Table o f Content s
0. Introductio n 1
0.1. Theorem s an d construction s an d ergodi c theor y 1
0.2. A little histor y 2
0.3. Th e stor y an d purpos e o f these note s 3
0.4. Acknowledgement s 4
Part I . Approximatio n an d Genericit y i n Ergodi c Theor y 5
1. Periodi c processe s 5
2. Genericit y o f approximation 10
3. Variou s types o f approximation 12
3.1. Cycli c approximatio n 12
3.2. Approximatio n o f type (n , n + 1) 15
3.3. a-wea k mixin g an d singularit y o f convolutions 15
4. Spectra l multiplicit y o f ergodic transformation s 18
4.1. Essentia l valu e of spectral multiplicit y 19
4.2. Transformation s wit h arbitrar y maxima l spectra l multiplicit y 19
4.3. Cartesia n power s an d multiplicitie s bounde d fro m below 2 2
4.4. Som e recent result s 2 6
5. Approximatio n an d codin g 2 7
6. Invarian t measure s fo r transformatio n wit h specificatio n 3 7
7. Generi c induce d map s 4 2
8. Combinatoria l approximatio n b y conjugatio n constructio n 4 4
8.1. Introductio n 4 4
8.2. Genera l framewor k 4 5
8.3. Ergodicit y an d rotatio n factor s 4 8
8.4. Non-standar d transformation s 5 0
Part II . Cocycles , Cohomolog y an d Combinatoria l Construction s 5 3
9. Definition s an d principa l construction s 5 4
9.1. Cocycles , coboundarie s an d Macke y rang e 5 5
9.2. Lipschit z cocycles , pseudo-isometrie s an d th e Ambrose-Kakutan i
theorem 5 8
m
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