Thermodynamic formalism for subsystems of expanding Thurston maps and large deviations asymptotics

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 some subgroups of Kleinian groups. We use thermodynamic formalism to prove the Variational Principle and the existence of equilibrium states for strongly irreducible subsystems and real-valued Hölder 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.