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I am looking for a good introductory book on ergodic theory. Can someone recommend some introductory texts on that?

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Could you expand a bit on your background and what you'd like to explore and what your interests are? The literature on ergodic theory and its applications is more than vast. Good and helpful recommendations must take your interests into account. There are some very basic theorems (e.g. Poincaré recurrence, von Neumann, Birkhoff) that you'll find in any book, but when it comes to applications, the focuses of the books quickly diverge. –  t.b. Jun 22 '11 at 12:28
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I intend a book with the basics (recurrence/ergodicity) for the first contact with subject for a grad student in math, with some background in measure theory –  j.o. Jun 22 '11 at 13:36
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I was about to post a similar question. I glanced over some books at google and I was thinking about purchasing Silva: Invitation to ergodic theory books.google.com/books?id=eCoES7HzrHQC or Nadkarni: Basic ergodic theory books.google.com/books?id=w4WPxmTqq-sC In case someone knows these books I would be grateful for your opinion about them. –  Martin Sleziak Jun 22 '11 at 13:37
    
Ergodic Theory by Karl Petersen. –  KCd Jun 22 '11 at 16:51
    
If you have a little background in probability theory then you might like "The ergodic theory of discrete sample paths" by Shields. –  Stéphane Laurent Feb 4 '13 at 12:40
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10 Answers

You could also try

Ergodic theory with a view towards number theory be Einsiedler and Ward.

Direct link to the online edition

The book is available on springerlink. I do have to warn you that the book can be experienced as quite chaotic but the good thing is that the writers are experts on the topic.

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Interesting assessment of a book about chaos theory :) I added a persistent link, since I don't know how stable yours is. Consider using DOI, which is guaranteed to be stable. –  t.b. Jun 23 '11 at 0:01
    
@Theo Buehler: Thanks. –  Jonas Teuwen Jun 23 '11 at 12:17
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You can try:

  • Paul Halmos – Introduction to ergodic theory

  • Harry Furstenberg - Recurrence in ergodic theory and combinatorial number theory

  • Dynamical systems and ergodic theory – Mark Pollicott, Michiko Yuri.

The last two are developed to be able to prove some combinatorial results such as van der Waerden's theorem and Szemeredi's theorem.

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Ben Green has made available notes from his Ergodic Theory class here. They are certainly worth a look.

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Have a look at "Randomness and Recurrence in Dynamical Systems" By Rodney Nillsen. Google books link http://books.google.com/books?id=NkzPSr-JpBwC&printsec=frontcover#v=onepage

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The most basic book on Ergodic theory that I have come across is,

Introduction to Dynamical Systems, By Brin and Stuck.

This book is actually used as an undergraduate text, but as a first contact with the subject, this will be perfect. The first few chapters deal with Topological and Symbolic Dynamics. Ch.4 is devoted to Ergodic theory, and is independent on earlier chapters. Subsequenct chapters deal with similar topics. Ergodic theory is notoriously difficult; once you have read through parts of this book, the other books on the subject will not be so intimidating.

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You can try with "An Introduction to Ergodic Theory" By Peter Walters, this is a Graduate Text of Mathematics but is really good. Good Luck!

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This is a nice book. –  ncmathsadist Aug 30 '11 at 1:48
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I know it's not exacly what you are looking for but it's really interesting. As long you start to see some basics definitions and some nice results of finite measure ergodic theory, you should read something about ergodic theory on $\sigma$-finite spaces. The theory is quite different and very beautiful!

Aaronson, J. - An Introduction to Infinite Ergodic Theory. Mathematical Surveys and Monographs, AMS, 1997.

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I recommend Silva: Invitation to ergodic theory –  Stefan Smith Jul 12 '12 at 22:24
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I like the book Statistical Properties of Deterministic systems by Jiu Ding and Aihui zhong.

You can read the book online here or can buy it really cheap here Or you can spend way more and buy it on amazon.

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I recommend Silva, "Invitation to Ergodic theory". This is a wonderful little book. He starts from the ground up, assuming no background except for some competence in analysis, and reaches what seem to be important issues in the theory (I am not an expert). Along the way your knowledge of measure theory should be solidified. For the total novice, the introduction to measure theory, Lebesgue measurable sets, etc., is the best I've seen.

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The Best Book of ergodic theory, that there, that shows the power of theory in all areas, the book is that of Ricardo Mane:

MAÑÉ, R. - Ergodic Theory and Differentiable Dynamics. Berlin, Springer-Verlag, 1987.

Another book is really interesting:

Peter Walters - An Introduction to Ergodic Theory. Graduate Text of Mathematics. Springer-Verlag

The rest of the books are variations of the above two.

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-1 For the last sentence. –  Michael Greinecker Feb 4 '13 at 6:43
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