Some Results in Thermodynamic Formalism

Abstract

This dissertation investigates several key questions at the intersection of dynamical systems, computability theory, and thermodynamic formalism. In the symbolic setting, we establish novel results regarding the statistical properties of equilibrium states, deriving bounds on probabilities of configurations and relating these bounds to the Gibbs property through the homoclinic relation. Additionally, we examine the computability of thermodynamic quantities such as pressure, ground state energy, and residual entropy. We show that topological pressure is computable from above for general subshifts and computable for strongly irreducible shifts, with similar results extending to ground state energy and residual entropy. Extending beyond subshifts, we explore freezing phase transitions in general dynamical systems. We show that for a given invariant measure 𝜇, it is no more restrictive that 𝜇 is the freezing state for some potential than it is for 𝜇 to be the equilibrium state for some potential. Additionally we show that the set of freezing potentials is dense in the space of all potentials when the entropy map is upper semi-continuous. Together, these results contribute to the existing literature on thermodynamic formalism, deepening our understanding of equilibrium and freezing states while advancing the study of computability in this context.