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.

中文翻译：

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离散动态和可微堆栈

在本文中，我们将连接流形上离散群的作用研究与其被视为可微叠的轨道空间的作用联系起来。我们表明，在简单连接的流形上的离散动力学系统的轨道堆栈将动力学编码为共轭和反演。我们还证明了该结果对任意离散组和非简单连通流形的推广，并将其与堆栈的覆盖理论联系起来。作为应用程序，我们获得了圆的无理旋转的Rieffel定理的几何形式，我们计算了双曲线环型自同构的堆叠理论基本组，并重新研究了晶状体空间的分类。