Local properties of self-similar measures
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.
scientific article; zbMATH DE number 1191164