Recurrent critical points and typical limit sets for conformal measures
Let \(f\) be a rational function. A measure \(\mu\) on the Riemann sphere is called conformal of exponent \(\alpha\) if \(\mu(f(A))=\int_{A}|f'(z)|^{\alpha}d\mu\) for all measurable sets \(A\) where \(f\) is injective. Suppose that \(\mu\) is a conformal measure of exponent \(\alpha\) and that the Julia set \(J(f)\) of \(f\) contains a subset \(L\) of positive \(\mu\)-measure such that the points in \(L\) are not preimages of critical or parabolic points and such that the \(\omega\)-limit sets of the points in \(L\) are not contained in the union of the \(\omega\)-limit sets of the recurrent critical points. It is shown that then \(f\) is exact, ergodic and conservative, that \(\mu\) is unique, non-atomic and supp\((\mu)=J(f)\), and that \(\omega(z)=J(f)\) for \(\mu\)-almost every point \(z\). Moreover, \(\alpha\) is the minimal exponent for which a conformal measure with support on \(J(f)\) exists. This result extends previous results by various other authors, e.g. results by \textit{M. Yu. Lyubich} [Dokl. Akad. Nauk SSSR 268; 29-32 (1983; Zbl 0595.30034)] and \textit{C. T. McMullen} [Complex Dynamics and Renormalization, Princeton, NJ (1994)] dealing with the case that \(\mu\) is the Lebesgue measure.
scientific article; zbMATH DE number 1545863
scientific article