A Ruelle operator for holomorphic correspondences

TL;DR

This paper extends thermodynamic formalism to complex holomorphic correspondences, defining topological entropy, pressure, and analyzing the Ruelle operator on the Riemann sphere, contributing to complex dynamics theory.

Contribution

It introduces a framework for thermodynamic formalism in holomorphic correspondences, including definitions of entropy and pressure, and studies the Ruelle operator in this context.

Findings

Defined topological entropy for holomorphic correspondences

Introduced pressure for continuous functions on the Riemann sphere

Analyzed the Ruelle operator on the support of the Dinh-Sibony measure

Abstract

In this paper, we extend the ideas of certain notions that one studies in thermodynamic formalism of maps to the context when the dynamics in the phase space evolves by complex holomorphic correspondences. Towards that end, we define the topological entropy of holomorphic correspondences using spanning sets. We then, define the pressure of a real-valued continuous function defined on the Riemann sphere and investigate the Ruelle operator with respect to the H\"{o}lder continuous function, however restricted on the support of the Dinh-Sibony measure.