A Lyapunov exponent attached to modular functions

Abstract To each weakly holomorphic modular function for , which is nonnegative on the geodesic arc , we attach a ‐invariant map that generalizes the Lyapunov exponent function introduced by Spalding and Veselov. We prove that it takes every value between 0 and and it gives an increasing convex function on the Markov irrationalities when ordered using their parameterization by Farey fractions in . In the case of quadratic irrationals with purely periodic continued fraction expansion, the value equals the real part of the cycle integral of along the associated geodesic on the modular surface, normalized with the word length of the associated hyperbolic matrix as a word in the generators and . These results are related to conjectures of Kaneko who observed several similar behavior for the cycle integrals of the modular function when normalized by the hyperbolic length of the geodesic .