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.