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Transactions of the American Mathematical Society

Published by the American Mathematical Society since 1900, Transactions of the American Mathematical Society is devoted to longer research articles in all areas of pure and applied mathematics.

ISSN 1088-6850 (online) ISSN 0002-9947 (print)

The 2020 MCQ for Transactions of the American Mathematical Society is 1.48.

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Periodic points and the measure of maximal entropy of an expanding Thurston map
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by Zhiqiang Li PDF
Trans. Amer. Math. Soc. 368 (2016), 8955-8999 Request permission

Abstract:

In this paper, we show that each expanding Thurston map $f\colon S^2\!\rightarrow S^2$ has $1+\deg f$ fixed points, counted with appropriate weight, where $\deg f$ denotes the topological degree of the map $f$. We then prove the equidistribution of preimages and of (pre)periodic points with respect to the unique measure of maximal entropy $\mu _f$ for $f$. We also show that $(S^2,f,\mu _f)$ is a factor of the left shift on the set of one-sided infinite sequences with its measure of maximal entropy, in the category of measure-preserving dynamical systems. Finally, we prove that $\mu _f$ is almost surely the weak$^*$ limit of atomic probability measures supported on a random backward orbit of an arbitrary point.
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Additional Information
  • Zhiqiang Li
  • Affiliation: Department of Mathematics, University of California Los Angeles, Los Angeles, California 90095-1555
  • MR Author ID: 842169
  • Email: lizq@math.ucla.edu
  • Received by editor(s): April 13, 2014
  • Received by editor(s) in revised form: December 3, 2014
  • Published electronically: March 18, 2016
  • Additional Notes: The author was partially supported by NSF grant DMS-1162471.
  • © Copyright 2016 American Mathematical Society
  • Journal: Trans. Amer. Math. Soc. 368 (2016), 8955-8999
  • MSC (2010): Primary 37D20; Secondary 37B99, 37F15, 37F20, 57M12
  • DOI: https://doi.org/10.1090/tran/6705
  • MathSciNet review: 3551595