In hyperbolic geometry, the **horocyclic flow** on a hyperbolic surface is a canonical flow on the unit tangent bundle of the surface that moves tangent vectors sideways along the horocycle whose center is in the direction of the tangent vector.

## Definition

More precisely, let \(S\) be a complete oriented hyperbolic surface, and let \(T^1(S)\) denote the unit tangent bundle of \(S\). Consider a point \(p = (x, v)\in T^1(S)\), where \(x\in S\) and \(v\) is a unit tangent vector based at \(x\). We define the horocyclic flow by describing where the flow takes \(p\) at time \(t\in \mathbb{R}\). We use \(h\) to denote the horocyclic flow on \(S\), and we let \(h_t: T^1(S)\to T^1(S)\) denote the mapping at time \(t\).

As shown in the figure above, lift \(p\) to a point \(\widetilde{p} = (\widetilde{x}, \widetilde{v})\) in the unit tangent bundle of the hyperbolic plane \(\mathbb{H}\). (The hyperbolic plane is the universal cover of \(S\).) Let \(c\) denote the point in the boundary at infinity of \(\mathbb{H}\) that \(\widetilde{p}\) points to. In other words, the point \(c\) is the endpoint of the geodesic ray that starts at \(\widetilde{x}\) and whose tangent vector is \(\widetilde{v}\). Consider the oriented horocycle \(C\) in \(\mathbb{H}\) with center \(c\) and that passes through \(\widetilde{x}\) from left to right when facing in the direction of \(\widetilde{v}\). (This is where we use the orientation of \(S\).)

Define \(\widetilde{p}_t\) to be the element \((\widetilde{x}_t, \widetilde{v}_t)\in T^1(\mathbb{H})\), where \(\widetilde{x}_t\) is the result of moving \(\widetilde{x}\) along \(C\) for time \(t\) at unit speed (measuring distance relative to the metric that \(C\) inherits from \(\mathbb{H}\)), and \(\widetilde{v}_t\) is the unit tangent vector based at \(\widetilde{x}_t\) pointing towards the same boundary point \(c\). Note that the distance between \(\widetilde{x}_t\) and \(\widetilde{x}\) as measured in \(\mathbb{H}\) will be strictly less than \(t\) because \(C\) is not a geodesic. (This distance in \(\mathbb{H}\) is \(2\sinh^{-1}(t / 2) < t\).) Project \(\widetilde{p}_t\) down to \(p_t\in T^1(S)\) to get \(h_t(p)\). This defines a well-defined flow because the construction doesn't depend on the choice of lift nor on the time \(t\).