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A new notion of dimension for dynamical systems and shift embeddability

Tom Meyerovitch

TL;DR

The work addresses the shift embeddability problem by introducing a new invariant $\dim(X,T)$ for dynamical systems, defined via a covers-and-measures framework that combines topological and measure-theoretic data. It establishes a coherent hierarchy with mean dimension, proves dimension invariance under embeddings, and provides exact calculations for finite-group actions and cubical shifts, while connecting to standard notions through the TM-pair category. A central achievement is the almost-embedding theorem, which shows finitary dimensional systems admit near-embeddings into shifts and yields consequences for distal systems and the Dranishnikov–Levin counterexamples. The results unify obstructions to shift embeddability, link SBP to dimension zero, and give concrete embedding thresholds across amenable and finite group actions, advancing the understanding of when dynamical systems can be sampled and embedded into shifts. Overall, the paper offers a robust, dimension-theoretic framework that captures all known obstructions to shift embeddability and guides future embedding results.

Abstract

A dynamical system $(X,T)$ is \emph{shift embaddable} if $(X,T)$ embeds continuously and equivariantly in the shift over $[0,1]^d$ for some finite $d$. Refuting a major conjecture in the field, in a recent result of Dranishnikov and Levin it was shown that Gromov's mean dimension and Lebesgue covering dimension of finite orbits are not the only obstructions for shift embaddability. We present a new notion of dimension for dynamical systems over any countable group. We show that this new notion of dimension accounts for all known obstructions for shift embaddability.

A new notion of dimension for dynamical systems and shift embeddability

TL;DR

The work addresses the shift embeddability problem by introducing a new invariant for dynamical systems, defined via a covers-and-measures framework that combines topological and measure-theoretic data. It establishes a coherent hierarchy with mean dimension, proves dimension invariance under embeddings, and provides exact calculations for finite-group actions and cubical shifts, while connecting to standard notions through the TM-pair category. A central achievement is the almost-embedding theorem, which shows finitary dimensional systems admit near-embeddings into shifts and yields consequences for distal systems and the Dranishnikov–Levin counterexamples. The results unify obstructions to shift embeddability, link SBP to dimension zero, and give concrete embedding thresholds across amenable and finite group actions, advancing the understanding of when dynamical systems can be sampled and embedded into shifts. Overall, the paper offers a robust, dimension-theoretic framework that captures all known obstructions to shift embeddability and guides future embedding results.

Abstract

A dynamical system is \emph{shift embaddable} if embeds continuously and equivariantly in the shift over for some finite . Refuting a major conjecture in the field, in a recent result of Dranishnikov and Levin it was shown that Gromov's mean dimension and Lebesgue covering dimension of finite orbits are not the only obstructions for shift embaddability. We present a new notion of dimension for dynamical systems over any countable group. We show that this new notion of dimension accounts for all known obstructions for shift embaddability.
Paper Structure (10 sections, 25 theorems, 133 equations)

This paper contains 10 sections, 25 theorems, 133 equations.

Table of Contents

  1. Introduction
  2. Order and dimension of covers
  3. Dimension in the category of topological-measure pairs
  4. Dynamical systems and their associated topological-measure pairs
  5. The small boundary property and dimension zero for dynamical systems
  6. Separating closed subsets and a Baire category lemma
  7. Dimension for free actions of finite groups
  8. Dimension of cubical shifts
  9. Dimension for actions of amenable groups
  10. An almost embedding theorem

Key Result

Lemma 2.1

Let $\{ U_1',\ldots,U_\ell'\}$ be a finite open cover of $X$. Then there exist open sets $U_1,\ldots, U_\ell$ so that $\overline{U_j} \subseteq U'_j$ and so that $\mathcal{U}=\left\{U_1,\ldots,U_\ell\right\}$ is still a cover of $X$.

Theorems & Definitions (45)

  • Lemma 2.1
  • Lemma 2.2: The "Brickwall" Cover
  • proof
  • Lemma 2.3: Simplicial complexes and nerve maps
  • proof
  • Lemma 2.4
  • proof
  • Lemma 2.5: Ostrand-Kolmogorov covers
  • proof
  • Proposition 3.1
  • ...and 35 more