Proximality and Equidistribution on the Furstenberg Boundary

Let G be a connected semisimple Lie group with finite center and without compact factor, and Γ a lattice in G, that is, a discrete subgroup of G such that Γ\G has finite volume. In this article we investigate the distribution of orbits of Γ acting on the Furstenberg boundary of G. Recall that the Furstenberg boundary can be identified with the factor space G/P , where P is a minimal parabolic subgroup of G. It is known that every orbit of Γ in G/P is dense (see [Mo]). We show that orbits of Γ are equidistributed with respect to the averages over Riemannian balls. Since we study the action of a nonamenable group on a space without a finite invariant measure, our result lies outside the scope of the classical ergodic theory. The published results about distribution of dense orbits of nonamenable groups are limited to a few special examples. Arnold and Krylov showed in [AK] that dense orbits of groups generated by two rotations acting on the 2-dimensional sphere are equidistributed. A similar problem was considered by Kazhdan in [Ka] where he studied the action of a group generated by two affine isometries on the plane R. Distribution of dense orbits of a lattice in SL(2,R) acting on R was investigated by Ledrappier [L] and Nogueira [N]. Let X be the symmetric space of G equipped with a right invariant Riemannian metric d. Note that X can be identified with L\G for a maximal compact subgroup L of G. Fix x, x̃ ∈ X and denote by K and K̃ the stabilizers of x and x̃ respectively. Let ν and ν̃ be the probability Haar measures on K and K̃ and mx̃ the harmonic measures at x̃ on G/P , that is, the unique K̃-invariant probability measure on G/P . For S ⊂ G