Entropy properties of rational endomorphisms of the Riemann sphere. (English) Zbl 0537.58035

Lyubich, M. Yu.

Ergodic Theory Dyn. Syst. 3, 351-385 (1983).

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