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Flexibility of Lyapunov exponents with respect to two classes of measures on the torus

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

Published online by Cambridge University Press:  31 May 2021

ALENA ERCHENKO
ALENA ERCHENKO*
Affiliation:
Mathematics Department, Stony Brook University, Simons Center for Geometry and Physics, Stony Brook, NY, USA
*
e-mail: alena.erchenko@stonybrook.edu
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Abstract

We consider a smooth area-preserving Anosov diffeomorphism $f\colon \mathbb T^2\rightarrow \mathbb T^2$ homotopic to an Anosov automorphism L of $\mathbb T^2$ . It is known that the positive Lyapunov exponent of f with respect to the normalized Lebesgue measure is less than or equal to the topological entropy of L, which, in addition, is less than or equal to the Lyapunov exponent of f with respect to the probability measure of maximal entropy. Moreover, the equalities only occur simultaneously. We show that these are the only restrictions on these two dynamical invariants.

Keywords

flexibilityLyapunov exponentsAnosov diffeomorphismMarkov partitionslow-down deformation

MSC classification

Primary: 37D20: Uniformly hyperbolic systems (expanding, Anosov, Axiom A, etc.) 37A35: Entropy and other invariants, isomorphism, classification
Secondary: 37C05: Smooth mappings and diffeomorphisms 37A05: Measure-preserving transformations
Type
Original Article
Information
Ergodic Theory and Dynamical Systems , Volume 42 , Issue 2: Anatole Katok Memorial Issue Part 1: Special Issue of Ergodic Theory and Dynamical Systems , February 2022 , pp. 737 - 776
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Copyright
© The Author(s), 2021. Published by Cambridge University Press

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Footnotes

Dedicated to Anatole Katok

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