On the dimensions of Cantor Julia sets of rational maps
Let \(f\) be a rational map of degree \(d \geq 2\), and denote by \(f^n\) the \(n\)-th iterate of \(f\). The Fatou set \(\mathcal{F}(f)\) is the maximal open set in which the sequence \((f^n)\) is a normal family in the sense of Montel, while the complement of \(\mathcal{F}(f)\) in \(\widehat{\mathbb C}\) is the Julia set \(\mathcal{J}(f)\). A probability measure \(\mu\) supported on \(\mathcal{J}(f)\) is called a conformal measure with exponent \(\alpha\) (\(0 < \alpha \leq 2\)) for \(f\) if the equation \[ \mu(f(A)) = \int_A |f'(z)|^\alpha\,d\mu \] holds for every Borel set \(A \subset \mathcal{J}(f)\) such that \(f_{|A}\) is injective. The existence of a conformal measure for a rational map was proved by \textit{D.~Sullivan} [Lect. Notes Math. 1007, 725--752 (1983; Zbl 0524.58024)]. All the conformal measures of \(f\) form a closed set in the sense of weak convergence. So there exists a conformal measure whose exponent is the infimum of all the exponents of conformal measures of \(f\). This minimal exponent is called the conformal dimension of \(\mathcal{J}(f)\) and is denoted by \(\alpha_*(f)\). Sullivan also proved that for any hyperbolic rational map \(f\) there is a unique conformal measure \(\mu\) supported on \(\mathcal{J}(f)\), whose exponent \(\delta\) equals the Hausdorff dimension of \(\mathcal{J}(f)\). This implies that the conformal dimension of \(\mathcal{J}(f)\) equals the Hausdorff dimension of \(\mathcal{J}(f)\) in the expanding case. Therefore, it is natural to ask whether this also holds for a general rational map. This problem has been studied for several classes of rational maps in the past. In this paper the author proves that \(\alpha_*(f) = \dim_H(\mathcal{J}(f))\) for every rational map \(f\) with a Cantor Julia set satisfying the following two properties: (1) \(\mathcal{F}(f)\) is an attracting domain, that means that \(\mathcal{F}(f)\) contains an attracting fixed point of \(f\). (2) \(\mathcal{J}(f)\) contains no persistently recurrent critical points. The main tools of the proof are Branner-Hubbard puzzles and some distortion lemmas which are concluded from the famous Koebe distortion theorem.
scientific article; zbMATH DE number 6156167
scientific article