SELF-SIMILAR RANDOM MEASURES ARE LOCALLY SCALE-INVARIANT

• 成果类型: Article
• 署名作者: PATZSCHKE, N; ZAHLE, M
• 刊物名称: PROBABILITY THEORY AND RELATED FIELDS
• ISSN/ISSBN: 0178-8051
• DOI: 10.1007/BF01192964
• 发表日期: 1993
• 页码: 559-574
• 关键词: recursive construction; similar fractals

摘要

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 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 Mauldin and Williams [9] as a corollary.