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

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.