Reference for Ergodic Theory

Question

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

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
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
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 http://books.google.com/books?id=eCoES7HzrHQC or Nadkarni: Basic ergodic theory http://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
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 04 '13 at 12:40

score 7 · Answer 1 · edited Jun 22 '11 at 23:59

7

You could also try

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

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.

edited Jun 22 '11 at 23:59

t.b.

78,116

answered Jun 22 '11 at 19:40

JT_NL

14,514

3

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 00:01
@Theo Buehler: Thanks. – JT_NL Jun 23 '11 at 12:17

score 5 · Answer 2 · answered Feb 06 '13 at 12:24

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.

score 5 · Answer 3 · answered Aug 29 '11 at 23:59

5

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!

answered Aug 29 '11 at 23:59

Bastian Galasso-Diaz

345

This is a nice book. – ncmathsadist Aug 30 '11 at 01:48

Beni Bogosel · Answer 4 · 2011-06-22T13:18:20.950

4

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.

edited Jun 22 '11 at 13:18

answered Jun 22 '11 at 12:42

Beni Bogosel

23,381

score 4 · Answer 5 · answered Aug 30 '11 at 00:05

4

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

answered Aug 30 '11 at 00:05

JavaMan

13,153

score 3 · Answer 6 · edited Jun 23 '11 at 02:53

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.

This is a hard and very dense book. – user56202 Jan 18 '21 at 17:43 — user56202, Jan 18 '21 at 17:43

score 3 · Answer 7 · edited Jun 22 '11 at 18:08

3

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

edited Jun 22 '11 at 18:08

t.b.

78,116

answered Jun 22 '11 at 17:15

PeterR

835

score 2 · Answer 8 · edited Feb 04 '13 at 11:29

2

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.

edited Feb 04 '13 at 11:29

Martin Sleziak

53,687

answered Jul 12 '12 at 21:51

user35682

474

I recommend Silva: Invitation to ergodic theory – Stefan Smith Jul 12 '12 at 22:24

score 1 · Answer 9 · answered Mar 25 '12 at 17:36

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.

-1 For the last sentence. – Michael Greinecker Feb 04 '13 at 06:43 — Michael Greinecker, Feb 04 '13 at 06:43

score 1 · Answer 10 · answered Feb 04 '13 at 06:36

1

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.

answered Feb 04 '13 at 06:36

yiyi

7,352

score 0 · Answer 11 · answered Jul 19 '20 at 02:25

0

Two good books are:

Viana - Foundations of Ergodic Theory;
Mañé - Ergodic Theory and Differetiable Dynamics

answered Jul 19 '20 at 02:25

Tiago Emilio Siller

1,116

score 0 · Answer 12 · answered Mar 12 '21 at 03:15

Arguably this is not a book but Terence Tao's lecture-blogs (the very first of many as I understand) may be worth looking into: https://terrytao.wordpress.com/category/teaching/254a-ergodic-theory/ .

Rudolph's Fundamentals of measurable dynamics. Ergodic theory on Lebesgue spaces is also noteworthy in my humble opinion.

Reference for Ergodic Theory

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