A Lipschitz refinement of the Bebutov--Kakutani dynamical embedding theorem

TL;DR

This paper refines the classical Bebutov--Kakutani embedding theorem by establishing conditions under which an tion on a compact metric space can be embedded into the space of one-Lipschitz functions, provided its fixed point set embeds in the interval.

Contribution

The authors provide a Lipschitz refinement of the Bebutov--Kakutani theorem, linking fixed point set embeddings to dynamical embeddings in Lipschitz function spaces.

Findings

Embedding of ctions with fixed points in [0,1] into Lipschitz function space

Refinement of classical embedding theorem with Lipschitz conditions

Conditions for equivariant embedding based on fixed point set topology

Abstract

We prove that an $\mathbb{R}$-action on a compact metric space embeds equivariantly in the space of one-Lipschitz functions $\mathbb{R}\to[0,1]$ if its fixed point set can be topologically embedded in the unit interval. This is a refinement of the classical Bebutov--Kakutani theorem (1968).