Symbolic dynamics for surface diffeomorphisms with positive entropy

Sarig, Omri¹

¹ Faculty of Mathematics and Computer Science, The Weizmann Institute of Science, 234 Herzl Street, POB 26, Rehovot 76100, Israel

Journal of the American Mathematical Society, Tome 26 (2013) no. 2, pp. 341-426

Résumé

Let $f$ be a $C^{1+\varepsilon }$ diffeomorphism on a compact smooth surface with positive topological entropy $h$. For every $0\delta $, 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).

Détail

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

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