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

TL;DR

This paper determines the almost sure $ ext{Phi}$-dimensions of random measures supported on non-homogeneous Moran sets, revealing a threshold phenomenon and extending previous homogeneous cases to unequal scaling factors.

Contribution

It generalizes earlier work on homogeneous Moran sets to non-homogeneous cases, providing explicit $ ext{Phi}$-dimension values and threshold behavior for random Moran measures.

Findings

Identifies threshold phenomena in $ ext{Phi}$-dimensions

Extends results to non-homogeneous Moran sets

Provides applications with fixed scaling factors and uniform probabilities

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.