## Ergodic and Bernoulli properties of analytic maps of complex projective space

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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.## References

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## Additional Information

**Lorelei Koss**- Affiliation: Department of Mathematics and Computer Science, Dickinson College, P.O. Box 1773, Carlisle, Pennsylvania 17013
- MR Author ID: 662937
- Email: koss@dickinson.edu
- 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
- © Copyright 2002 American Mathematical Society
- 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
- MathSciNet review: 1885659