Self-similar random measures are locally scale invariant
In an earlier paper the first author and \textit{U. Zähle} [Math. Nachr. 149, 285-302 (1990; Zbl 0719.60001)] have proved the existence of a fractional tangent measure at the typical point of a self-similar random measure \(\Phi\) 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 \(\Phi\) on the boundary of the open set with probability 1. The tangent measure is \(D\)-scale-invariant, where \(D\) is the similarity dimension of \(\Phi\). Moreover, we approximate the tangential distribution by means of \(\Phi\) and use this in order to prove that the Hausdorff dimension of the tangent measure equals \(D\). Since the former coincides with the Hausdorff dimension of \(\Phi\) we obtain an earlier result of \textit{R. D. Mauldin} and \textit{S. C. Williams} [Trans. Am. Math. Soc. 295, 325-346 (1986; Zbl 0625.54047)] as a corollary.
existence of a fractional tangent measure
scientific article; zbMATH DE number 569003
scientific article