Harmonic measure and polynomial Julia sets
Let \(\Omega\) be a domain in the Riemann sphere \(\widehat{\mathbb C}\), let \(E\subset \widehat{\mathbb C}\), and let \(a\in \Omega\). Let \(\omega(a,E,\Omega)\) denote the harmonic measure in \(\Omega\) of the set \(E\) evaluated at \(a\). An important theorem of \textit{P. W. Jones} and \textit{T. H. Wolff} [Acta Math. 161, 131--144 (1988; Zbl 0667.30020 )] asserts that the Hausdorff dimension of the support of \(\omega\) is at most \(1\). In the paper under review the authors study stronger estimates involving the dimension of harmonic measure. Let \[ \alpha_{a,\Omega}(z)=\liminf_{\delta\to 0}\frac{\log \omega(a,\{w:| w-z| \leq \delta\},\Omega)}{\log \delta}. \] The universal dimension spectrum of harmonic measure is the function \[ \Phi(\alpha)=\sup \text{dim} \{z:\alpha_{a,\Omega}(z)\leq \alpha\}, \] where dim denotes Hausdorff dimension and the supremum is taken over all planar domains \(\Omega\) and all evaluation points \(a\in \Omega\). Let \(\Phi_{\text{sc}}(\alpha)\) be the corresponding quantity with the supremum taken over all simply connected domains. The authors conjecture that for \(\alpha\geq 1\), \(\Phi(\alpha)=\Phi_{\text{sc}}(\alpha)\). They propose the application of methods of Complex Dynamics for the study of the conjecture. The main result of the paper is that the conjecture is true when \(\partial \Omega\) is the Julia set of a polynomial such that the orbits of all critical points escape to \(\infty\). The proof of this deep theorem uses earlier results of \textit{N. G. Makarov} [Algebra Anal. 10, 1--62 (1998); translation in St. Petersbg. Math. J. 10, 217--268 (1999; Zbl 0909.30016)] (involving the pressure function of a polynomial and the universal integral means spectrum), and a construction of \textit{B. Branner} and \textit{J. H. Hubbard} [Acta Math. 160, 143--206 (1988; Zbl 0668.30008)]. Recently I. Binder and P. Jones proved the conjecture mentioned above.
scientific article; zbMATH DE number 1933168
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