Dynamical Complexity and Controlled Operator $K$-Theory

In this volume, the authors introduce a property of topological dynamical systems that they call finite dynamical complexity. For systems with this property, one can in principle compute the $ K$-theory of the associated crossed product $C^{?}$-algebra by splitting it up into simpler pieces and using the methods of controlled $K$-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. The authors have tried to keep the volume as self-contained as possible and hope the main part will be accessible to someone with the equivalent of a first course in operator $K$-theory. In particular, they do not assume prior knowledge of controlled $K$-theory and use a new and concrete model for the Baum-Connes conjecture with coefficients that requires no bivariant $K$-theory to set up.