Connections between Complex Dynamics and Ergodic Theory

Abstract: While the Julia sets of rational maps of the sphere usually conjure up images of interesting topological features, they also possess many measure theoretic properties worth studying. Every rational map has several distinguished invariant measures: one is the unique invariant measure of maximal entropy and the other is a more geometric measure called conformal measure. Only in rare instances do they coincide. There is often a nonatomic invariant measure equivalent to conformal measure, sometimes infinite and sometimes finite. We give families of examples of these. We also mention one-sided Bernoulli properties and which maps rule out one-sided Bernoulli behavior.