Ergodic and Bernoulli properties of analytic maps of complex projective space

Koss, Lorelei

doi:10.1090/S0002-9947-02-02725-3

Ergodic and Bernoulli properties of analytic maps of complex projective space
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Trans. Amer. Math. Soc. 354 (2002), 2417-2459 Request permission

Abstract:

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.

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