  We study the uniformly recurrent subgroups of groups acting by homeomorphisms
on a topological space. We prove a general result relating uniformly recurrent
subgroups to rigid stabilizers of the action, and deduce a $C^*$-simplicity
criterion based on the non-amenability of rigid stabilizers. As an application,
we show that Thompson's group $V$ is $C^\ast$-simple, as well as groups of
piecewise projective homeomorphisms of the real line. This provides examples of
finitely presented $C^\ast$-simple groups without free subgroups. We prove that
a branch group is either amenable or $C^\ast$-simple. We also prove the
converse of a result of Haagerup and Olesen: if Thompson's group $F$ is
non-amenable, then Thompson's group $T$ must be $C^\ast$-simple. Our results
further provide sufficient conditions on a group of homeomorphisms under which
uniformly recurrent subgroups can be completely classified. This applies to
Thompson's groups $F$, $T$ and $V$, for which we also deduce rigidity results
for their minimal actions on compact spaces.
