Classification and orbit equivalence relations / Greg Hjorth.

Format:

Book

Author/Creator:

Hjorth, Greg, 1963- author.

Series:

Mathematical surveys and monographs ; v. 75.

Mathematical surveys and monographs ; volume 75

Language:

English

Subjects (All):

Equivalence classes (Set theory).

Equivalence relations (Set theory).

Classification.

Physical Description:

1 online resource (217 p.)

Place of Publication:

Providence, Rhode Island : American Mathematical Society, [2000]

Language Note:

English

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.

Contents:

""Contents""; ""Preface""; ""Chapter 1. An outline""; ""1.1. Some specific classification problems""; ""1.2. The form of this book (and one time paper)""; ""1.3. Acknowledgments""; ""Chapter 2. Definitions and technicalities""; ""2.1. Polish groups and Polish spaces""; ""2.2. Equivalence relations""; ""2.3. Spaces of countable structures""; ""2.4. Baire category methods""; ""Chapter 3. Turbulence""; ""3.1. Generic ergodicity""; ""3.2. The definition of turbulence""; ""3.3. Examples""; ""3.4. Historical remarks""; ""Chapter 4. Classifying homeomorphisms""; ""4.1. Definitions and remarks""

""4.2. Classification in dimension 1""""4.3. Non-classification in dimension 2""; ""4.4. Remarks and connections""; ""Chapter 5. Infinite dimensional group representations""; ""Chapter 6. A generalized Scott analysis""; ""6.1. A preliminary discussion of a specific case""; ""6.2. The general case""; ""6.3. A counterexample""; ""6.4. A different direction""; ""Chapter 7. GE groups""; ""7.1. More on Polish groups; Glimm-Effros ...""; ""7.2. Invariantly metrizable and nilpotent are GE""; ""7.3. Dynamic changes in topologies""; ""7.4. Products of locally compact groups""

""7.5. CLI groups have the weak Glimm-Effros property""""Chapter 8. The dark side""; ""Chapter 9. Beyond Borel""; ""9.1. Two theorems by transfinite changes in topologies""; ""9.2. Cardinality in L(R)""; ""Chapter 10. Looking ahead""; ""Appendix A. Ordinals""; ""Appendix B. Notation""; ""Bibliography""; ""Index""; ""A""; ""B""; ""C""; ""D""; ""E""; ""F""; ""G""; ""H""; ""I""; ""J""; ""K""; ""L""; ""M""; ""N""; ""O""; ""P""; ""R""; ""S""; ""T""; ""U""; ""V""; ""W""; ""X""; ""Y""; ""Z""

Notes:

Description based upon print version of record.

Includes bibliographical references (pages 189-191) and index.

Description based on print version record.

ISBN:

1-4704-1302-7