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.Comment: 30 pages; revised versio

Meromorphic continuation of Selberg zeta functions with twists having non-expanding cusp monodromy

“…In view of the already existing strict transfer operator approaches and the different methods for their construction [42,16,1,44,29,30,3,41,4,34,17,37,22,6,28,38,32,40], it might well be that this requirement is not a severe restriction on Γ at all. Moreover, it is most likely that with the methods we propose in this article the meromorphic continuability of (1) can also be shown starting with nonstrict transfer operator approaches as provided in [41,34,31] (for certain classes of cofinite Fuchsian groups). In the latter case, the representation of Z Γ,1 is of the form…”

Computations of Eigenvalues and Resonances on Perturbed Hyperbolic Surfaces with Cusps

“…There are various cases when this surface can be obtained as a quotient of H by a Hecke triangle group G q , q ≥ 3, namely when r −1 = 2 cos π q . In particular, for r = 1 √ 3 , r = 1 √ 2 , and r = 1, the resulting surfaces are arithmetic and correspond to the surfaces obtained from the Hecke triangle groups G q in the cases q = 6, 4, 3, respectively (see [26]). These are the only arithmetic cases.…”

Geometry and Analysis of Fractals

“…The earlier literature on transfer operators includes applications to physics [LR69], to the Selberg zeta function [FM12], to dynamical zeta functions [Rue02,Rue96,Nau12,MMS12]; to C * -dynamical systems [Kwa12,ABL11]; to the study of Hausdorff dimension [Hen12]; to spectral theory [ABL12]. Our paper has two aims: One is to unify, and extend earlier studies; and the other is to prove a number of theorems on measures, dynamical systems, stochastic processes built from infinite products.…”