# zbMATH — the first resource for mathematics

Entropy properties of rational endomorphisms of the Riemann sphere. (English) Zbl 0537.58035
In this paper the existence of a unique measure of maximal entropy for rational maps $$f(z)=P(z)/Q(z)$$ of the Riemann sphere $$S^ 2$$ is proved. The invariant measure is supported on the Julia set of f and is mutually singular with Lebesgue measure of the sphere. It is also proved the topological entropy of a rational map f equals ln deg f, and this fact is used to prove that the constructed measure is of maximal entropy.
Reviewer: I.P.Malta

##### MSC:
 37A99 Ergodic theory 28D20 Entropy and other invariants
##### Keywords:
maximal entropy; Julia set; topological entropy