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A thermodynamic approach to two-variable Ruelle and Selberg zeta functions via the Farey map

In this paper we consider the transfer operator approach to the Ruelle and Selberg zeta functions associated to continued fractions transformations and the geodesic flow on the full modular surface. We extend the results by Ruelle and Mayer to two-variable zeta functions, $ζ(q,z)$ and $Z(q,z)$. The $q$ variable plays the role of the inverse temperature and the introduction of the "geometric variable" $z$ is essential in the tentative to provide a general approach, based on the Farey map, to the correspondence between the analytic properties of the zeta functions themselves, the spectral properties of a class of generalised transfer operators and the theory of a generalisation of the three-term functional equations studied by Lewis and Zagier. The first step in this direction is a detailed study of the spectral properties of a family of signed transfer operators $\PP^{\pm}_{q}$ associated to the Farey map.