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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
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.- N. Aoki and K. Hiraide, Topological theory of dynamical systems, North-Holland Mathematical Library, vol. 52, North-Holland Publishing Co., Amsterdam, 1994. Recent advances. MR 1289410
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