Ergodic and Bernoulli Properties of Analytic Maps of Complex Projective Space

We examine the measurable ergodic theory of analytic maps F of complex projective space. We focus on two different classes of maps, Ueda maps of Pn, and rational maps of the sphere with parabolic orbifold and Julia set equal to the entire sphere. We construct measures which are invariant, ergodic, weakor strong-mixing, exact, or automorphically Bernoulli with respect to these maps. We discuss topological pressure and measures of maximal entropy (hμ(F ) = htop(F ) = log(deg F )). We find analytic maps of P1 and P2 which are one-sided Bernoulli of maximal entropy, including examples where the maximal entropy measure lies in the smooth measure class. Further, we prove that for any integer d > 1, there exists a rational map of the sphere which is one-sided Bernoulli of entropy log d with respect to a smooth measure.