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A Lipschitz Refinement of the Multidimensional Bebutov--Kakutani Dynamical Embedding Theorem

Yonatan Gutman, Qiang Huo, Masaki Tsukamoto

TL;DR

The paper proves a sharp Lipschitz refinement of the multidimensional Bebutov--Kakutani embedding theorem: a continuous $\mathbb{R}^n$-action on a compact metrizable space embeds equivariantly into the shift on $Lip_1(\mathbb{R}^n, I)$ if and only if the fixed-point set embeds into $I$. The authors develop a robust Lipschitz filter that converts continuous equivariant maps to $\mathbb{R}^n$-equivariant, 1-Lipschitz maps while preserving boundary data and fixed points, enabling a Baire-category strategy to produce an embedding. A filtration by stabilizer dimensions $X_k$ and corresponding target filtrations $Y_k$ guide iterative local perturbations that distinguish points with the same stabilizers, culminating in a dense $G_\delta$ set of embeddings. This removes the weak local freeness assumption from previous results and presents a compact, Lipschitz universal target for multidimensional dynamical embeddings, with potential generalizations to broader groups. The work thus connects fixed-point topological data to Lipschitz dynamical realizations in a compact, universal setting.

Abstract

We prove that a continuous action of $\mathbb{R}^n$ on a compact metrizable space equivariantly embeds into the shift action on the space of one-Lipschitz functions from $\mathbb{R}^n$ to $[0,1]$ if and only if the set of fixed points topologically embeds in $[0,1]$. This is a Lipschitz refinement of classical dynamical embedding theorems of Bebutov, Kakutani, Jaworski and Chen.

A Lipschitz Refinement of the Multidimensional Bebutov--Kakutani Dynamical Embedding Theorem

TL;DR

The paper proves a sharp Lipschitz refinement of the multidimensional Bebutov--Kakutani embedding theorem: a continuous -action on a compact metrizable space embeds equivariantly into the shift on if and only if the fixed-point set embeds into . The authors develop a robust Lipschitz filter that converts continuous equivariant maps to -equivariant, 1-Lipschitz maps while preserving boundary data and fixed points, enabling a Baire-category strategy to produce an embedding. A filtration by stabilizer dimensions and corresponding target filtrations guide iterative local perturbations that distinguish points with the same stabilizers, culminating in a dense set of embeddings. This removes the weak local freeness assumption from previous results and presents a compact, Lipschitz universal target for multidimensional dynamical embeddings, with potential generalizations to broader groups. The work thus connects fixed-point topological data to Lipschitz dynamical realizations in a compact, universal setting.

Abstract

We prove that a continuous action of on a compact metrizable space equivariantly embeds into the shift action on the space of one-Lipschitz functions from to if and only if the set of fixed points topologically embeds in . This is a Lipschitz refinement of classical dynamical embedding theorems of Bebutov, Kakutani, Jaworski and Chen.

Paper Structure

This paper contains 11 sections, 25 theorems, 92 equations, 2 figures.

Table of Contents

  1. Introduction
  2. The Bebutov--Kakutani theorem and its multidimensional generalization
  3. Lipschitz refinements and the main result
  4. Strategy of the proof
  5. Lipschitz filter and applications
  6. Preliminaries on Lipschitz functions
  7. Construction of a filter
  8. Applications
  9. Construction of local perturbation maps
  10. Local section
  11. Proofs of Propositions \ref{['proposition: C_k(A) is open and dense']} and \ref{['proposition: C_k(B, B^prime) is open and dense']}

Key Result

Theorem 1.1

Let $T\colon \mathbb{R}\times X\to X$ be a continuous action of $\mathbb{R}$ on a compact metrizable space $X$. It equivariantly embeds in $C(\mathbb{R},I)$ if and only if $\mathrm{Fix}(X, T)$ topologically embeds in the unit interval $I$.

Figures (2)

  • Figure 1: The schematic picture of the local section. The line segment in the center represents the local section $E$. The thick line segment inside $E$ represents the set $A_0$. The set $A_0$ contains the point $p$. The rectangle represents the “flow box” $A$.
  • Figure 2: The schematic picture of the local section $E_i$ and the flow box $B_i$. The line segment in the center represents the local section $E_i$, and the rectangle represents $B_i$. The point $p_i$ belongs to the set $A_i$ (the thick line segment inside $E_i$).

Theorems & Definitions (53)

  • Theorem 1.1: Bebutov 1940, Kakutani 1968
  • Theorem 1.2: Jaworski 1974, Chen 1975
  • Remark 1.3
  • Theorem 1.4: Jin--Gutman--Tsukamoto 2019
  • Theorem 1.6: Gutman--Huo
  • Theorem 1.7: Main theorem
  • Proposition 1.9
  • Proposition 1.10
  • proof : Proof of Theorem \ref{['theorem: main theorem']}, assuming Propositions \ref{['proposition: C_k(A) is open and dense']} and \ref{['proposition: C_k(B, B^prime) is open and dense']}
  • Lemma 2.1
  • ...and 43 more