Nonlinear thermodynamic formalism for correspondences

Abstract

We develop a nonlinear thermodynamic formalism for correspondences on compact metric spaces. A continuous energy is evaluated on empirical measures of successive pairs of states. The measure-theoretic pressure of a stationary transition pair is the sum of its kernel entropy and the energy of the joint distribution of two successive states. For forward expansive correspondences satisfying an ergodic-abundance condition, we prove a variational principle and the existence of nonlinear equilibrium pairs. We further prove that limits of the two-coordinate projections of asymptotically maximizing Gibbs ensembles are averages of the two-coordinate distributions of equilibrium pairs. When the energy depends on finitely many potentials on the space of admissible pairs, Legendre regularity assumptions allow us to identify nonlinear equilibrium pairs among linear equilibrium pairs through a self-consistency equation and a global maximization condition. Under these assumptions, real analyticity in one dimension and uniqueness of linear equilibrium pairs imply finiteness of the nonlinear equilibrium set. Finite-state examples exhibit continuous and first-order transitions, metastability, and phases that have identical one-coordinate marginals but distinct two-coordinate distributions. A criterion based on Taylor expansions determines the associated critical exponents