Pointwise equidistribution for one parameter diagonalizable group action on homogeneous space

Abstract

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