Pointwise equidistribution for one parameter diagonalizable group action on homogeneous space

Let Γ Γ be a lattice of a semisimple Lie group L L . Suppose that a one parameter Ad Ad -diagonalizable subgroup { g t } \{g_t\} of L L acts ergodically on L / Γ L/Γ with respect to the probability Haar measure μ μ . For certain proper subgroup U U of the unstable horospherical subgroup of { g t } \{g_t\} and certain x ∈ L / Γ x∈ L/Γ we show that for almost every u ∈ U u∈ U the trajectory { g t u x : 0 ≤ t ≤ T } \{g_tux: 0≤ t≤ T\} is equidistributed with respect to μ μ as <inline-formula content-type="