ERGODIC-THEORY OF EQUILIBRIUM STATES FOR RATIONAL MAPS

Let T be a rational map of degree d greater-than-or-equal-to 2 of the Riemann sphere CBAR = C union {infinity}. We develop the theory of equilibrium states for the class of Holder continuous functions closed-integral for which the pressure is larger than sup closed-integral. We 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 we show that the system is exact with respect to the equilibrium measure.