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Metric mean dimension, Hölder regularity and Assouad spectrum

Alexandre Baraviera, Maria Carvalho, Gustavo Pessil

TL;DR

The paper develops a comprehensive framework linking metric mean dimension $\overline{\mathrm{mdim}}_M$ to Hölder regularity and fractal geometry, and proves a Misiurewicz-type formula for a two-parameter interval map family built from accumulating horseshoes. It extends entropy notions to subshifts on general, Ahlfors regular alphabets via a sharp entropy formula and shows how the Assouad spectrum captures Hölder effects in this dynamical setting. An explicit computation of $\overline{\mathrm{mdim}}_M$ is achieved for the interval maps $T_{a,b}$, including a precise relation to the Hölder exponents and a demonstration of sharpness. The work also introduces a dynamical Minkowski–Bouligand framework for subshifts, yielding an entropy- mdim_M connection, and discusses consequences for non-homogeneous spaces, including applications to Weierstrass functions. Together, these results highlight how geometry, regularity, and dynamical complexity intertwine in noncompact or infinite-entropy regimes and provide tools for exact mdim_M calculations in complex settings.

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.

Metric mean dimension, Hölder regularity and Assouad spectrum

TL;DR

The paper develops a comprehensive framework linking metric mean dimension to Hölder regularity and fractal geometry, and proves a Misiurewicz-type formula for a two-parameter interval map family built from accumulating horseshoes. It extends entropy notions to subshifts on general, Ahlfors regular alphabets via a sharp entropy formula and shows how the Assouad spectrum captures Hölder effects in this dynamical setting. An explicit computation of is achieved for the interval maps , including a precise relation to the Hölder exponents and a demonstration of sharpness. The work also introduces a dynamical Minkowski–Bouligand framework for subshifts, yielding an entropy- mdim_M connection, and discusses consequences for non-homogeneous spaces, including applications to Weierstrass functions. Together, these results highlight how geometry, regularity, and dynamical complexity intertwine in noncompact or infinite-entropy regimes and provide tools for exact mdim_M calculations in complex settings.

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.
Paper Structure (17 sections, 13 theorems, 118 equations, 1 figure)

This paper contains 17 sections, 13 theorems, 118 equations, 1 figure.

Table of Contents

  1. Introduction
  2. Hölder regularity
  3. Horseshoes in the interval
  4. An entropy formula
  5. Preliminaries
  6. Metric mean dimension
  7. Other fractal dimensions
  8. Subshifts of compact type
  9. Spectral radius bounds
  10. Ahlfors regular spaces
  11. Minkowski–Bouligand dimension
  12. Proof of Theorem
  13. Proof of Corollary
  14. Proof of Theorem
  15. Proof of Theorem
  16. ...and 2 more sections

Key Result

Theorem A

Let $(X,d)$ be a compact metric space with finite Assouad dimension. If $\alpha\,\in\,(0,1)$ and $T\colon X\to X$ is $\alpha-$Hölder, then If, in addition, $\dim_A(X,d)\,>\,0$ then

Figures (1)

  • Figure 1: Graph of the map $T_{a,b}$.

Theorems & Definitions (24)

  • Theorem A
  • Corollary A
  • Corollary B
  • Example 1.1: Weierstrass functions
  • Example 1.2: Hypersensitive zipper maps
  • Theorem B
  • Theorem C
  • Corollary 1.3
  • Corollary 1.4
  • Definition 2.1
  • ...and 14 more