Density of the level sets of the metric mean dimension for homeomorphisms

Abstract

Let $N$ be an $n$-dimensional compact riemannian manifold, with $n\geq 2$. In this paper, we prove that for any $α\in [0,n]$, the set consisting of homeomorphisms on $N$ with lower and upper metric mean dimensions equal to $α$ is dense in $\text{Hom}(N)$. More generally, given $α,β\in [0,n]$, with $α\leq β$, we show the set consisting of homeomorphisms on $N$ with lower metric mean dimension equal to $α$ and upper metric mean dimension equal to $β$ is dense in $\text{Hom}(N)$. Furthermore, we also give a proof that the set of homeomorphisms with upper metric mean dimension equal to $n$ is residual in $\text{Hom}(N)$.

Figures (3)

• Figure 2.1: 2-dimensional $5$-horseshoe
• Figure 2.2: 3-dimensional 9-horseshoe
• Figure 2.3: $E$ is the first square, is a 2-dimensional 3-horseshoe for $\phi$

Theorems & Definitions (21)

• Definition 1.1
• Definition 1.2
• Remark 1.3
• Definition 1.4
• Theorem 1.5
• Theorem 1.6
• Definition 2.1: $n$-dimensional $(2k+1)^{n-1}$-horseshoe
• Lemma 2.2
• Lemma 2.3
• proof