  Following the recent advances in the study of groups of circle
diffeomorphisms, we describe an efficient way of classifying the topological
dynamics of locally discrete, finitely generated, virtually free subgroups of
the group $\mathsf{Diff}^\omega_+(\mathbb S^1)$ of orientation preserving
real-analytic circle diffeomorphisms, which include all subgroups of
$\mathsf{Diff}^\omega_+(\mathbb S^1)$ acting with an invariant Cantor set. An
important tool that we develop, of independent interest, is the extension of
classical ping-pong lemma to actions of fundamental groups of graphs of groups.
Our main motivation is an old conjecture by P. R. Dippolito [Ann. Math. 107
(1978), 403--453] from foliation theory, which we solve in this restricted but
significant setting: this and other consequences of the classification will be
treated in more detail in a companion work.
