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

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. Our argument builds on previous work of the author (http://arxiv.org/abs/1410.3380) and recent work of Barthelmé and Fenley (http://arxiv.org/abs/1505.07999). This result combined with the work of Foulon and Hasselblatt (http://www.tufts.edu/as/math/Preprints/FoulonHasselblattLegendrian.pdf) is then used to obtain the first examples of hyperbolic contact 3-manifolds on which every Reeb flow has positive topological entropy.

arXiv comment18 pages. Comments wellcome! arXiv admin note: text overlap with arXiv:1410.3380