Assouad-like dimensions of a class of random Moran measures. II. Non-homogeneous Moran sets

Abstract

In this paper, we determine the almost sure values of the $\Phi$-dimensions of random measures $\mu$ supported on random Moran sets in ${\mathbb{R}}^d$ that satisfy a uniform separation condition. This paper generalizes earlier work done on random measures on homogeneous Moran sets in Hare and Mendivil (2022) to the case of unequal scaling factors. The $\Phi$-dimensions are intermediate Assouad-like dimensions with the (quasi-)Assouad dimensions and the $\theta$-Assouad spectrum being special cases.

The almost sure value of ${\dim }_{\Phi }\mu$ exhibits a threshold phenomenon, with one value for “large” $\Phi$ (with the quasi-Assouad dimension as an example of a “large” dimension) and another for “small” $\Phi$ (with the Assouad dimension as an example of a “small” dimension). We give many applications, including both where the scaling factors are fixed and the probabilities are uniformly distributed, and also where the probabilities are fixed and the scaling factors are uniformly distributed.