  We define the notion of uniformly recurrent subgroup, URS in short, which is
a topological analog of the notion of invariant random subgroup (IRS),
introduced in a work of M. Abert, Y. Glasner and B. Virag. Our main results are
as follows. (i) It was shown by B. Weiss that for an arbitrary countable
infinite group G, any free ergodic probability measure preserving G-system
admits a minimal model. In contrast we show here, using URS's, that for the
lamplighter group there is an ergodic measure preserving action which does not
admit a minimal model. (ii) For an arbitrary countable group G, every URS can
be realized as the stability system of some topologically transitive G-system.
