A Lipschitz refinement of the Bebutov-Kakutani dynamical embedding theorem for connected Lie group actions

Abstract

Let $G$ be a connected Lie group equipped with a compatible right-invariant metric and let $G$ act continuously on a compact metrizable space $X$. We prove that $X$ admits a $G$-equivariant embedding into the right argument shift action on the space of one-Lipschitz maps from $G$ to $[0,1]$ if and only if the set of fixed points topologically embeds into $[0,1]$. This generalizes a theorem proved by Gutman, Huo and Tsukamoto for $\mathbb{R}^k$-flows.