Joint work with Lei Jin and Masaki Tsukamoto.
We prove that an $\mathbb{R}$-action on a compact metric space may be
equivariantly embedded
into the compact space of one-Lipschitz functions on the real line if
its fixed point set
can be topologically embedded into the unit interval.
This is a refinement of the classical Bebutov-Kakutani theorem (1968).