On uniformly distributed orbits of certain horocycle flows

Abstract

G = SL (2,R), Γ=SL(2,Z), u₁ = (¹₀^l₁)(where t ε R) and let μ be the G-invariant probability measure on G/Gamma. We show that if x is a non-periodic point of the flow given by the (u_t)-action on G/Gamma then the (u_t)-orbit of x is uniformly distributed with respect to μ ; that is, if Ω is an open subset whose boundary has zero measure, and l is the Lebesque measure on R then, as T→∞T^-1{0≤t≤ T\u_txεΩ} , converges to μ (Ω).