Some applications of thermodynamic formalism to manifolds with constant negative curvature

Abstract

The purpose of this article is to present a different approach to the analysis of the spectrum of a compact surface of constant negative curvature. To explain our ideas we recall that there exists a highly successful theory for analyzing zeta functions for closed geodesics, originated by A. Selberg. The zeros for this zeta function identify the eigenvalues of the Laplacian for the surface. The main tool in this analysis is the Selberg trace formula [7]. There exists an approach, having its roots in ergodic theory and statistical mechanics, for analyzing zeta functions for geodesic and more general flows. The principle here is quite different from that in the Selberg approach. The zeta function is some form of determinant of a family of “Perron-Frobenius type” operators parameterized by the complex domain. The zeta function behaves like a determinant in that zeros correspond to parameters for which the corresponding operator is not invertible [ 10, 111. Since both methods apply to surfaces of constant negative curvature this suggests that there is some mechanism involving a Perron-Frobenius type operator characterizing the Laplacian spectrum of the surface. 161 OOOI-8708/91 $7.50