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 ${\mathbb P}^{n}$, and rational maps of the sphere with parabolic orbifold and Julia set equal to the entire sphere. We construct measures which are invariant, ergodic, weak- or strong-mixing, exact, or automorphically Bernoulli with respect to these maps. We discuss topological pressure and measures of maximal entropy ( $h_{\mu}(F) = h_{top}(F)= \log(\deg F)$). We find analytic maps of ${\mathbb P}^1$ and ${\mathbb P}^2$ 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.