The equilibrium states for semigroups of rational maps

TL;DR

This paper studies the dynamics of semigroups of rational maps on the Riemann sphere, establishing conditions for the existence and uniqueness of equilibrium states and conformal measures related to these systems.

Contribution

It introduces new conditions under which unique equilibrium states and conformal measures exist for semigroups of rational maps.

Findings

Existence of a unique equilibrium state under certain conditions.

Existence of a unique conformal measure related to the topological pressure.

Conditions linking dynamics and potential functions for these measures.

Abstract

We consider the dynamics of skew product maps associated with finitely generated semigroups of rational maps on the Riemann sphere. We show that under some conditions on the dynamics and the potential function \psi, there exists a unique equilibrium state for \psi and a unique $\exp(\P(\psi)-\psi)$-conformal measure, where P(\psi) denotes the topological pressure of \psi.