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

In this paper, we show that each expanding Thurston map f:S2→S2f\colon S^2\!\rightarrow S^2 has 1+deg⁡f1+\deg f fixed points, counted with appropriate weight, where deg⁡f\deg f denotes the topological degree of the map ff. We then prove the equidistribution of preimages and of (pre)periodic points with respect to the unique measure of maximal entropy μf\mu _f for ff. We also show that (S2,f,μf)(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 μf\mu _f is almost surely the weak∗^* limit of atomic probability measures supported on a random backward orbit of an arbitrary point.