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