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Hello,

I'd like to hear your opinion for ergodic theory books which would suit a beginner (with background in measure theory, real analysis and topological groups). I am looking for something well structured, well motivated, and perhaps with application to other fields.

any such book exists?

I tried a book by nadkarni, and could not read through it, seemed to concise to me, and tried the book by Petersen which I felt was accessible but didn't follow a clear path, jumping from subject to subject with lots of different object or properties.

What are your recommendations on the subject?

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    $\begingroup$ @Blade 'nadkarni - this book: books.google.com.au/books/about/… ? 'Petersen' this one: books.google.com.au/books/about/… ? $\endgroup$
    – David Roberts ♦
    Commented Dec 5, 2011 at 1:07
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    $\begingroup$ When you say beginner, do you mean grad student or otherwise? $\endgroup$
    – David Roberts ♦
    Commented Dec 5, 2011 at 1:08
  • $\begingroup$ @David Roberts, Yes, these are the books. Next time I'll post more specific bibliography. As for me, I am a grad student, but uexperienced with ergodic theory. $\endgroup$
    – Yaniv Ganor
    Commented Dec 5, 2011 at 9:51

10 Answers 10

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I think another good choice is the book "Ergodic Theory: With a View Towards Number Theory" by Manfred Einsiedler and Thomas Ward,Graduate Texts in Mathematics 259.Besides basic concepts of ergodic theory,the book also discusses the connection between ergodic theory and number theory,which is a hot topic recently.And a forthcoming second volume will discuss about entropy,drafts of the book can be found on the homepage of Thomas Ward (https://tbward0.wixsite.com/books).

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For me the standard text is Peter Walters, "An Introduction to Ergodic Theory", Springer Graduate Texts in Mathematics.

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  • $\begingroup$ I second this, especially if Petersen has been tried. $\endgroup$
    – Steve Huntsman
    Commented Dec 5, 2011 at 2:33
  • $\begingroup$ I like this book, it is my first choice $\endgroup$
    – yaoxiao
    Commented Oct 8, 2013 at 16:04
  • $\begingroup$ An excellent suggestion indeed. $\endgroup$
    – user39719
    Commented Apr 29, 2014 at 23:29
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I really like (and recommend) Billingsley's Ergodic Theory and Information. It is a well-written book with very clear explanations. For example, his treatment of entropy tops those in both Walter's An Introduction to Ergodic Theory and Petersen's Ergodic Theory, both of which are also good books though.

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    $\begingroup$ This book makes for beautiful reading. The only problem is that it's hard to get a copy. $\endgroup$
    – Daniel Mansfield
    Commented Sep 24, 2015 at 5:57
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I second Siming Tu's recommendation for E-W book. It is a well balanced book (regarding theory vs applications), it has nice appendix contains relevant theory from functional analysis, and it contains a nice selection of subjects (although not addressing entropy, which one might say is a very big problem).

I think that overall, Petersen's book is good as well, maybe not as streamlined as one might expect, but still very through.

So apart from this, which are "standard references", and maybe also Walters' book (which is kind of dated, and the last chapters are biased towards entropy theory of continuous maps over compact spaces), there are few references which are good for specific subjects and maybe not as a whole standard reference book.

Dan Rudolph have a very nice book called - "Fundamentals of Measurable Dynamics: Ergodic Theory on Lebesgue Spaces", it is one of the most accurate books in the technical level (Lebesgue spaces, ergodic decomposition), plus it have nice treatments of the theory of joinings and entropy. He also actually proves Ornstein's theorem, kind of a rare thing.

Another nice option is the classic book by Furstenberg - "Recurrence in ergodic theory and combinatorial number theory" (Princeton). It is an extremely suitable book for students I think (because Furstenberg is a great teacher and lecturer), and it shows part of the motivations towards the modern development of ergodic theory, and it shows also topics in topological dynamics, which other books omit. Nevertheless, it does not as extensive as E-W or Petersen on the ergodic theoretic part, but it definitely worth your time after you got the hang of the basics.

The last option I have in mind is Shmuel (Eli) Glasner's book - "Ergodic Theory via Joinings" (AMS). This is a very extensive book, but it is kind of deep, and in my opinion, doesn't suitable fro students (although he for example discuss the general notion of ergodic group action, besides Z or R actions).

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What about the following?

Sinai, Ya. G. Introduction to ergodic theory. Translated by V. Scheffer. Mathematical Notes, 18. Princeton University Press, Princeton, N.J., 1976. 144 pp. ISBN: 0-691-08182-4

This seems to have the highest content-to-volume ratio. It treats, among others, invariant measures, translations on compact abelian groups, geodesic flows on Riemannian manifolds; it gives applications to number theory and discusses ergodic theory of ideal gas (as applications to "other fields" that you may be interested in). Two chapters deal with entropy.
The proofs are not always carried out in full detail, though.

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Let me suggest you a recent book by Steve Kalikow and Randall McCutcheon: "An Outline of Ergodic Theory". This is a nice book to get a solid background in isomorphism theory of measurable dynamical systems. I like the way proofs of theorems are presented through guided exercises.

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I have learned topological dynamics from this textbook.

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Personally, I like Mañe's book Teoría Ergódica. I do think it's a classical book full of exercises. On the other hand the book has loads of mistakes, which makes it interesting to read, you realise that you are understanding everything when you spot the mistakes. As far as I know, both versions (English and Portuguese) are sold out.

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I'd recommend "Introduction to Ergodic Theory" by Nathanial A. Friedman. The book is reasonably concrete and short and treats the important "cutting and stacking" constructions in detail. This only will help you with the measurable setting and is an older book, though.

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I recommend Foundations of Ergodic Theory by Marcelo Viana and Oliviera https://books.google.it/books/about/Foundations_of_Ergodic_Theory.html?id=vlRyCwAAQBAJ&printsec=frontcover&source=kp_read_button&redir_esc=y#v=onepage&q&f=false This book has almost everything explained.

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