Convergence of measures under diagonal actions on homogeneous spaces

Abstract

Let $\lambda$ be a probability measure on $\mathbb T^{n-1}$ where $n=2$ or 3. Suppose $\lambda$ is invariant, ergodic and has positive entropy with respect to the linear transformation defined by a hyperbolic matrix. We get a measure $\mu $ on $SL_n(\mathbb Z)\backslash SL_n(\mathbb R)$ by putting $\lambda$ on some unstable horospherical orbit of the right translation of $a_t=\mathrm{diag}(e^t,..., e^t, e^{-(n-1)t})$ $(t>0)$. We prove that if the average of $\mu$ with respect to the flow $a_t$ has a limit, then it must be a scalar multiple of the probability Haar measure. As an application we show that if the entropy of $\lambda$ is large, then Dirichlet's theorem is not improvable $\lambda$ almost surely.