Harmonic, Gibbs and Hausdorff measures on repellers for holomorphic maps. I (Q916179)

For a simply connected domain \(\Omega\subset {\mathbb{C}}\) whose boundary \(\partial \Omega\) is self-similar the following dichotomy is proved. If \(\omega\) denotes the harmonic measure on \(\partial \Omega\), then either (i) \(\partial \Omega\) is piecewise real-analytic, or (ii) \(\omega\) is singular with respect to the Hausdorff measure \(\Lambda_{\phi_ c}\) associated with the Makarov's function \(\phi_ c\) for some \(c=c(\omega)>0\) and is absolutely continuous with respect to \(\Lambda_{\phi_ c}\) for every \(c>c(\omega).\) This dichotomy is basically deduced from the dichotomy concerning Gibbs measures for Hölder continuous functions on a mixing repeller \(X\subset {\mathbb{C}}\) for a holomorphic mapping.