On the representation of measurable and continuous dynamical systems by Lipschitz functions

TL;DR

This paper introduces two theorems that represent measurable and continuous dynamical systems using Lipschitz functions, extending classical results and providing new embedding techniques.

Contribution

It generalizes Eberlein's 1973 theorem for $ $-flows and refines the Bebutov-Kakutani theorem with Lipschitz functions for $ ^k$ actions.

Findings

Any Borel $G$-action admits an injective Lipschitz embedding into $Lip_1(G)$.

Continuous $ ^k$-actions on compact spaces can be embedded into $Lip_1( ^k)$ under certain conditions.

The results extend classical theorems to Lipschitz function representations.

Abstract

Two representations theorems are presented: 1. Any Borel action of a second countable locally compact group $G$ on a standard Borel space $X$ admits an injective $G$-equivariant Borel map into the shift space of $1$-Lipschitz functions from $G$ to the unit interval $Lip_1(G)$. 2. Any continuous action of $\mathbb{R}^k$ ($k\in \mathbb{N}$) on a metrizable compact space $X$ admits an injective $G$-equivariant continuous map into $Lip_1(\mathbb{R}^k)$ if the fixed point set $Fix(X,\mathbb{R}^k)$ embeds into $[0,1]$ and $(X,\mathbb{R}^k)$ is \textit{weakly locally free}, that is $\mathbb{R}^k$ acts freely outside the fixed point set. The first theorem generalizes a theorem from 1973 by Eberlein for $\mathbb{R}$-flows. The second theorem generalizes a Lipschitz refinement of the Bebutov-Kakutani theorem proven by Gutman, Jin and Tsukamoto in 2019.