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Journal of the American Mathematical Society

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Symbolic dynamics for surface diffeomorphisms with positive entropy

Author: Omri M. Sarig
Journal: J. Amer. Math. Soc. 26 (2013), 341-426
MSC (2010): Primary 37D25; Secondary 37D35
Published electronically: November 26, 2012
MathSciNet review: 3011417
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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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Additional Information

Omri M. Sarig
Affiliation: Faculty of Mathematics and Computer Science, The Weizmann Institute of Science, 234 Herzl Street, POB 26, Rehovot 76100, Israel

Keywords: Markov partitions, symbolic dynamics, periodic points, Lyapunov exponents
Received by editor(s): January 21, 2011
Received by editor(s) in revised form: September 2, 2012
Published electronically: November 26, 2012
Additional Notes: This work was partially supported by the NSF grant DMS–0400687 and by the ERC award ERC-2009-StG no. 239885
Article copyright: © Copyright 2012 American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication.