Introduction .................................................... 7
Chapter 1 Preliminaries ....................................... 17
1.1 Measure-theoretic dynamical systems ....................... 17
1.2 Ergodic dynamical systems with discrete spectrum .......... 20
1.3 Measure-theoretical joinings .............................. 22
1.4 Group extensions of measure-theoretic dynamical systems ... 24
1.5 Rokhlin cocycle extensions ................................ 28
1.6 Gauss dynamical systems ................................... 29
1.7 Topological dynamics - definitions and notations .......... 31
1.8 Universal flows ........................................... 35
Chapter 2 Semisimple Automorphisms ............................ 43
2.1 Group and isometric extensions, joinings .................. 43
2.2 Furstenberg decomposition ................................. 46
2.3 Semisimplicity ............................................ 48
2.4 Natural factors and the structure of factors for
semisimple automorphisms .................................. 49
2.5 Joinings of ergodic group extensions of semisimple
automorphisms ............................................. 51
2.6 Applications of natural families .......................... 57
2.7 Final remarks ............................................. 60
Chapter 3 Semisimple Group Extensions of Rotations ............ 61
3.1 General backgrounds ....................................... 61
3.2 Self-joinings of Rokhlin cocycles extensions for regular
cocycles .................................................. 62
3.3 Cocycles over irrational rotations ........................ 69
3.1 Semisimple authomorphisms ................................. 79
3.5 Final remarks ............................................. 81
Chapter 4 Natural Families of Factors in Topological
Dynamics ............................................ 83
4.1 General backgrounds ....................................... 83
4.2 A natural family of factors defined by minimal joinings ... 85
4.3 A natural family of factors defined by B-joinings ......... 90
4.4 Group extensions of minimal rotations ..................... 94
Chapter 5 Real Cocycle Extensions of Minimal Rotations ....... 101
5.1 Existence of almost periodic oints ....................... 101
5.2 Essential values of a cocycle ............................ 102
5.3 Characterization of essential values for minimal
rotations ................................................ 107
5.4 Classification of continuous real cocycles over minimal
rotations ................................................ 108
5.5 Zero-time cocycle for Morse shifts ....................... 110
5.6 An application - a disjointness theorem .................. 113
Chapter 6 Essential Values of Topological Cocycles over
Minimal Rotations .................................. 121
6.1 Essential values of a cocycle ............................ 121
6.2 The groups of essential values for extensions of
minimal rotations ........................................ 127
6.3 Atkinson's theorem and regularity of cylinder flows ...... 131
6.4 A more general case ...................................... 137
Chapter 7 Cylinder Cocycle Extensions and Rotations .......... 141
7.1 The problem of minimality for cylinder extensions of
minimal rotations on a circle ............................ 142
7.2 The problem of minimality for cylinder extensions of
adding machines .......................................... 146
7.3 Existence of point transitive cocycles over compact
monothetic groups ........................................ 148
Chapter 8 Some Applications of Groups of Essential Values
of Cocycles in Topological Dynamics ................ 153
8.1 Preliminaries ............................................ 153
8.2 Counterexamples in topological dynamics .................. 155
8.3 Isomorphisms of Rokhlin cocycle extensions of point
transitive flows ......................................... 158
8.4 A remark on some recent results .......................... 166
Appendix A. Lebesgue Spaces and their Properties .............. 167
А.1 Point and set maps of measure spaces ..................... 167
A.2 Probability Lebesgue spaces .............................. 168
A.3 Spectral theory of unitary operators ..................... 169
Appendix B. Topological topics ................................ 177
B.l Uniform structures ....................................... 177
B.2 The Čech-Stone compactification of a discrete
topological group ........................................ 178
Bibliography .................................................. 183
Index ......................................................... 189
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