Asymptotically conformal similarity between Julia and Mandelbrot sets

Abstract

An almost conformal local similarity between the connectedness locus M d and the corresponding Julia set is true for almost every point of ∂ M d with respect to harmonic measure. The harmonic measure is supported on Lebesgue density points of the complement of M d which are not accessible from outside within John angles and at which the boundary of M d spirals infinitely often in both directions. A more general result can be obtained for d = 2 in terms of the renormalization property. Finally, we prove that for almost all c ∈ ∂ M d in the sense of harmonic measure the Lyapunov exponent of c under iterates of z d + c is equal to log d .