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→([0,1]^D)^ℤ whose fibers contain at most ⌊ D/(D−m)⌋ 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 ℝ whose Fourier transforms, understood in the sense of distributions, are supported in $[-a/2,a/2]$, with each fiber containing at most ⌊ a/(a−m)⌋ 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 ℤ-actions.

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BibTeX RIS

BibTeX

@misc{gutman2026finitetoone,
  title = {{Finite-to-One Maps into Band-Limited Function Spaces and Cubical Shifts with a Hurewicz Bound}},
  author = {Yonatan Gutman and Xulei Wang},
  howpublished = {arXiv},
  year = {2026},
  month = oct,
  doi = {10.48550/arxiv.2610.29509},
  eprint = {2610.29509},
  archivePrefix = {arXiv},
  url = {https://arxiv.org/abs/2610.29509},
}

RIS

TY  - UNPB
TI  - Finite-to-One Maps into Band-Limited Function Spaces and Cubical Shifts with a Hurewicz Bound
AU  - Yonatan Gutman
AU  - Xulei Wang
PY  - 2026
DA  - 2026/10/21
PB  - arXiv
DO  - 10.48550/arxiv.2610.29509
UR  - https://arxiv.org/abs/2610.29509
AB  - 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→([0,1]^D)^ℤ whose fibers contain at most ⌊ D/(D−m)⌋ 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 ℝ whose Fourier transforms, understood in the sense of distributions, are supported in $[-a/2,a/2]$, with each fiber containing at most ⌊ a/(a−m)⌋ 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 ℤ-actions.
ER  -

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