MA 346a: Homogeneous dynamics
Credits: 3:0
Pre-requisites :
- Measure theory (equivalent to MA222) will be assumed.
- Familiarity with basic notions of differential geometry (covered in MA235) will be helpful.
The course is about the ergodic theory of actions by (subgroups of) semisimple Lie groups which arise as groups of isometries of non-compact symmetric spaces. Some of the main topics include Howe-Moore’s theorem on vanishing of matrix coefficients at infinity for unitary actions on Hilbert spaces, Moore’s ergodicity theorem, ergodic aspects of the geodesic flow, the horocycle flow and classification of ergodic invariant measures of the horocycle flow. Dani-Margulis’ proof of a stronger version of Oppenheim’s conjecture will be discussed at the end of the course as an application of topics covered. Topics from the theory of non-compact semisimple Lie groups including Cartan involution, restricted root spaces, Weyl chambers, Iwasawa decomposition, Cartan decomposition and Bruhat decomposition will be discussed in some detail. Basic topics from ergodic theory like ergodicity, strong mixing and the pointwise ergodic theorem will also be recalled.
Suggested books and references:
- M. B. Bekka, M. Mayer, Ergodic Theory and Topological Dynamics of Group Actions on Homogeneous Spaces, Cambridge University Press 2013.
- Einsiedler, Ward, Homogeneous dynamics and applications