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Unique equilibrium states, large deviations and Lyapunov spectra for the Katok map

Part of: Dynamical systems with hyperbolic behavior

Published online by Cambridge University Press:  20 March 2020

TIANYU WANG
TIANYU WANG*
Affiliation:
Department of Mathematics, The Ohio State University, Columbus, OH 43210, USA email wang.7828@buckeyemail.osu.edu
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Abstract

We study the thermodynamic formalism of a $C^{\infty }$ non-uniformly hyperbolic diffeomorphism on the 2-torus, known as the Katok map. We prove for a Hölder continuous potential with one additional condition, or geometric $t$ -potential $\unicode[STIX]{x1D711}_{t}$ with $t<1$ , the equilibrium state exists and is unique. We derive the level-2 large deviation principle for the equilibrium state of $\unicode[STIX]{x1D711}_{t}$ . We study the multifractal spectra of the Katok map for the entropy and dimension of level sets of Lyapunov exponents.

Keywords

thermodynamical formalismlarge deviation principlemultifractal analysis

MSC classification

Primary: 37D35: Thermodynamic formalism, variational principles, equilibrium states
Secondary: 37D25: Nonuniformly hyperbolic systems (Lyapunov exponents, Pesin theory, etc.)

Information

Type
Original Article
Information
Ergodic Theory and Dynamical Systems , Volume 41 , Issue 7 , July 2021 , pp. 2182 - 2219
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Copyright
© The Author(s) 2020. Published by Cambridge University Press

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