Expanding curves in ${\mathrm{T}}^1({\mathbb{H}}^n)$ under geodesic flow and equidistribution in homogeneous spaces

Abstract

Let $H=\mathrm{SO}(n,1)$ and $A=\{a(t):t\in \mathbb{R}\}$ be a maximal $\mathbb{R}$-split Cartan subgroup of $H$. Let $G$ be a Lie group containing $H$ and $\hat{I}“$ be a lattice of $G$. Let $x=g\hat{I}“\in G/\hat{I}“$ be a point of $G/\hat{I}“$ such that its $H$-orbit $Hx$ is dense in $G/\hat{I}“$. Let $\ddot{I}•:I=[a,b]\to H$ be an analytic curve, then $\ddot{I}•(I)x$ gives an analytic curve in $G/\hat{I}“$. In this article, we will prove the following result: if $\ddot{I}•(I)$ satisfies some explicit geometric condition, then $a(t)\ddot{I}•(I)x$ tends to be equidistributed in $G/\hat{I}“$ as $t\to \infty$. It answers the first question asked by Shah in ~\cite{Shah_1} and generalizes the main result of that paper.

15 pages. The paper is rewritten according to referee's suggestions