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An invariant measure for rational maps. (English) Zbl 0568.58027
Let $${\bar {\mathbb{C}}}$$ be the Riemann sphere and $$f: {\mathbb{C}}\hookleftarrow$$ an analytic endomorphism of degree $$d\geq 2$$ (i.e. a rational function $$f=P/Q$$, where P and Q are polynomials without common roots). Given $$a\in {\bar {\mathbb{C}}}$$ denote $$z_ i^{(n)}(a)$$, $$i=1,...,d^ n$$, the roots of the equation $$f^ n(z)=a$$, and let $$\delta_ i^{(n)}(a)$$ be the Dirac probability supported at $$z_ i^{(n)}(a)$$. Define $$\mu_ n=d^{-n}\sum_{i}\delta_ i^{(n)}(a)$$. In this paper it is proved that there exists an f-invariant probability $$\mu$$, whose support is exactly the Julia set of f, satisfying $$\mu =\lim_{n\to +\infty}\mu_ n$$ in the weak topology, for all $$a\in {\bar {\mathbb{C}}}$$ with only two exceptions at most, that can be explicitly and easily characterized. Moreover (f,$$\mu)$$ is exact and $$\mu$$ is the unique f-invariant probability such that $$\mu (f(A))=d\mu (A)$$ for every Borel set A such that f/$$\Delta$$ is injective. The authors conjecture that (f,$$\mu)$$ is measure theoretically equivalent to the one-sided Bernoulli shift $$\sigma$$ : $$B^+(1/d,...,1/d)\hookleftarrow$$. It was later proved [the third author, Ergodic Theory Dyn. Syst. 5, 71-88 (1985)] that there exists $$m>0$$ such that $$(f^ m,\mu)$$ is equivalent to $$\sigma^ m: B^+(1/d,...,1/d)\hookleftarrow$$.

##### MSC:
 37A99 Ergodic theory 28D05 Measure-preserving transformations 58C35 Integration on manifolds; measures on manifolds
##### Keywords:
entropy; Dirac probability; Julia set
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##### References:
 [1] H. Brolin,Invariant sets under iteration of rational functions, Arkiv für Mathematik, Band G, nr. 6 (1966). [2] M. Fatou,Sur les fonctions satisfaisant certes equations fonctionelles, Bull. de la Soc. Math. de France 47–48 (1919–1920). [3] M. Gromov,On the entropy of holomorphic maps, Preprint I.H.E.S. [4] G. Julia,Sur l’iteration des fonctions rationelles, Journal de Mathematique Pure et Appliquées, 8 (1918).
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