Mayer's transfer operator and representations of \(\mathrm{GL}_{2}\)
Transfer operators, often called Frobenius-Perron operators, appear in statistical mechanics and general dynamical systems. In mathematics, the transfer operator encodes information about an iterated map such as the Gauss map and has been extensively studied by a number of people, among them Ruelle, Kuzmin, Wirsing and more recently Dieter Mayer. Hilgert studies a particular transfer operator \({\mathcal L}_\beta\) introduced by Mayer, depending on a complex parameter \(\beta\), for the geodesic flow of the modular surface. For \(\beta=1\) it corresponds to the Gauss map. Hilgert relates it to the representation theory of the semigroup \(\text{GL}(2,R)^+\) of real invertible \(2\times 2\)-matrices with non-negative entries. Mayer already constructed a Hilbert space realization of \({\mathcal L}_\beta\) as a self-adjoint operator on some (generalized) Hardy space. From the standpoint of representation theory however it seems more natural to use a weighted Bergman space of holomorphic functions on the halfplane \(C^+\). Then it is shown that the semigroup representation can be analytically extended to the simply connected covering of the group \(\text{SL}(2,R)\) as discrete series representations.
scientific article; zbMATH DE number 5313099