C 0 -Stability of Topological Entropy for Contactomorphisms
aa r X i v : . [ m a t h . S G ] J a n C -STABILITY OF TOPOLOGICAL ENTROPY FORCONTACTOMORPHISMS LUCAS DAHINDEN
Abstract.
Topological entropy is not lower semi-continous: small perturba-tion of the dynamical system can lead to a collapse of entropy. In this notewe show that for some special classes of dynamical systems (geodesic flows,Reeb flows, positive contactomorphisms) topological entropy at least is stablein the sense that there exists a nontrivial continuous lower bound, given thata certain homological invariant grows exponentially.
Contents
1. Introduction and results 12. Examples for total collapse of topological entropy 43. Proof of Theorem 1 54. Proof of Theorem 2 6References 111.
Introduction and results
Topological entropy.
Topological entropy h top p ϕ q is a good numerical measure forthe complexity of a self-map ϕ : M Ñ M of a compact metrizeable space M . Weuse the following definition of Bowen [4] and Dinaburg [6]: Fix a metric d generatingthe topology of M . A p K, δ q -separated set is a subset N Ď M such that for all n ‰ n P N there is a k P r , K s such that d p ϕ k p n q , ϕ k p n qq ě δ . Topological entropyis defined as the growth rate of maximal cardinality of p T, δ q -separated sets: h top p ϕ q “ sup δ ą Γ p maximal cardinality of a p T, δ q -separated set q , where for a sequence of non-negative numbers a k , Γ p a k q : “ lim sup k log a k is theexponential growth rate.Unfortunately, topological entropy is very hard to compute explicitly, see [15,20, 12]. One reason for this is the lack of lower semi-continuity of the function ϕ ÞÑ h top p ϕ q that associates to a map its entropy. The examples in Section 2 showthat even for smooth diffeomorphisms of 3-dimensional closed manifolds, h top cancollapse to 0 under perturbation, i.e. can jump up from 0 when passing to a limit.Only in very special situations the topological entropy is continuous. For example,in the class of diffeomorphisms of closed 2-dimensional surfaces, topological entropy Date : January 17, 2020.This work is supported by SFB/TRR 191 ‘Symplectic Structures in Geometry, Algebra andDynamics’ funded by the DFG. is continuous due to the existence and stability of hyperbolic sets, see e.g. [14, S5.6].For further stability results see [16] or [9] and the references therein.
Remark . On the class of smooth maps, h top is uppersemi-continuous in the C topology, see [21]. This fails on the class of C r maps. Stability of topological entropy.
Since there is no hope for lower semi-continuityof topological entropy, we aim at the next best thing: we look for an interestingclass of spaces and maps for which the topological entropy is “stable”, i.e. thereis a nontrivial continuous lower bound for h top . Given stability, it is still hard toexplicitly compute entropy, but one can estimate it from below, since total collapseof entropy under perturbation is prevented.To formalize this, let P be a parameter space which continuously parametrizesa subset of the space of smooth self-maps P Ñ D Ď C p M, M q equipped with the C -topology. Definition 1.2.
Topological entropy is stable on P if there exists a not identicallyvanishing continuous function γ : P Ñ R ě such that γ p p q ď h top p ϕ p q . Note that since topological entropy is upper semi-continuous in C p M, M q forsmooth maps, points where γ p p q “ h top p ϕ p q are points of continuity of topologicalentropy on D . Remark . Given stability of topological entropy, one still can collapse topologicalentropy by large perturbations. See e.g. [1], where it is proven that every contactstructure on a closed manifold admits a sequence of contact forms with fixed volumewhose Reeb flows have topological entropy converging to 0. This is complementaryto the present article, where we show stability under small perturbations.
A criterion for stability.
We work in the setting of maps ϕ which are time-1 mapsof twisted periodic flows, a class of flows containing autonomous flows: let ϕ t be asmooth 1-parameter family of maps such that ϕ “ id and ϕ t “ X t , where X t is a1-periodic vector field. Then, iterations p ϕ q k of ϕ : “ ϕ coincide with ϕ t at integertimes t “ k . We formulate a criterion that implies stability of topological entropyof ϕ . It is used in all the examples given at the end of the introduction.Let a class of twisted periodic flows ϕ t of a closed manifold M be parametrizedby P . Our main assumption is the abundance of chords from a fixed submanifoldto a generic fiber of some fibered region in the manifold: Criterion 1.4.
There is a submanifold A k Ď M of dimension k and a subset N Ď M , where π : N Ñ B is a fibration over a manifold B k ` of dimension k ` α : P Ñ R ą such that for almost all b P B the number n p A, b, T, ϕ t q of ϕ t -chords from A to thefiber over b of length ď T is finite and grows uniformly α -exponentially in T , i.e.for some T “ T p ϕ t q we have for T ě T : n p A, b, T, ϕ t q ě e αT . Note that then the exponential growth rate of n p A, b, T, ϕ t q is at least α . We assumethe number T to be uniform in b , but not necessarily continuous in P .The following result has been used in specific situations multiple times before.We give a proof for the following general formulation in Section 3. Theorem 1.
If a class of twisted periodic flows ϕ t of smooth self-maps of a compactmanifold M satisfies Criterion 1.4, then α constitutes a stability function for thetopological entropy of ϕ “ ϕ .Applications of Theorem 1. The following is to my knowledge the most generalclass of smooth 1-parameter families of smooth self-maps for which volume growthis stable. It is an extension of the result in [7].Let p M, Λ , α q be a closed contact manifold with closed Legendrian Λ which isfillable by the triple p W, L, λ q consisting of a Liouville domain with asymptoticallyconical exact Lagrangian L . This means that M “ B W, Λ “ B L “ L X M, α “ λ | T M .We parametrize the class of positive paths of contactomorphisms by the set ofpositive contact Hamiltonians P “ t h P C p M ˆ r , s , R q | ă h u . The functions h define the contact Hamiltonian vector fields by α p X h q “ h ; ι X h dα “ ´ dh ` dh p R α q α, and these vector fields generate the paths ϕ th by ϕ h “ id ; ϕ th “ X h p ϕ th q . A positive contactomorphism is the endpoint of a positive path of contactomor-phisms. We can extend ϕ t to a twisted periodic flow by convex combination witha Reeb flow. Note that the function h “ α . Thestability then comes from the growth of the positive part of the filtered LagrangianRabinowitz–Floer homology RFH T ` p W, L, h q , which is defined for the subset P reg ofHamiltonians for which Ť t ‰ ϕ th p Λ q & Λ (see Section 4 for more on this homology).
Theorem 2.
If for some h P P reg the positive Lagrangian Rabinowitz–Floer homol-ogy RFH T ` p W, L, h q has positive dimensional growth Γ RFH p h q : “ Γ p dim ι ˚ RFH ă T p h qq ,then for h P P reg we have h top p ϕ th q ě Γ RFH p h q . This lower bound extends continuously to all of P , is positive and stable in the C -topology on P : Γ RFH p h q ě min x,t h p x, t q ¨ Γ RFH p q ą . Remark . Our proof holds for the much more general class of (not necessarilytwisted periodic) positive paths of contactomorphisms, as long as we impose uniformbounds 0 ă c ď h ď C . However, the conclusions only hold for volume growthand we do not know the connection to topological entropy. For more on this, seeRemark 3.1.We conclude this introduction by a list of examples where the Criterion 1.4 issatisfied. Example 1.6.
As a first example let Q be a closed manifold with exponentiallygrowing fundamental group. Let P “ t Riemannian metrics on Q u and D “ t ϕ | ϕ t is a geodesic flow on SQ u . The sets in Criterion 1.4 are then A “ S p Q , B Ď Q is a neighbourhood of p and N “ π ´ p B q . The number of geodesic chords from A to π ´ p q q of length at most R is then at least the number of elements of π p Q, p, q q that are represented by paths that lift into the ball r B R p p q Ď r Q in the universalcover, since the minimizer of length is a geodesic. The function γ is then given LUCAS DAHINDEN by the growth of this number, which is independent of q and positive if the groupgrowth of π p Q q is positive. Continuity of γ with respect to g in C -norm comesfrom the fact that the balls r B R p p q vary continuously with g . This is a classicdiscussion: We define the ball volume growth as the growth of volume of a ballin the universal cover of Q : Γ ball p M, g q “ Γ p vol p r B R p p qq . For the relation betweenΓ ball and the group growth of π and for the fact that Γ ball is a lower bound fortopological entropy, see [18].A bit more intricate is the situation if the fundamental group is finite and thehomology of the based loop space grows exponentially. The function γ is then givenby the growth Γ p dim ι ˚ HM ď T p E qq of the Morse homology of the set of loops in M of energy at most T . As Gromov [13] showed, positivity of γ is a topologicalinvariant of Q .These examples can be generalized to P “ t Contact forms on p S ˚ Q, ξ std qu and D “ t ϕ | ϕ t is a Reeb flow on S ˚ Q u , as has been shown by Frauenfelder–Macarini–Schlenk [19, 11] using the Abbondandolo–Schwarz isomorphism from the Morse ho-mology of the based loop space to Lagrangian Floer homology. The function γ isgiven by γ p α q “ C min α α , where the constant C is the dimensional growth of theLagrangian Floer homology of a reference contact form α .A further extension of the above results was given by Alves–Meiwes [3] to bound-aries of Liouville domains with exponentially growing wrapped Floer homology (theopen string analog to symplectic homology). The role of A is then taken by someLegendrian sphere that is fillable by an exact Lagrangian, and N is a neighbour-hood of A which is fibered by Legendrian spheres. In this framework Alves andMeiwes managed to construct many examples of contact manifolds different fromcosphere bundles that admit a nonvanishing function γ . Among the examples are(non-standard) spheres of dimension ě S ˆ S , and non-standard contact struc-tures on any plumbing of cosphere bundles of base dimension ě Acknowledgments.
I wish to thank Leonid Polterovich and Felix Schlenk forinputs and advice. This work is supported by SFB/TRR 191 ‘Symplectic Structuresin Geometry, Algebra and Dynamics’ funded by the DFG.2.
Examples for total collapse of topological entropy
The following example by Milnor [20] shows that topological entropy is not lowersemi-continuous, even in very simple settings: it fails for smooth maps from theclosed unit disk to itself.
Example 2.1.
The family f t : D Ď C ÞÑ D ; z ÞÑ tz of maps is smooth. Lookingat the restriction f | B D : B D Ñ B D one sees that h top p f q “ log 2. However, f t haszero entropy for all t ă f t is the origin. In Riemanniann geometry this is called volume growth. We use the different term to avoidconfusion with Γ vol . If one imposes that the surface be closed and the map be a C r , r ě , diffeomor-phism, we have lower semi-continuity because of the existence and local stabilityof hyperbolic sets, as described in [14, S5.6]. However, this result is 2-dimensionalin essence and ceases to hold in higher dimensions. The following example showsthe lack of lower semi-continuity of topological entropy in the category of C r dif-feomorphisms p r ě r “ 8q of closed manifolds in three dimensions: Example 2.2.
Let Σ be a closed surface and f : Σ Ñ Σ be a C r diffeomorphismwith h top p f q ą f s of C r diffeomorphisms. We can assume that f s “ f for s near 0 and f s “ id for s near1. Let τ : T Ñ R be a bump function on the circle T “ R { Z supported in p , q with τ p q “
1. Further, let g t be the negative gradient flow of a Morse functionon T with critical points only in 0 and . Then F t : Σ ˆ T Ñ Σ ˆ T ; p x, θ q ÞÑp f τ p θ q p x q , g t p θ qq is a smooth family of C r diffeomorphisms such that F “ p f τ p θ q , id q has positive topological entropy in the fiber θ “ and such that F t has zerotopological entropy for every t ą ˆ t , u of F t is the identity.3. Proof of Theorem 1
Here, we give a proof of the Stability Theorem 1 under the assumption of Crite-rion 1.4. The proof is based on the proofs in the special situations, see e.g. [10, 7].In this paper, we always work with smooth maps ϕ on compact manifolds M ,so by a combined theorem of Yomdin [23] and Newhouse [21], topological entropycoincides with volume growth: h top p ϕ t q “ Γ vol p ϕ t q “ sup S Ă M Γ p vol ϕ t p S qq , where the supremum is taken over all compact submanifolds S of arbitrary codi-mension and vol is taken with respect to any Riemannian metric. Since volumegrowth is independent of the choice of Riemannian metric, we can choose a niceone. Let g B be a Riemannian metric on the base B k ` and vol k ` B its inducedvolume. Then we choose g on M such that the induced k ` g in N is larger than its shadow on B , i.e.vol k ` g ě π ˚ vol k ` B . (3.1)We denote the set of b P B for which the growth condition holds by B reg , and wedenote the trace left behind by ϕ t A by A T : “ Ť t Pr ,T s ϕ t A .The proof is completed in 6 steps:Step 1 Γ vol p ϕ t q ě Γ p vol k p ϕ t A qq ,Step 2 Γ p vol k p ϕ t A qq ě Γ p vol k ` p A T qq Step 3 Γ p vol k ` p A T qq ě Γ p vol k ` p A T q X N q Step 4 Γ p vol k ` p A T q X N q ě Γ p ş A T π ˚ d vol B q Step 5 Γ p ş A T π ˚ d vol B q ě Γ p ş B reg n p A, b, T, ϕ t q d vol B q Step 6 Γ p ş B reg n p A, b, T, ϕ t q d vol B q ě α .Step 1 holds since the supremum is an upper bound. LUCAS DAHINDEN
Step 2 holds since the vector field X generating ϕ t is bounded above since it isperiodic on a compact space:vol k ` p A T q “ ż T ż ϕ t A } pr p ϕ t A q K X } d vol k dt ď sup } X } ż T vol k p ϕ t A q dt. Then, we note that for f with lim sup t Ñ8 f p t q ě p ş T f p t q dt q ď Γ p f p T qq .Step 3 holds since volume is monotone under inclusion and since f ď g im-plies Γ p f q ď Γ p g q . Step 4 is an immediate consequence of (3.1). Step 5 is just areformulation.Step 6 is concluded since we assumed Criterion 1.4 and thus for some T we have n p A, b, T, ϕ t q ě e αT for all b P B and T ě T . This completes the proof. Remark . For volume growth Γ vol itmakes sense to work with the much more general class of non-autonomous dy-namical systems, i.e. for smooth families of smooth maps ϕ t with ϕ “ id , whichare not necessarily twisted periodic. The above proof goes through as long as weimpose that X t : “ ϕ t is bounded from above. Also, the proof of Theorem 2 inSection 4 does not rely on twisted periodicity, but only on the Hamiltonian h beinguniformly bounded for all time 0 ă c ď h ď C .One can also define topological entropy in this more general setting: A p T, δ q -separated set is a subset N Ď M such that for all n ‰ n P N there is a t P r , T s suchthat d p ϕ t p n q , ϕ t p n qq ě δ . Topological entropy is then defined as the exponentialgrowth rate of maximal cardinality of p T, δ q -separated sets: h top p ϕ t q “ sup δ ą Γ p maximal cardinality of a p T, δ q -separated set q . This generalized notion of topological entropy for non-autonomous systems is stud-ied only in a few papers. For example, in [17] it is shown that in the discrete timesetting this definition of generalized topological entropy (which is analogous to thedefinition of Bowen and Dinaburg) coincides with the definition that generalizesthe definition of topological entropy by Adler, Konheim and McAndrew.For this article it is of interest under what conditions the theorems of Yomdin [23]and Newhouse [21] generalize to non-autonomous dynamical systems.
Question . How are volume growth and topological entropy related for non-autonomous dynamical systems?4.
Proof of Theorem 2
In this section we show how to obtain Criterion 1.4 in the setup of [7], whichproves Theorem 2. Since this should serve as a model for similar situations, we referfor details to [7] and focus on the structure of the proof. The main point is thesetup of a persistence module for each element of P and proving that a perturba-tion of the parameter changes the persistence module Lipschitz–continuously withrespect to (logarithmized) interleaving distance. We refer to [22] for basic notionson persistence modules. Liouville domains and conical Lagrangians.
Let p M, Λ , α q be a contact manifoldwith Legendrian Λ which is fillable by the triple p W, L, λ q consisting of a Liouvilledomain p W, λ q with asymptotically conical exact Lagrangian L . This means that M “ B W, Λ “ B L “ L X M, α “ λ | T M . Example 4.1.
The stereotypical example for this setup is the cosphere bundle of acompact manifold p S ˚ Q, S ˚ q Q, λ q , where λ “ pdq is the tautological 1-form, togetherwith the Legendrian being a fiber over a point. We can realize the explicitly as thelevel set of a fiberwise starshaped Hamiltonian function H . We can then take thesublevel set as Liouville domain (i.e. the codisk bundle D ˚ Q ), where a codisk fiberfills S ˚ q Q asymptotically conically. Positive contactomorphisms.
In the introduction we parametrize the class of twistedperiodic positive paths of contactomorphisms by positive periodic contact Hamil-tonians. However, in view of Remark 3.1 it makes sense to directly parametrize theset of positive paths of contactomorphisms by the set of bounded positive contactHamiltonians P “ t h P C p M ˆ R , R q | D c, C P R ` : 0 ă c ď h ď C u . The Hamiltonian vector field of such a function generates the smooth family ofcontactomorphisms ϕ th , which are positively transverse to the contact structureker α (since h ą h ”
1, then X is the Reeb vector field of α . If h is autonomous, then X h is the Reeb vector field of h α . Nonautonomous Hamiltonians generate all pathsof contactomorphisms. This is a contrast to symplectic dynamics, where there isan obstruction in H p M ; R q for a path of symplectomorphisms to be Hamiltonian,called flux. The set Cont ` p M, ker α q of positive contactomorphisms consists of allcontactomorphisms that are reached through positive paths of contactomorphisms. Example 4.2.
Continuing the example of the cosphere bundle S ˚ Q of a com-pact manifold, we observe that the Hamiltonian flow of the defining starshapedHamiltonian is a reparametrization of the Reeb flow on the spherization. If theHamiltonian is } p } g with respect to some Riemannian metric g , then the inducedflow on the -level Σ g set is the co-geodesic flow on Q . Any other fiberwise star-shaped hypersurface Σ is graphical over Σ g by radial dilation: Σ “ f Σ g , where f : Σ g Ñ R ą and α Σ “ f α Σ g . Thus, studying characteristic flows of fiberwisestarshaped hypersurfaces amounts to the same as studying the set of autonomouscontact Hamiltonian flows. Persistent Rabinowitz–Floer homology.
Given a Liouville domain and asymptoti-cally conical exact Lagrangian p W, L, λ q that is bounded by a contact manifold witha Legendrian p M, Λ , α q , we say a Hamiltonian h P P is regular if Ť t ‰ ϕ th p Λ q & Λ.We denote the set of regular Hamiltonians by P reg . Note that P reg Ď P is comea-ger. For a regular Hamiltonian we can define the action filtered positive LagrangianRabinowitz–Floer chain complex RFC T ` p W, L ; h q . The induced homology is the ac-tion filtered positive Lagrangian Rabinowitz–Floer homologyRFH T ` p W, L ; h q . We drop
W, L from the notation if there is no possibility of confusion. For theanalytical details of this homology, especially the discussion of the differential, werefer to [7]. Note that the results below on interleavings require that h P P reg , LUCAS DAHINDEN whereas the results on topological entropy do not. The reason will become clear inthe last paragraph, where perturbations of the Legendrian are discussed.Positive Lagrangian Rabinowitz–Floer homology has the following properties,cf. [7] and the references therein.(1) The chain complex is a Z -vector space generated by chords of the contactHamiltonian vector field X h that start and end at Λ and have length in p , T q . Therefore,dim RFH T ` p h q ď t X h -chords from Λ to Λ of lenght ď T u . (2) The family of vector spaces t RFH T ` p h qu T P R ą Yt`8u is a persistence mod-ule: it is a direct system directed by morphisms induced by inclusion ofgenerators for T ď T ,RFH T ` p h q ι T,T ÝÝÝÑ
RFH T ` p h q , which satisfy ι T ,T ˝ ι T,T “ ι T,T , and for each finite T the chain complexRFC T ` is finite dimensional. We define the total homology as the directlimit RFH ` p h q : “ lim ÝÑ RFH T ` p h q . (3) For h, k P P with h ď k pointwise, we have continuation morphismsRFC T ` p h q φ Th,k
ÝÝÝÑ
RFC T ` p k q . Given h ď k ď g the morphisms satisfy φ Tk,g ˝ φ Th,k “ φ Th,g . These mor-phisms commute with the morphisms induced by inclusion and induce anisomorphism in the total homology. In other words, the following diagramcommutes.RFH T ` p h q RFH T ` p h q RFH ` : “ RFH ` p h q – RFH ` p k q RFH T ` p k q RFH T ` p k q φ Th,k ι T,T φ T h,k ι ι T,T ι (4) There is a cofinal subfamily of C Ď P reg that is closed under scaling, i.e. c P C , λ ą ñ λc P C , such that continuation morphisms with respect toscaling coincide with persistent morphisms, i.e. for λ ą T ` p λc q – RFH λT ` p c q . Cofinal means that @ h, k P P there exists c P C such that c ě h and c ě k . Remark . In the following we assume that 1 P P reg , so that every positiveconstant is in P reg and we can choose C “ R ą . If 1 R P reg , then we chooseany regular autonomous Hamiltonian and scalings thereof, for they also admit thescaling property. Alternatively we can choose C to be the set of regular autonomousHamiltonians. In dynamical terms, autonomous Hamiltonians parametrize positivepaths of contactomorphisms that are flows. Interleaving stability of persistence modules.
We can combine the diagram in (3)with the property in (4) to obtain for 0 ă c ď h ď C and for every T RFH T ` p c q RFH T ` p h q RFH T ` p C q – RFH Cc T ` p c q φ Tc,h φ Th,C
If we logarithmize the persistence parameter T “ e τ , concatenating the abovediagram for T “ T and T “ Cc T yields an interleaving of the two persistencemodules ι RFH e τ ` p h q and ι RFH e τ ` p c q with interleaving distance log Cc :RFH T ` p c q RFH Cc T ` p c q RFH T ` p h q RFH Cc T ` p h q . Thus, our choice of P implies that every induced persistence module is interleavedwith the module of a Reeb flow, with interleaving distance given by the logarith-mized oscillation of the Hamiltonian. Infinite persistence module.
From the persistence module RFH T ` p h q we can definea new one by taking the image in the total homology ι RFH T ` p h q Ď RFH ` , wherethe persistence morphisms are given by ι T,T ˝ ι “ ι ˝ ι T,T and where continuationmorphisms for different h, k are given by φ ,Th,k ˝ ι “ ι ˝ φ Th,k . This amounts todeleting all finite bars from the associated barcodes. Since all morphisms commutewith ι , the properties above also hold for ι RFH T ` p h q . The interleaving diagramfrom before becomes ι RFH T ` p c q ι RFH Cc T ` p c q ι RFH T ` p h q ι RFH Cc T ` p h q . ĎĎ All persistence morphisms ι T,T are inclusions as subspaces of RFH ` and thusfor any h P P the dimension of ι RFH T ` p h q is monotone in T . Since the aboveinterleaving diagram commutes and since the horizontal arrows are inclusions, thevertical and diagonal arrows must be injections. We conclude thatdim ι RFH T ` p h q P ” dim ι RFH T ` p c q , dim ι RFH Cc T ` p c q ı , which implies thatΓ p dim ι RFH T ` p h qq ě Γ p dim ι RFH T ` p c qq “ c Γ p dim ι RFH T ` p qq as claimed. Remark . Note that the bound from above is not interesting since there mightbe homologically invisible chords, and since the volume growth of the Legendriansubmanifold might not realize the supremum in the definition of volume growth.In rare cases this is not the case, as for geodesic flows of hyperbolic manifolds.On the way we have reproved the following classical corollary for the logarith-mized persistence parameter:
Corollary 3.
The persistence modules
RFH e τ p h q and RFH e τ p min h q are log Cc -interleaved. Likewise, the persistence modules ι RFH e τ p h q and ι RFH e τ p min h q are log Cc -interleaved.Connection with wrapped homology. If 1 P P reg , then the positive LagrangianRabinowitz–Floer homology of the constant Hamiltonian 1 (which induces the Reebflow of α ) is isomorphic to the positive part of wrapped Floer homology, cf. [7]. Thatis, if 0 ď a , then WH p ,a q p W, L q –
RFH a ` p q . This isomorphism is analogous to the isomorphism between Rabinowitz–Floer ho-mology and symplectic homology [5].This is of much interest to our situation, since in some cases wrapped Floerhomology is more computable than Lagrangian Rabinowitz–Floer homology. Inparticular, wrapped Floer homology admits a Pontrjagin product, which can beused to study the dimensional growth Γ symp p W, L q of WH p ,a q p W, L q . This is usedin [3] to find examples of contact manifolds different from cotangent bundles suchthat every Reeb flow has positive topological entropy. By the above isomorphism,also all positive contactomorphisms on these spaces have positive topological en-tropy. Stability of chord counting under perturbations of the Legendrian.
As the last step inthe construction, we want to make a statement about a generic nearby Lagrangianof Λ. For this we take the following result from [7, Proposition 1.7].
Proposition 4.5.
Let Λ be a Legendrian that is isotopic through Legendrians to Λ . Let ψ be a contactomorphism that takes Λ to Λ so that p ψ ´ q ˚ α “ f α . Thenthe exponential growth of the number of ϕ t -chords from Λ to Λ of length ď T is atleast min f ¨ min h ¨ Γ symp p W, L q . We sketch the proof. Suppose that a Legendrian Λ is perturbed through anisotopy of Legendrians to a nearby Legendrian Λ . The Legendrian isotopy can beextended to a path of contactomorphisms t ψ t u t Pr , s . The X h chords from Λ to Λ are in 1-1 correspondence with D p ψ q ´ X h chords from p ψ q ´ Λ to Λ.We can define a new Hamiltonian g which generates ψ t for time t P r , s and thustakes Λ to p ψ q ´ Λ in time 1, and then generates D p ψ q ´ X h . The transformationformula of the contact Hamiltonian tells us that g p x, t q| t ě “ α x p D p ψ q ´ X h p ψ p x q , t ´ qq “ f p x q h p ψ p x q , t ´ q and therefore g | t ě ě min f ¨ min h . Unfortunately, the Hamiltonian defined thisway is not nessessarily smooth at 1 and is not necessarily positive in r , s .Both problems can be solved by convex combination with a Reeb flow, whichstrongly flows forward in time r , s and then is reverted in time r , T s for T largeenough such that combination with g affects the lower bounds only a little. Thechords from Λ to Λ of the resulting path of contactomorphisms of length ě T arein 1-1 correspondence with the chords of length ě T ´ , where thecorrespondence shifts the period by 1, which proves the proposition.If the deformation of the Legendrian is small, then min f is close to 1. A conve-nient consequence of this perturbative stability is that for almost all deformationsΛ of Λ the Hamiltonian g produced in the proof of Proposition 4.5 is regular, g P P reg . Thus, the conclusion of Theorem 2 holds for all Hamiltonians h P P . References [1] A. Abbondandolo, M. Alves, M. Sa˘glam, F. Schlenk. Entropy collapse versus entropyrigidity for Reeb and Finsler flows.
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