Distribution of Kaneko's val function

Abstract

Kaneko's val function is defined as the normalized cycle integral of the elliptic modular $j$-function along closed geodesics on the modular surface. We prove that, when primitive hyperbolic conjugacy classes are ordered by geodesic length, its values concentrate at the single point 720. More generally, an analogous concentration result holds for every weakly holomorphic modular function $f$ of weight 0, with the concentration point given by Atkin's inner product $(f, 1)_{\mathrm{At}}$. The proof combines an equidistribution theorem following Pollicott with the ergodicity of a continued-fraction suspension flow and uses the decomposition formula of Bengoechea-Imamo\={g}lu to construct a bounded continuous observable.