We exhibit the first examples of contact structures on S 2n−1 with n ≥ 4 and on S 3 × S 2 , all equipped with their standard smooth structures, for which every Reeb flow has positive topological entropy. As a new technical tool for the study of the volume growth of Reeb flows we introduce the notion of algebraic growth of wrapped Floer homology. Its power stems from its stability under several geometric operations on Liouville domains. 2010 Mathematics Subject Classification. Primary 37J05, 53D40. Key words and phrases. Topological entropy, contact structure, Reeb dynamics, Floer homology. Marcelo R.R. Alves supported by the Swiss National Foundation. Matthias Meiwes supported by German-Israeli Foundation (GIF).. 1.2. Main results. The main result of this paper is the existence of contact structures with positive entropy on high dimensional manifolds. Theorem 1.1.A) Let S 2n−1 be the (2n − 1) -dimensional sphere with its standard smooth structure. For n ≥ 4 there exists a contact structure on S 2n−1 with positive entropy.B) There exists a contact structure on S 3 × S 2 with positive entropy.Recall that a contact manifold is said to be exactly fillable if it is the boundary of a Liouville domain. From Theorem 1.1 and the methods developed in this paper we obtain the following more general result.Theorem 1.2.♣ If V is a manifold of dimension 2n − 1 ≥ 7 that admits an exactly fillable contact structure, then V admits a contact structure with positive entropy.♦ If V is a 5-manifold that admits an exactly fillable contact structure, then the connected sum V #(S 3 × S 2 ) admits a contact structure with positive entropy.Note that the standard contact structure on spheres as well as the canonical contact structure on S * S 3 ∼ = S 3 × S 2 have a contact form with periodic Reeb flow. In particular these are not
In this article we study persistence features of topological entropy and periodic orbit growth of Hamiltonian diffeomorphisms on surfaces with respect to Hofer's metric. We exhibit stability of these dynamical quantities in a rather strong sense for a specific family of maps introduced by Polterovich and Shelukhin. A crucial ingredient comes from some enhancement of lower bounds for the topological entropy and orbit growth forced by a periodic point, formulated in terms of the geometric self-intersection number and a variant of Turaev's cobracket of the free homotopy class that it induces. Those bounds are obtained within the framework of Le Calvez and Tal's forcing theory. 2020 Mathematics Subject Classification. 37E30, 37J46. A. Chor was partially supported by the Israel Science Foundation grant 667/18. M. Meiwes was partially supported by the Israel Science Foundation grant 2026/17. 1 In fact a similar result holds for C 0 -perturbations of T , see [45].1 * (φ) α for r ∈ Ê, and whose linear maps π r,s : HF r * (φ) α → HF s * (φ) α are induced by the inclusion maps CF r * (M, H) α → CF s * (M, H) α , where H is a Hamiltonian that generates φ.
In this article we show that the braid type of a set of 1periodic orbits of a non-degenerate Hamiltonian diffeomorphism on a surface is stable under perturbations which are sufficiently small with respect to the Hofer metric d Hofer . We call this new phenomenon braid stability for the Hofer metric.We apply braid stability to study the stability of the topological entropy h top of Hamiltonian diffeomorphisms on surfaces with respect to small perturbations with respect to d Hofer . We show that h top is lower semicontinuous on the space of Hamiltonian diffeomorphisms of a closed surface endowed with the Hofer metric, and on the space of compactly supported diffeormophisms of the two-dimensional disk D endowed with the Hofer metric. This answers the two-dimensional case of a question of Polterovich.En route to proving the lower semicontinuity of h top with respect to d Hofer , we prove that the topological entropy of a diffeomorphism φ on a compact surface can be recovered from the topological entropy of the braid types realised by the periodic orbits of φ.
Abstract. A contact manifold admittting a supporting contact form without contractible Reeb orbits is called hypertight. In this paper we construct a Rabinowitz Floer homology associated to an arbitrary supporting contact form for a hypertight contact manifold Σ, and use this to prove versions of conjectures of Sandon [San13] and Mazzucchelli [Maz15] on the existence of translated points and invariant Reeb orbits, and to show that positive loops of contactomorphisms give rise to non-contractible Reeb orbits.
In this paper, we study the regularity of topological entropy, as a function on the space of Riemannian metrics endowed with the C 0 topology. We establish several instances of entropy robustness (persistence of entropy non-vanishing after small C 0 perturbations).A large part of this paper is dedicated to metrics on the 2-dimensional torus, for which our main results are that metrics with a contractible closed geodesic have robust entropy (thus generalizing and quantifying a result of Denvir-Mackay) and that metrics with robust positive entropy on the torus are C 8 generic. Moreover, we quantify the asymptotic behavior of volume entropy in the Teichmüller space of hyperbolic metrics on a punctured torus, which bounds from below the topological entropy for these metrics.For general closed manifolds of dimension at least 2 we prove that the set of metrics with robust and high positive entropy is C 0 -large in the sense that it is dense, contains cones and arbitrarily large balls. Contents 17 4.3. Robustness of entropy via ribbons 19 4.4. Ribbons exist for C 8 generic metrics 21 5. Robustness by retractable neck on general manifolds 23 Appendix A. Robustness of non-degenerate length spectrum 28 References 29
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