# Content

28.05.2018: - -Proof of the theorem on the relations between entropy and leafwise measures (3). * References:* Pisa notes and Chapter 10 of Volume 3.

-Proof of Oppenheim's conjecture (2).

21.05.2018: -Whit Monday.

14.05.2018: -Proof of the theorem on the relations between entropy and leafwise measures (2). * References:* Pisa notes and Chapter 10 of Volume 3.

-Proof of Oppenheim's conjecture (1).

07.05.2018: -Proof of the theorem on the relations between entropy and leafwise measures. * References:* Pisa notes and Chapter 10 of Volume 3.

-Dani's theorem and Ratner's joining theorem revisited. * References:* Chapter 6 of Volume 3.

30.04.2018: -Some properties of leafwise measures.

-Entropy & leafwise measures and entropy contribution. * References:* Pisa notes and Chapter 9 & 10 of Volume 3.

23.04.2018: -Construction of leafwise measures. * References:* The sent draft of Einsiedler-Lindenstrauss or Pisa notes for an alternative construction.

16.04.2018: -Happy sechseläuten!

09.04.2018: -Rudolph's theorem (2). * References:* Same as previous week.

02.04.2018: -Happy easter!

26.03.2018: -Pinsker algebra. * References:* Chapter 2 of Einsiedler-Lindenstrauss-Ward's Volume 2.

-Rudolph's theorem (1). * References:* Chapter 9 of Einsiedler-Lindenstrauss-Ward's Volume 2.

19.03.2018: -Entropy and conditional entropy of a transformation. * References:* Same as previous week.

12.03.2018: -End of the proof of Ratner's joining theorem.

- Entropy and conditional entropy of sigma-algebras. * References:* Chapter 1 and Chapter 2 of Einsiedler-Lindenstrauss-Ward's Volume 2.

05.03.2018: -Ratner's theorem on joinings of horocycle flow (2). * References:* Section 6.7 of the Volume 3 draft of Einsiedler-Ward.

26.02.2018: -Measure classification for the horocycle flow on SL(2,R) quotients by discrete subgroups. * References:* Section 6.6 of the Volume 3 draft of Einsiedler-Ward.

-Ratner's theorem on joinings of horocycle flow (1). * References:* Section 6.7 of the Volume 3 draft of Einsiedler-Ward.

19.02.2018: - Conditional measures, ergodic decomposition and joinings. * References:* (1) Chapters 5, 6 and Section 8.7 from Einsiedler-Ward's "Ergodic theory with a view towards number theory", (2) Part 4 of Yves Coudene's "Ergodic theory and dynamical systems".
(Note that, you can freely download the reference (1) using this this link through ETH net.)

## Exercises

See our online and collaborative exercise sheet.You can also find the exercise sheet of the course "Homogeneous dynamics and applications 1".

### References for topics of the advanced part of the exam

- Rudolph's theorem.*References:*Section 9.3 Volume 2

- Ratner's measure classification for joinings of the horocycle flow on SL(2,R)xSL(2,R) quotients (only the proof using entropy theory).

*References:*Section 6.7 of the Volume 3

- Leafwise measures and entropy for the case of hyperbolic toral automorphisms.

*References: 7.26-7.33 in Pisa notes and related material.*