Counting intersections of closed orbits of the horocyclic flow

Consider a finite-volume hyperbolic surface $X$ with cusp(s), and let $Y$ denote its unit tangent bundle. It is well-known that a closed horocycle $c_l$, of period $l$, becomes equidistributed in $Y$ as $l\to\infty$. Given a second closed horocycle $c_0$, of fixed period, it can therefore be expected that $c_l$ and $c_0$ would intersect for $l$ large enough. Indeed, it is the case, and this leads us to some natural follow-up questions: Is the set of intersection points finite for $l\leq X$ ? Does it become equidistributed along the two closed horocycles ? If so, how fast, and what can be said about the joint equidistribution ?
We will discuss these questions, and how they can be parametrized in terms of some families of exponential sums that are prevalent in number theory.