  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.
