Limiting modular symbols and the Lyapunov spectrum

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.