Self-similar random measures. I: Notion, carrying Hausdorff dimension, and hyperbolic distribution
The notion of self-similarity is of increasing interest in many theoretical and applied settings, such as random processes, turbulence, and dynamical systems. Self-similar sets are usually fractal sets, i.e., sets of non-integer Hausdorff (-Besicovich) dimension. Moreover, the parameters of self-similarity determine the Hausdorff dimension: If a compact set K is the union of sets \(K_ 1,...,K_ N\), which are similar to K in the ratio \(r<1\), and if \(K_ i\cap K_ j\) is ``almost empty'', then dim K\(=\ln N/(-\ln r).\) The mentioned notion of strong deterministic self-similarity is restrictive and does not apply to many phenomena, such as graphs and level sets of random processes. Thus it is naturally to seek weaker notions, which preserve the very essence of self-similarity (small parts are the ``same'', up to a scale factor, as the whole) and the connection of some self-similarity index with Hausdorff dimension.
scientific article; zbMATH DE number 4040973
scientific article