Moments of the boundary hitting function for the geodesic flow on a hyperbolic manifold

TL;DR

This paper studies the distribution of boundary hitting times for geodesic flow on hyperbolic manifolds with geodesic boundary, deriving formulas for moments in terms of the orthospectrum and explicit results for surfaces.

Contribution

It provides new formulas for moments of boundary hitting times in hyperbolic manifolds, connecting them to the orthospectrum and deriving explicit expressions for surfaces.

Findings

First two moments match known orthospectrum identities

Explicit trilogarithm formula for average hitting time in surfaces

Derived moments relate to boundary hitting distribution

Abstract

In this paper we consider finite volume hyperbolic manifolds X with non-empty totally geodesic boundary. We consider the distribution of the times for the geodesic flow to hit the boundary and derive a formula for the moments of the associated random variable in terms of the orthospectrum. We show that the the first two moments correspond to two cases of known identities for the orthospectrum. We further obtain an explicit formula in terms of the trilogarithm functions for the average time for the geodesic flow to hit the boundary in the surface case, using the third moment.