A sharp embedding theorem for topological dynamical systems

Abstract

Let $(X,T)$ be a topological dynamical system, and let $\dim (X,T)$ denote Meyerovitch's dynamical dimension. We prove that, for every integer $n\ge 1$, if $\dim (X,T)<n/2$, then the set of maps $f\in C(X,[0,1{]}^n)$ for which the orbit map $$I_f:X⟶([0,1{]}^n{)}^{\mathbb{Z}},I_f(x)=(f(T^{k}x){)}_{k\in \mathbb{Z}},$$ is a topological embedding is a dense $G_{\delta }$ set. Consequently, $(X,T)$ embeds into $(([0,1{]}^n{)}^{\mathbb{Z}},\sigma )$ for some integer $n\ge 1$ if and only if $\dim (X,T)<\infty$. The constant $1/2$ and the strict inequality are optimal.

arXiv comment32 pages