Manfred Einsiedler (Princeton)

Rigidity of measures - the high entropy case and joinings

Abstract. In this talk we will discuss two separate joint projects with A. Katok and E. Lindenstrauss respectively.

For measure rigidity of the Cartan action on $SL(3,\mathbb{R})/SL(3,\mathbb{Z})$ two separately developed techniques have been used: the high and low entropy argument. In a recent joint work with A. Katok we have generalized the former to non-split groups where we also include locally homgeneous spaces defined by products of real and $p$ -adic Lie groups. Moreover, we do not assume that our action is defined by the full Cartan subgroup. This is important in the application to joinings of Cartan actions which is a joint work with E. Lindenstrauss: any ergodic joining between two Cartan actions on $SL(3,\mathbb{R})/\Gamma$ is algebraic.