  In this paper, we use the thermodynamical formalism to show that there exists
a unique equilibrium state $\mu_\phi$ for each expanding Thurston map $f:
S^2\rightarrow S^2$ together with a real-valued H\"older continuous potential
$\phi$. Here the sphere $S^2$ is equipped with a natural metric induced by $f$,
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 $(S^2,\mu_\phi)$
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.
