Variational principles of topological pressure for correspondences

TL;DR

This paper extends the variational principles of topological pressure for correspondences, establishing new formulas and conditions, and explores the differentiability and equilibrium states of the pressure.

Contribution

It introduces two types of variational principles for correspondences, broadening the theoretical framework beyond previous forward expansive cases.

Findings

Derived variational principles for a class of correspondences.

Established an abstract variational principle without forward expansiveness.

Investigated differentiability and equilibrium states of topological pressure.

Abstract

Recently, Li, Li and Zhang introduced the topological pressure for correspondences and measure-theoretic entropy for transition probability kernels. Building thereon, they established a variational principle for correspondences satisfying the forward expansiveness condition. In this work, we extend this research by deriving two types of variational principles: (i) For a class of correspondences, the topological pressure equals the supremum of the measure-theoretic pressures over extreme points of invariant measures. (ii) An abstract variational principle holds for general correspondences without requiring forward expansiveness. Furthermore, the differentiability and equilibrium states of the topological pressure for correspondences are also investigated.