Finite-to-One Maps into Band-Limited Function Spaces and Cubical Shifts with a Hurewicz Bound

Abstract

Let $(X,T)$ be a dynamical system with the marker property and finite mean dimension $m$. We prove that, for every integer $D>m$, there exists an equivariant continuous map $f:X\to([0,1]^D)^{\mathbb Z}$ whose fibers contain at most $\lfloor D/(D-m)\rfloor$ points; in particular, $f$ is an embedding when $D>2m$. This is the dynamical analogue of the fiber bound in Hurewicz's classical theorem (1933) for continuous maps into Euclidean cubes. For every real number $a>m$, we also construct an equivariant continuous map from $X$ into the closed unit ball of the real Bernstein space of bounded continuous functions on $\mathbb R$ whose Fourier transforms, understood in the sense of distributions, are supported in $[-a/2,a/2]$, with each fiber containing at most $\lfloor a/(a-m)\rfloor$ points. The proof adapts the signal-analysis method of Gutman--Qiao--Tsukamoto (2019) to the finite-to-one setting and simplifies its analytic and geometric ingredients for $\mathbb Z$-actions.