Representation of measurable multidimensional flows by Lipschitz functions

TL;DR

The article proves that the shift space $Lip_1(\mathbb{R}^k)$, with its $\mathbb{R}^k$-action, serves as a topological model for all free measurable $\mathbb{R}^k$-flows, extending Eberlein's classic $k=1$ result to higher dimensions. The authors develop a multidimensional Lipschitz refinement of the Bebutov--Kakutani embedding framework, combining Lipschitz extension techniques, density arguments, and orbit-structure control through complete lacunary cross-sections. A separate expository proof for $k=1$ is provided, and a robust embedding theorem (GJT) for multidimensional flows is established via a countable family of orbital Lipschitz functions and a perturbation scheme. The culmination is a generalization showing that the Lipschitz shift model captures all free measurable $\mathbb{R}^k$-flows, with potential implications for ergodic theory and the Warsaw Jewett--Krieger program in higher dimensions. The work integrates Ambrose--Kakutani-type representations, Lipschitz extension results, and Baire-category arguments to achieve a unifying multidimensional model.

Abstract

We show that the shift space of $1$-Lipschitz functions from $\mathbb{R}^k$ to the unit interval is a topological model for all free measurable $\mathbb{R}^k$-flows. This generalizes a theorem from 1973 by Eberlein for $\mathbb{R}$-flows. For $k\geq 2$ the proof relies on an $\mathbb{R}^k$-generalization of a Lipschitz refinement of the Bebutov--Kakutani theorem proven by Gutman, Jin and Tsukamoto. A separate expository proof of Eberlein's theorem ($k=1$) is included (differing from the original proof).

Theorems & Definitions (69)

• Theorem 1.1
• Theorem 1.2
• Definition 1.3
• Theorem 1.4
• Theorem 1.5
• Remark 2.1
• Remark 2.2
• Definition 2.3
• Definition 2.4
• Definition 2.5