Assouad type spectra for some fractal families

TL;DR

This paper introduces an Assouad type spectrum to analyze the scaling and homogeneity of fractal sets, explicitly computing it for various well-known fractals and exploring its applications as a bi-Lipschitz invariant.

Contribution

It defines and computes a new dimension spectrum for fractals, revealing diverse behaviors across different models and providing new invariants and bounds for metric space analysis.

Findings

Spectrum varies significantly across fractal models

Provides explicit spectrum calculations for several fractals

Introduces new bi-Lipschitz invariants and bounds

Abstract

In a previous paper we introduced a new `dimension spectrum', motivated by the Assouad dimension, designed to give precise information about the scaling structure and homogeneity of a metric space. In this paper we compute the spectrum explicitly for a range of well-studied fractal sets, including: the self-affine carpets of Bedford and McMullen, self-similar and self-conformal sets with overlaps, Mandelbrot percolation, and Moran constructions. We find that the spectrum behaves differently for each of these models and can take on a rich variety of forms. We also consider some applications, including the provision of new bi-Lipschitz invariants and bounds on a family of `tail densities' defined for subsets of the integers.