Mayer's transfer operator and representations of \(\mathrm{GL}_{2}\)
type: Item
instance of: scholarly article
title: Mayer's transfer operator and representations of \(\mathrm{GL}_{2}\)
zbMATH Open document ID: 1146.43006
author: Joachim Hilgert
published in: Semigroup Forum
publication date: 2008-08-19T00:00:00Z
review text: 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.
reviewed by: Gert Roepstorff
Mathematics Subject Classification ID: 43A80
Mathematics Subject Classification ID: 20M30
Mathematics Subject Classification ID: 47D03
Mathematics Subject Classification ID: 47G10
zbMATH DE Number: 5313099
zbMATH Keywords: transfer operator
zbMATH Keywords: Frobenius-Perron-Ruelle operator
zbMATH Keywords: Bergman space
zbMATH Keywords: semi-group representation
MaRDI profile type: Publication
full work available at URL: s00233-008-9076-2
OpenAlex ID: W2086999839
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DOI: 10.1007/S00233-008-9076-2
description: scientific article; zbMATH DE number 5313099