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Ergodic and Bernoulli properties of analytic maps of complex projective space


Author: Lorelei Koss
Journal: Trans. Amer. Math. Soc. 354 (2002), 2417-2459
MSC (2000): Primary 37A25, 37A35, 37F10
DOI: https://doi.org/10.1090/S0002-9947-02-02725-3
Published electronically: February 7, 2002
MathSciNet review: 1885659
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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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Additional Information

Lorelei Koss
Affiliation: Department of Mathematics and Computer Science, Dickinson College, P.O. Box 1773, Carlisle, Pennsylvania 17013
Email: koss@dickinson.edu

DOI: https://doi.org/10.1090/S0002-9947-02-02725-3
Received by editor(s): March 22, 1999
Received by editor(s) in revised form: March 14, 2000
Published electronically: February 7, 2002
Additional Notes: Supported in part by GAANN (Graduate Assistance in Areas of National Need) Fellowship
Article copyright: © Copyright 2002 American Mathematical Society

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