Limit distributions of expanding translates of certain orbits on homogeneous spaces

Let L be a Lie group and Lambda a lattice in L. Suppose G is a non-compact simple Lie group realized as a Lie subgroup of L, and the image of G on L/Lambda is dense. Let c be a diagonalizable element of G not contained in a compact subgroup. Let U be the expanding horospherical subgroup of G associated to c. Let Omega be a nonempty open subset of U, and {n_i} be a sequence of natural numbers tending to infinity. It is shown that the image of the union of the sets c^{n_i}.Omega in L/Lambda is dense. A stronger measure theoretic formulation of this result is also obtained. Among other applications of the above result, we describe G-equivariant topological factors of the product space L/Lambda \times G/P, where the real rank of G is greater than 1, P is a parabolic subgroup of G, and G acts diagonally. We also describe equivariant topological factors of unipotent flows on finite volume homogeneous spaces of Lie groups.