We establish a relation between the growth of the cylindrical contact homology of a contact manifold and the topological entropy of Reeb flows on this manifold. We show that if a contact manifold $(M,\xi)$ admits a hypertight contact form $\lambda_0$ for which the cylindrical contact homology has exponential homotopical growth rate, then the Reeb flow of every contact form on $(M,\xi)$ has positive topological entropy. Using this result, we provide numerous new examples of contact 3-manifolds on which every Reeb flow has positive topological entropy.Comment: 42 pages, 2 figures. Exposition improved. Comments are welcome

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 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.

We develop a forcing theory of topological entropy for Reeb flows in dimension three. A transverse link L in a closed contact $3$ -manifold $(Y,\xi )$ is said to force topological entropy if $(Y,\xi )$ admits a Reeb flow with vanishing topological entropy, and every Reeb flow on $(Y,\xi )$ realizing L as a set of periodic Reeb orbits has positive topological entropy. Our main results establish topological conditions on a transverse link L, which imply that L forces topological entropy. These conditions are formulated in terms of two Floer theoretical invariants: the cylindrical contact homology on the complement of transverse links introduced by Momin [A. Momin. J. Mod. Dyn.5 (2011), 409–472], and the strip Legendrian contact homology on the complement of transverse links, introduced by Alves [M. R. R. Alves. PhD Thesis, Université Libre de Bruxelles, 2014] and further developed here. We then use these results to show that on every closed contact $3$ -manifold that admits a Reeb flow with vanishing topological entropy, there exist transverse knots that force topological entropy.

Let (M, ξ) be a compact contact 3-manifold and assume that there exists a contact form α 0 on (M, ξ) whose Reeb flow is Anosov. We show this implies that every Reeb flow on (M, ξ) has positive topological entropy, giving a positive answer to a question raised in [1]. Our argument builds on previous work of the author [1] and recent work of Barthelmé and Fenley [4]. This result combined with the work of Foulon and Hasselblatt [14] is then used to obtain the first examples of hyperbolic contact 3-manifolds on which every Reeb flow has positive topological entropy.