On the spectrum of Farey and Gauss maps

In this paper we introduce Hilbert spaces of holomorphic functions given by generalized Borel and Laplace transforms which are left invariant by the transfer operators of the Farey map and its induced transformation, the Gauss map, respectively. By means of a suitable operator-valued power series we are able to study simultaneously the spectrum of both these operators along with the analytic properties of associated dynamical zeta functions. This construction establishes an explicit connection between previously unrelated results of Mayer and Rugh (see [Ma1] and [Rug]).