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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