  In this paper we relate the study of actions of discrete groups over
connected manifolds to that of their orbit spaces seen as differentiable
stacks. We show that the orbit stack of a discrete dynamical system on a simply
connected manifold encodes the dynamics up to conjugation and inversion. We
also prove a generalization of this result for arbitrary discrete groups and
non-simply connected manifolds, and relate it to the covering theory of stacks.
As applications, we obtain a geometric version of Rieffel's theorem on
irrational rotations of the circle, we compute the stack-theoretic fundamental
group of hyperbolic toral automorphisms, and we revisit the classification of
lens spaces.
