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.
Best Answer
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.