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.