Dynamical complexity and controlled operator K-theory
Abstract
In this paper, we introduce a property of topological dynamical systems that we call finite dynamical complexity. For systems with this property, one can in principle compute the -theory of the associated crossed product -algebra by splitting it up into simpler pieces and using the methods of controlled -theory. The main part of the paper illustrates this idea by giving a new proof of the Baum-Connes conjecture for actions with finite dynamical complexity. We have tried to keep the paper as self-contained as possible: we hope the main part will be accessible to someone with the equivalent of a first course in operator -theory. In particular, we do not assume prior knowledge of controlled -theory, and use a new and concrete model for the Baum-Connes conjecture with coefficients that requires no bivariant -theory to set up.
Cite
@article{arxiv.1609.02093,
title = {Dynamical complexity and controlled operator K-theory},
author = {Erik Guentner and Rufus Willett and Guoliang Yu},
journal= {arXiv preprint arXiv:1609.02093},
year = {2022}
}
Comments
Second version has extensive revisions: chiefly, the main argument has been reworked based on a new notion of `finite dynamical complexity'. This is partly to correct an error, and partly as the new version seemed simpler and more natural. A new appendix has been added comparing finite dynamical complexity to other properties. Third version corrects a minor error in Definition 7.4