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Hölder continuous maps on the interval with positive metric mean dimension

Jeovanny M. Acevedo, Sergio Romaña, Raibel Arias

Abstract

Fix a compact metric space $X$ with finite topological dimension. Let $C^{0}(X)$ be the space of continuous maps on $X$ and $ H^α(X)$ the space of $α$-Hölder continuous maps on $X$, for $α\in (0,1].$ $H^{1}(X)$ is the space of Lipschitz continuous maps on $X$. We have $$H^{1}(X)\subset H^β(X) \subset H^α(X) \subset C^{0}(X),\quad\text{ where }0<α<β<1.$$ It is well-known that if $φ\in H^{1}(X)$, then $φ$ has metric mean dimension equal to zero. On the other hand, if $X$ is a finite dimensional compact manifold, then $C^{0}(X)$ contains a residual subset whose elements have positive metric mean dimension. In this work we will prove that, for any $α\in (0,1)$, there exists $φ\in H^α([0,1]) $ with positive metric mean dimension.

Hölder continuous maps on the interval with positive metric mean dimension

Abstract

Fix a compact metric space with finite topological dimension. Let be the space of continuous maps on and the space of -Hölder continuous maps on , for is the space of Lipschitz continuous maps on . We have It is well-known that if , then has metric mean dimension equal to zero. On the other hand, if is a finite dimensional compact manifold, then contains a residual subset whose elements have positive metric mean dimension. In this work we will prove that, for any , there exists with positive metric mean dimension.
Paper Structure (5 sections, 7 theorems, 100 equations, 3 figures)

This paper contains 5 sections, 7 theorems, 100 equations, 3 figures.

Table of Contents

  1. Introduction
  2. Metric mean dimension for continuous maps
  3. Horseshoes and metric mean dimension
  4. Hölder continuous maps with positive metric mean dimension
  5. Some conjectures about this research

Key Result

Lemma 3.1

Suppose that $I_{k}=[a_{k-1},a_{k}]\subseteq [0,1]$ is an $s_{k}$-horseshoe for $\phi:[0,1]\rightarrow [0,1]$ consisting of $s_{k}$ subintervals with the same length $I_{k}^{1},\dots, I_{k}^{s_{k}}$. Setting $\varepsilon_{k}=\frac{|I_{k}|}{s_{k}}$, we have

Figures (3)

  • Figure 3.1: $J$ is an $3$-horseshoe
  • Figure 3.2: $s$-horseshoes
  • Figure 4.1: Each $I_{n}$ is a $(2n+1)$-horseshoe

Theorems & Definitions (22)

  • Definition 2.1
  • Definition 2.2
  • Definition 2.3
  • Remark 2.4
  • Remark 2.5
  • Remark 2.6
  • Lemma 3.1
  • Theorem 3.2
  • Proposition 3.3
  • proof
  • ...and 12 more