Metric mean dimension, H\"older regularity and Assouad spectrum
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.
Keywords
Cite
@article{arxiv.2407.15774,
title = {Metric mean dimension, H\"older regularity and Assouad spectrum},
author = {Alexandre Baraviera and Maria Carvalho and Gustavo Pessil},
journal= {arXiv preprint arXiv:2407.15774},
year = {2025}
}
Comments
Accepted version in Journal of Fractal Geometry. We have clarified the role of Theorem 4.3, which has become Theorem C in the introduction, since its proof also provides a formula for the topological entropy of any subshift on an Ahlfors alphabet in terms of the size of the set of its admissible words, thus generalizing the well known formula for subshifts on finite alphabets