Self-similar random measures are locally scale invariant

Abstract

In an earlier paper Patzschke and U. Zahle [11] have proved the existence of a fractional tangent measure at the typical point of a self-similar random measure Φ under rather special technical assumptions. In the present paper we remove the most restrictive one. Here we suppose the open set condition for the similarities, a constant positive lower bound for the random contraction ratios, and vanishing Φ on the boundary of the open set with probability 1. The tangent measure isD-scale-invariant, whereD is the similarity dimension of Φ. Moreover, we approximate the tangential distribution by means of Φ and use this in order to prove that the Hausdorff dimension of the tangent measure equalsD. Since the former coincides with the Hausdorff dimension of Φ we obtain an earlier result of Mauldin and Williams [9] as a corollary.