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

TL;DR

This paper investigates how expanding curves in the unit tangent bundle of hyperbolic space behave under geodesic flow and proves their equidistribution in homogeneous spaces, extending previous results and answering open questions.

Contribution

It establishes conditions under which expanding curves in hyperbolic space become equidistributed under geodesic flow, generalizing Shah's earlier work.

Findings

Proves equidistribution of expanding curves under specific geometric conditions.

Answers a question posed by Shah regarding distribution of curves in homogeneous spaces.

Generalizes previous results on dynamics of curves in hyperbolic 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 $\Gamma$ be a lattice of $G$. Let $x = g\Gamma \in G/\Gamma$ be a point of $G/\Gamma$ such that its $H$-orbit $Hx$ is dense in $G/\Gamma$. Let $\phi: I= [a,b] \rightarrow H$ be an analytic curve, then $\phi(I)x$ gives an analytic curve in $G/\Gamma$. In this article, we will prove the following result: if $\phi(I)$ satisfies some explicit geometric condition, then $a(t)\phi(I)x$ tends to be equidistributed in $G/\Gamma$ as $t \rightarrow \infty$. It answers the first question asked by Shah in ~\cite{Shah_1} and generalizes the main result of that paper.