Mayer’s transfer operator approach to Selberg’s zeta function
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- by A. Momeni and A. B. Venkov
- St. Petersburg Math. J. 24 (2013), 529-553
- DOI: https://doi.org/10.1090/S1061-0022-2013-01252-0
- 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.References
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Bibliographic 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
- 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)
- © Copyright 2013 American Mathematical Society
- Journal: St. Petersburg Math. J. 24 (2013), 529-553
- MSC (2010): Primary 11M36, 11M41
- DOI: https://doi.org/10.1090/S1061-0022-2013-01252-0
- MathSciNet review: 3088005