A Ruelle operator for holomorphic correspondences

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.