Discrete dynamics and differentiable stacks
Abstract
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.
Cite
@article{arxiv.1804.00220,
title = {Discrete dynamics and differentiable stacks},
author = {Alejandro Cabrera and Matias del Hoyo and Enrique Pujals},
journal= {arXiv preprint arXiv:1804.00220},
year = {2020}
}
Comments
Revised version, 24 pages, a section about the characteristic class of a dynamics was added