Introduction | p. 1 |
Theorems and constructions and ergodic theory | p. 1 |
A little history | p. 2 |
The story and purpose of these notes | p. 3 |
Acknowledgements | p. 4 |
Approximation and Genericity in Ergodic Theory | p. 5 |
Periodic processes | p. 5 |
Genericity of approximation | p. 10 |
Various types of approximation | p. 12 |
Cyclic approximation | p. 12 |
Approximation of type (n, n + 1) | p. 15 |
a-weak mixing and singularity of convolutions | p. 15 |
Spectral multiplicity of ergodic transformations | p. 18 |
Essential value of spectral multiplicity | p. 19 |
Transformations with arbitrary maximal spectral multiplicity | p. 19 |
Cartesian powers and multiplicities bounded from below | p. 22 |
Some recent results | p. 26 |
Approximation and coding | p. 27 |
Invariant measures for transformation with specification | p. 37 |
Generic induced maps | p. 42 |
Combinatorial approximation by conjugation construction | p. 44 |
Introduction | p. 44 |
General framework | p. 45 |
Ergodicity and rotation factors | p. 48 |
Non-standard transformations | p. 50 |
Cocycles, Cohomology and Combinatorial Constructions | p. 53 |
Definitions and principal constructions | p. 54 |
Cocycles, coboundaries and Mackey range | p. 55 |
Lipschitz cocycles, pseudo-isometries and the Ambrose-Kakutani theorem | p. 58 |
Cohomological equations for measure-preserving transformations and flows | p. 59 |
Structure of equivalence classes | p. 62 |
Majorization and density in L[superscript 1] | p. 62 |
Continuous and almost differentiable representations | p. 66 |
Rigidity and stability | p. 68 |
Definitions | p. 68 |
Translations of the torus and smooth rigidity | p. 70 |
Stability of Holder cocycles for transformations with specification | p. 74 |
Livshitz theory | p. 78 |
Invariant distributions and stability of partially hyperbolic systems | p. 81 |
Stability determined by invariant distributions in parabolic systems | p. 85 |
Wild cochains with tame coboundaries | p. 89 |
Continuous cocycles over measure-preserving homeomorphisms | p. 90 |
Fast approximation and C[superscript infinity] cocycles | p. 94 |
Minimal nonergodic diffeomorphisms of T[superscript 2] | p. 97 |
Minimal nonergodic interval exchange transformations | p. 98 |
Non-trivial cocycles | p. 102 |
Two general criteria | p. 102 |
The case of fast C[superscript infinity] approximation | p. 105 |
Weakly mixing flows on T[superscript 2] | p. 107 |
Ergodicity of analytic cylindrical cascades | p. 112 |
Weak mixing of special flows over interval exchange transformations | p. 114 |
References | p. 117 |
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