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Ergodic Theory: with a view towards Number Theory (Graduate Texts in Mathematics) (Englisch) Gebundene Ausgabe – 23. September 2010


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From the reviews:

“The book is an introduction to ergodic theory and dynamical systems. … The book is intended for graduate students and researchers with some background in measure theory and functional analysis. Definitely, it is a book of great interest for researchers in ergodic theory, homogeneous dynamics or number theory.” (Antonio Díaz-Cano Ocaña, The European Mathematical Society, January, 2014)

“A book with a wider perspective on ergodic theory, and yet with a focus on the interaction with number theory, remained a glaring need in the overall context of the development of the subject. … The book under review goes a long way in fulfilling this need. … it covers a good deal of conventional ground in ergodic theory … . a very welcome addition and would no doubt inspire interest in the area among researchers as well as students, and cater to it successfully.” (S. G. Dani, Ergodic Theory and Dynamical Systems, Vol. 32 (3), June, 2012)

“The book under review is an introductory textbook on ergodic theory, written with applications to number theory in mind. … it aims both to provide the reader with a solid comprehensive background in the main results of ergodic theory, and of reaching nontrivial applications to number theory. … The book should also be very appealing to more advanced readers already conducting research in representation theory or number theory, who are interested in understanding the basis of the recent interaction with ergodic theory.” (Barak Weiss, Jahresbericht der Deutschen Mathematiker-Vereinigung, Vol. 114, 2012)

“This introductory book, which goes beyond the standard texts and allows the reader to get a glimpse of modern developments, is a timely and welcome addition to the existing and ever-growing ergodic literature. … This book is highly recommended to graduate students and indeed to anyone who is interested in acquiring a better understanding of contemporary developments in mathematics.” (Vitaly Bergelson, Mathematical Reviews, Issue 2012 d)

“The book contains a presentation of the ergodic theory field, focusing mainly on results applicable to number theory. … of interest for researchers, specialists, professors and students that work within some other areas than precisely the ergodic theory. … ‘Ergodic Theory. With a view toward number theory’ is now an indispensable reference in the domain and offers important instruments of research for other theoretical fields.” (Adrian Atanasiu, Zentralblatt MATH, Vol. 1206, 2011)

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This text is a rigorous introduction to ergodic theory, developing the machinery of conditional measures and expectations, mixing, and recurrence.

Beginning by developing the basics of ergodic theory and progressing to describe some recent applications to number theory, this book goes beyond the standard texts in this topic. Applications include Weyl's polynomial equidistribution theorem, the ergodic proof of Szemeredi's theorem, the connection between the continued fraction map and the modular surface, and a proof of the equidistribution of horocycle orbits.

Ergodic Theory with a view towards Number Theory will appeal to mathematicians with some standard background in measure theory and functional analysis. No background in ergodic theory or Lie theory is assumed, and a number of exercises and hints to problems are included, making this the perfect companion for graduate students and researchers in ergodic theory, homogenous dynamics or number theory.

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Amazon.com: 5.0 von 5 Sternen 3 Rezensionen
3 von 3 Kunden fanden die folgende Rezension hilfreich
5.0 von 5 Sternen Precise theorem statements, careful proofs, and fascinating content 5. Mai 2015
Von Jordan Bell - Veröffentlicht auf Amazon.com
Format: Gebundene Ausgabe
Whether or not you care about number theory, this is the best introduction to ergodic theory and continued fractions I have seen. This is an uncommonly good book. First, the notation is consistent and unambiguous. Second, some authors define a symbol to mean something in the middle of a paragraph and then continue to use this symbol without ever repeating the definition, and that does not happen here. Third, the proofs are clearly written and do not lazily avoid the technicalities a learner needs to see but which feel below the eminence of many established mathematicians. This book should be a first choice both as a text from which to learn ergodic theory and as a reference.

I came to this book for Chapter 3, on continued fractions. Continued fractions are a one-to-one correspondence between sequences of positive integers and the irrational real numbers in the unit interval. I assert that this chapter by itself is the best exposition that has ever been written on measure theoretic results for continued fractions, in particular the Gauss map and the Khinchin-Lévy formula (Corollary 3.8).

Chapter 2 is a complete presentation of the elements of ergodic theory. And aside from doing a better job than most authors at presenting the standard equipment of the theory, the authors also prove Kac's return time formula (Theorem 2.44), a quantitative version of the Poincare recurrence theorem that I have not seen in other books.

Chapter 4 is about invariant measures on a compact metric space, and includes, for example, the Krylov–Bogolyubov theorem and unique ergodicity. Chapter 5 is on conditional expectations and martingales, and this is valuable for being an entirely measure theoretic exposition of objects that are usually described in the language of probability theory.

Even the appendices are unusually good. An appendix does not need to prove or even to motivate the results that it states, but it ought to give precise statements and to define all the objects in these statements, and not to make tacit assumptions (like saying something about Hilbert spaces that is only true for separable Hilbert spaces).

Finally, each chapter has notes making historical statements, discussing other forms of theorems, and introducing the reader to theorems and objects that are not expounded in this book, and also should precisely define objects which different authors define differently, like what one means by a Borel measure and what it means for a Borel measure to be regular.
1 von 1 Kunden fanden die folgende Rezension hilfreich
5.0 von 5 Sternen Great text for a graduate student 29. Mai 2015
Von Taylor - Veröffentlicht auf Amazon.com
Format: Gebundene Ausgabe Verifizierter Kauf
This is an excellent text for graduate students interested in dynamical systems and number theory. There is a lot of explication and motivation, which makes the book great for self-study (or with a professor or reading group). I highly recommend it.
0 von 6 Kunden fanden die folgende Rezension hilfreich
5.0 von 5 Sternen Strongly recommend 23. August 2013
Von jean - Veröffentlicht auf Amazon.com
Format: Gebundene Ausgabe Verifizierter Kauf
Nice book for studying ergodic theory. The shipping is fast and the book is in good condition, so it is worth the money.
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