Dynamics on Lie groups over local fields and its applications

Abstract

We study the structure of reductive algebraic groups over ultra-metric local fields K and explore some dynamical properties of associated homogeneous spaces. To investigate the behavior of orbits in such spaces, we use the geometry of Euclidean Bruhat-Tits building and relate those dynamical systems to certain shift spaces with countably many alphabets.

First, we prove the exponential mixing property of the action of Cartan subgroups of K-rank 1 algebraic K-groups under certain conditions, which includes the case when the quotient spaces are geometrically finite graphs of groups. This result can be applied to counting the number of closed paths in graphs of groups with an error rates.

We also show when K is the field of formal Laurent series, on the space of unimodular lattices in K^n the Birkhoff type Ergodic theorem holds for almost everywhere in every orbit of unipotent groups for maximal singular rays. Using this result, we obtain the finer version of quantitative Khintchine-Groshev type theorem in K^n, which concerns the asymptotic of the number of solutions of certain Diophantine inequalities with weights and directions.