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

In this paper, we show that each expanding Thurston map f : S → S 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 μf for f . We also show that (S , f, μf ) 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 is almost surely the weak∗ limit of atomic probability measures supported on a random backward orbit of an arbitrary point.