Limiting modular symbols and the Lyapunov spectrum

Abstract

This paper consists of variations upon a theme, that of limiting modular symbols introduced in [17]. We show that, on level sets of the Lyapunov exponent for the shift map T of the continued fraction expansion, the limiting modular symbol can be computed as a Birkhoff average. We show that the limiting modular symbols vanish almost everywhere on T-invariant subsets for which a corresponding transfer operator has a good spectral theory, thus improving the weak convergence result proved in [17]. We also show that, even when the limiting modular symbol vanishes, it is possible to construct interesting non-trivial homology classes on modular curves that are associated to non-closed geodesics. These classes are related to "automorphic series", defined in terms of successive denominators of continued fraction expansion, and their integral averages are related to certain Mellin transforms of modular forms of weight two considered in [17]. We discuss some variants of the Selberg zeta function that sum over certain classes of closed geodesics, and their relation to Fredholm determinants of transfer operators. Finally, we argue that one can use T-invariant subsets to enrich the picture of non-commutative geometry at the boundary of modular curves presented in [17].