Dynamics and Orbits on Homogeneous Spaces

Course Description

Starting with dynamical systems on hyperbolic surfaces we will study the action of subgroups on quotients of Lie groups. Closed orbits have here often intimate connections to number theory and algebraic geometry: Borel density theorem and Borel Harish-Chandra theorem. The proofs given will be dynamical and the second part of the lectures will concern some cases of Ratners theorems.

List of Concepts

• Lattices in Lie groups
• Mixing properties of  Lie group actions
• Algebraic groups defined over Q
• Borel density theorem
• Non-divergence properties of unipotent flows
• Almost unique ergodicity of horospherical actions
• Ratner's measure classification
• Counting Lattice Points
  

There will be a script for the lecture.

Problem Sheets