The distribution of geodesic excursions into the neighborhood of a cone singularity on a hyperbolic 2-orbifold (Q2427577)

Let \(G\) be a cocompact Fuchsian group acting on the hyperbolic plane \(\mathbb H\). We suppose that \(G\) is not a triangle group. We call an elliptic fixed point of \(\mathbb H\) a cone point if the order \(k\) of the stabilizer is \(>2\). We consider then an \(r\) neighbourhood \(B(p,r)\) of a cone point \(p\). Let next \(\gamma\) be a geodesic half--ray. For \(z\) with \(0<z<r\) we define \(\mathrm{dist}_k (r,z)(\gamma)\) for almost all \(\gamma\) as follows. For almost all \(\gamma\) the ray will meet \(G B(p,r)\) infinitely often. The author proves that again for almost all \(\gamma\) the proportion of encounters where the ray comes within \(z\) of \(Gp\) exists. This proportion, defined for almost all \(\gamma\) is \(\mathrm{dist}_k (r,z)(\gamma)\). The author then gives a formula with a fairly complicated structure for \(\mathrm{dist}_k (r,z)(\gamma)\). The proof of this and some further theorems of the same type is by combining arguments from hyperbolic gometry and the ergodic theory of Fuchsian groups and the Birkoff ergodic theorem.