Symbolic dynamics for the geodesic flow on Hecke surfaces
Dieter Mayer Fredrik Strömberg
Journal of Modern Dynamics 2008, 2(4): 581-627 doi: 10.3934/jmd.2008.2.581
In this paper we discuss a coding and the associated symbolic dynamics for the geodesic flow on Hecke triangle surfaces. We construct an explicit cross-section for which the first-return map factors through a simple (explicit) map given in terms of the generating map of a particular continued-fraction expansion closely related to the Hecke triangle groups. We also obtain explicit expressions for the associated first return times.
keywords: symbolic dynamics Hecke triangle groups continued fractions geodesic flow
The transfer operator for the Hecke triangle groups
Dieter Mayer Tobias Mühlenbruch Fredrik Strömberg
Discrete & Continuous Dynamical Systems - A 2012, 32(7): 2453-2484 doi: 10.3934/dcds.2012.32.2453
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.
keywords: $\lambda_q$-continued fractions Ruelle and Selberg zeta function. Hecke triangle groups transfer operator

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