On uniformly distributed orbits of certain horocycle flows

Let G=SL(2,R), T = SL(2,Z), u, ' l (where t e R) and let fi. be the G-invariant probability measure on G/Y. We show that if x is a non-periodic point of the flow given by the (w,)-action on G/Y then the («,)-orbit of x is uniformly distributed with respect to JU. ; that is, if SI is an open subset whose boundary has zero measure, and / is the Lebesque measure on R then, as T -> oo, T~l{0 < t < T\u,x e SI} converges to ̂ (SI). Let G = SL (2, R), the special linear group of 2 x 2 matrices, and let Y = SL (2, Z) be the subgroup consisting of integral matrices in G. The homogeneous space G/Y carries a unique G-invariant probability measure which we shall denote by fi. Let («,) be the one-parameter subgroup of G denned by u, = (I {) for all t e R. Let P be the subgroup of G consisting of all upper triangular matrices in G. Consider the action of (u,) on G/T. It is well-known that for any g e P F the («,)-orbit of gT in G/T is periodic. Further, if giPT then the («,)-orbit of gT is dense in G/F. The object of this paper is to show that each of these dense orbits is uniformly distributed on G/Y with respect to n; that is, if g&PY and SI is an open subset of G/Y whose boundary has zero /x-measure then as T -* oo,