Ergodic theory of equilibrium states for rational maps

Abstract

Let T be a rational map of degree d>or=2 of the Riemann sphere C=C union ( infinity ). The authors develop the theory of equilibrium states for the class of Holder continuous functions f for which the pressure is larger than sup f. They show that there exist a unique conformal measure (reference measure) and a unique equilibrium state, which is equivalent to the conformal measure with a positive continuous density. The associated Perron-Frobenius operator acting on the space of continuous functions is almost periodic and they show that the system is exact with respect to the equilibrium measure.