Mayer Transfer Operator Approach to Selberg Zeta Function

TL;DR

This paper surveys Mayer's transfer operator approach to expressing the Selberg zeta function for hyperbolic surfaces as a Fredholm determinant, linking dynamical systems, spectral theory, and number theory.

Contribution

It presents a detailed account of how the Selberg zeta function can be represented via transfer operators and their matrix forms involving the Riemann zeta and gamma functions.

Findings

Selberg zeta function expressed as a Fredholm determinant of a transfer operator

Transfer operator acts on holomorphic function spaces with matrix entries involving zeta and gamma functions

Connection established between dynamical zeta functions and spectral properties of hyperbolic surfaces

Abstract

These notes are based on three lectures given by the second author at Copenhagen University (October 2009) and at Aarhus University, Denmark (December 2009). We mostly present here a survey of results of Dieter Mayer on relations between Selberg and Smale-Ruelle dynamical zeta functions. In a special situation the dynamical zeta function is defined for a geodesic flow on a hyperbolic plane quotient by an arithmetic cofinite discrete group. More precisely, the flow is defined for the corresponding unit tangent bundle. It turns out that the Selberg zeta function for this group can be expressed in terms of a Fredholm determinant of a classical transfer operator of the flow. The transfer operator is defined in a certain space of holomorphic functions and its matrix representation in a natural basis is given in terms of the Riemann zeta function and the Euler gamma function.