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A Homology Theory for Smale Spaces

Ian F. Putnam University of Victoria, Victoria, British Columbia, Canada
Available Formats:
Electronic ISBN: 978-1-4704-1897-7
Product Code: MEMO/232/1094.E
List Price: $76.00 MAA Member Price:$68.40
AMS Member Price: $45.60 Click above image for expanded view A Homology Theory for Smale Spaces Ian F. Putnam University of Victoria, Victoria, British Columbia, Canada Available Formats:  Electronic ISBN: 978-1-4704-1897-7 Product Code: MEMO/232/1094.E  List Price:$76.00 MAA Member Price: $68.40 AMS Member Price:$45.60
• Book Details

Memoirs of the American Mathematical Society
Volume: 2322014; 122 pp
MSC: Primary 37;

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.

• 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
• Requests

Review Copy – for reviewers who would like to review an AMS book
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Volume: 2322014; 122 pp
MSC: Primary 37;

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.

• 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
Review Copy – for reviewers who would like to review an AMS book
Permission – for use of book, eBook, or Journal content
Accessibility – to request an alternate format of an AMS title
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