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Thermodynamics of smooth models of pseudo–Anosov homeomorphisms

Part of: Dynamical systems with hyperbolic behavior Low-dimensional dynamical systems Smooth dynamical systems: general theory Ergodic theory

Published online by Cambridge University Press:  18 May 2021

DOMINIC VECONI
DOMINIC VECONI*
Affiliation:
Department of Mathematics, Penn State University, State College, Pennsylvania16802, USA
*
e-mail: [email protected]
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Abstract

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We develop a thermodynamic formalism for a smooth realization of pseudo-Anosov surface homeomorphisms. In this realization, the singularities of the pseudo-Anosov map are assumed to be fixed, and the trajectories are slowed down so the differential is the identity at these points. Using Young towers, we prove existence and uniqueness of equilibrium states for geometric t-potentials. This family of equilibrium states includes a unique SRB measure and a measure of maximal entropy, the latter of which has exponential decay of correlations and the central limit theorem.

Keywords

nonuniform hyperbolicitypseudo-Anosov diffeomorphismsthermodynamic formalismsmooth ergodic theory

MSC classification

Primary: 37C05: Smooth mappings and diffeomorphisms 37C40: Smooth ergodic theory, invariant measures 37D25: Nonuniformly hyperbolic systems (Lyapunov exponents, Pesin theory, etc.) 37D35: Thermodynamic formalism, variational principles, equilibrium states
Secondary: 37A50: Relations with probability theory and stochastic processes 37E30: Homeomorphisms and diffeomorphisms of planes and surfaces
Type
Original Article
Information
Ergodic Theory and Dynamical Systems , Volume 42 , Issue 3: Anatole Katok Memorial Issue Part 2: Special Issue of Ergodic Theory and Dynamical Systems , March 2022 , pp. 1284 - 1326
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Creative Commons
Creative Common License - CCCreative Common License - BY
This is an Open Access article, distributed under the terms of the Creative Commons Attribution licence (https://creativecommons.org/licenses/by/4.0/), which permits unrestricted re-use, distribution, and reproduction in any medium, provided the original work is properly cited.
Copyright
© The Author(s), 2021. Published by Cambridge University Press

Footnotes

Dedicated to the memory of Anatole Katok

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