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

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.

arXiv comment35 pages. Version 2: We add subsection 5.6 to explain how infinite-dimensional Bedford-McMullen carpet systems are related to finite-dimensional self-affine carpets. Remark 5.3 in Version 1 is replaced by Corollary 5.3. We add discussions on mean Hausdorff dimension of carpet systems in Corollary 5.3. Comments are welcome!