Convergence of measures under diagonal actions on homogeneous spaces

TL;DR

This paper investigates the convergence of measures on homogeneous spaces under diagonal actions, demonstrating that measures with positive entropy tend to the Haar measure and applying this to Diophantine approximation.

Contribution

It establishes measure convergence results for measures with positive entropy on homogeneous spaces and connects these results to Diophantine approximation limits.

Findings

Measures with positive entropy converge to Haar measure under diagonal flows.

High entropy measures imply non-improvability of Dirichlet's theorem almost surely.

The convergence result applies to measures supported on unstable horospherical orbits.

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.