# American Institute of Mathematical Sciences

2020, 16: 1-36. doi: 10.3934/jmd.2020001

## The degree of Bowen factors and injective codings of diffeomorphisms

 Laboratoire de Mathématiques d'Orsay, Université Paris-Sud, F-91405 Orsay Cedex, France

Received  August 2018 Revised  July 10, 2019 Published  December 2019

We show that symbolic finite-to-one extensions of the type constructed by O. Sarig for surface diffeomorphisms induce Hölder-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 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.

Citation: Jérôme Buzzi. The degree of Bowen factors and injective codings of diffeomorphisms. Journal of Modern Dynamics, 2020, 16: 1-36. doi: 10.3934/jmd.2020001
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##### References:
The subshift of finite type in Example 4.9
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