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9780821820025

Classification and Orbit Equivalence Relations

by
  • ISBN13:

    9780821820025

  • ISBN10:

    0821820028

  • Format: Hardcover
  • Copyright: 1999-11-01
  • Publisher: Amer Mathematical Society

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Summary

Actions of Polish groups are ubiquitous in mathematics. In certain branches of ergodic theory and functional analysis, one finds a systematic study of the group of measure-preserving transformations and the unitary group. In logic, the analysis of countable models intertwines with results concerning the actions of the infinite symmetric group. This text develops the theory of Polish group actions entirely from scratch, ultimately presenting a coherent theory of the resulting orbit equivalence classes that may allow complete classification by invariants of an indicated form. The book concludes with a criterion for an orbit equivalence relation classifiable by countable structures considered up to isomorphism. This self-contained volume offers a complete treatment of this active area of current research and develops a difficult general theory classifying a class of mathematical objects up to some relevant notion of isomorphism or equivalence.

Table of Contents

Preface xi
An outline
1(4)
Some specific classification problems
1(1)
The form of this book (and one time paper)
2(1)
Acknowledgments
3(2)
Definitions and technicalities
5(32)
Polish groups and Polish spaces
5(10)
Equivalence relations
15(3)
Spaces of countable structures
18(5)
Baire category methods
23(14)
Turbulence
37(22)
Generic ergodicity
37(4)
The definition of turbulence
41(11)
Examples
52(3)
Historical remarks
55(4)
Classifying homeomorphisms
59(16)
Definitions and remarks
59(2)
Classification in dimension 1
61(5)
Non-classification in dimension 2
66(8)
Remarks and connections
74(1)
Infinite dimensional group representations
75(6)
A generalized Scott analysis
81(34)
A preliminary discussion of a specific case
81(12)
The general case
93(15)
A counterexample
108(3)
A different direction
111(4)
GE groups
115(34)
More on Polish groups; Glimm-Effros; G+δ orbits
115(13)
Invariantly metrizable and nilpotent are GE
128(6)
Dynamic changes in topologies
134(5)
Products of locally compact groups
139(6)
CLI groups have the weak Glimm-Effros property
145(4)
The dark side
149(6)
Beyond Borel
155(22)
Two theorems by transfinite changes in topologies
155(14)
Cardinality in L(R)
169(8)
Looking ahead
177(4)
Appendix A. Ordinals 181(4)
Appendix B. Notation 185(4)
Bibliography 189(4)
Index 193

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