Combinatorial Constructions in Ergodic Theory and Dynamics

By: Anatole Katok

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Paperback | 1 January 2003

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Ergodic theory studies measure-preserving transformations of measure spaces. These objects are intrinsically infinite, and the notion of an individual point or of an orbit makes no sense. Still there are a variety of situations when a measure preserving transformation (and its asymptotic behavior) can be well described as a limit of certain finite objects (periodic processes). The first part of this book develops this idea systematically. Genericity of approximation in various categories is explored, and numerous applications are presented, including spectral multiplicity and properties of the maximal spectral type. The second part of the book contains a treatment of various constructions of cohomological nature with an emphasis on obtaining interesting asymptotic behavior from approximate pictures at different time scales. The book presents a view of ergodic theory not found in other expository sources. It is suitable for graduate students familiar with measure theory and basic functional analysis.

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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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Combinatorial Constructions in Ergodic Theory and Dynamics

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Paperback

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