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Quelques propriétés ergodiques des applications rationnelles (Some ergodic properties of rational maps). (French) Zbl 0567.58016
The author shows how dimension, entropy and exponent are related for rational maps of \({\bar {\mathbb{C}}}\) which have a finite invariant measure; a variational principle is used to characterize the absolutely continuous invariant measure which arises. For such a map f with invariant probability measure m satisfying \(| f^{n'}| \to +\infty\) m almost surely he proves the equivalence of: (i) m is absolutely continuous with respect to Lebesgue measure, (ii) \(\lim_{\delta \to 0} \log m(U(z,\delta))/\log \delta =2\), m a. s., (iii) \(h_ m(f)=2\int \log | f'| dm\), (iv) m is equivalent to Lebesgue measure, (v) \(0<\inf_{\delta}(m(U(z,\delta))/\delta^ 2)\leq \sup (m(U(z,\delta))/\delta^ 2)<\infty\), m a. s. In particular (\({\bar {\mathbb{C}}},m,f)\) is ergodic. The proof is sketched with frequent reference to his earlier work and that of A. Manning. A related result is stated without proof.
Reviewer: G.R.Goodson

MSC:
37A99 Ergodic theory
28D20 Entropy and other invariants
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