Horocycle flow

Example

Let $G = \mathrm{PSL}_2(\R)$, the quotient of the special linear group $\mathrm{SL}_2(\R)$ by $\set{\pm I}$. As in geodesic flow on a locally symmetric space, identify $G$ with the unit tangent bundle of the upper half-plane $\mathbb{H}$. Then the geodesic flow is $g \mapsto g a_t$, with $a_t$ the class of $\operatorname{diag}(e^{t/2}, e^{-t/2})$. Let $h_s$ be the class of $\begin{pmatrix} 1 & s \\ 0 & 1 \end{pmatrix}$. The (stable) horocycle flow is $\psi_s(g) = g h_s$, for $s \in \R$. For a lattice $\Gamma \subseteq G$, it descends to $\Gamma \backslash G$, because left and right multiplications commute.

Check of the name and of the link with the geodesic flow. A matrix computation gives $a_{-t} h_s a_t = h_{e^{-t} s}$. Hence $g h_s a_t = g a_t h_{e^{-t}s}$, so the geodesic orbits of $g$ and of $g h_s$ approach each other as $t \to +\infty$. So the orbits of $\psi$ are the stable leaves of the geodesic flow. They project to horocycles in $\mathbb{H}$: circles tangent to the boundary, and horizontal lines.