Xianghui Shi (Peking University)
Expanding Thurston maps were introduced by M. Bonk and D. Meyer with motivation from complex dynamics and Cannon’s conjecture from geometric group theory via Sullivan’s dictionary. In this talk, we explore the ergodic theory surrounding these maps and present some recent advancements. We demonstrate that the entropy map is upper semi-continuous if and only if the map has no periodic critical points. Furthermore, we show that ergodic measures are entropy-dense and derive level-2 large deviation principles for Birkhoff averages, periodic points, and iterated preimages. The main tools used in the proof are called subsystems of expanding Thurston maps, which naturally emerge in the study of dynamics on subsets. We develop the thermodynamic formalism for subsystems and establish the existence, uniqueness, and ergodic properties of equilibrium states for Hölder continuous potentials. This is joint work with Zhiqiang Li.