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
Related
- Thermodynamic Formalism on SRB Measures of Compact Convex Projective manifolds, Geometry and Counting
- Thermodynamic formalism for partially hyperbolic endomorphisms with critical set
- Thermodynamic formalism for non-compact systems with expansivity and specification
- Thermodynamic Formalism Out of Equilibrium Part III: Local Limit Theorem and Statistical Properties
- Thermodynamic formalism and multifractal analysis of Birkhoff averages for parabolic rational maps
BibTeX
@misc{tang2026nonlinear,
title = {{Nonlinear thermodynamic formalism for correspondences}},
author = {Dingxuan Tang and Rui Yang and Zhiming Li},
howpublished = {arXiv},
year = {2026},
month = sep,
doi = {10.48550/arxiv.2609.25915},
eprint = {2609.25915},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2609.25915},
note = {CC0},
}
RIS
TY - UNPB
TI - Nonlinear thermodynamic formalism for correspondences
AU - Dingxuan Tang
AU - Rui Yang
AU - Zhiming Li
PY - 2026
DA - 2026/09/22
PB - arXiv
DO - 10.48550/arxiv.2609.25915
UR - https://arxiv.org/abs/2609.25915
AB - 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
ER -