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El. knyga: Homology Theory for Smale Spaces

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Putnam defines a type of homology theory for Smale spaces, which include the basic sets for Smale's Axiom A diffeomorphism.. His approach is founded on two pillars. One 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 other is Krieger's dimension group invariant for shifts of finite type. He covers dynamics, dimensional groups, the complexes of an s/u-bijective factor map, the double complexes of an s/u-bijective pair, a Lefschetz formula, examples, and questions. Annotation ©2015 Ringgold, Inc., Portland, OR (protoview.com)
Preface vii
Chapter 1 Summary
1(10)
Chapter 2 Dynamics
11(26)
2.1 Smale spaces
11(4)
2.2 Shifts of finite type
15(3)
2.3 Maps
18(1)
2.4 Self-products of a map
19(2)
2.5 s/u-resolving and s/u-bijective maps
21(7)
2.6 s/u-bijective pairs
28(4)
2.7 Technical results
32(5)
Chapter 3 Dimension groups
37(18)
3.1 Free abelian groups
37(1)
3.2 The dimension group of a graph
38(2)
3.3 The dimension group of a shift of finite type
40(1)
3.4 The dimension group as a covariant functor
41(6)
3.5 The dimension group as a contravariant functor
47(8)
Chapter 4 The complexes of an s/u-bijective factor map
55(30)
4.1 Definitions of the complexes
56(4)
4.2 Symbolic presentations
60(9)
4.3 Equivalence of the complexes
69(9)
4.4 Functorial properties
78(1)
4.5 Independence of resolution
79(6)
Chapter 5 The double complexes of an s/u-bijective pair
85(20)
5.1 Definitions of the complexes
85(6)
5.2 Symbolic presentations
91(6)
5.3 Equivalence of the complexes
97(2)
5.4 Functorial properties
99(1)
5.5 Independence of s/u-bijective pair
100(5)
Chapter 6 A Lefschetz formula
105(8)
6.1 The statement
105(1)
6.2 The periodic point side
106(5)
6.3 The homological side
111(2)
Chapter 7 Examples
113(4)
7.1 Shifts of finite type
113(1)
7.2 Totally disconnected stable sets
113(1)
7.3 Solenoids
114(1)
7.4 A hyperbolic toral automorphism
115(2)
Chapter 8 Questions
117(4)
8.1 Order on homology groups
117(1)
8.2 Machinery
118(1)
8.3 Relation with Cech (co)homology
119(1)
8.4 C*-algebras
119(2)
Bibliography 121
Ian F. Putnam, University of Victoria, British Columbia, Canada.