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A homology theory for Smale spaces

About this Title

Ian F. Putnam, Department of Mathematics and Statistics, University of Victoria, Victoria, B.C. V8W 3R4 Canada

Publication: Memoirs of the American Mathematical Society
Publication Year: 2014; Volume 232, Number 1094
ISBNs: 978-1-4704-0909-8 (print); 978-1-4704-1897-7 (online)
DOI: https://doi.org/10.1090/memo/1094
Published electronically: March 17, 2014
Keywords: Smale space, homology, Lefschetz formula
MSC: Primary 37D20, 37D45

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Table of Contents

Chapters

  • Preface
  • 1. Summary
  • 2. Dynamics
  • 3. Dimension groups
  • 4. The complexes of an $s/u$-bijective factor map
  • 5. The double complexes of an $s/u$-bijective pair
  • 6. A Lefschetz formula
  • 7. Examples
  • 8. Questions

Abstract

We develop 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. We prove 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 1970’s.

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