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A Homology Theory for Smale Spaces
Ian F. Putnam, University of Victoria, British Columbia, Canada

Memoirs of the American Mathematical Society
2014; 122 pp; softcover
Volume: 232
ISBN-10: 1-4704-0909-7
ISBN-13: 978-1-4704-0909-8
List Price: US$76
Individual Members: US$45.60
Institutional Members: US$60.80
Order Code: MEMO/232/1094
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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.

Table of Contents

  • Summary
  • Dynamics
  • Dimension groups
  • The complexes of an \(s/u\)-bijective factor map
  • The double complexes of an \(s/u\)-bijective pair
  • A Lefschetz formula
  • Examples
  • Questions
  • Bibliography
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