부산대학교 도서관

It was shown by the reviewer [Bull. Amer. Math. Soc. (N.S.) {\bf 25} (1991), no.~1, 55--60; MR1080004 (91j:58130)] that Selberg's zeta functionfor ${\rm PSL}(2,\bold Z)$ can be expressed in terms of the Fredholmdeterminants ${\rm det}(1\pm L_s)$ of the generalized transferoperator $L_s$ for the Gauss map $Tx=(1/x)\bmod 1$. This mapserves to describe the geodesic flow on the modular surface $\scrH/{\rm PSL}(2,\bold Z)$ symbolically in the Bowen-Series approach. Thethermodynamic formalism thus makes possible a completely new access toSelberg's theory without invoking the trace formula. In the physicsliterature the approach sketched above is the prime example forquantum chaos, which relates classical and quantum behaviour forchaotic systems. In the present paper the authors give a heuristicderivation of the above results. They show how the eigenfunctions ofthe hyperbolic Laplacian can be related to the eigenfunctions of thetransfer operator $L_s$ for the eigenvalues $\pm 1$ via a certain Helgasontransform. They also try to extend the approach to more general chaoticHamiltonian systems (see also an earlier paper of Bogomolʹnyĭ [Nonlinearity {\bf 5} (1992), no.~4, 805--866; MR1174220 (93k:81057)]). Itshould be mentioned that J. Lewis hasgiven a rigorous derivation of the relation between Maass waveformsand eigenfunctions of $L_s$ for eigenvalue $\lambda=1$. Thesefunctions turn out to be so-called period functions, already known in thetheory of modular forms.