Uniqueness and exponential mixing for the measure of maximal entropy for piecewise hyperbolic maps

Mark F. Demers

doi:10.3934/dcds.2020217

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2021, Volume 41, Issue 1: 217-256. Doi: 10.3934/dcds.2020217

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Uniqueness and exponential mixing for the measure of maximal entropy for piecewise hyperbolic maps

Mark F. Demers

Department of Mathematics, Fairfield University, Fairfield CT 06824, USA

Received: December 2019

Early access: May 2020

Published: January 2021

MD was partly supported by NSF grant DMS 1800321

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Abstract

For a class of piecewise hyperbolic maps in two dimensions, we propose a combinatorial definition of topological entropy by counting the maximal, open, connected components of the phase space on which iterates of the map are smooth. We prove that this quantity dominates the measure theoretic entropies of all invariant probability measures of the system, and then construct an invariant measure whose entropy equals the proposed topological entropy. We prove that our measure is the unique measure of maximal entropy, that it is ergodic, gives positive measure to every open set, and has exponential decay of correlations against Hölder continuous functions. As a consequence, we also prove a lower bound on the rate of growth of periodic orbits. The main tool used in the paper is the construction of anisotropic Banach spaces of distributions on which the relevant weighted transfer operator has a spectral gap. We then construct our measure of maximal entropy by taking a product of left and right maximal eigenvectors of this operator.

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Mathematics Subject Classification: Primary: 37D35, 37D50; Secondary: 37A35, 37B40, 37C30.

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Figure 1. A possible intersection between $ \mathcal{S}_j^+ $ (dashed line) and $ \mathcal{S}^- $ (solid lines). $ \mathcal{S}^- $ is the boundary between two domains $ M_i^- $ and $ M_{i+1}^- $, while $ \mathcal{S}_j^+ $ is the boundary of elements of $ \mathcal{M}_0^j $. The local stable manifold $ V \subset T^{-j}W $ is contained in a single element of $ \mathcal{M}_0^j $, yet the intersection $ V \cap M_i^- $ has two connected components whose images under $ T^{-1} $ will both lie in $ M_i^+ $ and be within distance $ \varepsilon $ of one another in the metric $ \bar d $

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