A homology theory for smale spaces: A summary

Abstract

We consider Smale spaces, a particular class of hyperbolic topological dynamical systems, which include the basic sets for Smale's Axiom A systems. We present an algebraic invariant for such systems which is based on Krieger's dimension group for the special case of shifts of finite type. This theory provides a Lefschetz formula relating trace data with the number of periodic points of the system, answering a question posed by R. Bowen. The key ingredient is the existence of Markov partitions with special properties.