Work

Metric mean dimension, Hölder regularity and Assouad spectrum

Year: 2025

Type: article

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 Hölder 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.

Source: Journal of Fractal Geometry Mathematics of Fractals and Related Topics

DOI: 10.4171/jfg/169

Authors: Alexandre Baraviera, Maria Carvalho, Gustavo Pessil

Institutions: Universidade Federal do Rio Grande do Sul, Universidade do Porto

Language: English

Topic: Mathematical Dynamics and Fractals

Subfield: Mathematical Physics

Field: Mathematics

Domain: Physical Sciences

Open Access status: diamond

Funders: Fundação para a Ciência e a Tecnologia, Centro de Matemática Universidade do Porto