Speaker: Ian F. Putnam

Title: Smale's Axiom A systems, C*-algebras and homology

Abstract: Stephen Smale initiated an ambitious program to describe dynamical systems which he called Axiom A. These are smooth maps on manifolds which possess a certain hyperbolicity. David Ruelle gave a purely topological description and later showed how one could associated various C^*-algebras to such systems. The most basic examples are the shifts of finite type (SFT), and the associated C^*-algebras in this case include the Cuntz-Krieger algebras and their AF cores. In this talk, I will give a brief introduction to the construction of these dynamical systems the construction of the C^*-algebras and what is known about them. Finally, I will discuss how attempts to compute the K-theory of the C^*-algebras lead to a natural homology theory for the systems, answering a question of Rufus Bowen.