Seminar on Homogeneous Dynamics and Applications Fall 2017

Seminars

Week 14 (22.12.2017) (Last session, presence is optional)

1. Y.K.: Closing and Shadowing of geodesic flow orbits
2. A.W.: Number fields and closed geodesics

Week 13 (15.12.2017)

1. A.G.: Dani-Margulis recurrence
2. S.J.: Dirichlet unit theorem

Week 12 (08.12.2017)

1. J.V.H.: Siegel domains
2. J.A.O.: Dani correspondance in Diophantine approximation

Week 11 (01.12.2017)

1. C.K.: Unique ergodicity of nilrotations
2. R.B.: Mahler compactness criterion

Week 10 (24.11.2017)

1. F.R.: Closed linear subgroups
2. J.F.F.: Proof of Minkowski's theorems

Week 9 (17.11.2017)

1. D.H.: Hopf's proof of ergodicity of geodesic flow
2. J.F.F.: Lattices in Rd and Minkowski's theorems

Week 8 (10.11.2017)

R.P.: Oseledets' multiplicative ergodic theorem

Week 7 (03.11.2017)

1. M.W.L.: Ledrappier's three dots example and mixing properties of x2x3 system (E-W Section 8.2)
2. E.R.: Banach-Tarski and Hausdorff paradoxes, different characterizations of amenability (References: for the paradoxes, see Juschenko's notes and Tao's book; for generalities on amenability, see Juschenko's notes or Breuillard's lecture notes or the appendix of the book by Bekka-de la Harpe-Valette)

Week 6 (27.10.2017)

1. J.V.H.:Unique ergodicity, Furstenberg's Theorem and equidistribution of irrational polynomials (E-W Section 4.4)
2. M.W.L.: Ledrappier's three dots example and mixing properties of x2x3 system (E-W Section 8.2)

Week 5 (20.10.2017)

1. J.A.O.: Properties of Haar measure and examples ( Notes )
2. D.H.: Fourier analysis on compact abelian groups and Pontryagin duality

Week 4 (13.10.2017)

1. R.B.: Birkhoff's ergodic theorem (references: A proof can be found in this study guide by Einsiedler-Ward)
2. A.G.: Weak-mixing and its characterizations (references: Einsiedler-Ward (E-W) "Ergodic Theory with a View towards Number Theory" Sections 2.7, 2.8, Walters "An introduction to ergodic theory" Section 1.7)

Week 3 (06.10.2017)

1. C.K.: Continued fractions, Gauss map and Diophantine approximations (part 2) (references: E-W Sections 3.2, 3.3)
2. M.A.G.: Continued fractions and Farey tesselation of the hyperbolic plane (This article of Caroline Series (you may also see her lecture notes), and Hatcher's book entitled "Topology of numbers")

Week 2 (29.09.2017)

1. J.F.F.: Gauss-Bonnet formula and the Poincaré disk model (part 2)
2. C.K.: Continued fractions, Gauss map and Diophantine approximations (part 1) (references: E-W Sections 3.1, 3.2)

Week 1 (22.09.2017)

1. J.F.F.: Gauss-Bonnet formula and the Poincaré disk model (references: any book on hyperbolic geometry, in particular lecture notes of Caroline Series)
2. F.R.: Flows on the tori (references: Parry "Topics in ergodic theory" Section 1.1, Einsiedler-Ward "Ergodic Theory with a View towards Number Theory" Section 1.1)

Literature

Our main reference is Einsiedler-Ward's book entitled "Ergodic Theory with a View towards Number Theory". You can freely download it using this link through ETH net.