If $\alpha : X \to X$ is a minimal homeomorphism on a compact Hausdorff space $X$, then if $X$ is connected, $\alpha$ is totally minimal, that is $\alpha^k$ is minimal for every $k \in \mathbb{Z}$.

I am interested in a minimal homeomorphisms on locally compact but not necessarily compact spaces.

Is there a similar result relating connected spaces to total minimality in the not necessarily compact case?

In general, it seems to be difficult to find results on minimal homeomorphisms on noncompact spaces, so in addition to the above question, any good reference here would be very useful.

Thank you, Nimh