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Genericity of homeomorphisms with full mean Hausdorff dimension

Jeovanny de Jesus Muentes Acevedo

Abstract

It is well known that the presence of horseshoes leads to positive entropy. If our goal is to construct a continuous map with infinite entropy, we can consider an infinite sequence of horseshoes, ensuring an unbounded number of legs. Estimating the exact values of both the metric mean dimension and mean Hausdorff dimension for a homeomorphism is a challenging task. We need to establish a precise relationship between the sizes of the horseshoes and the number of appropriated legs to control both quantities. Let $N$ be an $n$-dimensional compact Riemannian manifold, where $n \geq 2$, and $α\in [0, n]$. In this paper, we construct a homeomorphism $φ: N \rightarrow N$ with mean Hausdorff dimension equal to $α$. Furthermore, we prove that the set of homeomorphisms on $N$ with both lower and upper mean Hausdorff dimensions equal to $α$ is dense in $\text{Hom}(N)$. Additionally, we establish that the set of homeomorphisms with upper mean Hausdorff dimension equal to $n$ contains a residual subset of $\text{Hom}(N).$

Genericity of homeomorphisms with full mean Hausdorff dimension

Abstract

It is well known that the presence of horseshoes leads to positive entropy. If our goal is to construct a continuous map with infinite entropy, we can consider an infinite sequence of horseshoes, ensuring an unbounded number of legs. Estimating the exact values of both the metric mean dimension and mean Hausdorff dimension for a homeomorphism is a challenging task. We need to establish a precise relationship between the sizes of the horseshoes and the number of appropriated legs to control both quantities. Let be an -dimensional compact Riemannian manifold, where , and . In this paper, we construct a homeomorphism with mean Hausdorff dimension equal to . Furthermore, we prove that the set of homeomorphisms on with both lower and upper mean Hausdorff dimensions equal to is dense in . Additionally, we establish that the set of homeomorphisms with upper mean Hausdorff dimension equal to contains a residual subset of
Paper Structure (5 sections, 10 theorems, 103 equations, 3 figures)

This paper contains 5 sections, 10 theorems, 103 equations, 3 figures.

Table of Contents

  1. Introduction
  2. Continuous maps on the interval with positive mean Hausdorff dimension
  3. Density of continuous maps on the interval with positive mean Hausdorff dimension
  4. Homeomorphisms on manifold with positive mean Hausdorff dimension
  5. Tipical homeomorphism on a manifold has maximal mean Hausdorff dimension

Key Result

Lemma 2.3

Suppose for each $k\in\mathbb{N}$ there exists a $s_{k}$-horseshoe for $\phi\in C^{0}([0,1])$, $I_{k}=[a_{k-1},a_{k}]\subseteq [0,1]$, consisting of sub-intervals $I_{k}^{1}, I_{k}^{2},\dots, I_{k}^{s_{k}}$ with the same length, where $s_{k}\geq 2$ for all $k\geq 1$. We have

Figures (3)

  • Figure 4.1: 3-dimensional 9-horseshoe
  • Figure 4.2: 3-dimensional 9-horseshoe
  • Figure 4.3: $E_{i}\subset C_{i}\subset D_{i}$.

Theorems & Definitions (28)

  • Definition 2.1
  • Remark 2.2
  • Lemma 2.3
  • proof
  • Lemma 2.4
  • proof
  • Example 2.5
  • Example 2.6
  • Example 2.7
  • Theorem 3.1
  • ...and 18 more