The degree of Bowen factors and injective codings of diffeomorphisms

Abstract

<p style='text-indent:20px;'>We show that symbolic finite-to-one extensions of the type constructed by O. Sarig for surface diffeomorphisms induce Hölder-continuous <i>conjugacies</i> on large sets. We deduce this from their <i>Bowen property</i>. This notion, introduced in a joint work with M. Boyle, generalizes a fact first observed by R. Bowen for Markov partitions. We rely on the notion of degree from finite equivalence theory and magic word isomorphisms. <p style='text-indent:20px;'>As an application, we give lower bounds on the number of periodic points first for surface diffeomorphisms (improving a result of Sarig) and for Sinaï billiards maps (building on a result of Baladi and Demers). Finally we characterize surface diffeomorphisms admitting a Hölder-continuous coding of all their aperiodic hyperbolic measures and give a slightly weaker construction preserving local compactness.