On compact uniformly recurrent subgroups

Abstract Let a group Γ act on a paracompact, locally compact, Hausdorff space M by homeomorphisms and let 2 M denote the set of closed subsets of M . We endow 2 M with the Chabauty topology, which is compact and admits a natural Γ-action by homeomorphisms. We show that for every minimal Γ-invariant closed subset $\mathcal{Y}$ of 2 M consisting of compact sets, the union $\bigcup \mathcal{Y}\subset M$ has compact closure. As an application, we deduce that every compact uniformly recurrent subgroup of a locally compact group is contained in a compact normal subgroup. This generalizes a result of Ušakov on compact subgroups whose normalizer is compact.