Brownian Motion and the Equilibrium Measure on the Julia Set of a Rational Mapping

Abstract

It is proved that if a rational mapping has $\infty$ as a fixed point in its Fatou set, then its Julia set has positive capacity and the equilibrium measure is invariant. If $\infty$ is attracting or superattracting, then the equilibrium measure is strongly mixing, whereas if $\infty$ is neutral, then the equilibrium measure is ergodic and has entropy zero. Lower bounds for the entropy are given in the attracting and superattracting cases. If the Julia set is totally disconnected, then the equilibrium measure is Gibbs and therefore Bernoulli. The proofs use an induced action by the rational mapping on the space of Brownian paths started at $\infty$.