Metric mean dimension, H\"older regularity and Assouad spectrum

TL;DR

This paper explores the relationship between metric mean dimension, fractal structures, and regularity in dynamical systems, providing new formulas and applications including classical functions and subshift dimensions.

Contribution

It introduces a dynamical analogue of Minkowski-Bouligand dimension and computes metric mean dimension for interval maps and Weierstrass functions, sharpening existing bounds.

Findings

Derived sharp bounds for metric mean dimension in interval maps.

Computed metric mean dimension for classical Weierstrass functions.

Developed a dynamical Minkowski-Bouligand dimension for subshifts on Ahlfors regular alphabets.

Abstract

Metric mean dimension is a geometric invariant of dynamical systems with infinite topological entropy. We relate this concept with the fractal structure of the phase space and the H\"older regularity of the map. Afterwards we improve our general estimates in a family of interval maps by computing the metric mean dimension in a way similar to the Misiurewicz formula for the entropy, which in particular shows that our bounds are sharp. As an application, we determine the metric mean dimension of the classical Weierstrass functions. Of independent interest, we develop a dynamical analogue of the Minkowski-Bouligand dimension for subshifts on Ahlfors regular alphabets, which also provides an entropy formula in terms of the size of the set of admissible words, generalizing the classical result for subshifts on finite alphabets.