Mean Assouad dimension and spectrum, with applications to infinite dimensional fractals
Qiang Huo, Adam Śpiewak
TL;DR
Mean Assouad dimension and mean Assouad spectrum extend fractal dimension interpolation to dynamical systems, yielding robust bi-Lipschitz invariants that capture both thick and thin parts across scales. The authors define $mdim_A$ and $mdim_A^\theta$ via a scale-interpolating function $S(X,r,\rho)$ and establish fundamental properties, including bi-Lipschitz invariance and bounds relative to metric mean dimension. They compute explicit formulae for infinite-dimensional Bedford–McMullen carpet systems and verify sharpness through full-shift and band-limited function examples, illustrating the framework's reach. The results advance infinite-dimensional fractal geometry in dynamics and pave the way for dynamical embedding implications and deeper connections with mean-dimension theory.
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.