The equilibrium states for semigroups of rational maps

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 $\exp (\mathrm{\P }(\psi )-\psi )$-conformal measure, where P(\psi) denotes the topological pressure of \psi.