  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 study subsystems of
expanding Thurston maps motivated via Sullivan's dictionary as analogs of some
subgroups of Kleinian groups. We prove the uniqueness and various ergodic
properties of the equilibrium states for strongly primitive subsystems and
real-valued H\"older continuous potentials, and establish the equidistribution
of preimages of subsystems with respect to the equilibrium states. Here, the
sphere $S^{2}$ is equipped with a natural metric, called a visual metric,
introduced by M. Bonk and D. Meyer. As a result, for strongly primitive
subsystems of expanding Thurston maps without periodic critical points, we
obtain a level-$2$ large deviation principle for Birkhoff averages and iterated
preimages.
