Nearest 𝜆_{𝑞}-multiple fractions

Abstract

We discuss the nearest λ q -multiple continued fractions and their duals for λ q = 2 cos π q which are closely related to the Hecke triangle groups G q , q = 3, 4, . . .. They have been introduced in the case q = 3 by Hurwitz and for even q by Nakada. These continued fractions are generated by interval maps f q respectively f ⋆ q which are conjugate to subshifts over infinite alphabets. We generalize to arbitrary q a result of Hurwitz concerning the G qand f q -equivalence of points on the real line. The natural extension of the maps f q and f ⋆ q can be used as a Poincaré map for the geodesic flow on the Hecke surfaces G q \H and allows to construct the transfer operator for this flow. 15 3.2. λ q -CF's and their generating interval maps 16 3.3. Markov partitions for f q and f ⋆ q 17 4. The maps f q and f ⋆ q and regular respectively dual regular λ q -CF's 23 1991 Mathematics Subject Classification. Primary 11A55, Secondary 11J70, 30B70 . Key words and phrases. Continued fractions, Hecke triangle groups, discrete dynamical systems, natural extension, subshifts of infinite type, sofic systems. The second named author was supported in part by the Deutsche Forschungsgemeinschaft through the DFG Research Project "Transfer operators and non arithmetic quantum chaos" (Ma 633/16-1). 1 I R 3 = [−R 3 , R 3 ] = ε=+,− ∞ m=2 J ⋆ εm with J ⋆ m • ∩J ⋆ n • = ∅ for all m = n.

The transfer operator for the Hecke triangle groups

“…In [12,Lemmas 17 and 33] it is shown that these two algorithms lead to the regular and dual regular λ q -CF's…”

“…e.g. [12]). This means that there exist partitions of the intervals I q and I Rq with the property that the set of boundary points is preserved by the map f q and f ⋆ q , respectively.…”

“…. , ±κ q } and S • denotes the interior of the set S. As in [12] we introduce next a finer partition which is compatible with the intervals of monotonicity for f q .…”

“…One possibility to obtain such a generalization is via a cohomological approach [1], which has recently been considered for the case G 3 = PSL(2, Z) in [2]. We concentrate on the transfer operator approach to this circle of problems and started to work out this approach in [12,13] for the Hecke triangle groups which, contrary to modular groups studied up to now, are mostly non-arithmetic.…”

Symmetries of the transfer operator for Γ0(N) and a character deformation of the Selberg zeta function for Γ0(4)

“…In the transfer operator approach to Selberg's zeta function for a Fuchsian group Γ this function gets expressed in terms of the Fredholm determinant of this operator which is constructed from the symbolic dynamics of the geodesic flow on the corresponding surface of constant negative curvature. Even if this approach has been carried out up to now only for certain groups like the modular subgroups of finite index [2], [3], [4], or the Hecke triangle groups [16], [14], [15] it has lead for instance to new points of view on this function [22] or the theory of period functions [12]. Another application of this method is a precise numerical calculation of the Selberg zeta function [20], which seems to be impossible by other means at the moment.…”