The degree of Bowen factors and injective codings of diffeomorphisms

TL;DR

This paper investigates symbolic codings of surface diffeomorphisms, establishing conditions for H"older-continuous conjugacies, and providing bounds on periodic points, with applications to Sinai billiards and characterizations of hyperbolic measures.

Contribution

It introduces the Bowen property for finite-to-one extensions, derives H"older-continuous conjugacies, and improves bounds on periodic points for surface diffeomorphisms and Sinai billiards.

Findings

H"older-continuous conjugacies exist on large sets for certain symbolic extensions.

Lower bounds on periodic points are established, improving previous results.

Characterization of surface diffeomorphisms with H"older-continuous coding of hyperbolic measures.

Abstract

We show that symbolic finite-to-one extensions of the type constructed by O. Sarig for surface diffeomorphisms induce H\"older-continuous conjugacies on large sets. We deduce this from their Bowen property. 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. As an application, we give lower bounds on the number of periodic points first for surface diffeomorphisms (improving a result of Sarig) and for Sinai billiards maps (building on a result of Baladi and Demers). Finally we characterize surface diffeomorphisms admitting a H\"older-continuous coding of all their aperiodic hyperbolic measures and give a slightly weaker construction preserving local compactness.