A variational principle for holomorphic correspondences

TL;DR

This paper develops a variational principle for holomorphic correspondences on the Riemann sphere by defining measure-theoretic entropy and pressure, extending classical dynamical systems concepts to set-valued holomorphic maps.

Contribution

It introduces a measure-theoretic entropy and pressure framework for holomorphic correspondences, establishing a variational principle analogous to classical dynamics.

Findings

Defined measure-theoretic entropy for holomorphic correspondences

Formulated pressure for continuous functions in this context

Established a variational principle for these dynamical systems

Abstract

In this paper, we consider a dynamical system on the Riemann sphere that evolves through a set-valued map, namely a holomorphic correspondence. Analogous to the investigation of the dynamics effected by a continuous map defined on a compact metric space, wherein the concept of measure-theoretic entropy of the map and its utility in defining the pressure of a function are well-studied, we define the measure-theoretic entropy of a holomorphic correspondence and use the same to define the pressure of continuous functions. These ideas naturally lead to the formulation of a variational principle in the context of the dynamics of a holomorphic correspondence.