Conformal Measures For Rational Functions Revisited

. We show that the set of conical points of a rational function of the Riemann sphere supports at most one conformal measure. We then study the problem of existence of such measures and their ergodic properties constructing Markov partitions on increasing subsets of sets of conical points and applying ideas o f the thermodynamic formalism. 1 Research supported by NSF Grant DMS 9502952 2 Research supported by NSF Grant DMS 9502952 x1. Conical Points. The interesting ergodic properties of a rational map take place on its Julia set, see the books [Be] and [CG] for definitions and some basic properties. ?From now on let f : CI ! CI be a rational function of degree d 2. Following [U2], by analogy with the theory of Kleinian groups, we call a point z 2 J(f ), the Julia set of f , a conical point of f if there exists ffi ? 0 and an infinite increasing sequence n k 1 of positive integers such that for each k there exists f \Gamman k z , holomorphic inverse branches of f nk , defin..