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Expanding Thurston maps were introduced by M. Bonk and D. Meyer withmotivation from complex dynamics and Cannon's conjecture from geometric grouptheory via Sullivan's dictionary. In this paper, we introduce subsystems ofexpanding Thurston maps motivated via Sullivan's dictionary as analogs ofcertain subgroups. We develop thermodynamic formalism to prove the VariationalPrinciple and the existence of equilibrium states for strongly irreduciblesubsystems and real-valued H\"older continuous potentials. Here, the sphere$S^{2}$ is equipped with a natural metric, called a visual metric, introducedby M. Bonk and D. Meyer. As an application, we establish large deviationasymptotics for expanding Thurston maps.

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