Thurston maps are a class of branched covering maps on the 2-sphere that arose in W. Thurston's characterization of postcritically finite rational maps. By imposing a natural expansion condition, M. Bonk and D. Meyer investigated a subclass of Thurston maps known as expanding Thurston maps, which turned out to enjoy nice topological, metric, and dynamical properties.

Thermodynamical formalism has been a powerful tool, for many classical dynamical systems, to investigate invariant measures whose Jacobian functions have strong regularity properties.

In this talk, we will first introduce expanding Thurston maps with some motivation from their connection to other topics of mathematics. We will then use thermodynamical formalism to sketch a proof for the existence, uniqueness, and exactness of equilibrium states for expanding Thurston maps and Hölder continuous potentials.

If time permits, we will also show that an expanding Thurston map is asymptotically $h$-expansive if and only if it has no periodic critical points, which suggests the subtlety of our notion of expansion.