The transfer operator for the Hecke triangle groups

Dieter Mayer; Tobias Mühlenbruch; Fredrik Strömberg

doi:10.3934/dcds.2012.32.2453

Article Contents

2012, Volume 32, Issue 7: 2453-2484. Doi: 10.3934/dcds.2012.32.2453

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The transfer operator for the Hecke triangle groups

1.
Lower Saxony Professorship, Institute for Theoretical Physics, TU Clausthal, D-38678 Clausthal-Zellerfeld
2.
Department of Mathematics and Computer Science, FernUniversität in Hagen, D-58084 Hagen
3.
Department of Mathematics, TU Darmstadt, D-64289 Darmstadt

Received: November 30, 2009

Revised: February 28, 2010

Published: June 2012

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Abstract

In this paper we extend the transfer operator approach to Selberg's zeta function for cofinite Fuchsian groups to the Hecke triangle groups $G_q,\, q=3,4,\ldots$, which are non-arithmetic for $q\not= 3,4,6$. For this we make use of a Poincar\'e map for the geodesic flow on the corresponding Hecke surfaces, which has been constructed in [13], and which is closely related to the natural extension of the generating map for the so-called Hurwitz-Nakada continued fractions. We also derive functional equations for the eigenfunctions of the transfer operator which for eigenvalues $\rho =1$ are expected to be closely related to the period functions of Lewis and Zagier for these Hecke triangle groups.

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Mathematics Subject Classification: Primary: 11M36, 37C30; Secondary: 37B10, 37D35, 37D40, 37D20.

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References

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