Dynamics on locally homogeneous spaces

Manfred Einsiedler

Abstract.

Let $ G$ be a closed subgroup of $ \operatorname{SL}(n,\mathbb{R})$, and let $ \Gamma<G$ be a discrete subgroup. Then any subgroup $ H<G$ defines a dynamical system on $ X=\Gamma\backslash G$ by right translation. These systems are often considered because of their intimate connection to problems in number theory. E.g. the dynamics of $ \operatorname{SO}(2,1)$ on $ X_3=\operatorname{SL}(3,\mathbb{Z})\backslash\operatorname{SL}(3,\mathbb{R})$ was used by Margulis in his solution to the Oppenheim conjecture. Marina Ratner later showed many strong structure theorems about invariant measures, orbit closures, and equidistribution of a general class of subgroups (thus solving the Raghunathan-Dani conjectures). In the first part of the course we will discuss these results as well as their connection to some problems in number theory, and prove some special cases.

The above result says nothing about the dynamics of the diagonal subgroup $ A$ in $ \operatorname{SL}(n,\mathbb{R})$ acting on $ X_n=\operatorname{SL}(n,\mathbb{Z})\backslash\operatorname{SL}(n,\mathbb{R})$. Furstenberg, Katok and Spatzier, as well as Margulis conjectured that some of the above results should also hold for this action. Currently there is only a partial classification of invariant measures known. However, this is already enough for several applications to number theoretic problems. In the second part of the course we will discuss some of these more recent results.

Background assumed:

We will assume only minimal knowledge about Lie groups, e.g. Chapter 0 of Knapp's book "Lie groups beyond an introduction" and the basic theorems from ergodic theory: Poincare recurrence, mean and pointwise ergodic theorems. Especially for the second part prior knowledge of entropy would be desirable, but this we will discuss briefly. As a preparation we recommend Walters "An Introduction to Ergodic Theory", or whatever chapters of Einsiedler-Ward "Ergodic Theory: with a view towards Number Theory" are already available (see http://www.mth.uea.ac.uk/ergodic/).