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

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 $\lfloor D\mathrm{/}(D-m)\rfloor$ 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 $[-a\mathrm{/}2,a\mathrm{/}2]$, with each fiber containing at most $\lfloor a\mathrm{/}(a-m)\rfloor$ 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.