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Journal of the American Mathematical Society

Published by the American Mathematical Society, the Journal of the American Mathematical Society (JAMS) is devoted to research articles of the highest quality in all areas of pure and applied mathematics.

ISSN 1088-6834 (online) ISSN 0894-0347 (print)

The 2020 MCQ for Journal of the American Mathematical Society is 4.79.

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Symbolic dynamics for surface diffeomorphisms with positive entropy
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by Omri M. Sarig PDF
J. Amer. Math. Soc. 26 (2013), 341-426 Request permission

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
  • Email: omsarig@gmail.com
  • 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
  • © Copyright 2012 American Mathematical Society
    The copyright for this article reverts to public domain 28 years after publication.
  • Journal: J. Amer. Math. Soc. 26 (2013), 341-426
  • MSC (2010): Primary 37D25; Secondary 37D35
  • DOI: https://doi.org/10.1090/S0894-0347-2012-00758-9
  • MathSciNet review: 3011417