Homogeneous Space Dynamics
Student No.:40
Time:Tue & Thu 13:30-15:05, Sep.18-Dec.13
Instructor:薛金鑫Xue Jinxin  
Place:Conference Room 1, Jin Chun Yuan West Bldg.
Starting Date:2018-9-18
Ending Date:2018-12-13

This is a branch of dynamical systems studying dynamics on homogeneous spaces such as Lie groups. It is deeply related to other fields such as number theory, ergodic theory, hyperbolic dynamics, etc. We will talk about 1. the structure theory of Lie group and its dynamical properties such as stable and unstable manifolds, mixing rate. 2. Diophantine approximation problems in terms of homogeneous space dynamics. 3. Ratner’s classification of orbit closures of unipotent subgroups. 4. Margulis’s proof of Oppenheim conjecture. 5. Entropy method and the work of Einsiedler, Katok, Lindenstrauss on Littlewood conjecture. Time permits, we will also talk about some links with the dynamics on Teichmuller space.

Prerequisite: Real analysis, group theory, complex analysis, linear algebra.

[1] Bekka, Mayer, ergodic theory and topological dynamics of group actions on homogeneous spaces;
[2] Einsiedler, Ward, ergodic theory with a view towards number theory;
[3] Einsiedler, Katok, Lindenstrauss, Invariant measures and the set of exceptions to Littlewood's conjecture, Annals of mathematics.