Manfred Einsiedler ETH Zürich

Effective Equdistribution of Closed Orbits

Abstract:

We discuss equidistribution results of closed orbits on homogeneous spaces. The original results due to Mozes and Shah from 1995 relied heavily on the fundamental work of Ratner in unipotent dynamics. However, Ratner’s theorems are ineffective — they do not provide any insights on the error rate. In various collaborations and increasing generality we have attacked that problem for closed orbits of noncompact semisimple subgroups. The latest results concerns "adelic quotients" and have interesting number theoretical applications. We outline the general method relying on (uniform) spectral gap, an effective ergodic theorem, and the shearing property of unipotent flows used by Ratner. The talk is based on several joint works in various combinations with Lindenstrauss, Margulis, Mohammadi, Wieser, and Venkatesh.