An application of Jacquet-Langlands correspondence to transfer operators for geodesic flows on Riemann surfaces

In the paper as a new application of the Jacquet-Langlands correspondence we connect the transfer operators for different cofinite Fuchsian groups by comparing the corresponding Selberg zeta functions. 1 Transfer operator for cofinite Fuchsian groups and Selberg’s zeta function In this section we give a short summary of the generalization of Mayer’s theory [1] by following Morita [2]. In Mayer’s theory a transfer operator Ls is introduced as a special case of Ruelle’s operator for a dynamical system, for instance for the group SL(2,Z) this is the twice iterated Gauss map T = T 2 G acting on the unit interval and the weight function −βlog|T (z)|. Then, as a result of the one-to-one correspondence between the closed geodesics on the modular surface M = H \ SL(2,Z) and the primitive periodic orbits of T the Selberg zeta function can be written as a Fredholm determinant of the transfer operator [3]. Morita in[2] takes instead of PSL(2,R) and the Poincare half plane H the isomorphic group PSU(1, 1) and as homogeneous space the unit disc D which is isomorphic toH . Let Γ be a cofinite Fuchsian group. The canonical fundamental domain of Γ is a polygon with a finite even number of sides si, each of which extends to a circular arc C(si) perpendicular to the unit circle S , the boundary of D. To every side si a generator g(si) is assigned which identifies the sides mutually. Now the action of the generators on the boundary points S defines a Markov map [4] TΓx = gix x ∈ S 1 (1)