Entropy density and large deviation principles without upper semi-continuity of entropy

Abstract: 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 paper, we show that the entropy map of an expanding Thurston map is upper semi-continuous if and only if the map has no periodic critical points. For all expanding Thurston maps, even in the presence of periodic critical points, we show that ergodic measures are entropy-dense and establish level-2 large deviation principles for the distribution of Birkhoff averages, periodic points, and iterated preimages. It follows that iterated preimages and periodic points are equidistributed with respect to the unique equilibrium state for an expanding Thurston map and a potential that is H\"older continuous with respect to a visual metric on $S^2$. In particular, our results answer two questions in [Li15]. The main technical tools in this paper are called subsystems of expanding Thurston maps, inspired by a translation of the notion of subgroups via Sullivan's dictionary.