Mean Assouad dimension and spectrum, with applications to infinite dimensional fractals

TL;DR

This paper introduces the mean Assouad dimension and spectrum as new bi-Lipschitz invariants for dynamical systems, providing explicit formulas for infinite-dimensional fractals like Bedford--McMullen carpets, advancing fractal geometry in infinite dimensions.

Contribution

It defines and studies the properties of mean Assouad dimension and spectrum, and computes these invariants for infinite-dimensional fractals, extending fractal analysis.

Findings

Explicit formulas for mean Assouad dimension and spectrum of infinite-dimensional Bedford--McMullen carpets.

Introduction of mean Assouad spectrum as a new bi-Lipschitz invariant.

Application to infinite-dimensional fractals, advancing the understanding of their geometric complexity.

Abstract

We introduce the mean Assouad dimension of a dynamical system, motivated by the Assouad dimension in fractal geometry. Using dimension interpolation, we further define the mean Assouad spectrum. This provides a new family of bi-Lipschitz invariants of dynamical systems. We study its basic properties and calculate it for several classes of dynamical systems. As an application, we determine explicit formulae for the mean Assouad dimension and spectrum of infinite-dimensional Bedford--McMullen carpet systems, contributing to the program of studying infinite dimensional fractals, initiated recently by Tsukamoto.