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Ergodic theory of expanding Thurston maps


Thurston maps are topological generalizations of postcritically-finite rational maps. More precisely, a Thurston map $f\colon S^2\rightarrow S^2$ is a (non-homeomorphic) branched covering map on a topological 2-sphere $S^2$ whose critical points are all preperiodic. This thesis provides a comprehensive study of the measure of maximal entropy, as well as a more general class of invariant measures, called equilibrium states, for expanding Thurston maps. In particular, given an expanding Thurston map $f\colon S^2 \rightarrow S^2$, we present a large class of equidistribution results for iterated preimages and (pre)periodic points with respect to the unique measure of maximal entropy by first establishing a formula for the number of fixed points. The formula states that $f$ has exactly $1+\operatorname{deg} f$ fixed points, counted with appropriate weights, where $\operatorname{deg} f$ denotes the topological degree of the map $f$. We then use the thermodynamical formalism to show that there exists a unique equilibrium state $\mu_\phi$ for $f$ together with a real-valued H\"{o}lder 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 a version of equidistribution of a random backward orbit with respect to the equilibrium state. After proving that $f$ is asymptotically $h$-expansive if and only if it has no periodic critical points, and that no expanding Thurston map is $h$-expansive, we obtain certain large deviation principles for iterated preimages and periodic points under the additional assumption that $f$ has no periodic critical points. This enables us to obtain general equidistribution results for iterated preimages and periodic points with respect to the equilibrium states under the same assumption on $f$. We also get the existence of equilibrium states for such $f$ and any continuous real-valued potential.

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