Equilibrium States for Expanding Thurston Maps

In this paper, we use the thermodynamical formalism to show that there exists a unique equilibrium state ${\mu }_{\varphi }$ for each expanding Thurston map $f: S^2\rightarrow S^2$togetherwithareal-valuedH\ddot{o}ldercontinuouspotential$\phi$\mathrm{.}Herethesphere$S^2$isequippedwithanaturalmetricinducedby$f$, called a visual metric. We also prove that identical equilibrium states correspond to potentials which are co-homologous upto a constant, and that the measure-preserving transformation $f$ of the probability space $(S^2,{\mu }_{\varphi })$ is exact, and in particular, mixing and ergodic. Moreover, we establish versions of equidistribution of preimages under iterates of $f$, and a version of equidistribution of a random backward orbit, with respect to the equilibrium state. As a consequence, all the above results hold for a postcritically-finite rational map with no periodic critical points on the Riemann sphere equipped with the chordal metric.