2018/04/30 by Alejandro Cabrera, Matias L. del Hoyo, Matias del Hoyo +1 · 1 citation
Mathematics · #Covering space #Differentiable function #Discrete group #Dynamical systems theory #Generalization #Geometric and Algebraic Topology #Group (periodic table) #Homotopy and Cohomology in Algebraic Topology #Manifold (fluid mechanics) #Mathematical Dynamics and Fractals #Orbit (dynamics) #Topological dynamics #math.AT #math.DG #math.DS #msc:22A22 #msc:37C15 #msc:57M12
paper · pdf · doi:10.4171/rmi/1194
published as Revista Matemática Iberoamericana (2020) · Revised version, 24 pages, a section about the characteristic class of a dynamics was added
openalex created_date 2018/04/13 · arxiv created 2019/07/30 · openalex publication_date 2020/04/03 · arxiv updated 2020/08/04 · openalex updated_date 2026/08/05
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.