Abstract

Several problems in number theory when reformulated in terms of homogeneous dynamics involve the study of limiting distributions of translates of algebraically defined measures on orbits of reductive groups. The general non-divergence and linearization techniques, in view of Ratner’s measure classification for unipotent flows, reduce such problems to dynamical questions about linear actions of reductive groups on finite-dimensional vector spaces.

We will limit ourselves to the treatment of the archimedean place (i.e. over the field of real numbers), taken from the eponymous article https://arxiv.org/abs/1305.6557 (Ergodic Theory and Dynamical Systems (this month issue)).

We will address these finite-dimensional problems, namely: in which directions does the trajectory of a bounded subset get arbitrarily small, or stay bounded. This involves the geometry of the concerned real algebraic groups (Mostow decomposition, convexity arguments on the associated symmetric spaces), and some representation theory.

The problem treated and the methods used give an example of the relevance, in homogeneous dynamics, of the notion of stability (in the Mumford sense) taken in an archimedean context and in an arithmetic context.