# Horocyclic flow

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 $$t$$.