I am currently studying basic ergodic theory:

  • Invariant measures
  • Poincaré recurrence theorem
  • Invariant measure for continuous transformations
  • The ergodic theorems and applications
  • Ergodic transformations
  • Unique ergodicity
  • Mixing transformations
  • Ergodic decomposition
  • Metric entropy
  • Variational principle

I already had a course in this subject. I want to deepen my understanding of this subject,I know the importance they play the examples and counter-examples for each topic listed above to achieve this goal.

Throughout the course I had to ergodic theory, I had some canonical examples in each of these topics. But in my view the list of examples was rather sparse. I wish you could talk me a list of examples, even a single example of his predilection, that you think might enhance my understanding of this subject.

An example: A transformation topologically mixing, which is not mixing or an example of the space of ergodic measures is not closed in the weak-* topology... Things like this

I hope you can help me.

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    $\begingroup$ As it stands this is a very broad question and I'm not sure it's possible to give a good answer. You'll probably get more helpful responses if you ask about specific examples/counterexamples that you'd like to see or understand better. $\endgroup$ Jan 12, 2013 at 20:05
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    $\begingroup$ I already know some basic examples, which are usually given in the course notes. I would like examples of unusual or curious, which I do not know. So I can not leave the question less vague. $\endgroup$
    – user11178
    Jan 12, 2013 at 20:24
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    $\begingroup$ Good question. I'd like to meet some interesting examples too. I believe that Juan does not necessarily want a list of examples, but examples of knowing one's personal preference. $\endgroup$
    – Klein
    Jan 12, 2013 at 21:08

3 Answers 3


The best list of classic examples are in the book Ergodic Problems of Classical Mechanics by V. I. Arnold and A. Avez. You can find a huge list of examples in the book Introduction to the Modern Theory of Dynamical Systems by Anatole Katok, Boris Hasselblatt.


A productive way to gain the intuition you are seeking is to work through a structured set of exercises in ergodic theory. I would recommend:

Charles Walkden's exercises, based on these lecture notes

You'll find some of the items on your wish list, such as an example that Poincaré's recurrence theorem does not hold on infinite measure spaces.


The "examples" in the "ergodic" literature don't seem very useful for physicists. For the latter I strongly recommend "Canonical Ensembles from Chaos" by Kusnezov, Bulgac, Bauer, Annals of Physics 204, 155-185 ( 1990 ), which has plenty of examples of the useful sort.

Bill Hoover, Ruby Valley NV


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