Continuity of the j $j$ -function on the Markov tree

Abstract

One way of defining the values of the modular $j$-function at real quadratic irrationalities is by its cycle integrals along geodesics in the upper half plane whose endpoints are the roots of an indefinite binary quadratic form. We show that the restriction of the modular $j$-function to Markov irrationalities (an important subset of real quadratic irrationalities in diophantine approximation) is continuous with respect to the topology induced by the euclidean norm.