In this paper we will study the statistics of the unit geodesic flow normal to the
boundary of a hyperbolic manifold with nonempty totally geodesic boundary.
Viewing the time it takes this flow to hit the boundary as a random variable, we
derive a formula for its moments in terms of the orthospectrum. The first moment
gives the average time for the normal flow acting on the boundary to again reach the
boundary, which we connect to Bridgeman’s identity (in the surface case), and the
zeroth moment recovers Basmajian’s identity. Furthermore, we are able to give
explicit formulae for the first moment in the surface case as well as for manifolds of
odd dimension. In dimension two, the summation terms are dilogarithms. In
dimension three, we are able to find the moment generating function for this length
function.
Keywords
Basmajian's identity, identities on hyperbolic manifolds,
length function, moments
