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Ergodic Theory : Cambridge Studies in Advanced Mathematics (Paperback) - Karl E. Petersen

Ergodic Theory

Cambridge Studies in Advanced Mathematics (Paperback)

Paperback Published: 22nd January 1990
ISBN: 9780521389976
Number Of Pages: 344

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The author presents the fundamentals of the ergodic theory of point transformations and several advanced topics of intense research. The study of dynamical systems forms a vast and rapidly developing field even when considering only activity whose methods derive mainly from measure theory and functional analysis. Each of the basic aspects of ergodic theory--examples, convergence theorems, recurrence properties, and entropy--receives a basic and a specialized treatment. The author's accessible style and the profusion of exercises, references, summaries, and historical remarks make this a useful book for graduate students or self study.

Industry Reviews

'What the contents list does not prepare you for is the very high standard of exposition. The scholarship involved in this work will be appreciated by workers in the field and by generations of research students. I personally think this is an excellent book. It is a book which can be explored at leisure and has a place in the library of anyone with a more than passing interest in ergodic theory.' Bulletin of the London Mathematical Society ' ... an excellent introduction to several areas which are of interest both from the point of view of the mathematical scholar and from that of the research mathematician.' American Scientist

Prefacep. ix
Introduction and preliminariesp. 1
The basic questions of ergodic theoryp. 1
The basic examplesp. 5
Hamiltonian dynamics
Stationary stochastic processes
Bernoulli shifts
Markov shifts
Rotations of the circle
Rotations of compact abelian groups
Automorphisms of compact groups
Gaussian systems
Geodesic flows
Horocycle flows
Flows and automorphisms on homogeneous spaces
The basic constructionsp. 10
Skew products
Flow under a function
Induced transformations
Inverse limits
Natural extensions
Some useful facts from measure theory and functional analysisp. 13
Change of variables
Proofs by approximation
Measure algebras and Lebesgue spaces
Conditional expectation
The Spectral Theorem
Topological groups, Haar measure, and character groups
The fundamentals of ergodic theoryp. 23
The Mean Ergodic Theoremp. 23
The Pointwise Ergodic Theoremp. 27
Recurrencep. 33
Ergodicityp. 41
Strong mixingp. 57
Weak mixingp. 64
More about almost everywhere convergencep. 74
More about the Maximal Ergodic Theoremp. 74
Positive contractions
The maximal equality
Sign changes of the partial sums
The Dominated Ergodic Theorem and its converse
More about the Pointwise Ergodic Theoremp. 90
Maximal inequalities and convergence theorems
The speed of convergence in the Ergodic Theorem
Differentiation of integrals and the Local Ergodic Theoremp. 100
The Martingale convergence theoremsp. 103
The maximal inequality for the Hilbert transformp. 107
The ergodic Hilbert transformp. 113
The filling schemep. 119
The Chacon-Ornstein Theoremp. 126
More about recurrencep. 133
Construction of eigenfunctionsp. 133
Existence of rigid factors
Almost periodicity
Construction of the eigenfunction
Some topological dynamicsp. 150
Topological ergodicity and mixing
Equicontinuous and distal cascades
Uniform distribution mod 1
Structure of distal cascades
The Szemeredi Theoremp. 162
Furstenberg's approach to the Szemeredi and van der Waerden Theorems
Topological multiple recurrence, van der Waerden's Theorem, and Hindman's Theorem
Weak mixing implies weak mixing of all orders along multiples
Outline of the proof of the Furstenberg-Katznelson Theorem
The topological representation of ergodic transformationsp. 186
Recurrence along IP-sets
Perturbation to uniformity
Uniform polynomials
Conclusion of the argument
Two examplesp. 209
Metric weak mixing without topological strong mixing
A prime transformation
Entropyp. 227
Entropy in physics, information theory, and ergodic theoryp. 227
Information theory
Ergodic theory
Information and conditioningp. 234
Generators and the Kolmogorov-Sinai Theoremp. 243
More about entropyp. 249
More examples of the computation of entropyp. 249
Entropy of an automorphism of the torus
Entropy of a skew product
Entropy of an induced transformation
The Shannon-McMillan-Breiman Theoremp. 259
Topological entropyp. 264
Introduction to Ornstein Theoryp. 273
Finitary coding between Bernoulli shiftsp. 281
Sketch of the proof
Reduction to the case of a common weight
Framing the code
What to put in the blanks
Construction of the isomorphism
Referencesp. 302
Indexp. 322
Table of Contents provided by Syndetics. All Rights Reserved.

ISBN: 9780521389976
ISBN-10: 0521389976
Series: Cambridge Studies in Advanced Mathematics (Paperback)
Audience: Professional
Format: Paperback
Language: English
Number Of Pages: 344
Published: 22nd January 1990
Country of Publication: GB
Dimensions (cm): 22.8 x 15.2  x 2.0
Weight (kg): 0.48