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St. Petersburg Mathematical Journal
St. Petersburg Mathematical Journal
ISSN 1547-7371(online) ISSN 1061-0022(print)

 

Mayer's transfer operator approach to Selberg's zeta function


Authors: A. Momeni and A. B. Venkov
Original publication: Algebra i Analiz, tom 24 (2012), nomer 4.
Journal: St. Petersburg Math. J. 24 (2013), 529-553
MSC (2010): Primary 11M36, 11M41
Published electronically: May 24, 2013
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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). Mostly, a survey of the results of Dieter Mayer on relationships between Selberg and Smale-Ruelle dynamical zeta functions is presented. 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.


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Additional Information

A. Momeni
Affiliation: Department of Statistical Physics and Nonlinear Dynamics, Institute of Theoretical Physics, Clausthal University of Technology, 38678, Clausthal-Zellerfeld, Germany
Email: arash.momeni@tu-clausthal.de

A. B. Venkov
Affiliation: Institute for Mathematics and Centre of Quantum Geometry QGM, University of Aarhus, 8000, Aarhus C, Denmark
Email: venkov@imf.au.dk

DOI: http://dx.doi.org/10.1090/S1061-0022-2013-01252-0
PII: S 1061-0022(2013)01252-0
Keywords: Mayer's transfer operator, Selberg's zeta function
Received by editor(s): September 22, 2011
Published electronically: May 24, 2013
Additional Notes: The authors would like to thank Dieter Mayer for several important remarks and we would like to say also that all possible mistakes in the text belong to us but not to Mayer’s theory we presented in this paper. This work was supported by DAAD, the International Center of TU Clausthal, and the Danish National Research Foundation Center of Excellence, Center for Quantum Geometry of Moduli Spaces(QGM)
Article copyright: © Copyright 2013 American Mathematical Society