Tangent measure distributions of fractal measures

Abstract

Tangent measure distributions provide a natural tool to study the local geometry of fractal sets and measures in Euclidean spaces. The idea is, loosely speaking, to attach to every point of the set a family of random measures, called the $\alpha$ -dimensional tangent measure distributions at the point, which describe asymptotically the $\alpha$ -dimensional scenery seen by an observer zooming down towards this point. This tool has been used by Bandt [BA] and Graf [G] to study the regularity of the local geometry of self similar sets, but in this paper we show that its scope goes much beyond this situation and, in fact, it may be used to describe a strong regularity property possessed by every measure: We show that, for every measure $\mu$ on a Euclidean space and any dimension $\alpha$ , at $\mu$ -almost every point, all $\alpha$ -dimensional tangent measure distributions are Palm measures. This means that the local geometry of every dimension of general measures can be described – like the local geometry of self similar sets – by means of a family of statistically self similar random measures. We believe that this result reveals a wealth of new and unexpected information about the structure of such general measures and we illustrate this by pointing out how it can be used to improve or generalize recently proved relations between ordinary and average densities.