Lower complexity bounds for positive contactomorphisms

Dahinden, Lucas

doi:10.1007/s11856-018-1651-y

Cited by 12 publications

(17 citation statements)

References 22 publications

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“…This approach is reminiscent of the Entropy Conjecture and other results where algebraic topological features of a map give a lower bound on the topological entropy; see [Ka07,KH,Yo] for further references. The case of symplectomorphisms or contactomorphisms is very similar in spirit and closely related to the Reeb setting, and the results usually rely again on exponential growth of a variant of Floer homology; see, e.g., [Da18,Da21,FS06]. On the other hand, for compactly supported Hamiltonian diffeomorphisms, there is no Floer homology growth.…”

Section: Resultsmentioning

confidence: 99%

Topological entropy of Hamiltonian diffeomorphisms: a persistence homology and Floer theory perspective

Çineli¹,

Ginzburg²,

Gürel³

2021

Preprint

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We study topological entropy of compactly supported Hamiltonian diffeomorphisms from a perspective of persistence homology and Floer theory. We introduce barcode entropy, a Floer-theoretic invariant of a Hamiltonian diffeomorphism, measuring exponential growth under iterations of the number of not-too-short bars in the barcode of the Floer complex. We prove that the barcode entropy is bounded from above by the topological entropy and, conversely, that the barcode entropy is bounded from below by the topological entropy of any hyperbolic locally maximal invariant set, e.g., a hyperbolic horseshoe. As a consequence, we conclude that for Hamiltonian diffeomorphisms of surfaces the barcode entropy is equal to the topological entropy.

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Section: Resultsmentioning

confidence: 99%

Topological entropy of Hamiltonian diffeomorphisms: a persistence homology and Floer theory perspective

Çineli¹,

Ginzburg²,

Gürel³

2021

Preprint

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“…In a recent work [19] Dahinden has obtained an extension of the results of [45] to positive contactomorphisms. He showed that if the homology of the based loop space of a manifold Q is rich then every positive contactomorphism in (T 1 Q, ξ geo ) has positive topological entropy.…”

Section: Discussionmentioning

confidence: 97%

Legendrian contact homology and topological entropy

Alves

2019

J. Topol. Anal.

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In this paper we study the growth rate of a version of Legendrian contact homology, which we call strip Legendrian contact homology, in 3-dimensional contact manifolds and its relation to the topological entropy of Reeb flows. We show that: if for a pair of Legendrian knots in a contact 3-manifold (M, ξ) the strip Legendrian contact homology is defined and has exponential homotopical growth with respect to the action, then every Reeb flow on (M, ξ) has positive topological entropy. This has the following dynamical consequence: for all Reeb flows (even degenerate ones) on (M, ξ) the number of hyperbolic periodic orbits grows exponentially with respect to the period. We show that for an infinite family of 3manifolds, infinitely many different contact structures exist that possess a pair of Legendrian knots for which the strip Legendrian contact homology has exponential growth rate.1991 Mathematics Subject Classification. Primary 37J05, 53D42.

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“…It was applied to proving Theorem 7.5 by Macarini and the second author in [131], based on earlier applications of Lagrangian Floer homology to volume growth in [82,83]. Theorem 7.5 generalizes work of Dinaburg, Gromov, Paternain and Petean on geodesic flows, see [157], and in turn can be generalized to those contactomorphisms on spherisations which can be reached by a contact isotopy that is everywhere positively transverse to the contact structure, see [58]. The proper setting of the classical results on the topological entropy of geodesic flows is thus contact topology.…”

Section: Psfrag Replacementsmentioning

confidence: 99%

Floer Homologies, with Applications

Abbondandolo

Schlenk

2018

Jahresber. Dtsch. Math. Ver.

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Floer invented his theory in the mid eighties in order to prove the Arnol'd conjectures on the number of fixed point of Hamiltonian diffeomorphisms and Lagrangian intersections. Over the last thirty years, many versions of Floer homology have been constructed. In symplectic and contact dynamics and geometry they have become a principal tool, with applications that go far beyond the Arnol'd conjectures: The proof of the Conley conjecture and of many instances of the Weinstein conjecture, rigidity results on Lagrangian submanifolds and on the group of symplectomorphisms, lower bounds for the topological entropy of Reeb flows and obstructions to symplectic embeddings are just some of the applications of Floer's seminal ideas. Other Floer homologies are of topological nature. Among their applications are Property P for knots and the construction of compact topological manifolds of dimension greater than five that are not triangulisable. This is by no means a comprehensive survey on the presently known Floer homologies and their applications. Such a survey would take several hundred pages. We just describe some of the most classical versions and applications, together with the results that we know or like best. The text is written for non-specialists and the focus is on ideas rather than generality. Two intermediate sections recall basic notions and concepts from symplectic dynamics and geometry. 14 4.

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Lower complexity bounds for positive contactomorphisms

Cited by 12 publications

References 22 publications

Topological entropy of Hamiltonian diffeomorphisms: a persistence homology and Floer theory perspective

Topological entropy of Hamiltonian diffeomorphisms: a persistence homology and Floer theory perspective

Legendrian contact homology and topological entropy

Floer Homologies, with Applications

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