On Arnol’d’s and Kazhdan’s equidistribution problems

Abstract

AbstractWe consider isometric actions of lattices in semisimple algebraic groups on (possibly non-compact) homogeneous spaces with (possibly infinite) invariant Radon measure. We assume that the action has a dense orbit, and demonstrate two novel and non-classical dynamical phenomena that arise in this context. The first is the existence of a mean ergodic theorem even when the invariant measure is infinite; this implies the existence of an associated limiting distribution, which can be different from the underlying invariant measure. The second is uniform quantitative equidistribution of all orbits in the space, which follows from a quantitative mean ergodic theorem for such actions. In turn, these results imply quantitative ratio ergodic theorems for isometric actions of lattices. This sheds some unexpected light on certain equidistribution problems posed by Arnol’d [Arnol’d’s Problems. Springer, Berlin, 2004] and also on the ratio equidistribution conjecture for dense subgroups of isometries formulated by Kazhdan [Uniform distribution on a plane. Tr. Mosk. Mat. Obs. 14 (1965), 299–305]. We briefly mention the general problem regarding ergodic theorems for actions of lattices on homogeneous spaces and its solution given by Gorodnik and Nevo [Duality principle and ergodic theorems, in preparation], and present a number of examples to demonstrate our results. Finally, we also prove results on quantitative equidistribution for absolutely continuous averages in transitive actions.