PreprintarXiv licencearXiv
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≥1, if dim(X,T)<n/2, then the set of maps f∈ C(X,[0,1]n) for which the orbit map If: X⟶([0,1]n)ℤ, If(x)=(f(Tk x))k∈ℤ, is a topological embedding is a dense Gδ set. Consequently, (X,T) embeds into (([0,1]n)ℤ,σ) for some integer n≥1 if and only if dim(X,T)<∞. The constant 1/2 and the strict inequality are optimal.
Related
- A Lipschitz refinement of the Bebutov-Kakutani dynamical embedding theorem for connected Lie group actions
- Entropy Scales in Topological Dynamical Systems
- Topological Characteristics for the Analysis of Vector Fields. New stable characteristics for discrete and continuous dynamical systems
- Exact dg categories II: The embedding theorem
- Quandles from group actions and a Cayley-type embedding theorem
BibTeX
@misc{shi2026sharp,
title = {{A sharp embedding theorem for topological dynamical systems}},
author = {Ruxi Shi},
howpublished = {arXiv},
year = {2026},
month = sep,
doi = {10.48550/arxiv.2609.05664},
eprint = {2609.05664},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2609.05664},
}
RIS
TY - UNPB
TI - A sharp embedding theorem for topological dynamical systems
AU - Ruxi Shi
PY - 2026
DA - 2026/09/04
PB - arXiv
DO - 10.48550/arxiv.2609.05664
UR - https://arxiv.org/abs/2609.05664
AB - 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≥1, if dim(X,T)<n/2, then the set of maps f∈ C(X,[0,1]^n) for which the orbit map I_f: X⟶([0,1]^n)^ℤ, I_f(x)=(f(T^k x))_k∈ℤ, is a topological embedding is a dense G_δ set. Consequently, (X,T) embeds into (([0,1]^n)^ℤ,σ) for some integer n≥1 if and only if dim(X,T)<∞. The constant 1/2 and the strict inequality are optimal.
ER -