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.