Harmonic, Gibbs and Hausdorff Measures on Repellers for Holomorphic Maps, I

Abstract

We prove that for a simply connected domain Q C C whose boundary a Q is self-similar there is the following dichotomy, concerning the harmonic measure X on a Q viewed from 0: Either a Q is (piecewise) real-analytic or else o is singular with respect to the Hausdorff measure A,, (notation o I A,,,) using Makarov's function 4Dc(t) = t exp(c log 1/t log log log 1/t) for some c = c(X) > 0, and X is absolutely continuous with respect to A,:, (notation X c(@). So if Q has fractal boundary then the boundary compression and the radial growth of j logIR'l j for a Riemann mapping R: D -* are as strong, respectively as fast, as permitted by Makarov's theory. We prove that c(X) = ,2a 2/X for some asymptotic variance a 2 for a sequence of weakly dependent random variables and a Lyapunov characteristic exponent X. This includes the case where a Q is a mixing repeller (in Ruelle's sense) for a holomorphic map f defined on its neighbourhood, the case a Q is a quasi-circle, invariant under the action of a quasi-Fuchsian group (for a pair of isomorphic, compact surface, Fuchsian groups) and the cases of the boundary of the snowflake and, more generally of Carleson's fractal Jordan curves. This dichotomy is partially deduced from the dichotomy concerning Gibbs measures for Holder continuous functions on an arbitrary mixing repeller X C C for a holomorphic map. Either i I AK where K is the Hausdorff dimension of y