Section 1: Introduction

This paper is an attempt to survey some recent results in the theory of topological dynamics which have been obtained using C*-algebras, K-theory and tools which are native to those branches of mathematics. For certain topological dynamical systems, one may construct a C*-algebra and consider its K-theory. This K-theory is then interpreted in purely dynamical terms. The results we have in mind, show how this K-theoretic invariant may be used to classify the system up to orbit equivalence, or other natural equivalences in dynamics.

In addition to surveying recent results, we will also present an introduction to the construction of the C*-algebras involved. It is aimed at readers with a background in dynamics rather than C*-algebras. The pace is quite pedestrian and the intent is to convey the basic ideas. It is not meant to be thorough; we quickly evade technical issues which would challenge these aims. The hope is to give readers in dynamics the notion of these C*algebras as geometric/topological/dynamic objects.

The idea of associating a C*-algebra to a topological dynamical system of some sort goes back to Murray and von Neumann. The actual construction is due to many people (depending on the sort of system involved): Zeller-Mayer, Effros and Hahn, Krieger, …. The most general construction is due to Renault [Ren]. We will consider two sorts of dynamical systems: an equivalence relation satisfying some conditions on a locally compact Hausdorff space X and a countable group г acting as homeomorphisms of such a space X.