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 $\R^d$ that satisfy a uniform separation condition. This paper generalizes earlier work done on random measures on homogeneous Moran sets \cite{HM} 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 phenomena, 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 where the scaling factors are fixed and the probabilities are uniformly distributed. The almost sure $\Phi$ dimension of the underlying random set is also a consequence of our results.