Noncommutative Geometry Seminar

Smale introduced the notion of an Axiom A system in the 1960's as part of an ambitious program to study the dynamics of smooth maps of manifolds. The interesting part of the dynamics occurs on invariant subsets, called basic sets. Typically, these are not submanifolds, but have some kind of fractal nature. Ruelle then gave a definition of a Smale space to axiomatize these systems. Among the most important of these are shifts of finite type. Bowen (following work of Sinai and others) showed that every basic set was the quotient of a shift of finite type. Manning used this to count the periodic points of a Smale space and proved that the Artin-Mazur zeta function was rational. This led Bowen to conjecture the existence of a homology theory for Smale spaces having a Lefschetz formula. Such a theory was given by Krieger for shifts of finite type by using ideas from C*-algebras and K-theory. In this talk, I will show how to combine Krieger's and Bowen's results to provide a homology theory, as proposed by Bowen.