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Symbolic Dynamics for the Geodesic Flow on Hecke Surfaces

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Symbolic Dynamics for the Geodesic Flow on Hecke Surfaces JOURNAL OF MODERN DYNAMICS doi: 10.3934/jmd.2008.2.581 VOLUME 2, NO. 4, 2008, 581–627 SYMBOLIC DYNAMICS FOR THE GEODESIC FLOW ON HECKE SURFACES DIETER MAYER AND FREDRIK STRÖMBERG (Communicated by Jens Marklof) ABSTRACT. In this paper we discuss a coding and the associated symbolic dy- namics for the geodesic flow on Hecke triangle surfaces. We construct an ex- plicit cross-section for which the first-returnmap factors through a simple (ex- plicit) 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. CONTENTS 1. Introduction 581 2. The geodesic flow on T 1M 583 3. λ-Continued fraction expansions 584 4. Construction of the cross-section 607 5. Construction of an invariant measure 619 6. Lemmas on continued-fraction expansions and reduced geodesics 622 1. INTRODUCTION Surfaces of negative curvature and their geodesics have been studied since the 1898 work of Hadamard [15] (see in particular the remark at the end of §58). Inspired by the work of Hadamard and Birkhoff [6], Morse [31] introduced a cod- ing of geodesics essentially corresponding to what is now known as “cutting se- quences” and used this coding to show the existence of a certain type of recur- rent geodesics [32]. Further ergodic properties of the geodesic flow on surfaces of constant neg- ative curvature given by Fuchsian groups were shown by, e.g., Artin [5], Nielsen [36], Koebe [26], Löbell [29], Myrberg [33], Hedlund [16, 17, 18, 19], Morse and Received January 23, 2008, revised June 17, 2008. 2000 Mathematics Subject Classification: Primary: 37D40, Secondary: 37E05, 11A55, 11K50. Key words and phrases: geodesic flow, symbolic dynamics, Hecke triangle groups, continued fractions. DM: This work has been supportedby the German Research Foundation (DFG) under contract Ma 633/16-1. IN THE PUBLIC DOMAIN AFTER 2036 581 ©2008AIMSCIENCES 582 DIETER MAYER AND FREDRIK STRÖMBERG Hedlund [30] and Hopf [20, 21]. In this sequence of papers one can see the sub- ject of symbolic dynamics emerging. For a more up-to-date account of the er- godic properties of the geodesic flow on a surface of constant negative curvature formulated in a modern language see, e.g., the introduction in Series [47]. Artin’s[5] approach was novel in thathe used continued fractions to code geo- desics on the modular surface. Since Artin, coding and symbolic dynamics on the modular surface have been studied by e.g., Adler and Flatto [1, 2, 3] and Se- ries [48]. For a recent review of different aspects of coding of geodesics on the modular surface, see for example the expository papers by Katok and Ugarcovici [24, 25]. Other important references for the theory of symbolic dynamics and coding of the geodesic flow on hyperbolic surfaces are, e.g., Adler–Flatto [4], Bowen and Series [7] and Series [47]. In the present paper, we study the geodesic flow on a family of hyperbolic surfaces with one cusp and two marked points, the so-called Hecke triangle sur- faces, generalizing the modular surface. Symbolic dynamics for a related billiard have also been studied by Fried [12]. We now give a summary of the paper. Sec- tions 1 and 2 contain preliminary facts about hyperbolic geometry and geodesic flows. In Section 3 we develop the theory of λ-fractions connected to the coding of the geodesic flow on the Hecke triangle surfaces. The explicit discretization of the geodesic flow in terms of a Poincaré section and Poincaré map is developed in Section 4. As an immediate application we derive invariant measures for cer- tain interval maps in Section 5. Some rather technical lemmas are confined to the end in Section 6. 1.1. Hyperbolic geometry and Hecke triangle surfaces. Recall that any hyper- bolic surface of constant negative curvature 1 is given as a quotient (orbifold) − dz M H /Γ. Here H z x i y y 0, x R together with the metric ds | | = = = + | > ∈ = y is the hyperbolic upper half-plane and Γ PSL2(R) SL2(R)/{ I } is a Fuchsian © ⊆ ª =∼ ± group. Here SL2(R) is the group of real two-by-two matrices with determinant 1, I 1 0 and PSL (R) is the group of orientation-preserving isometries of H . = 0 1 2 The boundary of H is ∂H R R { }, H H ∂H . If g a b PSL (R). ¡ ¢ ∗ ∗ c d 2 az b = = ∪ ∞ = ∪ = ∈ then g z + H for z H , g x ∂H for x ∂H and we say that g is elliptic, = cz d ∈ ∈ ∈ ∈ ¡ ¢ hyperbolic or+parabolic depending on whether Tr g a d 2, 2 or 2. = | + | < > = The same notation applies for fixed points of g. In the following we identify ¯ ¯ the element g PSL2(R) with the map it defines¯ on¯ H ∗. Note that the type of ∈ 1 fixed point is preserved under conjugation g Ag A− by A PSL (R). A para- 7→ ∈ 2 bolic fixed point is a degenerate fixed point, belongs to ∂H , and is usually called a cusp. Elliptic points appear in pairs, z, z¯ with z H and z¯ belonging to the ∈ lower half-plane H −, and Γz , the stabilizer subgroup of z in Γ, is cyclic of finite order. Hyperbolic fixed points appear also in pairs with x, x∗ ∂H , where x∗ is ∈ said to be the conjugate point of x. A geodesic γ on H is either a half-circle orthogonal to R or a line parallel to the imaginary axis and the endpoints of γ are denoted by γ ∂H . We identify ± ∈ the set of geodesics on H with G ξ,η ξ η R∗ and use γ ξ,η to denote = | 6= ∈ JOURNAL OF MODERN DYNAMICS ©¡ ¢ VªOLUME 2, NO.¡ 4 (2008),¢ 581–627 SYMBOLIC DYNAMICSFOR THE GEODESIC FLOW ON HECKE SURFACES 583 the oriented geodesic on H with γ ξ and γ η. Unless otherwise stated + = − = all geodesics are assumed to be parametrized with hyperbolic arc length with γ(0) either at height 1 if γ is vertical or the highest point on the half-circle. The tangent of γ at γ(t) is denoted by γ˙ (t). Itis knownthat z H and θ [ π,π) S1 ∈ ∈ − =∼ determine a unique geodesic (cf. Lemma 62) passing through z whose tangent at z makes an angle θ with the positive real axis. This geodesic is denoted by γz,θ. It is also well known that a geodesic γ ξ,η is closed if and only if ξ and η ξ∗ are = conjugate hyperbolic fixed points. 1¡ ¢ The unit tangent bundle of H , T H z H ~v TzH ~v 1 is the collec- = ∈ ∈ | | | = tion of all unit vectors in the tangent planes of H with base points z H , which 1H ~ F © ª ∈ we denote by Tz . By identifying v with its angle θ with respect to the positive real axis, we can view T 1H as the collection of all pairs (z,θ) H S1. We may ∈ × 2 also view this as the set of geodesics γz,θ on H or equivalently as G R∗ . ⊆ 1 Let π: H M be the natural projection map z Γz, and let π∗ : T H 1 → 1 7→ → T M be the extension of π to T H . Then γ∗ πγ is a closed geodesic on M if = and only if γ and γ are fixed points of the same hyperbolic map gγ Γ. For a + − ∈ more comprehensive introduction to hyperbolic geometry and Fuchsian groups see e.g., [23, 27, 41]. DEFINITION 1. For an integer q 3, the Hecke triangle group G PSL (R) is ≥ q ⊆ 2 the group generated by the maps S : z 1/z and T : z z λ, where λ 7→ − 7→ + = λ 2cos π/q [1,2). The corresponding orbifold (Riemann surface) is M q = ∈ q = G \H , which we sometimes identify with the standard fundamental domain of q ¡ ¢ Gq , F {z H z λ/2, z 1}, q = ∈ | |ℜ | ≤ | | ≥ πi with sides pairwise identified. Let ρ ρ e q and ρ ρ. We define the = + = − = − following oriented boundary components of Fq : L0 is the circular arc from ρ − to ρ . L1 is the vertical line from ρ to i and L 1 is the vertical line from i to + + ∞ − ∞ ρ . Thus ∂Fq L 1 L0 L1 is the positively oriented boundary of Fq . − = − ∪ ∪ REMARK 2. G is a realization of the Schwarz triangle group π/ ,π/q,π/2 , and q ∞ it is not hard to show (see, e.g., [27, VII]) that G , for q 3, is a cofinite Fuchsian q ≥ ¡ ¢ group with fundamental domain Fq and the only relations (1) S2 (ST )q Id – the identity in PSL (R). = = 2 Hence Gq has one cusp, that is the equivalence class of parabolic points, and two elliptic equivalence classes of orders 2 and q respectively. Note that G 3 = PSL2(Z)—the modular group—and that G4, G6 are conjugate to congruence sub- groups of the modular group. For q 3,4,6 the group G is nonarithmetic (cf. 6= q [23, pp. 151–152]), but in the terminology of [9, 46] it is semiarithmetic, meaning that it is possible to embed Gq as a subgroup of a Hilbert modular group. 2. THE GEODESIC FLOW ON T 1M We briefly recall the notion of the geodesic flow on a Riemann surface M 1 1 = Γ\H with Γ PSL2(R) a Fuchsian group. To any (z,θ) T H H S we can ⊂ ∈ =∼ × JOURNAL OF MODERN DYNAMICS VOLUME 2, NO. 4 (2008), 581–627 584 DIETER MAYER AND FREDRIK STRÖMBERG associate a unique geodesic γ γ on H such that γ(0) z and γ˙ (0) eiθ.
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