Hölder regularity for metric Anosov flows

Abstract

We define a class of flows on compact metric spaces that we call \emph{$C^α$ metric Anosov flows}, $0 < α< 1$ or $α= \mathrm{Lip}$. We give verifiable criteria for a metric Anosov flow to be $C^α$. We show that $C^α$ metric Anosov flows have Hölder continuous local product structure, and in particular Hölder continuous holonomies. We deduce that topological conjugacies between $C^α$ metric Anosov flows are automatically Hölder continuous. We show that any $C^α$ metric Anosov flow admits a strong (i.e. Hölder continuous) Markov coding. This yields strong ergodic properties for equilibrium states for Hölder potentials for $C^α$ metric Anosov flows. The regularity of the local product structure is proved in a non-compact setting which applies for the geodesic flow on the universal cover of a locally CAT(-1) space. This allows us to obtain Hölder regularity results for the holonomy maps for pinched negative curvature manifolds with respect to the Sasaki metric. This contrasts with the result of Ballmann, Brin and Burns that there are examples in this class for which the dynamical distributions are not Hölder.