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 $n\geq1$, if $\dim(X,T)<n/2$, then the set of maps $f\in C(X,[0,1]^n)$ for which the orbit map $$ I_f: X\longrightarrow([0,1]^n)^{\mathbb{Z}}, \qquad I_f(x)=\bigl(f(T^k x)\bigr)_{k\in\mathbb{Z}}, $$ is a topological embedding is a dense $G_\delta$ set. Consequently, $(X,T)$ embeds into $(([0,1]^n)^{\mathbb{Z}},\sigma)$ for some integer $n\geq1$ if and only if $\dim(X,T)<\infty$. The constant $1/2$ and the strict inequality are optimal.