Symbolic dynamics for surface diffeomorphisms with positive entropy

Sarig, Omri

doi:10.1090/S0894-0347-2012-00758-9

Symbolic dynamics for surface diffeomorphisms with positive entropy
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by Omri M. Sarig

J. Amer. Math. Soc. 26 (2013), 341-426

DOI: https://doi.org/10.1090/S0894-0347-2012-00758-9

Published electronically: November 26, 2012

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Abstract:

Let $f$ be a $C^{1+\varepsilon }$ diffeomorphism on a compact smooth surface with positive topological entropy $h$. For every $0<\delta <h$, we construct an invariant Borel set $E$ and a countable Markov partition for the restriction of $f$ to $E$ in such a way that $E$ has full measure with respect to every ergodic invariant probability measure with entropy greater than $\delta$. The following results follow: $f$ has at most countably many ergodic measures of maximal entropy (a conjecture of J. Buzzi), and in the case when $f$ is $C^\infty$, $\limsup \limits _{n\to \infty }e^{-n h}\#\{x:f^n(x)=x\}>0$ (a conjecture of A. Katok).

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