Attainable Assouad-like dimensions of randomly generated Moran sets and measures

Abstract

In this paper we study the Assouad-like $\Phi$ dimensions of sets and measures that are constructed by a random weighted iterated function system of similarities. These dimensions are distinguished by the depth of the scales considered and thus provide more refined infomation about the local geometry/behaviour of a set or measure. The Assouad dimensions are important well-known examples. We determine the almost sure value of the upper and lower Assouad dimension of the random set. We also determine the range of attainable upper and lower (small) $\Phi$ dimensions of the measures, in both the situation where the probability weights can depend on the scaling factors and when they cannot. In the later case we find that there is a ``gap'' between the dimension of the set and the dimensions of the associated family of random measures.