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.
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"For an introduction and general survey, Petersen's book is an excellent choice." Nathaniel Friedman, Mathematical Reviews
Book Description
The fundamentals of the ergodic theory of point transformations are presented along with several advanced topics of intense research in this well documented study of dynamical systems.
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This review is from: Ergodic Theory (Cambridge Studies in Advanced Mathematics) (Paperback)
This text is a must have for anyone studying measure theoretic ergodic theory. The introductory chapter is broad (and difficult) but it does show the vast breadth of the subject. I especially liked the historical references and background material there. The treatment of the mean ergodic theorem is also very nice and worked well with my MSc. students. The inclusion of very new material, like section 4.5, was very helpful for me as a Ph.D. student struggling to understand the research papers.
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First Sentence:
Without going into the details (to which the rest of the book is devoted), we mention some of the basic questions, examples, and constructions of ergodic theory, in order to provide an indication of the content and flavor of the subject as well as to establish reference points for terminology and notation. Read the first page Key Phrases - Statistically Improbable Phrases (SIPs): (learn more)
local ergodic theorem, maximal ergodic theorem, distal cascade, dominated ergodic theorem, filling scheme, mean ergodic theorem, countable measurable partition, pointwise ergodic theorem, positive upper density, ergodic automorphisms, horocycle flows, finite measurable partition, mixing transformations, long arithmetic progressions, uniquely ergodic, recurrence theorem, topological dynamics, topological entropy, measure algebra, positive contraction, ergodic case, orbit closure, weak mixing, finite measure space, maximal inequality Key Phrases - Capitalized Phrases (CAPs): (learn more)
Hindman's Theorem, Marriage Lemma, Waerden's Theorem, Chacon-Ornstein Theorem, Furstenberg-Katznelson Theorem, Spectral Theorem, Ajo Ajo, Bounded Convergence Theorem, T-invariant Borel, Dominated Convergence Theorem, Filler Lemma, Baire Category Theorem, Fatou's Lemma, Monotone Convergence Theorem, New York New!
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Without going into the details (to which the rest of the book is devoted), we mention some of the basic questions, examples, and constructions of ergodic theory, in order to provide an indication of the content and flavor of the subject as well as to establish reference points for terminology and notation. Read the first page Key Phrases - Statistically Improbable Phrases (SIPs): (learn more)
local ergodic theorem, maximal ergodic theorem, distal cascade, dominated ergodic theorem, filling scheme, mean ergodic theorem, countable measurable partition, pointwise ergodic theorem, positive upper density, ergodic automorphisms, horocycle flows, finite measurable partition, mixing transformations, long arithmetic progressions, uniquely ergodic, recurrence theorem, topological dynamics, topological entropy, measure algebra, positive contraction, ergodic case, orbit closure, weak mixing, finite measure space, maximal inequality Key Phrases - Capitalized Phrases (CAPs): (learn more)
Hindman's Theorem, Marriage Lemma, Waerden's Theorem, Chacon-Ornstein Theorem, Furstenberg-Katznelson Theorem, Spectral Theorem, Ajo Ajo, Bounded Convergence Theorem, T-invariant Borel, Dominated Convergence Theorem, Filler Lemma, Baire Category Theorem, Fatou's Lemma, Monotone Convergence Theorem, New York New!
Books on Related Topics | Concordance | Text Stats Browse Sample Pages:
Front Cover | Table of Contents | First Pages | Index | Back Cover | Surprise Me!
Citations (learn more)
This book cites 35
books:
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100
books
cite this book:
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- Ergodic Theory (Grundlehren der mathematischen Wissenschaften) by I. P. Kornfeld in Back Matter (1), Back Matter (2), and Back Matter (3)
- Real Analysis (3rd Edition) by Halsey Royden in Front Matter, and Back Matter
- Uniform Distribution of Sequences (Dover Books on Mathematics) by L. Kuipers on page 161, and Back Matter
- Recurrence in Ergodic Theory and Combinatorial Number Theory (Porter Lectures) by Harry Furstenberg on page 161, and Back Matter
- Extensions of Minimal Transformation Groups by I.U. Bronstein on page 161, and Back Matter
See all 35 books this book cites
- Modern Dynamical Systems and Applications by Michael Brin on 13 pages
- Dynamical Systems, Ergodic Theory and Applications (Encyclopaedia of Mathematical Sciences) by L.A. Bunimovich on 5 pages
- Convergence in Ergodic Theory and Probability (Ohio State University Mathematical Research Institute Publications, 5) by Conference on Convergence in Ergodic Theory and Probability (1993 : Ohio State University) on 4 pages
- Rigidity in Dynamics and Geometry by Marc Burger on 4 pages
- Ergodic Theory and Harmonic Analysis: Proceedings of the 1993 Alexandria Conference (London Mathematical Society Lecture Note Series) by Karl E. Petersen on 4 pages
See all 100 books citing this book
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