On the dimensions of conformal repellers. Randomness and parameter dependency

Abstract

Bowen's formula relates the Hausdorff dimension of a conformai repeller to the zero of a 'pressure' function. We present an elementary, self-contained proof to show that Bowen's formula holds for C1 conformai repellers. We consider time-dependent conformai repellers obtained as invariant subsets for sequences of conformally expanding maps within a suitable class. We show that Bowen's formula generalizes to such a repeller and that if the sequence is picked at random then the Hausdorff dimension of the repeller almost surely agrees with its upper and lower box dimensions and is given by a natural generalization of Bowen's formula. For a random uniformly hyperbolic Julia set on the Riemann sphere we show that if the family of maps and the probability law depend realanalytically on parameters then so does its almost sure Hausdorff dimension. 1. Random Julia sets and their dimensions Let {U,djj) be an open, connected subset of the Riemann sphere avoiding at least three points and equipped with a hyperbolic metric. Let K C U be a compact subset. We denote by £(AT, U) the space of unramified conformai covering maps / : Vf - » U with the requirement that the covering domain Vf C K. Denote by Df : Vf - ► M+ the conformai derivative of /, see equation (2.4), and by \\Df = supf-iK Df the maximal value of this derivative over the set f~1K. Let T = (fn) C £(K, U) be a sequence of such maps. The intersection