Chapter 11 Dynamics of subgroup actions on homogeneous spaces of lie groups and applications to number theory

Abstract

Publisher Summary This chapter presents an exposition of homogeneous dynamics—that is, the dynamical and ergodic properties of actions on the homogeneous spaces of Lie groups. Many concepts of the modern theory of dynamical systems appeared in connection with the study of the geodesic flow on a compact surface of constant negative curvature. The algebraic nature of the phase space and the action itself allows obtaining much more advanced results as compared to the general theory of smooth dynamical systems. This can be seen in the example of smooth flows with polynomial divergence of trajectories. Homogeneous actions are discussed in the chapter and some basic examples include rectilinear flow on a torus, solvable flows on a three-dimensional locally Euclidean manifold, suspensions of toral automorphisms, nilflows on homogeneous spaces of the three-dimensional Heisenberg group, the geodesic and horocycle flows on a constant negative curvature surface, and geodesic flows on locally symmetric Riemannian spaces. The chapter presents the main link between homogeneous actions and number theory (Mahler's criterion and its consequences).