Description:

Title: Holder continuity in metric Anosov flows

Abstract: Similarly to how an Anosov flow is defined on a manifold, one can define a metric Anosov flow on a compact metric space by requiring exponential contraction of stable and unstable sets which locally are "transverse." Examples of such flows include Lipschitz roof suspensions of shifts of finite type, geodesic flows of compact locally CAT(-1) spaces, and certain flows associated to Anosov representations of word hyperbolic groups. Given a metric Anosov flow satisfying a few mild assumptions, we show that its local product structure, which is defined by locally intersecting pieces of the stable and unstable sets, must be Holder continuous, and the flow has a Holder continuous Markov coding. Our proof works for C^1 Anosov flows: in particular the holonomies along (un)stable foliations are still Holder continuous without a C^(1+alpha) assumption. Our result also extends to some noncompact settings. In particular, we consider an example by Ballmann, Brin, and Burns of a smooth complete pinched negatively curved surface whose strong stable distribution is not Holder continuous. Applying our result, we obtain that the holonomies along the strong stable foliation in this example are nonetheless Holder continuous. This is counterintuitive, and we give some explanation for why this is possible. Joint work with Daniel Thompson.