ in U . We denote by Λ := T t ∈ R X t ( U ) the maximal invariantset of the plug ( U, X ). For i = 1 ,
2, the quotient space M i := U/ψ i is a closed three-dimensionalmanifold, and X induces a vector field Y i on M i . We assume that the hypotheses of Theorem 1.2are satisfied, that is0. the manifolds U , M and M are orientable,1. for i = 1 ,
2, the flow ( Y ti ) of the vector field Y i is a transitive Anosov flow,2. the gluing maps ψ and ψ are strongly isotopic, i.e. that there exists an isotopy ( ψ s ) s ∈ [1 , such that, for every s , the laminations L s and ψ s ( L uX ) are strongly transverse.In order to prove Theorem 1.2, we have to construct a homeomorphism H : M → M map-ping the oriented orbits of the Anosov flow ( Y t ) to the orbits of the Anosov flow ( Y t ). Theconstruction will be divided into several steps. φ in , φ out : S → S For i = 1 ,
2, we denote by π i the projection of U on the closed three-dimensional manifold M i = U/ψ i . We denote by S i = π i ( ∂ in U ) = π i ( ∂ out U )the projection of the boundary of U . The surface S i is endowed with the strongly transverselaminations L si := π i ( L sX ) and L ui := π i ( L uX ) . The maps π i | ∂ in U : ∂ in U → S i and π i | ∂ out U : ∂ out U → S i are invertible. This provides us withtwo diffeomorphisms φ in := π | ∂ in U ◦ ( π | ∂ in U ) − : S → S and φ out := π | ∂ out U ◦ ( π | ∂ out U ) − : S → S . The diffeomorphisms φ in and φ out are the starting point of our construction. Observe that,at this step, we are very far from getting an orbital equivalence. Indeed, φ in and φ out are in16o way compatible with the actions of the flows ( Y t ) and ( Y t ) ( i.e. they do not conjugate thePoincar´e return maps of ( Y t ) and ( Y t ) on the surface S and S ).Nevertheless, the definition of the diffeomorphisms φ in and φ out imply that they satisfy: φ in ( L s ) = L s and φ out ( L u ) = L u . Remark . Be careful: in general φ in ( L u ) = L u and φ out ( L s ) = L s .On the other hand, the strong isotopy connecting the gluing maps ψ and ψ can be used toconstruct an isotopy between the diffeomorphisms φ in and φ out : Proposition 3.2.
There exists a continuous family ( φ t ) t ∈ [0 , of diffeomorphisms from S to S , such that φ = φ out , such that φ = φ in , and such that the laminations φ t ( L u ) and L s arestrongly transverse for every t .Proof. By assumption, the gluing maps ψ and ψ are connected by a continuous path ( ψ s ) s ∈ [1 , of diffeomorphisms from ∂ out U to ∂ in U , such that the laminations ψ s ( L u ) and L s are stronglytransverse for every s . For t ∈ [0 , φ t := π | ∂ out U ◦ ψ − ◦ ψ − t ◦ ( π | ∂ out U ) − . From this formula, we immediately get φ = π | ∂ out U ◦ ( π | ∂ out U ) − = φ out . Plugging the equality π i | ∂ in U ◦ ψ i = π i | ∂ out U into the definition of φ , we get φ = π | ∂ out U ◦ ψ − ◦ ψ ◦ ( π | ∂ out U ) − = π | ∂ in U ◦ ( π | ∂ in U ) − = φ in . We know that the laminations L sX and ψ − t ( L uX ) are strongly transverse for every t . As aconsequence, the laminations π | ∂ out U ◦ ψ − ( L sX ) = π | ∂ in U ( L sX ) = L s and π | ∂ out U ◦ ψ − ◦ ψ − t ( L uX ) = φ t ◦ π | ∂ out U ( L uX ) = φ t ( L u )are strongly transverse for every t . This completes the proof.It is important to observe that the diffeomorphism φ in can be obtained as the restriction ofa diffeomorphism from M to M : Proposition 3.3.
The diffeomorphism φ in : S → S is the restriction of a diffeomorphism Φ in : M → M .Proof. Once again, we use the existence of a continous path ( ψ s ) s ∈ [1 , of diffeomorphisms from ∂ out U to ∂ in U connecting the gluing maps ψ and ψ . We consider a collar neighbourhood V of ∂ out U in U , and a diffeomorphism ξ : ∂ out U × [0 , → V of V such that ξ ( ∂ out U × { } ) = ∂ out U .We define a diffeomorphism ¯Φ in : U → U by setting ¯Φ in ( ξ ( x, t )) := ψ − − t ◦ ψ ( x ) for every( x, t ) ∈ ∂ out U × [0 , in = Id on U \ V . By construction, this diffeomorphism satisfies¯Φ in = Id on ∂ in U = ψ − ◦ ψ on ∂ out U. As a consequence, the relation π ◦ ¯Φ in = ¯Φ in ◦ π holds, and therefore ¯Φ in induces a diffeomor-phism Φ in : M → M . Since ¯Φ in = Id on ∂ in U , it follows that Φ in | S = π | ∂ in U ◦ ( π | ∂ in U ) − = φ in , as desired. 17ow, we introduce the return maps on the surface S and S . We first consider the crossingmap of the plug ( U, X ) θ X : ∂ in U \ L s → ∂ out U \ L u By definition, θ X ( x ) is the unique intersection point of the forward ( X t )-orbit of the point x with the surface ∂ out U . For i = 1 ,
2, the map θ X induces a map θ i := π i | ∂ out U ◦ θ X ◦ ( π i | ∂ in U ) − : S i \ L si → S i \ L ui . This map θ i is just the Poincar´e return map of the flow ( Y ti ) on the surface S i . Proposition 3.4.
The diffeomorphisms θ , θ , φ in and φ out are related by the following equality θ ◦ φ in = φ out ◦ θ . Proof.
This is an immediate consequence of the formulas defining θ , θ , φ in and φ out .Now we lift all the objects to the universal covers of M and M . We pick a point x ∈ M which will serve as the base point of the fundamental group of the manifold M . The point x := f ( x ) will be used as the base point of fundamental group of the manifold M . Thediffeomorphism Φ in provides us with an isomorphism (Φ in ) ∗ between the fundamental groups π ( M , x ) and π ( M , x ). For i = 1 ,
2, we denote by p i : f M i → M i the universal cover of themanifold M i . We denote by e Y i the lift of the vector field Y i on f M i . Observe that e Y i is equivariantunder the action of π ( M i , x i ): for γ ∈ π ( M i , x i ), one has e Y i ( γ ˜ x ) = D ˜ x γ. e Y i (˜ x ). We denote by e S i the complete lift of the surface S i ( i.e. e S i := p − i ( S i )).We denote by e L si and e L ui the complete lifts of the laminations L si and L ui . We denote by e θ i : e S i \ e L si → e S i \ L ui the first return map of the flow of the vector field e Y i on the surface e S i . Clearly, e θ i is a lift of themap θ i . Moreover, e θ i commutes with the deck transformations:˜ θ i ◦ γ = γ ◦ ˜ θ i for every γ ∈ π ( M i , x i ) . (1)This commutation relation is an immediate consequence of the equivariance of e Y i (see above).Now we fix a lift e Φ in : f M → f M of the diffeomorphism Φ in (note that, unlike what happensfor θ and θ , there is no canonical lift of Φ in ). Recall that the diffeomorphism Φ in maps thesurface S to the surface S , and that the restriction of Φ in to S coincides with φ in . As aconsequence, the lift ˜Φ in maps the surface e S to e S , and the restriction of ˜Φ in to e S is a lift ˜ φ in of the diffeomorphism φ in . By construction, this lift satisfies˜ φ in ◦ γ = (Φ in ) ∗ ( γ ) ◦ ˜ φ in for every γ ∈ π ( M , x ) (2)Now recall that, according to Proposition 3.2, there exists a continuous arc ( φ t ) t ∈ [0 , of diffeo-morphisms from S to S , such that φ = φ in and φ = φ out , and such that the laminations φ t ( L u ) and L s are strongly transverse for every t . We lift this isotopy, starting at the lift ˜ φ in of φ in = φ . This yields a continuous arc ( ˜ φ t ) t ∈ [0 , of diffeomorphisms from e S to e S , such that˜ φ = ˜ φ in and such that the laminations ˜ φ t ( e L u ) and e L s are strongly transverse for every t . Thedifffeomorphism ˜ φ out := ˜ φ is a lift of the diffeomorphism φ out . By continuity, the relation (2)remains true if we replace ˜ φ in = ˜ φ by ˜ φ t for any t ∈ [0 , φ out satisfies ˜ φ out ◦ γ = (Φ in ) ∗ ( γ ) ◦ ˜ φ out for every γ ∈ π ( M , x ) . (3)18 roposition 3.5. The diffeomorphisms ˜ θ , ˜ θ , ˜ φ in and ˜ φ out are related by the equality ˜ θ ◦ ˜ φ in = ˜ φ out ◦ ˜ θ . Proof.
According to Proposition 3.4, the diffeomorphisms θ ◦ φ in and φ out ◦ θ coincide. Hencethe diffeomorphisms ˜ θ ◦ ˜ φ in and ˜ φ out ◦ ˜ θ are two lifts of the same diffeomorphism. It followsthat there exists a deck transformation γ ∈ π ( M , y ) such that˜ θ ◦ ˜ φ in = γ ◦ ˜ φ out ◦ ˜ θ . Now consider a deck transformation γ ∈ π ( M , x ). On the one hand, using (2) and (1), we get˜ θ ◦ ˜ φ in γ = ˜ θ ◦ (Φ in ) ∗ ( γ ) ◦ ˜ φ in = (Φ in ) ∗ ( γ ) ◦ ˜ θ ◦ ˜ φ in = ((Φ in ) ∗ ( γ ) · γ ) ◦ ˜ φ out ◦ ˜ θ . On the other hand, using (1) and (3), we get˜ θ ◦ ˜ φ in ◦ γ = γ ◦ ˜ φ out ◦ ˜ θ ◦ γ = γ ◦ ˜ φ out ◦ γ ◦ ˜ θ = ( γ · (Φ in ) ∗ ( γ )) ◦ ˜ φ out ◦ ˜ θ . Hence (Φ in ) ∗ ( γ ) · γ = γ · (Φ in ) ∗ ( γ ) . Since (Φ in ) ∗ ( γ ) ranges over the whole fundamental group π ( M , y ), it follows that γ is inthe center of the fundamental group π ( M , y ). If γ = Id, this implies that π ( M , y ) hasa non-trivial center. Then, a (easy generalization of) well-known theorem of ´E. Ghys impliesthat, up to finite cover, the Anosov flow ( X t ) must be topologically equivalent to the geodesicflow on the unit tangent bundle of a closed hyperbolic surface (see [9], or [3, Th´eor`eme 3.1]).This is clearly impossible, since X admits a transverse torus (any connected component of thesurface S is such a torus). As a consequence, γ must be the identity, and the desired relation˜ θ ◦ ˜ φ in = ˜ φ in ◦ ˜ θ is proved. ∆ s : f s, ∞ → f s, ∞ and ∆ u : f u, ∞ → f u, ∞ In section 2, we have defined some symbolic spaces which allow to code certain orbits of certainAnosov flows. Let us introduce these symbolic space in our particular setting. For i = 1 ,
2, weconsider the alphabet A i := { connected components of e S i \ e L si } , and the symbolic spaceΣ i := { ( D p ) p ∈ Z such that D p ∈ A i and ˜ θ i ( D p ) ∩ D p +1 = ∅ for every p } . In order to code stable and unstable leaves, we consider the subspaces Σ si and Σ ui of Σ i definedby Σ si := { ( D p ) p ≥ such that D p ∈ A i and ˜ θ i ( D p ) ∩ D p +1 = ∅ for every p } and Σ ui := { ( D p ) p< such that D p ∈ A i and ˜ θ i ( D p ) ∩ D p +1 = ∅ for every p } . Proposition 3.6.
Let D and D ′ be two elements of A . Let D := e φ in ( D ) and D ′ := e φ in ( D ′ ) .Then e θ ( D ) intersects D ′ if and only if e θ ( D ) intersects D ′ . roof. We have the following sequence of equivalences. e θ ( D ) ∩ D ′ = ∅ (1) ⇐⇒ e φ in ( e θ ( D )) ∩ e φ in ( D ′ ) = ∅ (2) ⇐⇒ e φ out ( e θ ( D )) ∩ e φ in ( D ′ ) = ∅ (3) ⇐⇒ e θ ( e φ in ( D )) ∩ e φ in ( D ′ ) = ∅ (4) ⇐⇒ e θ ( D ) ∩ D ′ = ∅ . The first equivalence is straightforward. The last one is nothing but the definition of the con-nected components D and D ′ . Equivalence (3) follows from Proposition 3.5. It remains toprove equivalence (2). For that purpose, observe that e θ ( D ) is a strip bounded by two leavesof e L u , and e φ in ( D ′ ) is a strip bounded by two leaves of e L s . Now recall that there exists anisotopy ( e φ t ) t ∈ [0 , joining e φ in to e φ out , such that the lamination e φ t ( e L u ) is strongly transverse tothe lamination e L s . It follows that e φ out ( e θ ( D )) intersects e φ in ( D ′ ) if and only if e φ in ( e θ ( D ))intersects e φ in ( D ′ ).As an immediate consequence of Proposition 3.6, we get: Corollary 3.7. ( ˜ φ in ) ⊗ Z : A Z → A Z maps Σ to Σ . Corollary 3.7 entails that ( ˜ φ in ) ⊗ Z ≥ maps Σ s to Σ s , and ( e φ in ) ⊗ Z < maps Σ u to Σ u . Hence,the map ˜ φ in builds a bridge between the symbolic spaces associated to the vector filed Y andthose associated to the vector filed Y .Let us recall the definition of the coding maps constructed in Section 2.3. For i = 1 ,
2, wedenote by F si and F ui the weak stable and the weak unstable foliations of the Anosov flow ( Y ti )on the manifold M i . These two-dimensional foliations induce two one-dimensional foliations F si and F ui on the surface S i . We denote by e F si and e F ui the lifts of F si and F ui on e S i . We denoteby f si and f ui the leaf spaces of the foliations e F si and e F ui . We denote by f s, ∞ i the subset of f si made of the leaves that are not in f W s (Λ i ) (recall that f W s (Λ i ) is a union of leaves of e F si andtherefore f W s (Λ i ) ∩ e S i is a union of leaves of F si ). Similarly, we denote by f u, ∞ i the subset of f ui made of the leaves that are not in f W u (Λ i ). The construction of subsection 2.3 provides twobijective coding maps χ si : ˜ f s, ∞ i −→ Σ si ℓ ( D p ) p ≥ where ˜ θ pi ( ℓ ) ⊂ D p for every p ≥ χ ui : ˜ f u, ∞ i −→ Σ ui ℓ ( D p ) p< where ˜ θ pi ( ℓ ) ⊂ D p for every p < s := ( χ s ) − ◦ ( ˜ φ in ) ⊗ Z ≥ ◦ χ s : ˜ f s, ∞ −→ ˜ f s, ∞ and ∆ u := ( χ u ) − ◦ ( ˜ φ in ) ⊗ Z < ◦ χ u : ˜ f u, ∞ −→ ˜ f u, ∞ ∆ s and ∆ u We wish to extend the map ∆ s in order to obtain a bijective map between the leaf spaces ˜ f s and˜ f s . In view to that goal, we will prove that ∆ s preserves the order of the leaves of the foliations e F s and e F s . Our first task is to write a precise definition of these order. First we choose an orientation20f the lamination L uX ⊂ ∂ out U . Pushing this orientation by the maps π and π , this definessome orientations of the laminations L u = ( π ) ∗ ( L uX ) ⊂ S and L u = ( π ) ∗ ( L uX ) ⊂ S . Since L ui is a sublamination of the foliation F ui (and since L ui intersects every connected component of S i ),the orientations of the lamination L u and L u define some orientations of the foliations F u and F u . Finally, these orientation can be lifted, providing orientations of the lifted foliations e F u and e F u . It is important to notice that our choice of orientations for e F u and e F u are not independentfrom each other. More precisely, the orientation are chosen so that φ out = π | ∂ out U ◦ ( π | ∂ out U ) − maps the orientation of the lamination L u to the orientation of the lamination L u , and therefore:˜ φ out maps the orientated lamination e L u to the orientated lamination e L u . (4)As explained in Subsection 2.4, the orientation of the foliation e F ui induces a partial order ≺ i onthe leaf space ˜ f si defined as follows: given two leaves ℓ i , ℓ ′ i ∈ ˜ f si satisfy ℓ i ≺ i ℓ ′ i if there exists anarc segment of an oriented leaf of e F ui going from a point of ℓ i to a point of ℓ ′ i . Proposition 2.22proves that this indeed defines an order on ˜ f si . Moreover, this order on ˜ f si induces a partial orderon the alphabet A i : given two elements D i and D ′ i of A i , we write D i ≺ i D ′ i if there exists a leaf˜ α i of e F si included in D i and a leaf ˜ α ′ i of e F si included in D ′ i such that ˜ α i ≺ i ˜ α ′ i . Proposition 2.24shows that we can replace “there exists” by ”for every” in this definition. It follows that ≺ i isindeed a partial order on A i . Now comes the technical result which will allow us to extend themap ∆ s : Proposition 3.8.
The map ∆ s : ( f s, ∞ , ≺ ) −→ ( f s, ∞ , ≺ ) is order-preserving. In order to prove Proposition 3.8, we need several intermediary results.
Lemma 3.9.
The map ˜ φ in : ( A , ≺ ) −→ ( A , ≺ ) is order-preserving.Proof. Consider two elements D , D ′ of A . Assume that D ≺ D ′ . This means that thereexists a leaf ℓ of the oriented lamination e L u which crosses D before crossing D ′ . As a con-sequence, if we endow ˜ φ in ( ℓ ) with the image under ˜ φ in of the orientation of α , then ˜ φ in ( ℓ )crosses ˜ φ in ( D ) before crossing ˜ φ in ( D ′ ). Now recall that: • ˜ φ in ( D ) and ˜ φ in ( D ′ ) are strips bounded by leaves of the lamination ˜ φ in ( e L s ) = e L s , • there exists an isotopy ( ˜ φ t ) joining ˜ φ in to ˜ φ out such that the lamination ˜ φ t ( e L u ) is stronglytransverse to the lamination e L s for every t .We deduce that, if we endow ˜ φ out ( ℓ ) with the image under ˜ φ out of the orientation of ℓ , then˜ φ out ( ℓ ) crosses ˜ φ in ( D ) before crossing ˜ φ in ( D ′ ). According to (4), this means that there is aleaf of the oriented lamination e L u which crosses ˜ φ in ( D ) before crossing ˜ φ in ( D ′ ). By definitionof the partial order ≺ , this means that ˜ φ in ( D ) ≺ ˜ φ in ( D ′ ). Lemma 3.10.
Let D be a connected component of e S \ e L s . Set D := ˜ φ in ( D ) . Then thefollowing are equivalent:1. the map ˜ θ restricted to the strip D preserves the orientation of the foliation e F u ,2. the map ˜ θ restricted to the strip D preserves the orientation of the foliation e F u .Proof. The proof is a bit intricate, because we need to introduce no less than six leaves andcompare their orientations. Recall that we have chosen some orientations for the foliations e F u and e F u . In the sequel, we will also consider the foliations ( ˜ φ in ) ∗ e F u , ( ˜ φ out ) ∗ e F u and ( ˜ φ t ) ∗ e F u ; weendow them with the images under ˜ φ in , ˜ φ out and ˜ φ t of the orientation of e F u .21e pick a leaf ℓ of the lamination e L u so that ℓ ∩ D = ∅ (such a leaf always exists sincethe laminations e L s and e L u are strongly transverse). Then we set ℓ := ˜ φ out ( ℓ ) ˆ ℓ := ˜ φ in ( ℓ ) ℓ ′ := ˜ θ ( ℓ ∩ D ) ℓ ′ := ˜ θ ( ℓ ∩ D ) ˆ ℓ ′ := ˜ θ (ˆ ℓ ∩ D ) . Observe thatˆ ℓ ′ = ˜ θ ( ˜ φ in ( ℓ ) ∩ D ) = ˜ θ ◦ ˜ φ in ( ℓ ∩ D ) = ˜ φ out ◦ ˜ θ ( ℓ ∩ D ) = ˜ φ out ( ℓ ′ ) (5)(the third equality follows from Proposition 3.5). Now recall that, for i = 1 ,
2, both e L ui and(˜ θ i ) ∗ ( e L ui ∩ D si ) are sublaminations of the foliation e F ui . Also recall that ˜ φ out ( e L u ) = e L u . Thisprovides some natural orientations on ℓ , ℓ ′ , ℓ , ℓ ′ , ˆ ℓ , ˆ ℓ ′ : • ℓ and ℓ ′ are leaves of the foliation e F u , hence inherit of the orientation of e F u ; • ℓ and ℓ ′ are leaves of the foliation e F u , hence inherit of the orientation of e F u ; we endowthem with the orientation of this foliation; • ˆ ℓ is a leaf of the foliation ( ˜ φ in ) ∗ e F u , hence inherits of the orientation of ( ˜ φ in ) ∗ e F u ; • ˆ ℓ ′ is a leaf of the foliation ( ˜ φ out ) ∗ e F u , hence inherits of the orientation of ( ˜ φ out ) ∗ e F u .By symmetry, it is enough to prove the implication 1 ⇒
2. So, we assume that the restrictionof ˜ θ to D s preserves the orientation of e F u ; in particular:˜ θ maps the orientation of ℓ to those of ℓ ′ . (6)According to (4), ˜ φ out maps the orientation of ℓ to those of ℓ . (7)The orientations of ℓ , ℓ , ˆ ℓ , ˆ ℓ ′ are chosen in such a way that ˜ φ − in maps the orientation of ˆ ℓ tothose of ℓ , and ˜ φ out maps the orientation of ℓ ′ to those of ˆ ℓ ′ . Puting this together with (6), weobtain that ˜ φ out ◦ ˜ θ ◦ ˜ φ − in maps the orientation of ˆ ℓ to those of ˆ ℓ ′ . Using proposition 3.5, weobtain ˜ θ maps the orientation of ˆ ℓ to those of ˆ ℓ ′ . (8)Our final goal is to prove that ˜ θ maps the orientation of ℓ to those of ℓ ′ . So, in view of (8),we need to compare the orientations of ℓ and ˆ ℓ on the one hand, and the orientations ℓ ′ andˆ ℓ ′ on the other hand. We start by ℓ and ˆ ℓ .Recall that D is a strip in e S bounded by two leaves of the stable lamination e L s . We denotethese two leaves by α and β , in such a way that oriented unstable leaf ℓ enters in D by crossing α and exits D by crossing β . According to (7), the orientation of ℓ = ( ˜ φ out ) ∗ ℓ as a leaf of e L u ⊂ e F u coincides with the orientation as a leaf of ( ˜ φ out ) ∗ e L u ⊂ ( ˜ φ out ) ∗ F u . Moreover, recall thatthere exists an isotopy ( ˜ φ t ) t ∈ [0 , joining ˜ φ = ˜ φ in to ˜ φ = ˜ φ out , such that the lamination ˜ φ t ( e L u )is strongly transverse to the lamination e L u for every t . We deduce that ˆ ℓ = ( ˜ φ in ) ∗ ( ℓ ) crosses D in the same direction as ℓ = ( ˜ φ out ) ∗ ℓ . In other words,both ℓ and ˆ ℓ enter in D by crossing α and exits D by crossing β . (9)Let U and V be some disjoint neighborhoods of the stable leaves α and β in the strip D .Assertion (9) can be reformulated as followsthe arcs of oriented leaves ℓ ∩ D and ˆ ℓ ∩ D both go from U to V . (10)22e are left to compare the orientations of ℓ ′ and ˆ ℓ ′ . First observe that ˜ θ ( D ) is an openstrip in e S , bounded by two leaves of the unstable lamination e L u = ( ˜ φ out ) ∗ e L u . The closureCl(˜ θ ( D )) of ˜ θ ( D ) is the union of the open strip ˜ θ ( D ) and its two boundary leaves. Theboundary components of ˜ θ ( D ) are leaves of both the foliations F u and ( ˜ φ out ) ∗ F u . Moreover, F u and ( ˜ φ out ) ∗ F u induce two trivial oriented foliations on the closed strip Cl(˜ θ ( D )). In particular,the leaves of F u and ( ˜ φ out ) ∗ F u in Cl(˜ θ ( D )) go from one end of Cl(˜ θ ( D )) to the other end.In order to distinguish the two ends of the closed strip Cl(˜ θ ( D )), we use the set Cl(˜ θ ( U ))and Cl(˜ θ ( V )). These sets are disjoint neighbourhoods of the two ends of Cl(˜ θ ( D )). So wejust need to decide if the leaves go from Cl(˜ θ ( U )) and Cl(˜ θ ( V )), or the contrary. On the onehand, putting (8) and (10) together, we obtain that ˆ ℓ goes from Cl(˜ θ ( U )) to Cl(˜ θ ( V )). Onthe other hand, F u and ( ˜ φ out ) ∗ F u are trivial oriented foliations on Cl(˜ θ ( D )), and, accordingto (4), they induce the same orientation on the boundary leaves of D ′ . So we conclude that allthe leaves of both the oriented foliations F u and ( ˜ φ out ) ∗ F u go from Cl(˜ θ ( U )) to Cl(˜ θ ( V )). Inparticular, the oriented leaves ℓ ′ and ˆ ℓ ′ go from ˜ θ ( U ) to ˜ θ ( V ). (11)From (10) and (11), we deduce that ˜ θ | D maps the orientation of ℓ to those of ℓ ′ . By definitionof the orientations of ℓ and ℓ ′ , this means that the restriction of ˜ θ to the strip D preservesthe orientation of the foliation e F u . This completes the proof of the implication 1 ⇒ orollary 3.11. Let D , , . . . , D ,p − be connected components of e S \ e L s , so that T p − p =0 e θ p ( D ,p ) is non-empty. For p = 1 , . . . , p − , let D ,p := ˜ φ in ( D ,p ) . Then the following are equivalent:1. the map e θ p restricted to T p − p =0 e θ p ( D ,p ) preserves the orientation of the foliation e F u ,2. the map e θ p restricted to T p − p =0 e θ p ( D ,p ) preserves the orientation of the foliation e F u .Proof. For i = 1 ,
2, consider the set J i := n j ∈ { , . . . , p − } s. t. the restriction of e θ i to D i,p preserves the orientation of e F ui o . On the one hand, Lemma 3.10 implies that the sets J and J coincide. On the other hand, it isclearly that the restriction of e θ i to T p − j =0 e θ pi ( D i,p ) preserves the orientation of the leaves of e F ui if and only if the cardinality of J i is even. The corollary follows. Proof of Proposition 3.8.
We consider two leaves γ and γ ′ in f s, ∞ , we denote γ := ∆ s ( γ ) and γ ′ := ∆ s ( γ ′ ), and we assume that γ ≺ γ ′ . We aim to prove γ ≺ γ ′ . Let χ s (˜ γ ) = ( D ,p ) p ≥ χ s (˜ γ ′ ) = (cid:0) D ′ ,p (cid:1) p ≥ χ s (˜ γ ) = ( D ,p ) p ≥ χ s (˜ γ ′ ) = (cid:0) D ′ ,p (cid:1) p ≥ . By defintion of the map χ si , this means that, for i = 1 , γ i = \ p ≥ ˜ θ − pi ( D i,p ) and ˜ γ ′ i = \ p ≥ ˜ θ − pi ( D ′ i,p ) . And since ˜ γ = ∆ s (˜ γ ) and ˜ γ ′ = ∆ s (˜ γ ′ ), we have D ,p = φ in ( D ,p ) and D ′ ,p = φ in (cid:0) D ′ ,p (cid:1) for every p ≥
0. We denote by p the smallest integer p such that D ,p = D ′ ,p .Let us consider the case where the map e θ p restricted to T p − p =0 ˜ θ − p ( D ,p ) preserves theorientation of the foliation e F u . • Proposition 2.28 implies that D ,p ≺ D ′ ,p . • Since φ in : A → A is order-preserving (Lemma 3.9), it follows that D ,p ≺ D ′ ,p . • Corollary 3.11 implies that the map e θ p , restricted to T p − p =0 ˜ θ − p ( D ,p ) preserves the ori-entation of the foliation e F u . • Using again Proposition 2.28, we deduce from the two last items above that ˜ γ ≺ ˜ γ ′ , asdesired.The case where the map e θ p restricted to T p − p =0 ˜ θ − p ( D ,p ) reverses the orientation of the foliation e F u follows from the very same arguments. Corollary 3.12.
The map ∆ s : f s, ∞ −→ f s, ∞ extends in a unique way to an order-preservingbijection ∆ s : f s −→ f s .Proof. This is an immediate consequence of the following facts: • ∆ s : f s, ∞ −→ f s, ∞ is an order-preserving map (Proposition 3.8); • for i = 1 , f s, ∞ i is a dense subset of the (non-separated) one-dimensional manifold f si (Proposition 2.7); 24 for i = 1 ,
2, each leaf ℓ ∈ f si has a neighborhood U ℓ in f si so that the leaves in U ℓ aretotally ordered (Proposition 2.23).Of course, the stable and the unstable direction play some symmetric roles, hence the samearguments as above alow to prove the following analog of Corollary 3.12: Corollary 3.13.
The map ∆ u : f u, ∞ −→ f u, ∞ extends in a unique way to an order-preservingbijection b ∆ u : f u −→ f u . b ∆ s and b ∆ u : construction of the map b ∆ Now, we will mate the maps b ∆ s and b ∆ u to obtain a b ∆ : e S → e S . In view to that goal, we needthe following lemma: Lemma 3.14.
Consider a leaf ℓ s of the stable foliation e F s and a leaf ℓ u of the unstable foliation e F u . Then ℓ s intersects ℓ u if and only if b ∆ s ( ℓ s ) intersects b ∆ u ( ℓ u ) .Proof. The case where the leaves ℓ s and ℓ u belong to f s, ∞ and f u, ∞ is a consequence of Proposi-tion 3.6 (together with the definitions of the maps ∆ s , ∆ u and ∆): the leaves ℓ s and ℓ u intersectat x if and only if the leaves b ∆ s ( ℓ s ) = ∆ s ( ℓ s ) and b ∆ u ( ℓ u ) = ∆ u ( ℓ u ) intersect at ∆( x ). Thegeneral case follows by density of f s, ∞ i and f u, ∞ i in f s, ∞ i and f u, ∞ i .Now we define a map b ∆ : e S → e S . Let ˜ x be any point in e S . Denote by ℓ s (resp. ℓ u ) theleaf of the stable foliation e F s (resp. the unstable foliation e F u ) passing through x . Recall that x is the unique intersection point of ℓ s and ℓ u . According to the preceding lemma, the stable leaf b ∆ s ( ℓ s ) and the unstable leaf b ∆ u ( ℓ u ) do intersect. According to Proposition 2.2, the intersectionis a single point. We define b ∆(˜ x ) to be the unique intersection point of the leaves b ∆ s ( ℓ s ) and b ∆ u ( ℓ u ). In other words, b ∆ is defined by b ∆( ℓ s ∩ ℓ u ) = b ∆ s ( ℓ s ) ∩ b ∆ u ( ℓ u ) . (12)By construction, the map b ∆ is bijective, maps the foliations e F s , e F u to the foliations e F s , e F u ,preserving the orders on the leaf spaces. Since the leaf spaces are locally totally ordered (Propo-sition 2.23), it follows that b ∆ is continuous. Hence b ∆ is a homeomorphism. Proposition 3.15.
The map b ∆ : e S → e S is equivariant with respect to the actions of thefundamental groups: for every γ of π ( M ) , b ∆ ◦ γ = ( ˜Φ in ) ∗ ( γ ) ◦ b ∆ . Proof.
This is a rather immediate consequence of the construction of b ∆. First recall that b ∆ isa continuous extension of the map ∆ : e S ∞ → e S ∞ and recall that e S ∞ , e S ∞ are dense subsets of e S , e S . As a consequence, it is enough to prove that ∆ is equivariant with respect to the actionsof the fundamental groups. Now recall that ∆ is defined as the composition of three maps:∆ = ( χ ) − ◦ ( ˜ φ in ) ⊗ Z ◦ χ . But we know that: • the map χ i commutes with the action of the fundamental group π ( M i ) for i = 1 , the map ˜ φ in satisfies the following equivariance ˜ φ in ◦ γ = ( ˜Φ in ) ∗ ( γ ) ◦ ˜ φ in (equation (2)).This shows that the map ∆ satisfies the equivariance relation ∆ ◦ γ = ( ˜Φ in ) ∗ ( γ ) ◦ ∆, andcompletes the proof. Proposition 3.16.
The map b ∆ : e S → e S conjugates the Poincar´e maps ˜ θ and ˜ θ , that is b ∆ ◦ ˜ θ = ˜ θ ◦ b ∆ . Proof.
On the one hand, for i = 1 ,
2, the coding map χ si conjugates the Poincar´e map ˜ θ i on e S i to the shift map on the symbolic space Σ si (Proposition 2.17). On the other hand, themap ( ˜ φ in ) ⊗ Z ≥ obviously conjugates the shift map on Σ s to the shift map on Σ s . Hence,∆ s = ( χ s ) − ◦ ( ˜ φ in ) ⊗ Z ≥ ◦ χ s conjugates the action ˜ θ on f s, ∞ to the action of ˜ θ on f s, ∞ . Bydensity of f s, ∞ in f si , it follows that b ∆ s conjugates the action ˜ θ on f s to the action of ˜ θ on f s . Similalrly, b ∆ u conjugates the action ˜ θ on f u to the action of ˜ θ on f u . Finally, since e ∆ isdefined by mating b ∆ s and b ∆ u (see (12)), this implies that e ∆ conjugates ˜ θ to ˜ θ . b ∆ to the orbital equivalence To conclude the proof of Theorem 1.2, we need to introduce the orbit spaces of the Anosov flows( Y t ) and ( Y t ). The orbit space of ( Y ti ) is by definition the quotient of the manifold f M i by theaction of the flow ( Y ti ). We denote it by O i , and we denote by pr i the natural projection of f M i on O i . The action of the fundamental group π ( M i ) on f M i induces an action of this groupon O i . The two dimensional foliations e F si and e F ui are leafwise invariant under the flow ( Y ti )and therefore can be projected in the orbit space O i . They induce a pair ( g si , g ui ) of transverse1-dimensional foliations on O i .The orbit space O i by itself does not carry much information: indeed, O i is always separatedmanifold diffeomorphic to R (see [8, Proposition 2.1] or [2, Theorem 3.2]). The pair of transversefoliations ( g si , g ui ) carries a much more interesting information (see the work of Barbot and Fenleyon the subject; good references are. Barbot’s habilitation memoir [3] and Barthelm´e’s lecturenotes [4]). The action of π ( M i ) on O i carries an even richer dynamical information: actually,this action characterizes the flow ( Y ti ) up to topological equivalence (see Theorem 3.22 below).Recall that Λ denotes the maximal invariant set of the initial hyperbolic plug ( U, X ), thatΛ i denotes the projection of Λ in the manifold M i = U/ψ i , and that e Λ i the complete lift of Λ i in the universal cover f M i . Now, we denote by L i the projection of the set e Λ i in O i . Lemma 3.17.
The projection pr i ( e S i ) of the surface e S i in the orbit space O i is exactly thecomplement of the set L i in O i .Proof. The set Λ is the union of the orbits of the vector field X which remain in U forever, i.e. which do not intersect ∂U . Hence the set Λ i = π i (Λ) is the union of the orbits of the vector field Y i = ( π i ) ∗ X which do not intersect the surface S i = π i ( ∂U ). As a further consequence, e Λ i is theunion of the orbits of the vector field e Y i which do not intersect the surface e S i . This means thatthe projection of e S i in the orbit space O i is exactly the complement of the projection of the set e Λ i . Proposition 3.16 can be rephrased as follows: two points x, x ′ ∈ e S belong to the same orbitof the flow ( e Y t ) if and only if the points b ∆( x ) and b ∆( x ′ ) belong to the same orbit of the flow ( e Y t ).As a consequence, the homeomorphism b ∆ : e S → e S induces a homeomorphism δ : pr ( e S ) = O \ L −→ pr ( e S ) = O \ L . δ is also equivariant: for every γ ∈ π ( M ), δ ◦ γ = ( ˜Φ in ) ∗ ( γ ) ◦ δ. Our next step is to extend the map η to the whole orbit spaces. Proposition 3.18.
The homeomorphism δ : O \ L → O \ L can be extended in a uniqueway to a homeomorphism δ : O → O , which is equivariant with respect to the actions of thefundamental groups of M and M . We shall use the following general lemma of planar topology.
Lemma 3.19.
Let A and B be totally discontinuous subsets of R , and h : R \ A → R \ B .Assume that, for every compact subset K of R , the set h ( K \ A ) is relatively compact in R .Then h can be extended to a homeomorphism of ¯ h : R → R . This lemma is easy and certainly well-known by people working in planar topology, but wewere not able to find it in the literature. We provide a proof for sake of completeness.
Proof.
We proceed to the definition of ¯ h . Let x be a point in A . We pick a decreasing sequence( X n ) n ≥ of compact connected subsets of R so that X n = { x } for every n , and so that T n X n = { x } . For every n ≥
0, let Y n be the closure in R of the set h ( X n \ A ). Our assumptions implythat ( Y n ) n ≥ is a decreasing sequence of non-empty compact connected subsets of R . As aconsequence, the intersection T n Y n must be a non-empty compact connected subset of R .Moreover, since T n X n = { x } ⊂ A , the intersection T n Y n must be included in B . Since B is totally disconnected, it follows that T n Y n must be a singleton { y } . Standard argumentsshow that the point y does not depend on the choice of the sequence ( X n ). We set h ( x ) := y .Repeating the same procedure for each point x ∈ A , we get an extension ¯ h : R → R of h . Thecontinuity of ¯ h follows easily from its definition.Of course, the same procedure yields a continuous extension h − : R → R of the map h − : R \ B → R \ A . Since R \ A and R \ B are dense in R , the equalities h ◦ h − = Id R \ B and h − ◦ h = Id R \ A extend to ¯ h ◦ h − = h − ◦ ¯ h = Id R . This shows that that h is ahomeomorphism and completes the proof. Lemma 3.20.
For i = 1 , , the set L i is totally discontinuous in O i ≃ R . Let us introduce some terminology that will be used in the proof of Lemma 3.20. By a local section of a vector field Z on a 3-manifold P , we mean a compact surface with boundaryembedded in P and transverse to Z . A ( Z t )-invariant set Ω ⊂ P is said to be transversallytotally discontinuous if Ω ∩ Σ is totally discontinuous for every local section Σ of Z . Proof.
By our assumptions, the maximal invariant set Λ X of the hyperbolic plug ( U, X ) containsneither attractors nor repellors. Since Λ X is a hyperbolic set, it follows that Λ X is transversallytotally discontinuous. Hence the projection Λ i of Λ X in the manifold M i is also transversallytotally discontinuous (recall that Λ X sits in the interior of U and that the projection p i : U → M i is a homeomorphism in restriction to the interior of U ). As a further consequence, the completelift e Λ i of Λ i in the universal cover f M i is also transversally totally discontinuous.Now recall that ( f M i , e Y i ) is topologically equivalent to R equipped with the trivial verticalunit vector field. As a consequence, for every point x ∈ f M i , we can find a local section Σ of e Y i ,so that x ∈ Σ, and so that no orbit of e Y i intersects Σ twice. This implies that the restriction to Σof the projection pr : f M i → O i is one-to-one, hence a homeomorphism onto its image. Since e Λ i is transversally totally discontinuous, it follows that the set L i = pr( e Λ i ) is totally discontinuousin O i . 27 emma 3.21. For every compact set K ⊂ O ≃ R , the set η ( K \ L ) has compact closure in O ≃ R .Proof. For i = 1 ,
2, the surface O i \ L i has infinitely many ends. One of them is the end of O i ≃ R , that we denote by ∞ i . The other ends are in one to one correspondance with thepoints of L i (since L i is totally discontinuous). Proving lemma 3.21 is equivalent to proving thatthe homeomophism η : O \ L → O \ L maps the end ∞ to the end ∞ .From the viewpoint of the topology of the surface O i \ L i , nothing distinguishes ∞ i from theother ends. Hence we need to introduce some dynamical invariants to prove that η necessarilymaps ∞ to ∞ .For i = 1 ,
2, the foliation e F si induces a 1-dimensional foliation g si on the space O i . Wedenote by g si, the restriction of the foliation g si to O i \ L i . According to Lemma 3.17, g si, canbe obtained as the projection on O i of the foliation e F si ∩ e S i = e F si . As a consequence, η mapsthe foliation g s , to the foliation g s , .Since O i is a plane, every leaf of the foliation g si is a properly embedded line, going from ∞ i to ∞ i (recall that ∞ i is the unique end of O i ). The leaves of g si = (pr i ) ∗ e F si that intersect L i = pr i (Λ i ) are the projections of the leaves of the lamination W s ( e Λ i ). In particular, thereexist leaves of g si that do not intersect L i . As a consequence, there exist leaves of g si, going from ∞ i to ∞ i . On the other hand, if x is an end of O i \ L i corresponding to a point of L i , thenthere does not exist any leaf of g si, going from x to x (because every leaf ℓ of g si, is a connectedcomponent of ˆ ℓ \ L i where ˆ ℓ a line in O i going from ∞ i to ∞ i ). So, the foliation g si, allows todistinguish ∞ i from the other ends of O i \ L i . Since η maps g s , to g s , , it follows that η mustmap ∞ to ∞ . Since ∞ i is the unique end of O i , this exactly means that, for a compact set K ⊂ O ≃ R , the set η ( K \ L ) has compact closure in O ≃ R . Proof of Proposition 3.18.
Lemmas 3.20 and 3.21, together with the fact that O and O arehomeomorphic to R , show that we are exactly in the situation of Lemma 3.19. Applying thisLemma, we get a homeomophism ¯ δ : O → O extending η . The equivariance of ¯ η follows fromthose of δ , by continuity and by density of O i \ L i in O i .We will now conclude the proof of Theorem 1.2 by using a result of Barbot. Theorem 3.22 (See Theorem 3.4 of [2], or Proposition 1.36 and Corollaire 1.42 of [3]) . Twotransitive Anosov flows are topologically equivalent if and only if there exist a homeomorphismbetween their orbit spaces, which is equivariant with respect to the actions of the fundamentalgroups, and which does not exchange the stable/unstable directions.Proof of Theorem 1.2.
The Theorem is an immediate consequence of Proposition 3.18 and The-orem 3.22.
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E-mail: [email protected]
Bin Yu
School of Mathematical SciencesTongji University, Shanghai 200092, CHINA