An invariant measure for rational maps. (English) Zbl 0568.58027

Freire, Alexandre; Lopes, Artur; Mañe, Ricardo

Bol. Soc. Bras. Mat. 14, No. 1, 45-62 (1983).

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

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).