A consequence of the growth of rotation sets for families of diffeomorphisms of the torus
AA consequence of the growth of rotationsets for families of diffeomorphisms ofthe torus
Salvador Addas-Zanata
Instituto de Matem´atica e Estat´ısticaUniversidade de S˜ao PauloRua do Mat˜ao 1010, Cidade Universit´aria,05508-090 S˜ao Paulo, SP, Brazil
Abstract
In this paper we consider C ∞ -generic families of area-preserving dif-feomorphisms of the torus homotopic to the identity and their rotationsets. Let f t : T → T be such a family, (cid:101) f t : IR → IR be a fixed familyof lifts and ρ ( (cid:101) f t ) be their rotation sets, which we assume to have inte-rior for t in a certain open interval I. We also assume that some rationalpoint ( pq , lq ) ∈ ∂ρ ( (cid:101) f t ) for a certain parameter t ∈ I and we want to un-derstand consequences of the following hypothesis: For all t > t, t ∈ I, ( pq , lq ) ∈ int ( ρ ( (cid:101) f t )) . Under these very natural assumptions, we prove that there exists a f qt -fixed hyperbolic saddle P t such that its rotation vector is ( pq , lq ) and, thereexists a sequence t i > t, t i → t, such that if P t is the continuation of P t with the parameter, then W u ( (cid:101) P t i ) (the unstable manifold) has quadratictangencies with W s ( (cid:101) P t i )+( c, d ) (the stable manifold translated by ( c, d )) , where (cid:101) P t i is any lift of P t i to the plane, in other words, (cid:101) P t i is a fixed pointfor ( (cid:101) f t i ) q − ( p, l ) , and ( c, d ) (cid:54) = (0 ,
0) are certain integer vectors such that W u ( (cid:101) P t ) do not intersect W s ( (cid:101) P t ) + ( c, d ) . And these tangencies becometransverse as t increases.As we also proved that for t > t, W u ( (cid:101) P t ) has transverse intersectionswith W s ( (cid:101) P t ) + ( a, b ) , for all integer vectors ( a, b ) , one may consider thatthe tangencies above are associated to the birth of the heteroclinic inter-sections in the plane that did not exist for t ≤ t. e-mail: [email protected] The author is partially supported by CNPq, grant: 306348/2015-2 a r X i v : . [ m a t h . D S ] S e p Introduction
In this paper, in a certain sense, we continue the study initiated in [3]. Therewe looked at the following problem: Suppose f : T → T is a homeomorphismhomotopic to the identity and its rotation set, which is supposed to have inte-rior, has a point ρ in its boundary with both coordinates rational. The questionstudied was the following: is it possible to find two different arbitrarily small C -perturbations of f, denoted f and f in a way that ρ does not belong tothe rotation set of f and ρ is contained in the interior of the rotation set of f ? In other words we were asking if the rational mode locking found by A.de Carvalho, P. Boyland and T. Hall [4] in their particular family of homeo-morphisms was, in a certain sense, a general phenomenon or not. Our maintheorems and examples showed that the answer to this question depends onthe set of hypotheses assumed. For instance, regarding C -generic families, weproved that if ρ ∈ ∂ρ ( (cid:101) f t ) for some t and for t < t, close to t, ρ / ∈ ρ ( (cid:101) f t ) , then forall sufficiently small t − t > , ρ / ∈ int ( ρ ( (cid:101) f t )) . Now we are interested in the dynamical consequences of a situation, whichmay be obtained as a continuation of the previous one: Suppose f t : T → T is a one parameter family of diffeomorphisms of the torus homotopic to theidentity, for which the rotation set ρ ( (cid:101) f t ) at a certain parameter t = t, hasinterior, some rational vector ρ ∈ ∂ρ ( (cid:101) f t ) and for all sufficiently small t > t,ρ ∈ int ( ρ ( (cid:101) f t ))) . We want to understand what happens for the family f t , t > t. In other words, we are assuming that at t = t the rotation set is ready to locallygrow in a neighborhood of ρ. This is the usual situation for (generic) families: As the parameter changes,the rotation set hits a rational vector, this vector stays for a while in the bound-ary of the rotation set and finally, it is eaten by the rotation set, that is, itbecomes an interior point.Here we consider C ∞ -generic area preserving families in the sense of Meyer[19] and also satisfying other generic conditions, and the theorem proved goes in1he following direction: If for instance, f t : T → T is such a family for whichthe rotation set at t = t has interior, (0 , ∈ ∂ρ ( (cid:101) f t ) and for all sufficiently small t > t, (0 , ∈ int ( ρ ( (cid:101) f t )) , then (cid:101) f t : IR → IR has a hyperbolic fixed saddle (cid:101) P t ,for which W u ( (cid:101) P t i ) ( (cid:101) P t is the continuation of (cid:101) P t with the parameter t and t i > t is a certain sequence converging to t ) has heteroclinic tangencies with certainspecial integer translates of the stable manifold, W s ( (cid:101) P t i ) + ( a, b ) , ( a, b ) ∈ ZZ , which unfold (=become transversal) as t increases. The integer vectors ( a, b )mentioned above belong to a set K ZZ ⊂ ZZ and satisfy the following: for t ≤ t,W u ( (cid:101) P t ) can not have intersections with W s ( (cid:101) P t ) + ( a, b ) whenever ( a, b ) ∈ K ZZ . Moreover, the sequence t i → t depends on the choice of ( a, b ) ∈ K ZZ . So increasing the parameter until the critical value at t = t is reached (themoment when the rotation set is ready to locally grow), this critical parameteris accumulated from the other side by parameters at which there are heteroclinictangencies in the plane (homoclinic in the torus) not allowed to exist when t ≤ t. In fact, as we will prove, for any t > t,W u ( (cid:101) P t ) has transverse intersections with W s ( (cid:101) P t ) + ( a, b ) , ∀ ( a, b ) ∈ ZZ . (1)In this way, the creation of the heteroclinic intersections for integers ( a, b ) in (1)which did not exist for t ≤ t, also produces tangencies.In order to state things clearly and to precisely present our main result, afew definitions are necessary.
1. Let T = IR / ZZ be the flat torus and let p : IR −→ T be the associatedcovering map. Coordinates are denoted as (cid:101) z ∈ IR and z ∈ T .
2. Let
Dif f r (T ) be the set of C r diffeomorphisms ( r = 0 , , ..., ∞ ) of thetorus homotopic to the identity and let Dif f r (IR ) be the set of lifts ofelements from Dif f r (T ) to the plane. Maps from Dif f r (T ) are denoted f and their lifts to the plane are denoted (cid:101) f .
3. Let p , : IR −→ IR be the standard projections, respectively in thehorizontal and vertical components;2. Given f ∈ Dif f (T ) (a homeomorphism) and a lift (cid:101) f ∈ Dif f (IR ) , theso called rotation set of (cid:101) f , ρ ( (cid:101) f ) , can be defined following Misiurewicz andZiemian [20] as: ρ ( (cid:101) f ) = (cid:92) i ≥ (cid:91) n ≥ i (cid:40) (cid:101) f n ( (cid:101) z ) − (cid:101) zn : (cid:101) z ∈ IR (cid:41) (2)This set is a compact convex subset of IR (see [20]), and it was provedin [12] and [20] that all points in its interior are realized by compact f -invariant subsets of T , which can be chosen as periodic orbits in therational case. By saying that some vector ρ ∈ ρ ( (cid:101) f ) is realized by a compact f -invariant set, we mean that there exists a compact f -invariant subset K ⊂ T such that for all z ∈ K and any (cid:101) z ∈ p − ( z )lim n →∞ (cid:101) f n ( (cid:101) z ) − (cid:101) zn = ρ. (3)Moreover, the above limit, whenever it exists, is called the rotation vectorof the point z, denoted ρ ( z ) . If U ⊂ IR is an open topological disk whose boundary is a Jordan curve and (cid:101) f : IR → IR is an orientation preserving homeomorphism such that (cid:101) f ( U ) = U, it is easy to see that (cid:101) f : ∂U → ∂U is conjugate to a homeomorphism of thecircle, and so a real number ρ ( U ) = rotation number of (cid:101) f | ∂U can be associatedto this problem. Clearly, if ρ ( U ) is rational, there exists a periodic point in ∂U and if it is not, then there are no such points. This is known since Poincar´e.The difficulties arise when we do not assume ∂U to be a Jordan curve.The prime ends compactification is a way to attach to U a circle called thecircle of prime ends of U, obtaining a space U (cid:116) S with a topology that makesit homeomorphic to the closed unit disk. If, as above we assume the existenceof a planar orientation preserving homeomorphism (cid:101) f such that (cid:101) f ( U ) = U, then (cid:101) f | U extends to U (cid:116) S . The prime ends rotation number of (cid:101) f | U , still denoted3 ( U ) , is the usual rotation number of the orientation preserving homeomorphisminduced on S by the extension of (cid:101) f | U . But things may be quite different inthis setting. In full generality, it is not true that when ρ ( U ) is rational, there areperiodic points in ∂U and for some examples, ρ ( U ) is irrational and ∂U is notperiodic point free. Nevertheless, in the area-preserving case which is the caseconsidered in this paper, many interesting results were obtained. We refer thereader to [18], [13], [15] and [16]. To conclude, we present some results extractedfrom these works, adapted to our hypotheses.Assume h : T → T is an area-preserving diffeomorphism of the torus ho-motopic to the identity such that for each integer n > , h has finitely many n -periodic points. Moreover, we also assume more technical conditions on h :for each n > , at all n -periodic points a Lojasiewicz condition is satisfied, see[10]. And if the eigenvalues of Dh n at such a periodic point are both equal to1 , then the point is topologically degenerate; it has zero topological index. Asexplained in section 2 of [3], the dynamics near such a point is similar to theone in figure 2. In particular h has one stable separatrix (like a branch of a hy-perbolic saddle) and an unstable one at such a periodic point, both h -invariant.Topologically, the local dynamics in a neighborhood of the periodic point isobtained by gluing exactly two hyperbolic sectors.Fix some (cid:101) h : IR → IR , a lift of h to the plane. Given a (cid:101) h -invariant contin-uum K ⊂ IR , if O is a connected component of K c ( O is a topological opendisk in the sphere S def. = IR (cid:116) ∞ , the one point compactification of the plane,that is, O is a connected simply connected open subset of S ), which is alsoassumed to be (cid:101) h -invariant (it could be periodic with period larger than 1), let α be the rotation number of the prime ends compactification of O. From thehypothesis on h, we have: Theorem A. If α is rational, then ∂O has accessible (cid:101) h -periodic points. Andif such a point has period n, the eigenvalues of D (cid:101) h n at this periodic point mustbe real and can not be equal to − . So from the above properties assumed on h, in ∂O we either have accessible hyperbolic periodic saddles or periodic pointswith both eigenvalues equal to , whose local dynamics is as in figure 2. And hen, there exist connections between separatrices of the periodic points, theseseparatrices being either stable or unstable branches of hyperbolic saddles, or theunstable or the stable separatrix of a point as in figure 2. The existence of accessible (cid:101) h -periodic points can be found in [7]. The in-formation about the eigenvalues is a new result from [16] and the existence ofconnections in the above situation can be found in [18], [13] and also [16]. Theorem B. If α is irrational and O is bounded, then there is no periodicpoint in ∂O. This is a result from [15].
As the rotation set of a homeomorphism of the torus homotopic to the identity isa compact convex subset of the plane, there are three possibilities for its shape:1. it is a point;2. it is a linear segment;3. it has interior;We consider the situation when the rotation set has interior.Whenever a rational vector ( p/q, l/q ) ∈ int ( ρ ( (cid:101) f )) for some (cid:101) f ∈ Dif f (IR ) , (cid:101) f q ( • ) − ( p, l ) has a hyperbolic periodic saddle (cid:101) P ∈ IR such that W u ( (cid:101) P ) , its unstable manifold, has a topologically transverseintersection with W s ( (cid:101) P ) + ( a, b ) , for all integer vectors ( a, b ) . (4)That is, the unstable manifold of (cid:101) P intersects all integer translations of its stablemanifold . This result is proved in [1]. Now it is time to precisely define what atopologically transverse intersection is:
Definition (Top. Trans. Intersections): If f : M → M is a C diffeomor-phism of an orientable boudaryless surface M and p, q ∈ M are f -periodicsaddle points, then we say that W u ( p ) has a topologically transverse in-tersection with W s ( q ) , whenever there exists a point z ∈ W s ( q ) ∩ W u ( p )5 z clearly can be chosen arbitrarily close to q or to p ) and an open topo-logical disk B centered at z, such that B \ α = B ∪ B , where α is theconnected component of W s ( q ) ∩ B which contains z, with the followingproperty: there exists a closed connected piece of W u ( p ) denoted β suchthat β ⊂ B, z ∈ β, and β \ z has two connected components, one containedin B ∪ α and the other contained in B ∪ α, such that β ∩ B (cid:54) = ∅ and β ∩ B (cid:54) = ∅ . Clearly a C transverse intersection is topologically trans-verse. See figure 1 for a sketch of some possibilities. Note that as β ∩ α may contain a connected arc containing z, the disk B may not be chosenarbitrarily small.In order to have a picture in mind, consider z close to q, so that z belongsto a connected arc in W s ( q ) containing q, which is almost a linear seg-ment. Therefore, it is easy to find B as stated above, it could be chosenas an Euclidean open ball. Clearly, this is a symmetric definition: we canconsider a negative iterate of z, for some n < f n ( z ) belongs toa connected piece of W u ( p ) containing p, which is also almost a linear seg-ment. Then, a completely analogous construction can be made, switchingstable manifold with unstable: choose an open Euclidean ball B (cid:48) centeredat f n ( z ) , such that B (cid:48) \ β (cid:48) = B (cid:48) ∪ B (cid:48) , where β (cid:48) is the connected componentof W u ( p ) ∩ B (cid:48) which contains z, with the following property: there exists aclosed connected piece of W s ( q ) denoted α (cid:48) such that α (cid:48) ⊂ B (cid:48) , f n ( z ) ∈ α (cid:48) , and α (cid:48) \ f n ( z ) has two connected components, one contained in B (cid:48) ∪ β (cid:48) and the other contained in B (cid:48) ∪ β (cid:48) , such that α (cid:48) ∩ B (cid:48) (cid:54) = ∅ and α (cid:48) ∩ B (cid:48) (cid:54) = ∅ . So, f − n ( B (cid:48) ) , f − n ( β (cid:48) ) and f − n ( α (cid:48) ) are the corresponding sets at z. The most important consequence of a topologically transverse intersectionfor us is a C λ -lemma: If W u ( p ) has a topologically transverse intersectionwith W s ( q ) , then W u ( p ) C -accumulates on W u ( q ) . As pointed out in [2], the following converse of (4) is true: if (cid:101) g def. = (cid:101) f q ( • ) − ( p, l ) has a hyperbolic periodic saddle (cid:101) P ∈ IR such that W u ( (cid:101) P ) has a topolog-ically transverse intersection with W s ( (cid:101) P ) + ( a i , b i ) , for integer vectors ( a i , b i ) , table manifold stable manifoldunstable manifoldunstable manifoldstable manifold stable manifoldunstable manifoldunstable manifold BB BB . .. . β ββ β z zz z a) b)c) d)
Figure 1: z is a odd order tangencyb) there is a segment in the intersection of the manifolds c) a C -transverse crossingd) z is accumulated on both sides by even order tangencies. i = 1 , , ..., k, such that(0 , ∈ ConvexHull { ( a , b ) , ( a , b ) , ..., ( a k , b k ) } , then (0 , ∈ int ( ρ ( (cid:101) g )) ⇔ ( p/q, l/q ) ∈ int ( ρ ( (cid:101) f )) . This follows from the following:
Lemma 0.
Let g ∈ Dif f (T ) and (cid:101) g : IR → IR be a lift of g whichhas a hyperbolic periodic saddle point (cid:101) P such that W u ( (cid:101) P ) has a topologicallytransverse intersection with W s ( (cid:101) P ) + ( a, b ) , for some integer vector ( a, b ) (cid:54) =(0 , . Then ρ ( (cid:101) g ) contains (0 , and a rational vector parallel to ( a, b ) with thesame orientation as ( a, b ) . In order to prove this lemma, one just have to note that if W u ( (cid:101) P ) has atopologically transverse intersection with W s ( (cid:101) P ) + ( a, b ) , then we can producea topological horseshoe for (cid:101) g (see [2]), for which a certain periodic sequence willcorrespond to points with rotation vector parallel and with the same orientationas ( a, b ) . So, when ( p/q, l/q ) ∈ ∂ρ ( (cid:101) f ) for some (cid:101) f ∈ Dif f (IR ) , it may be the7ase that (cid:101) f q ( • ) − ( p, l ) has a hyperbolic periodic saddle (cid:101) P such that W u ( (cid:101) P )has a topologically transverse intersection with W s ( (cid:101) P ) + ( a, b ) , for some integervectors ( a, b ) , but not for all.Moreover, if r is a supporting line at ( p/q, l/q ) ∈ ∂ρ ( (cid:101) f ) , which means that r is a straight line which contains ( p/q, l/q ) and does not intersect int ( ρ ( (cid:101) f )) , andif −→ v is a vector orthogonal to r, such that −−→ v points towards the rotation set,then W u ( (cid:101) P ) has a topologically transverse intersection with W s ( (cid:101) P ) + ( a, b ) , forsome integer vector ( a, b ) ⇒ ( a, b ) . −→ v ≤ . If ρ ( (cid:101) f ) intersects r only at ( p/q, r/q ) , then ( a, b ) . −→ v ≥ ⇒ ( a, b ) = (0 , . On the family f t and the rational ρ = ( p/q, l/q ) which is in the boundary ofthe rotation set ρ ( (cid:101) f t ) at the critical parameter t = t, without loss of generality,we can assume ρ to be (0 , . Instead of considering f t and its lift (cid:101) f t , we just haveto consider f qt and the lift (cid:101) f qt − ( p, l ) . This is a standard procedure for this typeof problem. We just have to be careful because we will assume some hypothesesfor the family f t , (cid:101) f t and we must see that they also hold for f qt , (cid:101) f qt − ( p, l ) . Letus look at this.
Assume f t ∈ Dif f ∞ (T ) is a generic C ∞ -family of area-preserving diffeomor-phisms ( t ∈ ] t − (cid:15), t + (cid:15) [ for some parameter t and (cid:15) > ρ ( (cid:101) f t ) has interior for all t ∈ ] t − (cid:15), t + (cid:15) [;2. ( p/q, l/q ) ∈ ∂ρ ( (cid:101) f t ) , r is a supporting line for ρ ( (cid:101) f t ) at ( p/q, l/q ) , −→ v is aunitary vector orthogonal to r, such that −−→ v points towards the rotationset;3. ( p/q, l/q ) ∈ int ( ρ ( (cid:101) f t )) for all t ∈ ] t, t + (cid:15) [;4. The genericity in the sense of Meyer implies that if for some parameter t ∈ ] t − (cid:15), t + (cid:15) [ , a f t -periodic point has 1 as an eigenvalue, then it is asaddle-elliptic type of point, one which is going to give birth to a saddleand an elliptic point when the parameter moves in one direction and it8s going to disappear if the parameter moves in the other direction. Asthe family is generic, for each period there are only finitely many periodicpoints. Moreover, as is explained in page 3 of the summary of [10], we canassume, that at all periodic points, in particular, at each n -periodic point(for all integers n >
0) which has 1 , n -periodic points and must have zero topological index), aLojasiewicz condition is satisfied. So, as explained in section 2 of [3], thedynamics near such a point is as in figure 2. . Figure 2:
Dynamics in a neighborhood of the degenerate periodic points.
5. Saddle-connections are a phenomena of infinity codimension (see [11]).Therefore, as we are considering C ∞ -generic 1-parameter families, we canalso assume that for all t ∈ ] t − (cid:15), t + (cid:15) [ , f t does not have connectionsbetween invariant branches of periodic points, which can be either hyper-bolic saddles or degenerate as in figure 2. This is not strictly containedin the literature on this subject, but a proof assuming this more generalsituation can be obtained exactly in the same way as is done when only9yperbolic saddles are considered.6. Moreover, as explained in section 6 of chapter II of [21], a much strongerstatement holds: for C ∞ f t , if a point z ∈ T belongs to the intersection of a stable and an unstable manifold of some f t -periodic hyperbolic saddles and the intersection is not C -transverse,then it is a quadratic tangency, that is, it is not topologically transverse.This implies that every time an unstable manifold has a topologicallytransverse intersection with a stable manifold of some hyperbolic periodicpoints, this intersection is actually C -transverse. And when a tangencyappears, it unfolds generically with the parameter (with positive speed,see remark 6.2 of [21]). This means that if for some hyperbolic periodicsaddles q t and p t such that W u ( q t (cid:48) ) and W s ( p t (cid:48) ) have a quadratic tangencyat a point z t (cid:48) ∈ T for some parameter t (cid:48) , then for t close to t (cid:48) , in suitablecoordinates near z t (cid:48) = (0 , , we can write W u ( q t ) = ( x, f ( x ) + ( t − t (cid:48) ))and W s ( p t ) = ( x, , where f is a C ∞ function defined in a neighborhoodof 0 such that f (0) = 0 , f (cid:48) (0) = 0 and f (cid:48)(cid:48) (0) > . This implies thatif the parameter varies in a neighborhood of the tangency parameter, toone side a C -transverse intersection is created and to the other side, theintersection disappears.This is stated for families of general diffeomorphisms in [21], but the sameresult holds for families of area-preserving diffeomorphisms. It is not hardto see that in order to avoid degenerate tangencies (of order ≥ f t , (cid:101) f t satisfy the above hypotheses, then f qt , (cid:101) f qt − ( p, l ) clearly satisfy thesame set of hypotheses with respect to ρ = (0 , . Note that the supportingline at (0 ,
0) for ρ ( (cid:101) f qt − ( p, l )) is parallel to r, the supporting line for ρ ( (cid:101) f t ) at( p/q, l/q ) . So there is no restriction in assuming ( p/q, l/q ) to be (0 , . Under the 6 hypotheses assumed in 1.3.3 for the family f t with ( p/q, l/q ) = (0 , , the following holds: (cid:101) f t has a hyperbolic fixed saddle (cid:101) P t such hat W u ( (cid:101) P t ) has a topologically transverse (and therefore a C -transverse) in-tersection with W s ( (cid:101) P t ) + ( a, b ) for some integer vector ( a, b ) (cid:54) = (0 , . And thereexists K ( f ) > such that for any ( c, d ) ∈ ZZ for which ( c, d ) . −→ v > K ( f ) , if (cid:101) P t is the continuation of (cid:101) P t for t > t, then there exists a sequence t i > t, t i i →∞ → t such that W u ( (cid:101) P t i ) has a quadratic tangency with W s ( (cid:101) P t i ) + ( c (cid:48) , d (cid:48) ) , for some ( c (cid:48) , d (cid:48) ) ∈ ZZ which satisfies | ( c (cid:48) − c, d (cid:48) − d ) . −→ v | ≤ K ( f ) / , and the tangencygenerically unfolds for t > t i . The vector ( c (cid:48) , d (cid:48) ) is within a bounded distancefrom ( c, d ) in the direction of −→ v , but may be far in the direction of −→ v ⊥ . Thesetangencies, are heteroclinic intersections for (cid:101) f t i which could not exist at t = t. Finally, we point out that for all t > t and for all integer vectors ( a, b ) , W u ( (cid:101) P t ) has a C transverse intersection with W s ( (cid:101) P t ) + ( a, b ) . Remarks: • As we said, the tangencies given in the previous theorem are precisely forsome integer vectors ( c (cid:48) , d (cid:48) ) for which at t = t, they could not exist. Thiswill become clear in the proof. • We were not able to produce tangencies at t = t, even when ( c (cid:48) , d (cid:48) ) . −→ v > p and W u ( p ) intersects W s ( p ) , then there is a topologicallytransverse intersection between W u ( p ) and W s ( p ) . So, for a generic familyof such maps, a horseshoe is not preceded by a tangency: the existence ofa tangency already implies a horseshoe. This is clearly not true out of thethe area preserving world and shows how subtle is the problem of birth ofa horseshoe in the conservative case. • Intuitively, as the rotation set becomes larger, one would expect the topo-logical entropy to grow, at least for a tight model. In fact, in Kwapisz [17],some lower bounds for the topological entropy related to the 2-dimensionalsize of the rotation set are presented (it is conjectured there that the area11f the rotation set could be used, but what is actually used is a moretechnical computation on the size of the rotation set). Our main theoremsays that every time the rotation set locally grows near a rational point,then nearby maps must have tangencies, which generically unfold as theparameter changes. And this is a phenomenon which is associated to thegrowth of topological entropy, see [5] and [23]. More precisely, in the twoprevious papers it is proved that generically, if a surface diffeomorphism f has arbitrarily close neighbors with larger topological entropy, then f has a periodic saddle point with a homoclinic tangency. Both these resultswere not stated in the area preserving case, in fact they may not be true inthe area preserving case, but they indicate that whenever the topologicalentropy is ready to grow, it is expected to find tangencies nearby. • The unfolding of the above tangencies create generic elliptic periodicpoints, see [9]. • An analytic version of the above theorem can also be proved. We have toassume that:1. the family has no connections between separatrices of periodic points;2. for each period, there are only finitely many periodic points;3. if a periodic point has negative topological index, then it is a hyper-bolic saddle;These conditions are generic among C ∞ -1-parameter families, but for an-alytic families I do not know. The tangencies obtained in this case havefinite order, but are not necessarily quadratic. To prove such a result, firstremember that all isolated periodic points for analytic area-preserving dif-feomorphisms satisfy a Lojasiewicz condition, see section 2 of [3]. Andmoreover, if an isolated periodic point has a characteristic curve (see [10]and again [3], section 2), then from the preservation of area, the dynamicsin a neighborhood of such a point is obtained, at least in a topologicalsense, by gluing a finite number of saddle sectors.12nother important ingredient is the main result of [16] quoted here astheorem A, which among other things, says that for an area-preservingdiffeomorphism f of the plane, which for every n > n -periodic points, if it has an invariant topological open disk U with compactboundary, whose prime ends rotation number is rational, then ∂U containsperiodic points, all of the same period k > Df k )at these periodic points contained in ∂U are real and positive. So, if sucha map f is analytic, the k -periodic points in ∂U satisfy a Lojasiewiczcondition. If some of these periodic points, for instance denoted P, hastopological index 1 , then the eigenvalues of ( Df k | P ) are both equal to1 . Under these conditions, the main result of [24] implies the existenceof periodic orbits rotating around P with many different velocities withrespect to an isotopy I t from the Id to f. And a technical result in [16] saysthat if a periodic point belongs to ∂U, then this can not happen. So, thetopological indexes of all periodic points in ∂U are less or equal to zero andthus from section 2 of [3], they all have characteristic curves. Therefore,locally, all periodic points in ∂U are saddle like. They may have 2 sectors(index zero) or 4 sectors (index -1). And from [18], if ∂U is bounded,connections must exist. Thus, the hypothesis that there are no connectionsbetween separatrices of periodic points imply the irrationality of the primeends rotation number for all open invariant disks whose boundaries arecompact.And finally, the last result we need is due to Churchill and Rod [8]. Itsays that, for analytic area preserving diffeomorphisms, the existence ofa topologically transverse homoclinic point for a certain saddle, impliesthe existence of a C -transverse homoclinic point for that saddle. Usingthese results in the appropriate places of the proof in the next section, ananalytic version of the main theorem can be obtained.In the next section of this paper we prove our main result.13 Proof of the main theorem
The proof will be divided in 2 steps.
Here we prove that (cid:101) f t has a hyperbolic fixed saddle such that its unstablemanifold has a topologically transverse intersection (therefore, C -transverse)with its stable manifold translated by a non-zero integer vector ( a, b ) . Clearly,from lemma 0, ( a, b ) . −→ v ≤ . First of all, note that as (0 , ∈ ∂ρ ( (cid:101) f t ) and for all t > t, (0 , ∈ int ( ρ ( (cid:101) f t )) , (cid:101) f t has (finitely many) fixed points up to ZZ translations, it can not be fixed pointfree. The finiteness comes from the generic assumptions. If all these fixed pointshad zero topological index, as is explained before the statement of theorem 1,the dynamics near each of them would be as in figure 2. And in this situation,theorem 1 of [3] implies that (0 , / ∈ int ( ρ ( (cid:101) f t )) for any t close to t. So there must be (cid:101) f t -fixed points with non-zero topological index. From theNielsen-Lefschetz index theorem, we obtain a fixed point for (cid:101) f t with negativeindex. From the genericity of our family, the only negative index allowed is − − k > , , denoted { (cid:101) P t , ..., (cid:101) P kt } . So, in [0 , (cid:101) f t has k hyperbolic fixed saddle points and otherperiodic points with topological index greater or equal to zero.Now let us choose a rational vector in int ( ρ ( (cid:101) f t )) . Without loss of generality,conjugating f with some adequate integer matrix if necessary, we can supposethat this rational vector is of the form (0 , − /n ) for some n > . By some results from [1], let (cid:101) Q ∈ IR be a periodic hyperbolic saddle pointfor (cid:16) (cid:101) f t (cid:17) n + (0 ,
1) such that W u ( (cid:101) Q ) has a topologically transverse intersec-tion with W s ( (cid:101) Q ) + ( a, b ) for all integer vectors ( a, b ) . In this case, W u ( (cid:101) Q ) = W s ( (cid:101) Q ) = R.I. ( (cid:101) f t ) = Region of instability of (cid:101) f t , a (cid:101) f t -invariant equivariantset such that, if (cid:101) D is a connected component of its complement, then (cid:101) D is aconnected component of the lift of a f t -periodic open disk in the torus and for14very f t -periodic open disk D ⊂ T , p − ( D ) ⊂ (cid:16) R.I. ( (cid:101) f t ) (cid:17) c . Also, for any ra-tional vector ( p/q, l/q ) ∈ int ( ρ ( (cid:101) f t )) , there exists a hyperbolic periodic saddlepoint for (cid:16) (cid:101) f t (cid:17) q − ( p, l ) , such that its unstable manifold also has topologicallytransverse intersections with all integer translates of its stable manifold and so,the closure of its stable manifold is equal the closure of its unstable manifoldand they are both equal R.I. ( (cid:101) f t ) . As we said, these results were proved in [1]and similar statements hold for homeomorphisms [14].If for some 1 ≤ i ∗ ≤ k, W u ( (cid:101) P i ∗ t ) and W s ( (cid:101) P i ∗ t ) are both unbounded subsetsof the plane, then it can be proved that W u ( (cid:101) P i ∗ t ) must have a topologicallytransverse intersection with W s ( (cid:101) Q ) and W s ( (cid:101) P i ∗ t ) must have a topologicallytransverse intersection with W u ( (cid:101) Q ) . But, as the rotation vector of (cid:101) Q is notzero, this gives what we want in step 1. More precisely, the following fact holds: Fact : If W u ( (cid:101) P i ∗ t ) and W s ( (cid:101) P i ∗ t ) are both unbounded subsets of the plane, then W u ( (cid:101) P i ∗ t ) has a topologically transverse intersection with W s ( (cid:101) P i ∗ t ) − (0 , . Proof: As W s ( (cid:101) Q ) has a topologically transverse intersection with W u ( (cid:101) Q ) + ( a, b )for all integer vectors ( a, b ) , this implies that if W u ( (cid:101) P i ∗ t ) is unbounded, then W u ( (cid:101) P i ∗ t ) has a topologically transverse intersection with W s ( (cid:101) Q ) . This followsfrom the following idea: There is a compact arc λ u in W u ( (cid:101) Q ) that contains (cid:101) Q and a compact arc λ s in W s ( (cid:101) Q ) that also contains (cid:101) Q, such that λ u hastopologically transverse intersections with λ s +(0 ,
1) and λ s +(1 , . This impliesthat the connected components of the complement of ∪ ( a,b ) ∈ ZZ λ u ∪ λ s + ( a, b )are all open topological disks, with diameter uniformly bounded from above. So,if W u ( (cid:101) P i ∗ t ) is unbounded, it must have a topologically transverse intersectionwith some translate of λ s . As W s ( (cid:101) Q ) C -accumulates on all its integer translates,we finally get that W u ( (cid:101) P i ∗ t ) has a topologically transverse intersection with W s ( (cid:101) Q ) . m > , (cid:16) (cid:101) f t (cid:17) m.n ( (cid:101) Q ) = (cid:101) Q − (0 , m ) ,W u ( (cid:101) P i ∗ t ) C -accumulates on compact pieces of W u ( (cid:101) Q ) − (0 , k j ) for a certainsequence k j → ∞ , that is, given a compact arc (cid:101) θ contained in W u ( (cid:101) Q ) , thereexists a sequence k j → ∞ such that for some arcs (cid:101) θ j ⊂ W u ( (cid:101) P i ∗ t ) , (cid:101) θ j +(0 , k j ) → (cid:101) θ in the Hausdorff topology as j → ∞ . An analogous argument implies that if W s ( (cid:101) P i ∗ t ) is unbounded, then W s ( (cid:101) P i ∗ t ) has a topologically transverse intersectionwith W u ( (cid:101) Q ) . So if we choose a compact arc κ u contained in W u ( (cid:101) Q ) whichhas a topologically transverse intersection with W s ( (cid:101) P i ∗ t ) , we get that W u ( (cid:101) P i ∗ t )accumulates on κ u − (0 , k j ) and thus it has a topologically transverse intersectionwith W s ( (cid:101) P i ∗ t ) − (0 , k j ) for some k j > . As we pointed out afterthe definition of topologically transverse intersections, all the above follows froma C -version of the λ -lemma that holds for topologically transverse intersections.Now consider a compact subarc of a branch of W u ( (cid:101) P i ∗ t ) , denoted α u , startingat (cid:101) P i ∗ t and a compact subarc of a branch of W s ( (cid:101) P i ∗ t ) , denoted α s , starting at (cid:101) P i ∗ t such that α u has a topologically transverse intersection with α s − (0 , k j ) for some k j > . (5)Using Brouwer’s lemma on translation arcs exactly as in lemma 24 of [1], weget that either α u has an intersection with α s − (0 ,
1) or α u − (0 ,
1) has anintersection with α s . If α u had only non topologically transverse intersectionswith α s − (0 ,
1) and α u − (0 ,
1) had only non topologically transverse intersectionswith α s , then we could C -perturb α u and α s in an arbitrarily small way suchthat ( α uper ∪ α sper ) ∩ (( α uper ∪ α sper ) − (0 , ∅ and α uper ∩ ( α sper − (0 , k j )) (cid:54) = ∅ (because of the topologically transverse assumption (5)). But this contradictsBrouwer ’s lemma [6]. So, either α u has a topologically transverse intersectionwith α s − (0 ,
1) or α u − (0 ,
1) has a topologically transverse intersection with α s . The first possibility is what we want and the second is not possible becauselemma 0 would imply that ρ ( (cid:101) f t ) contains a point of the form (0 , a ) for some a > . As (0 , − /n ) ∈ int ( ρ ( (cid:101) f t )) these two facts contradict the assumption that(0 , ∈ ∂ρ ( (cid:101) f t ) . (cid:50)
16n order to conclude this step, we need the following lemma:
Lemma 2 . There exists ≤ i ∗ ≤ k such that for any choice of λ i ∗ u and λ i ∗ s , one unstable and one stable branch at (cid:101) P i ∗ t , they are both unbounded.Proof: For each 1 ≤ i ≤ k, as (cid:101) P it has topological index equal − , the two stable andthe two unstable branches are each, (cid:101) f t -invariant. Fixed some unstable branch λ iu and some stable λ is , let K iu = λ iu and K is = λ is . Both are connected (cid:101) f t -invariant sets. Let K i be equal to either K iu or K is andassume it is bounded. Without loss of generality, suppose that K i = K iu . First,we collect some properties about K i . • If K i intersects a connected component (cid:101) D of the complement of R.I. ( (cid:101) f t ) , then from lemma 6.1 of [13], λ iu \ (cid:101) P it is contained in (cid:101) D, which then, isa (cid:101) f t -invariant bounded open disk (see [1] and [14]). As the family ofdiffeomorphisms considered is generic (in particular, it does not have con-nections between stable and unstable separatrices of periodic points), therotation number of the prime ends compactification of (cid:101) D, denoted β, mustbe irrational by theorem A. So, if (cid:101) P it ∈ ∂ (cid:101) D, as λ iu \ (cid:101) P it ⊂ (cid:101) D, this wouldbe a contradiction with the irrationality of β, because (cid:101) f t ( λ iu ) = λ iu . Thus, (cid:101) P it is contained in (cid:101) D and the topological index of (cid:101) f t with respect to (cid:101) D is+1 (because β is irrational, therefore not zero). By topological index of (cid:101) f t with respect to (cid:101) D we mean the sum of the indexes at all the (cid:101) f t -fixedpoints contained in (cid:101) D. This information will be used in the end of theproof. • Suppose now that K i = K iu is contained in R.I. ( (cid:101) f t ) . It is not possiblethat K iu ∩ λ is = (cid:101) P it because it would imply that the connected compo-nent M of ( K i ) c which contains λ is \ (cid:101) P it has rational prime ends rotationnumber and as we already said, this does not happen under our generic17onditions. So, from lemma 2 of Oliveira [22], we get that either K iu ⊃ λ is or λ iu intersects λ is . If λ iu intersects λ is , then λ is intersects R.I. ( (cid:101) f t ) , so itis contained in R.I. ( (cid:101) f t ) and we can find a Jordan curve τ contained in λ iu ∪ λ is , (cid:101) P it ∈ τ. Theorem B implies that there is no periodic point in theboundary of a connected component of the complement of
R.I. ( (cid:101) f t ) , be-cause such a component is f t -periodic when projected to the torus and ithas irrational prime ends rotation number by theorem A. So, interior( τ ) in-tersects R.I. ( (cid:101) f t ) , and thus interior( τ ) intersects both W u ( (cid:101) Q ) and W s ( (cid:101) Q ) , something that contradicts the assumption that K iu is bounded because λ iu must intersect W s ( (cid:101) Q ) , which implies that it is unbounded.And if K iu ⊃ λ is , then from our assumption that K iu is bounded, we get that K is = λ is is also bounded. Arguing as above, we obtain that K is ∩ λ iu (cid:54) = (cid:101) P it . Otherwise, if M ∗ is the connected component of ( K is ) c that contains λ iu \ (cid:101) P it , as (cid:101) f t ( λ iu ) = λ iu , the rotation number of the prime ends compact-ification of M ∗ would be rational. As this implies connections betweenseparatrices of periodic points, which do not exist under our hypothe-ses, K is must intersect λ iu \ (cid:101) P it . And again, from lemma 2 of Oliveira [22], K is ⊃ λ iu or λ iu intersects λ is and we are done.Thus, the almost final situation we have to deal is when K iu ⊃ λ is and K is ⊃ λ iu . But it is contained in the proof of the main theorem of [22] thatthese relations imply that λ iu intersects λ is . And as we explained above, thisis a contradiction with the assumption that K iu is bounded. If K i = K is and K is ⊂ R.I. ( (cid:101) f t ) , an analogous argument could be applied in order toarrive at similar contradictions.Thus, if K i is bounded, it must be contained in the complement of R.I. ( (cid:101) f t ) . In order to conclude the proof, we are left to consider the case when for every1 ≤ i ≤ k, we can choose K i either equal to K iu or K is , such that it is boundedand contained in a connected component (cid:101) D i of the complement of R.I. ( (cid:101) f t ) . As we already obtained, the topological index of (cid:101) f t restricted to (cid:101) D i is +1 . This clearly contradicts the Nielsen-Lefschetz index formula because the sum of18he indices of f t at its fixed points which have (0 ,
0) rotation vector would bepositive. So, for some 1 ≤ i ∗ ≤ k, both unstable and both stable branches at (cid:101) P i ∗ t are unbounded. (cid:50) This concludes step 1.
From the previous step we know that there exists a (cid:101) f t -fixed point denoted (cid:101) P t such that W u ( (cid:101) P t ) has a topologically transverse, and therefore a C -transverseintersection with W s ( (cid:101) P t ) − (0 , . So, there exists a compact connected piece of a branch of W u ( (cid:101) P t ) , starting at (cid:101) P t , denoted λ tu and a compact connected piece of a branch of W s ( (cid:101) P t ) , startingat (cid:101) P t , denoted λ ts , such that λ tu ∪ (cid:16) λ ts − (0 , (cid:17) contains a continuous curveconnecting (cid:101) P t to (cid:101) P t − (0 , . The end point of λ tu , denoted w, belongs to λ ts − (0 , C -transverse heteroclinic point. The main consequence of the aboveis the following: Fact : The curve (cid:101) γ tV connecting (cid:101) P t to (cid:101) P t − (0 ,
1) contained in λ tu ∪ (cid:16) λ ts − (0 , (cid:17) projects to a (not necessarily simple) closed curve in the torus, homotopicto (0 , −
1) and it has a continuous continuation for t ≥ t suff. small. Thatis, for t − t ≥ (cid:101) γ tV connecting (cid:101) P t to (cid:101) P t − (0 ,
1) made by a piece of an unstable branch of W u ( (cid:101) P t ) and a pieceof a stable branch of W s ( (cid:101) P t ) − (0 ,
1) such that t → (cid:101) γ tV is continuous for t − t ≥ Proof:
Immediate from the fact that w is a C -transversal heteroclinic point whichhas a continuous continuation for all diffeomorphisms C -close to (cid:101) f t . (cid:50) As (0 , − /n ) is contained in int ( ρ ( (cid:101) f t )) , there are rational points in int ( ρ ( (cid:101) f t ))with positive horizontal coordinates. Thus, if we defineΓ tV,a def. = ... ∪ (cid:101) γ tV + ( a, ∪ (cid:101) γ tV + ( a, ∪ (cid:101) γ tV + ( a, ∪ (cid:101) γ tV + ( a, − ∪ ..., for a ∈ ZZ ,
19e get that for all sufficiently large integer n > , ( (cid:101) f t ) n (Γ tV, ) intersects Γ tV, transversely and moreover, we obtain a curve (cid:101) γ tH connecting (cid:101) P t to (cid:101) P t + (1 ,
0) ofthe following form: it starts at (cid:101) P t , goes through the branch of W u ( (cid:101) P t ) whichcontains λ tu until it hits in a topologically transverse way (so in a C -transverseway) λ ts + (1 , b ) for some integer b. If b < , we add to this curve the followingone: (cid:101) γ tV + (1 , ∪ (cid:101) γ tV + (1 , − ∪ ... ∪ (cid:101) γ tV + (1 , b + 1) (6)and if b ≥ , we add the following curve: (cid:101) γ tV + (1 , ∪ (cid:101) γ tV + (1 , ∪ ... ∪ (cid:101) γ tV + (1 , b ) ∪ λ ts + (1 , b ) (7) .... . ... t P ~ t P + (1,0) ~ t P ~ t P + (1,0) ~ b)a) Figure 3:
How to construct (cid:101) γ tH . In both cases above, we omit a small piece of λ ts + (1 , b ) in order to geta proper curve connecting (cid:101) P t and (cid:101) P t + (1 , . This follows from the fact thatwhen we consider iterates ( (cid:101) f t ) n (Γ tV, ) for some large n > , the arcs containedin stable manifolds are shrinking and arcs contained in unstable manifolds aregetting bigger. As there are orbits moving to the right under positive iterates20f (cid:101) f t , the above holds. So, (cid:101) γ tH is a continuous curve whose endpoints are (cid:101) P t and (cid:101) P t + (1 ,
0) and it is made of a connected piece of an unstable branch of W u ( (cid:101) P t ) , added to either one of the two vertical curves above, (6) or (7), a smallpiece of λ ts + (1 , b ) deleted. As the intersection between the branch of W u ( (cid:101) P t )which contains λ tu with λ ts + (1 , b ) is C -transverse and t → (cid:101) γ tV is continuousfor t − t ≥ t → (cid:101) γ tH is also continuous for t − t ≥ r is the supporting line at (0 , ∈ ∂ρ ( (cid:101) f t ) and −→ v is anunitary vector orthogonal to r such that −−→ v points towards the rotation set.Now we state a more general version of lemma 6 of [2]. It does not appear likethis in that paper, but the proof presented there also proves this more abstractversion. Lemma 6 of [2].
Let K H and K V be two continua in the plane, such that K H contains (0 , and (1 , and K V contains (0 , and (0 , . For every vector −→ w , it is possible to construct a connected closed set M −→ w which is equal to theunion of well chosen integer translates of K H and K V such that:1. M −→ w intersects every straight line orthogonal to −→ w ; M −→ w is bounded in the direction orthogonal to −→ w , that is, M −→ w is containedbetween two straight lines r M −→ w and s M −→ w , both parallel to −→ w , and the dis-tance between these lines is less then . max { diameter ( K H ) , diameter ( K V ) } . So, in particular ( M −→ w ) c has at least two unbounded connected components,one containing r M −→ w , denoted U r ( M −→ w ) and the other containing s M −→ w , de-noted U s ( M −→ w );So, applying this lemma to the setting of this paper, it gives us a pathconnected closed set θ t −→ v ⊥ ⊂ IR which is obtained as the union of certain integertranslates of (cid:101) γ tV or (cid:101) γ tH in a way that:1. θ t −→ v ⊥ intersects every straight line parallel to −→ v ;2. θ t −→ v ⊥ is bounded in the direction of −→ v , that is, θ t −→ v ⊥ is contained between twostraight lines l − and l + , both parallel to −→ v ⊥ , and the distance between these21ines is less then 3+2 . max { diameter ( (cid:101) γ tV ) , diameter ( (cid:101) γ tH ) } . And ( θ t −→ v ⊥ ) c hasat least two unbounded connected components, one containing l − , denoted U and the other containing l + , denoted U + ;Assume that l + and U + were chosen in a way that if ( c, d ) is an integervector such that θ t −→ v ⊥ + ( c, d ) belongs to U + , then( c, d ) . −→ v > . (8)It is not hard to see that for integer vectors ( c, d ) for which θ t −→ v ⊥ + ( c, d ) belongseither to U + or U − , an inequality like (8) needs to hold.For this, note that if −→ v is a rational direction for which ( c, d ) . −→ v = 0 , thenfrom the way θ t −→ v ⊥ is constructed, we get that θ t −→ v ⊥ + ( c, d ) intersects θ t −→ v ⊥ , so θ t −→ v ⊥ + ( c, d ) does not belong to ( θ t −→ v ⊥ ) c . And in case −→ v is an irrational direction,( c, d ) . −→ v (cid:54) = 0 . In particular, an integer vector ( c, d ) satisfying the inequality in(8) has the property that any positive multiple of it does not belong to ρ ( (cid:101) f t ) . This will be important soon.Moreover, if ( c, d ) . −→ v > . max { diameter ( (cid:101) γ tV ) , diameter ( (cid:101) γ tH ) } , then θ t −→ v ⊥ +( c, d ) belongs to U + . Now we are ready to finish the proof of the main theorem.As both t → (cid:101) γ tV and t → (cid:101) γ tH are continuous for t − t ≥ t → θ t −→ v ⊥ . Given any integer vector ( c, d ) such that( c, d ) . −→ v > K ( f ) def. = 4 . (3 + 2 . max { diameter ( (cid:101) γ tV ) , diameter ( (cid:101) γ tH ) } ) + 10 , if a t ∗ > t, sufficiently close to t is fixed, we can assume that θ t −→ v ⊥ + ( c, d ) ∩ θ t −→ v ⊥ = ∅ for all t ∈ [ t, t ∗ ] . And as (0 , ∈ int ( ρ ( (cid:101) f t ∗ )) , there exists an integer N > (cid:101) f t ∗ ) N ( θ t ∗ −→ v ⊥ ) has a topologically transverse intersection with θ t ∗ −→ v ⊥ + ( c, d ) . But as (0 , / ∈ int ( ρ ( (cid:101) f t )) , and ( c, d ) . −→ v is sufficiently large, we get that( (cid:101) f t ) N ( θ t −→ v ⊥ ) ∩ θ t −→ v ⊥ + ( c, d ) = ∅ . a, b ) ∈ ZZ such that θ t −→ v ⊥ + ( a, b ) is contained between θ t −→ v ⊥ and θ t −→ v ⊥ + ( c, d )and( a, b ) . −→ v > . (3 + 2 . max { diameter ( (cid:101) γ tV ) , diameter ( (cid:101) γ tH ) } ) + 5 . If we prove that ( (cid:101) f t ) N ( θ t −→ v ⊥ ) cannot have a topologically transverse intersectionwith θ t −→ v ⊥ + ( a, b ) , it clearly cannot intersect θ t −→ v ⊥ + ( c, d ) . So, if there were such atopologically transverse intersection, as θ t −→ v ⊥ + ( a, b ) is disjoint from θ t −→ v ⊥ and arcsof stable manifolds shrink under positive iterates of (cid:101) f t , there would be some( a (cid:48) , b (cid:48) ) , ( a (cid:48)(cid:48) , b (cid:48)(cid:48) ) ∈ ZZ , such that (cid:101) P t + ( a (cid:48) , b (cid:48) ) belongs to θ t −→ v ⊥ and its unstablemanifold has a transverse intersection with the stable manifold of (cid:101) P t + ( a (cid:48)(cid:48) , b (cid:48)(cid:48) )which belongs to θ t (cid:48) −→ v ⊥ + ( a, b ) . As θ t −→ v ⊥ is bounded in the direction of −→ v by3+2 . max { diameter ( (cid:101) γ tV ) , diameter ( (cid:101) γ tH ) } , we get that ( a (cid:48)(cid:48) − a (cid:48) , b (cid:48)(cid:48) − b (cid:48) ) . −→ v > , solemma 0 implies the existence of a rotation vector outside ρ ( (cid:101) f t ) , a contradiction.Thus, from the continuity of t → θ t −→ v ⊥ and t → (cid:101) f t , there exists t (cid:48) ∈ ] t, t ∗ [ suchthat ( (cid:101) f t (cid:48) ) N ( θ t (cid:48) −→ v ⊥ ) has a non-topologically transverse intersection with θ t (cid:48) −→ v ⊥ +( c, d )and for all t ∈ ] t (cid:48) , t ∗ ] , the intersection is topologically transverse. As above, fromthe fact that stable manifolds shrink under positive iterates of (cid:101) f t (cid:48) , the intersec-tion that happens for t = t (cid:48) corresponds to a tangency between the unstablemanifold of some translate of (cid:101) P t (cid:48) which belongs to θ t (cid:48) −→ v ⊥ with the stable manifoldof some translate of (cid:101) P t (cid:48) which belongs to θ t (cid:48) −→ v ⊥ + ( c, d ) . In other words, there is atangency between W u ( (cid:101) P t (cid:48) ) and W s ( (cid:101) P t (cid:48) )+( c ∗ , d ∗ ) for some integer vector ( c ∗ , d ∗ )such that | ( c ∗ − c, d ∗ − d ) . −→ v | ≤ . max { diameter ( (cid:101) γ tV ) , diameter ( (cid:101) γ tH ) } . Thisestimate follows from the fact that if (cid:101) P t +( e, f ) belongs to θ t −→ v ⊥ , then | ( e, f ) . −→ v | ≤ . max { diameter ( (cid:101) γ tV ) , diameter ( (cid:101) γ tH ) } . Here we are using that (cid:101) P t ∈ θ t −→ v ⊥ . As t ∗ > t is arbitrary, if we remember that for a generic family as we areconsidering, topologically transverse intersections are C -transverse and tan-gencies are always quadratic, the proof of the main theorem is almost complete.We are left to deal with the last part of the statement, which says that for allparameters t > t,W u ( (cid:101) P t ) has transverse intersections with W s ( (cid:101) P t ) + ( a, b ) , ∀ ( a, b ) ∈ ZZ . As for t > t, (0 , ∈ int ( ρ ( (cid:101) f t )) , the main result of [1] implies that (for each23 > t ) (cid:101) f t has a hyperbolic periodic saddle (not necessarily fixed) (cid:101) Z t ∈ IR suchthat W u ( (cid:101) Z t ) has transverse intersections with W s ( (cid:101) Z t ) + ( a, b ) , ∀ ( a, b ) ∈ ZZ . So,as W u ( (cid:101) P t ) and W s ( (cid:101) P t ) are both unbounded, exactly as we did in the proof of thefact from step 1 of this proof, W u ( (cid:101) P t ) has transverse intersections with W s ( (cid:101) Z t )and W s ( (cid:101) P t ) has transverse intersections with W u ( (cid:101) Z t ) . Thus an application ofthe λ -lemma concludes the proof. Acknowledgments:
When finishing the writing of this paper, I noticed that if I did not assume C -transversal intersections between stable and unstable manifolds of hyper-bolic periodic saddles, but only topologically transversal intersections in certainparts of the arguments, the proofs would become much more complicated. So,I started looking for a result saying that for generic 1-parameter families of sur-face diffeomorphisms, homoclinic and heteroclinic points are either quadratictangencies or C -transverse. At first, as I could not find it, I got quite anxiousand asked several people who have worked on the subject about this. Here Ithank them for their answers, which were positive, as this kind of result wasknown to be true since [21]. My thanks go to Carlos Gustavo Moreira, EduardoColli, Enrique Pujals, Marcelo Viana and Pedro Duarte.I also would like to thank Andres Koropecki for conversations on the resultsof [22] and for explanations on his results in [15] and [16]. References [1] Addas-Zanata S. (2015): Area preserving diffeomorphisms of the toruswhose rotation sets have non empty interiors.
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