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Dieter Mayer

Arnold-Sommerfeld-Str. 6 38678 Clausthal-Zellerfeld
Theoretische Physik, Technische Universitat Clausthal


Chang C. H., Mayer D.
Thermodynamic Formalism and Selberg's Zeta Function for Modular Groups
2000, vol. 5, no. 3, pp.  281-312
In the framework of the thermodynamic formalism for dynamical systems [26] Selberg's zeta function [29] for the modular group $PSL(2,\mathbb{Z})$ can be expressed through the Fredholm determinant of the generalized Ruelle transfer operator for the dynamical system defined by the geodesic flow on the modular surface corresponding to the group $PSL(2,\mathbb{Z})$ [19]. In the present paper we generalize this result to modular subgroups $\Gamma$ with finite index of $PSL(2,\mathbb{Z})$. The corresponding surfaces of constant negative curvature with finite hyperbolic volume are in general ramified covering surfaces of the modular surface for $PSL(2,\mathbb{Z})$. Selberg's zeta function for these modular subgroups can be expressed via the generalized transfer operators for $PSL(2,\mathbb{Z})$ belonging to the representation of $PSL(2,\mathbb{Z})$ induced by the trivial representation of the subgroup $\Gamma$. The decomposition of this induced representation into its irreducible components leads to a decomposition of the transfer operator for these modular groups in analogy to a well known factorization formula of Venkov and Zograf for Selberg's zeta function for modular subgroups [34].
Citation: Chang C. H., Mayer D.,  Thermodynamic Formalism and Selberg's Zeta Function for Modular Groups, Regular and Chaotic Dynamics, 2000, vol. 5, no. 3, pp. 281-312

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