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Mayer Transfer Operator Approach to Selberg Zeta Function

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  • Arash Momeni, Clausthal University of Technology, Germany
  • Alexei Venkov, Denmark
  • Centre for Quantum Geometry of Moduli Spaces
  • Department of Mathematical Sciences
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.
Original languageEnglish
Number of pages29
Publication statusPublished - 2010

Bibliographical note

The proofs of Lemmas 1, 4, 5, Remark. 1 and section. 2.1 are corrected and modified; added reference (18) for section 2

    Research areas

  • math-ph, math.MP, Mathematical Physics

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