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On the Bernoulli property for rational maps

Published online by Cambridge University Press:  19 September 2008

Ricardo Mañé
Ricardo Mañé
Affiliation:
Instituto de MatemáticaPura e Aplicada, Rio de Janeiro, Brasil
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Abstract

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Every rational function ƒ with degree has a unique invariant probability ƒƒ that maximizes entropy. It has been conjectured that the system (ƒ, μƒ) is equivalent to the one sided Bernoulli shift . In this paper we prove that there exists m >0 such that (ƒm, ƒƒ) is equivalent to .

Type
Research Article
Information
Ergodic Theory and Dynamical Systems , Volume 5 , Issue 1 , March 1985 , pp. 71 - 88
Copyright
Copyright © Cambridge University Press 1985

References

REFERENCES

[1]Brolin, H.. Invariant sets under iteration of rational functions. Arkiv für Matematik, Band G, 6 (1966).Google Scholar
[2]Freire, A., Lopes, A. & Mañé, R.. An invariant measure for rational maps. Bol. Soc. Bras. Mat. 14 (1983) 45–62.Google Scholar
[3]Gromov, M.. On the entropy of holomorphic maps. Preprint, IHES.Google Scholar
[4]Hille, E.. Analytic Functions Theory.Google Scholar
[5]Ljubich, V.. Entropy properties of rational endomorphisms of the Riemann sphere. Ergod. Th. & Dynam. Sys. 3 (1983), 351–386.Google Scholar
[6]Mañé, R.. On the uniqueness of the maximizing measure for rational maps. Bol. Soc. Bras. Mat. 14 (1983) 27–43.CrossRefGoogle Scholar
[7]Misiurewicz, M. & Przyticki, F.. Topological entropy and degree of smooth mappings. Bull. Acad. Pol. Sci. 25 (1977), 573–574.Google Scholar
[8]Ornstein, D.. Factors of Bernoulli shifts are Bernoulli. Adv. in Math. 5 (1970), 349–364.Google Scholar