Thermodynamic formalism for correspondences

TL;DR

The paper extends the core ideas of thermodynamic formalism to the setting of correspondences, defining topological pressure and measure-theoretic entropy through transition kernels and orbit-shift dynamics. It establishes a Variational Principle for forward expansive correspondences and constructs equilibrium states, including a Ruelle–Perron–Frobenius framework with Gibbs properties, under two complementary sets of hypotheses: (i) specification and Bowen summability, and (ii) openness, distance-expansion, strong transitivity, and Hölder continuity. The results apply to holomorphic/anti-holomorphic correspondences, including Lee–Lyubich–Makarov–Mukherjee anti-holomorphic matings and the family f_c=z^{q/p}+c, and yield equidistribution of backward orbits with respect to the unique equilibrium states. By connecting correspondences to Markov kernels and shift spaces, the work unifies complex-dynamics matings with probabilistic thermodynamic techniques, providing a robust toolkit for invariant measures and statistical properties in multi-valued dynamical systems.

Abstract

In this article, we investigate the Variational Principle and develop thermodynamic formalism for correspondences. We define the measure-theoretic entropy for transition probability kernels and topological pressure for correspondences. Based on these two notions, we establish the following results: The Variational Principle holds and equilibrium states exist for continuous potential functions, provided that the correspondence satisfies some expansion property called forward expansiveness. If, in addition, the correspondence satisfies the specification property and the potential function is Bowen summable, then the equilibrium state is unique. On the other hand, for a distance-expanding, open, strongly transitive correspondence and a Hölder continuous potential function, there exists a unique equilibrium state, and the backward orbits are equidistributed. Furthermore, we investigate the Variational Principle for general correspondences. In complex dynamics, we establish the Variational Principle for the Lee-Lyubich-Markorov-Mukherjee anti-holomorphic correspondences, which are matings of some anti-ho\-lo\-mor\-phic rational maps with anti-Hecke groups and are not forward expansive. We also show a Ruelle-Perron-Frobenius theorem for a family of hyperbolic holomorphic correspondences of the form $\boldsymbol{f}_c (z)= z^{q/p}+c$.

Theorems & Definitions (152)

• Conjecture : Variational Principle for correspondences
• Theorem A
• Theorem B
• Theorem C
• Theorem D
• Theorem E
• Theorem F
• Theorem G
• Proposition 3.1: LMM24
• Lemma 3.2