The locally discrete groups conjecture for analytic circle actions

Let $G\leq \operatorname{Diff}_+^\omega(\mathbf{S}^1)$ be a finitely generated locally discrete group. A locally discrete group's structure conjecture predicts both that the action admits a Markov partition of its minimal set and that, if the action is minimal and expanding, $G$ is analytically conjugate to a finite central extension of a cocompact Fuchsian group, whereas $G$ is virtually free in any other case. This packages the proposed dynamical and algebraic descriptions of finitely generated locally discrete analytic circle groups; the supplied text presents it as a paradigm motivating the work, without establishing the full statement.