  In this paper we study the values of modular functions at the Markov
quadratics which are defined in terms of their cycle integrals along the
associated closed geodesics. These numbers are shown to satisfy two properties
that were conjectured by Kaneko. More precisely we show that the values of a
modular function f, along any branch B of the Markov tree, converge to the
value of f at the Markov number which is the predecessor of the tip of B. We
also prove an interlacing property for these values.
