Local properties of self-similar measures
type: Item
instance of: scholarly article
title: Local properties of self-similar measures
zbMATH Open document ID: 0910.28006
author: Manav Das
published in: Illinois Journal of Mathematics
publication date: 1999-04-19T00:00:00Z
review text: This paper studies the local dimension and local properties for the following fractal measures. Let \(X\) be the \(d\)-Euclidean space and let \(\mu\) be a graph directed self-similar measure satisfying the open set condition, associated to a Mauldin-Williams graph with probabilities. Such a measure is defined on a fractal set constructed as the limit of compact sets in which each piece is replaced by a finite number of geometrically similar subsets with no overlaps, the similarities being governed by a fixed directed multigraph [\textit{R. D. Mauldin} and \textit{S. C. Williams}, Trans. Am. Math. Soc. 309, No. 2, 811-829 (1988; Zbl 0706.28007)]. The fractal set is thus represented as the image of a certain model of infinite strings, which correspond to paths in the graph. Let \(B_\varepsilon(x)\) be the closed ball of radius \(\varepsilon\) centered at \(x\in X\). The local dimension of \(\mu\) at \(x\) is defined as the limit, if it exists, \[ \lim_{\varepsilon\to 0} {\log\mu B_\varepsilon(x)\over \log\text{diam }B_\varepsilon(x)}. \] The aim of this paper is to study the existence of this limit and to provide an expression of the local dimension in symbolic terms. Indeed, explicit connections are given between the balls over the metric space and the cylinders of the underlying string space, roughly speaking, the paths in the graph that have the same beginning. The author will extend this approach in forthcoming papers to Hausdorff and packing dimensions.
Mathematics Subject Classification ID: 28A80
Mathematics Subject Classification ID: 28A78
zbMATH DE Number: 1191164
zbMATH Keywords: fractal sets
zbMATH Keywords: local dimension
zbMATH Keywords: graph directed self-similar measure
zbMATH Keywords: open set condition
zbMATH Keywords: Mauldin-Williams graph
reviewed by: Valérie Berthé
MaRDI profile type: Publication
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description: scientific article; zbMATH DE number 1191164