  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 introduce subsystems of
expanding Thurston maps motivated via Sullivan's dictionary as analogs of
certain subgroups. We develop thermodynamic formalism to prove the Variational
Principle and the existence of equilibrium states for strongly irreducible
subsystems and real-valued H\"older continuous potentials. Here, the sphere
$S^{2}$ is equipped with a natural metric, called a visual metric, introduced
by M. Bonk and D. Meyer. As an application, we establish large deviation
asymptotics for expanding Thurston maps.
