The Kakutani-Bebutov Theorem (1968) says that if a compact metric real flow satisfies that the set of its fixed points is homeomorphic to a subset of the real line, then it is embeddable into the shift on the space of all continuous functions from the real line to the unit interval. An interesting fact is that this universal space is a function space; however, it is not compact, nor locally compact. We provide an explicit compact metric universal space for all compact metric real flows, with no restriction, which is a countable product of compact function spaces; namely, we construct a compact metric real flow into which we can embed all compact metric real flows.
This is a joint work with Yonatan Gutman (2016).