A Homology Theory for Smale Spaces (Memoirs of the American Mathematical Society) by Ian F. Putnam
2014 | ISBN: 1470409097 | English | 122 pages | PDF | 1 MB
2014 | ISBN: 1470409097 | English | 122 pages | PDF | 1 MB
The author develops a homology theory for Smale spaces, which include the basics sets for an Axiom A diffeomorphism. It is based on two ingredients. The first is an improved version of Bowen's result that every such system is the image of a shift of finite type under a finite-to-one factor map. The second is Krieger's dimension group invariant for shifts of finite type. He proves a Lefschetz formula which relates the number of periodic points of the system for a given period to trace data from the action of the dynamics on the homology groups. The existence of such a theory was proposed by Bowen in the 1970s.