On the dimension of harmonic measure of Cantor repellers (Q1310135)

Let \(U,U_ 1,\dots,U_ d\) be simply connected domains in the complex plane with real analytic boundaries such that the closure of \(U_ j\) is contained in \(U\) for \(j=1,\dots,d\). Let \(f:\cup^ d_{j=1} U_ j \to U\) be an analytic map whose restriction to \(U_ j\) is a bijection between \(U_ j\) and \(U\) for each \(j\). Denote by \(f^ n\) the \(n\)th iterate of \(f\). The set \(J = \{x:f^ n(x) \in \cup^ d_{j=1} U_ j\), \(n=0,1,\dots\}\) is called a Cantor repeller. (Julia sets of certain polynomials are Cantor repellers.) It is shown in this paper that if \(U\) is symmetric with respect to the real axis and \(f(\overline z) = \overline {f(z)}\), then the dimension of the harmonic measure on \(J\) is strictly less than the Hausdorff dimension of \(J\). For polynomials, this is due to \textit{A. Zdunik} [Invent. Math. 99, 627-649 (1990)].