Journal of Modern Dynamics (JMD)

Positive topological entropy for Reeb flows on 3-dimensional Anosov contact manifolds

Pages: 497 - 509, Volume 10, 2016      doi:10.3934/jmd.2016.10.497

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Marcelo R. R. Alves - Institut de Mathématiques, Université de Neuchâtel, Rue Émile Argand 11, CP 158, 2000 Neuchâtel, Switzerland (email)

Abstract: Let $(M, \xi)$ be a compact contact 3-manifold and assume that there exists a contact form $\alpha_0$ on $(M, \xi)$ whose Reeb flow is Anosov. We show this implies that every Reeb flow on $(M, \xi)$ has positive topological entropy, answering a question raised in [2]. Our argument builds on previous work of the author [2] and recent work of Barthelmé and Fenley [4]. This result combined with the work of Foulon and Hasselblatt [13] is then used to obtain the first examples of hyperbolic contact 3-manifolds on which every Reeb flow has positive topological entropy.

Keywords:  Dynamics of Reeb flows, contact homology, topological entropy, Anosov flows.
Mathematics Subject Classification:  Primary: 37J05, 37D20, 53D42; Secondary: 37B40, 53D35.

Received: January 2016;      Revised: August 2016;      Available Online: November 2016.