Abstract
We prove that for a simply connected $\Omega \subset \mathbf{C}$ whose boundary $\partial\Omega$ is self-similar there is the following dichotomy, concerning the harmonic measure $\omega$ on $\partial\Omega$ viewed from $\Omega$: Either $\partial\Omega$ is (piecewise) real-analytic or else $\omega$ is singular with respect to the Hausdorff measure
$\Lambda_{\Phi_c}$ (notation $\omega \perp \Lambda_{\Phi_c})$ using Makarov’s function $\Phi_c(t) = t\exp(c\sqrt{\log 1/t \log\log\log 1/t})$ for some $c=c(\omega)>0$, and $\omega$ is absolutely continuous with respect to $\Lambda_{\Phi_c}$ (notation $\omega\ll \Lambda_{\Phi_c})$ for every $c>c(\omega)$.
So if $\Omega$ has “fractal” boundary then the boundary compression and the “radial growth” of $|\log|R’| \mid$ for a Riemann mapping $R: \mathbf{D} \to \Omega$ are as strong, respectively as fast, as permitted by Markov’s theory. We prove that $c(\omega) = \sqrt{2\sigma^2/\chi}$ for some asymptotic variance $\sigma^2$ for a sequence of weakly dependent random variables and a Lyapunov characteristic exponent $\chi$.
This includes the case where $\partial\Omega$ is a mixing repeller (in Ruelle’s sense) for a holomorphic map $f$ defined on its neighbourhood, the case $\partial\Omega$ 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.
The dichotomy is partially deduced from the dichotomy concerning Gibbs measures for Hölder continuous functions on an arbitrary mixing repeller $X \subset \mathbf{C}$ for a holomorphic map. Either $\mu \perp \Lambda_\kappa$ where $\kappa$ is the Hausdorff dimension of $\mu$ and moreover $\mu\perp \Lambda_{\Phi_{c(\mu)}^{(\kappa)}}$ and $\mu \ll \Lambda_{\Phi_{c}^{(\kappa)}}$ for every $c>c(\mu)$ for $\Phi_c^{(\kappa}(t) = t^\kappa \exp c\sqrt{\log 1/t \log\log\log 1/t}$, or $\mu$ is equivalent to $\Lambda_{HD(X)}$.
Most of the theory is carried out in the technically much harder situation of a “quasi-repeller” $X$—a limit set of a “tree” of pre-images of a point under iterations of a holomorphic map—and for Gibbs measures transported from the shift space to $X$ with the use of the “tree”. This includes the example $X=\partial\Omega$, $\mu=\sigma$ for $\Omega$ a basin of attraction to a sink for a holomorphic map.
The clue is the Refined Volume Lemma which is a refinement of L.-S. Young’s Volume Lemma in Pesin theory, in the sense that the Strong Law of Large Numbers is replaced by the Law of the Iterated Logarithm.
