The Assouad dimension of randomly generated fractals

TL;DR

This paper investigates the Assouad dimension of various random fractal models, showing it is typically maximal and independent of measure or topology, contrasting with other fractal dimensions.

Contribution

It provides the first comprehensive analysis of the Assouad dimension in multiple random fractal models, revealing a universal maximality property.

Findings

Assouad dimension is almost surely maximal in all models

The Assouad dimension is independent of measure-theoretic or topological structure

Contrasts with Hausdorff and packing dimensions

Abstract

We consider several different models for generating random fractals including random self-similar sets, random self-affine carpets, and fractal percolation. In each setting we compute either the \emph{almost sure} or the \emph{Baire typical} Assouad dimension and consider some illustrative examples. Our results reveal a common phenomenon in all of our models: the Assouad dimension of a randomly generated fractal is generically as big as possible and does not depend on the measure theoretic or topological structure of the sample space. This is in stark contrast to the other commonly studied notions of dimension like the Hausdorff or packing dimension.