Hausdorff dimension of harmonic measure for self-conformal sets

Abstract

Under some technical assumptions it is shown that the Hausdorff dimension of the harmonic measure on the limit set of a conformal infinite iterated function system is strictly less than the Hausdorff dimension of the limit set itself if the limit set is contained in a real-analytic curve, if the iterated function system consists of similarities only, or if this system is irregular. As a consequence of this general result the same statement is proven for hyperbolic and parabolic Julia sets, finite parabolic iterated function systems and generalized polynomial-like mappings. Also sufficient conditions are provided for a limit set to be uniformly perfect and for the harmonic measure to have the Hausdorff dimension less than 1-40. Some results in the spirit of Przytycki et aL (Ann. of Math. 130 (1989), 1-40; Stud. Math. 97 (1991), 189-225) are obtained. © 2002 Elsevier Science (USA).