A thermodynamic formalism approach to period functions for Maass forms and generalisations

In this paper we study one-parameter families of signed transfer operators P± q associated to the Farey map, as well as two-parameter families of operators for the Gauss map, obtained from the Farey map by inducing. We characterise all the analytic eigenfunctions of P± q with eigenvalue λ not embedded in the continuous spectrum, which for λ = 1 are the period functions for Maass forms studied by Lewis-Zagier. We also give a correspondence between eigenfunctions of P± q and of the induced operators. Moreover, we generalise the Mayer approach to the Selberg zeta function to our two-parameter operators for the Gauss map. This entails a relation between eigenfunctions of P± q and the zeroes of a two-variable Selberg zeta function, which for λ = 1 is the classical relation obtained by Mayer and Lewis-Zagier. Finally, we extend our approach to a two-variable Ruelle zeta function for the Farey map and write Selberg and Ruelle zeta functions as power series related to the Farey tree.