Given a regular language $U$, when does there exist $V$ such that $U^\omega$ = $\lim V$?

Question

What I know is that $W = \lim V$ for some $V$ if and only if $W$ is the language of some deterministic buchi automata, namely that of $V$.

So, to attack this problem I tried to come up with some language of the form $U^\omega$ which cannot be determinized. But I keep failing trying to come up with something like this.

Is it true in general that given regular $U$, there is some regular $V$ such that $U^\omega$ = $\lim V$?