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

TL;DR

This paper proves that if a contact 3-manifold admits an Anosov Reeb flow, then all Reeb flows on it have positive topological entropy, leading to new examples of hyperbolic contact manifolds with this property.

Contribution

It establishes that the existence of an Anosov Reeb flow on a contact 3-manifold implies all Reeb flows have positive topological entropy, providing the first such examples.

Findings

All Reeb flows on the manifold have positive topological entropy.

Constructs the first examples of hyperbolic contact 3-manifolds with this property.

Builds on previous work to connect Anosov flows with entropy properties.

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 Barthelm\'e 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.