POSITIVE TOPOLOGICAL ENTROPY FOR REEB FLOWS ON 3-DIMENSIONAL ANOSOV CONTACT MANIFOLDS

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 Barthelme 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.