QUANTUM MAPS FROM TRANSFER OPERATORS

The Selberg zeta function zeta(s)(s) yields an exact relationship betw een the periodic orbits of a fully chaotic Hamiltonian system (the geo desic flow on surfaces of constant negative curvature) and the corresp onding quantum system (the spectrum of the Laplace-Beltrami operator o n the same manifold). It was found that for certain manifolds, zeta(s) (s) can be exactly rewritten as the Fredholm-Grothendieck determinant det (1 - T(s)), where T(s) is a generalization of the Ruelle-Perron-Fr obenius transfer operator. We present an alternative derivation of thi s result, yielding a method to find not only the spectrum but also the eigenfunctions of the Laplace-Beltrami operator in terms of eigenfunc tions of T(s). Various properties of the transfer operator are investi gated both analytically and numerically for several systems.