The Convex-Analytic Structure of Thermodynamic Equilibrium: Pressure, Subdifferentials, and Phase Transitions

Abstract

We develop the convex-analytic structure of the thermodynamic formalism for continuous maps on compact metric spaces. The pressure functional is the Legendre-Fenchel transform of the negative entropy, and the biconjugate recovery of the entropy from the pressure establishes a complete duality. Equilibrium states are elements of the subdifferential of the pressure, uniqueness of equilibrium states corresponds to G\^{a}teaux differentiability, and first-order phase transitions correspond to non-differentiability. For systems with specification and H\"{o}lder potentials, the pressure is Fr\'{e}chet differentiable in the H\"{o}lder norm, and the second derivative of the pressure equals the asymptotic variance of the Birkhoff sums. We prove a universal variational principle that unifies the classical additive, the subadditive, and the relative variational principles through a single theorem on convex functionals satisfying convexity, lower semi-continuity, coercivity, and cocycle invariance. Extensions to systems with the specification property and to non-compact spaces under coercivity conditions are included, with applications to countable Markov shifts via Sarig's recurrence classification. This Part constitutes Part II of a six-part series on the thermodynamic formalism for hyperbolic dynamical systems.