Limiting modular symbols and the Lyapunov spectrum

TL;DR

This paper explores the properties of limiting modular symbols, connecting them with Lyapunov exponents, spectral theory, homology classes, and zeta functions within the context of modular curves and continued fractions.

Contribution

It introduces new expressions for limiting modular symbols as Birkhoff averages and links them to spectral properties and non-trivial homology classes.

Findings

Limiting modular symbols can be expressed as Birkhoff averages on Lyapunov level sets.

A vanishing theorem relates spectral properties of a generalized Gauss-Kuzmin operator to modular symbols.

Construction of non-trivial homology classes associated with non-closed geodesics.

Abstract

This paper consists of variations upon the theme of limiting modular symbols. Topics covered are: an expression of limiting modular symbols as Birkhoff averages on level sets of the Lyapunov exponent of the shift of the continued fraction, a vanishing theorem depending on the spectral properties of a generalized Gauss-Kuzmin operator, the construction of certain non-trivial homology classes associated to non-closed geodesics on modular curves, certain Selberg zeta functions and C^* algebras related to shift invariant sets.