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Periodic points and the measure of maximal entropy of an expanding Thurston map

Author: Zhiqiang Li
Journal: Trans. Amer. Math. Soc. 368 (2016), 8955-8999
MSC (2010): Primary 37D20; Secondary 37B99, 37F15, 37F20, 57M12
Published electronically: March 18, 2016
MathSciNet review: 3551595
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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

Keywords: Expanding Thurston map, postcritically finite map, fixed point, periodic point, preperiodic point, visual metric, measure of maximal entropy, maximal measure, equidistribution.
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.
Article copyright: © Copyright 2016 American Mathematical Society

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