On Sullivan's conformal measures for rational maps of the Riemann sphere

Abstract

For any rational map T:C to C the topological pressure P(t) of the function -t log mod T' mod is well defined, although its Julia set may contain critical points. The authors extend the class of Sullivan's (1982, 1983) conformal measures on Julia sets to the class of measures which are conformal except on some finite set and show that the minimal exponent for which such measures exist, the supremum of Hausdorff dimensions of ergodic T-invariant measures with positive entropy and the minimal zero of the function P(t) are equal. This result may be regarded as a version of the Bowen-McCluskey-Manning formula. A large subclass of rational maps is found for which the same statement is true for conformal measures in the sense of Sullivan.