Assouad-like dimensions of random Moran measures

TL;DR

This paper calculates the almost sure values of intermediate Assouad-like dimensions for random measures on Moran sets, revealing how these dimensions depend on the function  and applying results to various self-similar and random measures.

Contribution

It provides a comprehensive determination of -dimensions for random Moran measures, extending understanding of fractal dimensions in random and self-similar contexts.

Findings

-dimensions depend on the size of , interpolating between Assouad and quasi-Assouad dimensions.

Explicit formulas for -dimensions of random Moran measures and sets.

Applications include self-similar and 1-variable random Moran measures, such as Cantor-like measures.

Abstract

In this paper, we determine the almost sure values of the $\Phi $-dimensions of random measures supported on random Moran sets that satisfy a uniform separation condition. The $\Phi $-dimensions are intermediate Assouad-like dimensions, the (quasi-)Assouad dimensions and $\theta $-Assouad spectrum being special cases. Their values depend on the size of $\Phi $, 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 $1$-variable random Moran measures such as Cantor-like measures with probabilities that are uniformly distributed. We can also deduce the $\Phi $-dimensions of the underlying random sets.