The transfer operator for the Hecke triangle groups

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,..., which are non-arithmetic for q \not= 3,4,6. For this we make use of a Poincare map for the geodesic flow on the corresponding Hecke surfaces which has been constructed in arXiv:0801.3951 and which is closely related to the natural extension of the generating map for the so called Hurwitz-Nakada continued fractions. We derive simple 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.

30 pages; revised version

References in corpus (5)

• Transfer operators for ${\Gamma }_0(n)$ and the Hecke operators for the period functions for $PSL(2,Z)$
• Eigenfunctions of transfer operators and cohomology
• A realization of the Hecke algebra on the space of period functions for Gamma_0(n)
• Computation of Selberg zeta functions on Hecke triangle groups
• Nearest lambda_q-multiple fractions

Cited by in corpus (6)

• Halving dynamical systems
• A thermodynamic formalism approach to the Selberg zeta function for Hecke triangle surfaces of infinite area
• Odd and even Maass cusp forms for Hecke triangle groups, and the billiard flow
• A transfer-operator-based relation between Laplace eigenfunctions and zeros of Selberg zeta functions
• Symbolic dynamics and transfer operators for Weyl chamber flows: a class of examples
• Dynamics of geodesics, and Maass cusp forms