Assouad-like dimensions of a class of random Moran measures

Abstract

In this paper, we determine the almost sure values of the Φ-dimensions of random measures supported on random Moran sets that have a homogeneity property and satisfy a uniform separation condition. The Φ-dimensions are intermediate Assouad-like dimensions, the (quasi-)Assouad dimensions and θ-Assouad spectrum being special cases. Their values depend on the size of Φ, with one size coinciding with the Assouad dimension and the other coinciding with the quasi-Assouad dimension. We give many applications, including to equicontractive self-similar measures and Cantor-like measures with probabilities that are uniformly distributed. We can also deduce the Φ-dimensions of the underlying random sets.