Abstract:

Expanding Thurston maps were introduced by Bonk and Meyer, motivated by complex dynamics and Cannon’s conjecture in geometric group theory via Sullivan’s dictionary. This talk focuses on the ergodic theory of these maps and presents recent advances regarding their statistical properties and the role of subsystems. We establish 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. Consequently, periodic points and iterated preimages are shown to be equidistributed with respect to the equilibrium state. The proofs employ a thermodynamic formalism extended to subsystems of expanding Thurston maps, which naturally arise when studying dynamics on subsets. This talk is based on joint work with Zhiqiang Li.