Periodic Points and the Measure of Maximal Entropy of an Expanding Thurston Map

Abstract

In this paper, we show that each expanding Thurston map $f:S^2\to S^2$ has $1+degf$ fixed points, counted with appropriate weight, where $degf$ 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${}^{\ast }$ limit of atomic probability measures supported on a random backward orbit of an arbitrary point.

62 pages, 5 figures. Quotes some technical statements from arXiv:1009.3647 in the background discussion

References in corpus (2)

• Equilibrium States for Expanding Thurston Maps
• Thurston maps and asymptotic upper curvature