Zero is a resonance of every Schottky surface
aa r X i v : . [ m a t h . SP ] A ug ZERO IS A RESONANCE OF EVERY SCHOTTKY SURFACE
ALEXANDER ADAM, ANKE POHL, AND ALEXANDER WEIßE
Abstract.
For certain spectral parameters we find explicit eigenfunctionsof transfer operators for Schottky surfaces. Comparing the dimension of theeigenspace for the spectral parameter zero with the multiplicity of topologicalzeros of the Selberg zeta function, we deduce that zero is a resonance of everySchottky surface. Introduction and statement of results
The location of the resonances of hyperbolic surfaces is of great interest for applica-tions, see, e. g., [4, 5]. Accordingly much effort has been and is still being spend ontheir investigation. For the elementary hyperbolic surfaces (i. e., those with cyclic fundamental group, namely, the hyperbolic plane, the hyperbolic cyclinders, andthe parabolic cyclinders) the precise values of all resonances including their multi-plicities are known. For arbitrary hyperbolic surfaces, asymptotic counting resultson the number of their resonances such as Weyl laws (for surfaces of finite area),Weyl-type bounds, fractal Weyl bounds and refinements could be established, seee. g., [27, 28, 22, 13, 14, 15, 12, 1, 16, 26]. A good reference for this field is [3].Results on precise numerical values for resonances of non-elementary hyperbolicsurfaces, however, are still rather sparse. For non-elementary Schottky surfaces X (surfaces which we consider in this note) the only explicitly known resonance is thecritical exponent δ or, equivalently, the Hausdorff dimension of the limit set of thefundamental group of X . We provide the exact value of an additional resonance. Theorem 1.1.
Zero is a resonance of every Schottky surface.
For elementary Schottky surfaces, i. e., the hyperbolic cylinders, the methods weuse for Theorem 1.1 allow us to recover the known result on the full resonance set,albeit with a new proof.
Theorem 1.2.
The resonance set of the hyperbolic cylinder C ℓ with central geodesicof (primitive) length ℓ is − N + πiℓ Z . Each resonance has multiplicity . The proof of Theorem 1.1 as well as our proof of Theorem 1.2 is based on trans-fer operator techniques in combination with known properties of the Selberg zetafunction. This approach allows us to prove even deeper and more general resultsthan stated in Theorems 1.1 and 1.2, as explained in the following.
Mathematics Subject Classification.
Primary: 58J50, 37C30; Secondary: 11F72, 11M36.
Key words and phrases. resonances, Schottky surfaces, transfer operator. To improve readibility we restrict here to hyperbolic surfaces in the strict sense, i. e., tomanifolds, even though many of the results are valid for (two-dimensional real hyperbolic good)orbifolds as well.
Let X be a Schottky surface, let R ( X ) denote the set of all resonances of X (withmultiplicities), let L ( X ) denote the primitive geodesic length spectrum of X (alsowith multiplicities), and let δ denote the Hausdorff dimension of the limit set ofthe fundamental group of X . We refer to Section 2 below for more details on theobjects used in this overview.As is well-known, the Selberg zeta function Z X ( s ) := Y ℓ ∈ L ( X ) ∞ Y k =0 (cid:16) − e − ( s + k ) ℓ (cid:17) (more precisely, the infinite product used in this definition) converges for Re s >δ , and extends to an entire function, which we also denote by Z X . The set ofresonances R ( X ) is contained in the zeros of (the holomorphic continuation of) Z X , respecting multiplicities. Thus, determining zeros of Z X may serve as a firststep towards finding resonances of X .Further, for (any choice of Schottky data of) X there exists a transfer operatorfamily ( L X,s ) s ∈ C which acts as trace class operators on a certain Hilbert Bergmanspace and whose Fredholm determinant represents the Selberg zeta function Z X ,thus Z X ( s ) = det (1 − L X,s ) . Finding zeros of Z X is therefore equivalent to determining parameters s for whichthe transfer operator L X,s has an eigenfunction with eigenvalue .For the case that X is elementary, thus, a hyperbolic cylinder, we can identify notonly all such parameters s but also the full eigenspaces of L X,s . Theorem 1.3.
Suppose that X = C ℓ is a hyperbolic cylinder with central geodesic of(primitive) length ℓ . Then is an eigenvalue of L C ℓ ,s if and only if s ∈ − N + πiℓ Z .In this case, the geometric multiplicity of the eigenvalue is .For s = − n + πiℓ Z , n ∈ N , the eigenspace of L C ℓ ,s with eigenvalue is essentiallyspanned by two copies of z n . We refer to Theorem 3.4 below for an exact formula for the eigenfunctions.For the case that X is non-elementary, certain hyperbolic cylinders C provide coversof X (of infinite degree). Even though eigenfunctions of L X,s with eigenvalue aresubject to more symmetries than the eigenfunctions of L C,s with eigenvalue , forcertain choices of C and certain parameters s we can construct eigenfunctions of L X,s from those of L C,s .Throughout let χ ( X ) denote the Euler characteristic of X . Theorem 1.4.
Let X be a Schottky surface and let s ∈ − N . Then is aneigenvalue of L X,s with geometric multiplicity being at least − χ ( X ) .For s = − n , n ∈ N , we can construct − χ ( X ) linear independent eigenfunctions of L X,s with eigenvalue from the eigenfunctions provided by Theorem 1.3 for certainhyperbolic cylinders covering X . Theorem 3.6 below provides exact formulas for these eigenfunctions.To facilitate distinguishing in the following between results known for elementarySchottky surfaces only and those valid for all Schottky surfaces, we denote the
ERO IS A RESONANCE OF EVERY SCHOTTKY SURFACE 3 (rather, any) hyperbolic cylinder with central geodesic of (primitive) length ℓ by C ℓ .Surfaces denoted by X refer to arbitrary (elementary or non-elementary) Schottkysurfaces.Theorem 1.3 yields that the set of zeros of the Selberg zeta function Z C ℓ for X = C ℓ is − N + πiℓ Z , each zero having multiplicity . Theorem 1.4 implies that for anarbitrary Schottky surface X , the Selberg zeta function Z X has zeros at − N , andthat the multiplicity of the zero s = − n , n ∈ N , is at least − χ ( X ) .For completeness we note that for the case X = C ℓ being a hyperbolic cylinder,Theorem 1.4 estimates the multiplicity of the zeros of Z C ℓ at − N from belowby (note that χ ( C ℓ ) = 0 ), and hence, for the elementary Schottky surfaces,Theorem 1.3 is a much stronger result.For a generic Schottky surface X , the zeros of the Selberg zeta function Z X are notnecessarily given only by the resonances of X ; they may also be topological zeros.These two types of zeros may overlap, causing a zero of Z X to be of rather highmultiplicity.Borthwick–Judge–Perry [2] showed that (for any Schottky surface X ) the Selbergzeta function Z X admits the factorization(1) Z X ( s ) = e p ( s ) G ∞ ( s ) − χ ( X ) P X ( s ) , where p is a certain polynomial of degree at most (which is of no concern for ourapplication), the function G ∞ is G ∞ ( s ) = (2 π ) − s Γ( s ) G ( s ) with G being the Barnes G-function, and P X is the entire function P X ( s ) := s m (0) Y r ∈R ( X ) r =0 (cid:16) − sr (cid:17) exp (cid:18) sr + s r (cid:19) , where m (0) denotes the multiplicity of s = 0 as a resonance of X .Since the resonances of X are exactly the zeros of P X , including multiplicities, thetopological zeros of Z X are determined (again including multiplicities) by the zerosof G ∞ ( · ) − χ ( X ) . The zeros of G ∞ are s = − n , n ∈ N , with multiplicity n + 1 .Theorem 1.1 then follows by observing that for s = 0 , the geometric multiplicity ofthe eigenvalue of L X,s exceeds the multiplicity of s = 0 as topological zero of Z X by .If X non-elementary and hence χ ( X ) = 0 , and s < then such a comparisondoes not yet provide any insight. However, as shown in [8, 23], for non-elementary Schottky surfaces all zeros at s ∈ − N are topological.Theorem 1.2 follows from combining Theorem 1.3 with (1) and the fact that theEuler characteristic of hyperbolic cylinders vanishes, and hence each zero of theSelberg zeta function is indeed a resonance of the same multiplicity.As mentioned above, the resonance set for hyperbolic cylinders is known for long.The strength of the proof method presented here is not to provide an alterna-tive (rather involved) proof of Theorem 1.2. Its strength is in establishing Theo-rem 1.3 which, in view of the power of transfer operators for dynamical approaches A. ADAM, A. POHL, AND A. WEIßE to Laplace eigenfunctions, is an important step towards dynamical and transfer-operator-based characterizations of resonant states as well as period functions forthem [19, 18, 17, 10, 11, 6, 7, 25, 24, 21].The structure of this note is as follows. In Section 2 below we provide the neces-sary background information on hyperbolic surfaces, resonances, Schottky surfacesand transfer operators. The precise formulas for the eigenfunctions of the transferoperators as well as the proofs of Theorems 1.1-1.4 are the content of Section 3.3below.
Acknowledgement.
AP wishes to thank the Max Planck Institute for Mathematicsin Bonn for hospitality and excellent working conditions during the preparation ofthis manuscript. Further, she acknowledges support by the DFG grant PO 1483/2-1. 2.
Preliminaries
In this section we provide the necessary background information on the entitiesused for the proofs of Theorems 1.1-1.4, with the exception of the Selberg zetafunction which is already surveyed in the Introduction.2.1.
Hyperbolic surfaces and resonances.
We use the upper half plane model H := { z ∈ C : Im z > } , ds z = dz dz (Im z ) for the hyperbolic plane. In this model, the geodesic boundary ∂ H of H is identifiedwith R := R ∪ {∞} .We take advantage of the standard embedding of the hyperbolic plane H into theRiemann sphere b C = C ∪{∞} ∼ = P C . In this embedding, ∂ H is indeed the topologicalboundary of H .We identify the group of orientation-preserving Riemannian isometries of H with PSL ( R ) = SL ( R ) / {± id } . If an element g ∈ PSL ( R ) is represented by the matrix (cid:18) a bc c (cid:19) ∈ SL ( R ) then we may denote g by g = (cid:20) a bc d (cid:21) . The action of
PSL ( R ) extends continuously (even smoothly) to all of b C . With allidentifications in place, for g = (cid:2) a bc d (cid:3) ∈ PSL ( R ) and z ∈ b C we have g.z = ∞ if z = ∞ , c = 0 , or z = ∞ , cz + d = 0 , ac if z = ∞ , c = 0 , az + bcz + d otherwise.An element g ∈ PSL ( R ) is called hyperbolic if it fixes exactly two points in ∂ H .The prototypical hyperbolic elements are a L := e L e − L ! , ( L > , ERO IS A RESONANCE OF EVERY SCHOTTKY SURFACE 5 which act on H as multiplication by e L . Any hyperbolic element in PSL ( R ) isconjugate to some (unique) a L with L > . For any hyperbolic element h ∈ PSL ( R ) we let x + ( h ) and x − ( h ) denote the attracting and repelling fixed point, respectively,under its iterated action. Thus, for any z ∈ H we have x + ( h ) = lim n →∞ h n .z and x − ( h ) = lim n →∞ h − n .z. Given a Fuchsian group Γ , that is, a discrete subgroup Γ of PSL ( R ) , we denote thespace of its orbits on H by Γ \ H . This quotient space inherits from H the structure ofa Riemannian orbifold. If Γ is torsion-free then Γ \ H is even a Riemannian manifold.As is well-known, hyperbolic surfaces are precisely those Riemannian manifoldsthat are isometric to a quotient manifold of the form Γ \ H for some torsion-freeFuchsian group Γ .The Laplace–Beltrami operator ∆ H on H is given by ∆ H := − y (cid:0) ∂ x + ∂ y (cid:1) ( z = x + iy ; x, y ∈ R , y > . For any hyperbolic surface X , the Laplace operator ∆ H induces the Laplace oper-ator ∆ X : L ( X ) → L ( X ) on X . Its resolvent R X ( s ) := (∆ X − s (1 − s )) − : L ( X ) → L ( X ) is defined for s ∈ C with Re s > / and s (1 − s ) not being in the L -spectrum of ∆ X . It extends to a meromorphic family R X ( s ) : L ( X ) → H ( X ) on C , see [20, 13]. The poles of R X are the resonances of X . We let R ( X ) denote the set of resonances, repeated according to multiplicities.2.2. Schottky surfaces and Schottky data.
Schottky surfaces are preciselythose hyperbolic surfaces that are convex cocompact and of infinite area (and with-out orbifold points). The fundamental group of a Schottky surface is (isomorphicto) a (Fuchsian) Schottky group, that is, a geometrically finite, non-cofinite, torsion-free Fuchsian group without parabolic elements.As shown by Button [9], every (Fuchsian) Schottky group arises from a certaingeometric construction that we recall in what follows, and all Fuchsian groupsgiven by this construction are indeed Schottky.To simplify a uniform presentation of the geometric construction we recall that weidentify the Riemann sphere b C with C ∪ {∞} . We say that a subset D ⊆ b C is a Euclidean disk if there exists an element g ∈ PSL ( R ) such that g. D is containedin C and is a Euclidean disk in the usual sense. One easily checks that the notionof Euclidean disk does not depend on the choice of g .To construct a Schottky group, choose r ∈ N and fix r open Euclidean disks in b C that are centered on ∂ H and have mutually disjoint closures, say(2) D , D − , . . . , D r , D − r . A. ADAM, A. POHL, AND A. WEIßE
For convenience we here deviate from the successive enumeration as used in thestandard presentation of the construction. For j ∈ { , . . . , m } let S j be an elementof PSL ( R ) that maps the exterior of D j to the interior of D − j . Then the subgroup Γ of PSL ( R ) generated by S , . . . , S r is Schottky. All the generating element arehyperbolic.We stress that the numbering of the disks in (2) does not reflect in any way thepositions of the disks to each other. In fact, the construction and the arising Schot-tky group Γ depend on the choice of this numbering. Moreover, the constructedSchottky group Γ depends on the choice of the elements γ j , j = 1 , . . . , r . For futurereference we set S − j := S − j ( j = 1 , . . . , r ) , and we refer to the tuple (cid:16) r, (cid:8) D ± j (cid:9) rj =1 , { S ± j } rj =1 (cid:17) as a Schottky data for Γ .To facilitate notation, we use here and throughout (cid:8) E ± j (cid:9) mj =1 := (cid:8) E j , E − j : j = 1 , . . . , m } for E ∈ {D , S } .One easily sees that for any fixed Schottky group Γ , there exist infinitely manychoices of Schottky data for Γ . The number r of generators is an invariant in thisconstruction.Starting with a Schottky surface X , an additional choice in this construction ispossible, namely the one of the precise Schottky group Γ such that X is isometricto Γ \ H . We call a tuple (cid:16) Γ , r, (cid:8) D ± j (cid:9) rj =1 , { S ± j } rj =1 (cid:17) a Schottky data for X if Γ is isomorphic to the fundamental group of X , and (cid:16) r, (cid:8) D ± j (cid:9) rj =1 , { S ± j } rj =1 (cid:17) is a Schottky data for Γ . As above, X admits infinitely many choices of Schottkydata, and the number r of generators of Γ is an invariant.For any Schottky surface X we can find Schottky data such that none of the Eu-clidean disks D ± j , j = 1 , . . . , r , contains ∞ . However, for the discussions in thefollowing sections it is rather convenient to refrain from a restriction to such specialSchottky data.Schottky surfaces X for which the number of generators in a (and hence any)Schottky data is r = 1 are called hyperbolic cylinder . The generator S in anySchottky data is conjugate to a L for some L > . The value of L is independentof the choice of the Schottky data. It coincides with the (primitive) length of theunique non-oriented geodesic on X . By slight abuse of notions, we call X (and anyhyperbolic surface isometric to X ) the hyperbolic cylinder with central geodesic oflength L and denote it by C L . ERO IS A RESONANCE OF EVERY SCHOTTKY SURFACE 7
Transfer operators.
To any Schottky surface X and any choice of Schottkydata for X there is associated a transfer operator family which is crucial for ourargumentation. In the following we provide a brief presentation of this transferoperator family, its domain of definition and its properties. We refer to [12, 3] forproofs and more details. Before we can state the definition of the transfer operatorfamily we need a few preparations.For s ∈ C , any subset U ⊆ R , any element g = (cid:2) a bc d (cid:3) ∈ PSL ( R ) and any function f : U → C we set τ s ( g − ) f ( x ) := (cid:0) g ′ ( x ) (cid:1) s f ( g.x ) = | cx + d | s f (cid:18) ax + bcx + d (cid:19) ( x ∈ U ) whenever this is well-defined.In the following we make use of holomorphic extensions of this definition to functionsdefined on certain open subsets U ⊆ b C and applied to certain subsets of PSL ( R ) .We stress that such holomorphic extensions are subject to choices, and that none ofthe possible holomorphic extensions can be extended to a group action of PSL ( R ) .However, for all applications arising here there is indeed at least one holomorphicextension. The precise possibilities, in particular the choice of the logarithm, heavilydepend on the chosen Schottky data. All results obtained are independent of thechoice of the holomorphic extension. Since in all situations arising here the possiblechoices need to obey only finitely many restrictions which are easily seen to besatisfiable, we refrain throughout from this rather tedious discussion and tacitlymake appropriate choices.After having fixed such choices we say that a function f : U → C defined on someneighborhood U of ∞ is holomorphic at ∞ (for the parameter s ) if there exists g ∈ PSL ( R ) such that g.U ⊆ C and the map(3) τ s ( g ) f : g.U → C is holomorphic at g. ∞ . We remark that for all g ∈ PSL ( R ) , the operator τ s ( g ) is holomorphy-preserving as long as it is well-defined. We remark further that thisnotion of holomorphy depends on the value of s .For any open bounded subset U ⊆ C we let H ( U ) denote the Hilbert Bergmanspace on U , that is the space H ( U ) := (cid:26) f : U → C holomorphic : Z U k f k dvol < ∞ (cid:27) of holomorphic square-integrable functions on U , endowed with the inner product h f, g i := Z U h f ( z ) , g ( z ) i dvol( z ) . We extend this definition to Euclidean disks of b C possibly containing ∞ by combin-ing it with the definition (3) of holomorphy at ∞ from above. In all our applications,the operator applied to H ( U ) depends on a parameter s which is then used for thenatural definition of holomorphy at ∞ . This has the effect that the space H ( U ) itself effectively does not depend on s , only the choice of the transformation mapin a manifold chart. A. ADAM, A. POHL, AND A. WEIßE
Let X be a Schottky surface and let (cid:0) Γ , r, {D ± j } mj =1 , { S ± j } mj =1 (cid:1) be a choice of Schottky data for X .Let H := m M j =1 H (cid:0) D j (cid:1) ⊕ m M k =1 H (cid:0) D − k (cid:1) denote the direct sum of the Hilbert Bergman spaces on the Euclidean disks fromthe Schottky data. We identify functions f ∈ H with the function vectors(4) m M j =1 f j ⊕ m M k =1 f − k with f ℓ ∈ H (cid:0) D ℓ (cid:1) ( ℓ ∈ {± , . . . , ± r } ) . The transfer operator L s with parameter s ∈ C associated to X and the Schottkydata is the operator L s : H → H which, with respect to the presentation (4), has the matrix presentation L s = τ s (cid:0) S (cid:1) τ s (cid:0) S (cid:1) . . . τ s (cid:0) S r (cid:1) τ s (cid:0) S − (cid:1) . . . τ s (cid:0) S − r (cid:1) τ s (cid:0) S (cid:1) τ s (cid:0) S (cid:1) . . . τ s (cid:0) S r (cid:1) τ s (cid:0) S − (cid:1) . . . τ s (cid:0) S − r (cid:1) ... ... . . . ... ... ... . . . ... τ s (cid:0) S (cid:1) τ s (cid:0) S (cid:1) . . . τ s (cid:0) S r (cid:1) τ s (cid:0) S − (cid:1) τ s (cid:0) S − (cid:1) . . . τ s (cid:0) S (cid:1) . . . τ s (cid:0) S r (cid:1) τ s (cid:0) S − (cid:1) τ s (cid:0) S − (cid:1) . . . τ s (cid:0) S − r (cid:1) τ s (cid:0) S (cid:1) . . . τ s (cid:0) S r (cid:1) τ s (cid:0) S − (cid:1) τ s (cid:0) S − (cid:1) . . . τ s (cid:0) S − r (cid:1) ... ... . . . ... ... ... . . . ... τ s (cid:0) S (cid:1) τ s (cid:0) S (cid:1) . . . τ s (cid:0) S − (cid:1) τ s (cid:0) S − (cid:1) . . . τ s (cid:0) S − r (cid:1) . This operator is of trace class. Its Fredholm determinant represents the Selbergzeta function Z X ( s ) = det(1 − L s ) . Proofs of precisions of Theorems 1.1-1.4
In this section we provide the precise formulas for the eigenfunctions as announcedin the Introduction, and we present proofs of Theorems 1.1-1.4. For convenience werefer to eigenfunctions and eigenspaces with eigenvalue by -eigenfunctions and -eigenspaces, respectively.Each transfer operator considered in this note is build up from operators of theform τ s ( h ) , h ∈ PSL ( R ) hyperbolic. For that reason, we first consider, for any L > , the prototypical hyperbolic element a L = (cid:20) e L/ e − L/ (cid:21) ERO IS A RESONANCE OF EVERY SCHOTTKY SURFACE 9 with attracting fixed point ∞ and repelling fixed point , and study, in Proposi-tion 3.1 below, for the operator τ s ( a L ) the -eigenspaces of functions holomorphicin . With Lemma 3.2 we then clarify how these -eigenfunctions relate to -eigenfunctions with corresponding regularity of τ s ( h ) for an arbitrary hyperbolicelement h ∈ PSL ( R ) .Since the transfer operator of a hyperbolic cylinder essentially is the direct sum oftwo of these prototypical operators, the proof of Theorem 1.3 follows then imme-diately from Proposition 3.1 and Lemma 3.2, see Theorem 3.4 below.Then we study for which parameters s we can construct from the -eigenfunctionsof the transfer operators of hyperbolic cylinders -eigenfunctions for the transferoperators of arbitrary Schottky surfaces. Since the latter transfer operators ingeneral do not enjoy a direct sum structure, the results obtained are less sharp thanfor hyperbolic cylinders, see Proposition 3.5 and Theorem 3.6 below. However, fornonpositive integers we can construct explicit -eigenfunctions.From the results on the existence and counting of -eigenfunctions, the proofs ofTheorems 1.1-1.2 then follow immediately, see Section 3.3 below.3.1. Eigenfunctions for transfer operators for hyperbolic cylinders.
In thefollowing proposition we consider τ s ( a L ) to act on a space of functions that aredefined on a suitable neighborhood of . The result and the proof of the propositionshows that the exact neighborhood does not need to be specified any further as longas it is chosen in such a way that τ s ( a L ) is well-defined. Proposition 3.1.
Let
L > . Then τ s ( a L ) has a -eigenfunction that is holomor-phic in if and only if s ∈ − N + πiL Z . For s ∈ − n + πiL Z , n ∈ N , the space ofsuch -eigenfunctions is spanned by z n .Proof. Let s ∈ C and let f be a -eigenfunction for τ s ( a L ) that is holomorphic in . We deduce properties on s and f . Holomorphy of f in yields a power seriesexpansion f ( z ) = ∞ X n =0 a n z n for suitable a n ∈ C , n ∈ N , that is valid in a certain neighborhood U of . Itfollows that for all z ∈ U , we have τ s ( a L ) f ( z ) = e − Ls ∞ X n =0 a n e − Ln z n = ∞ X n =0 a n e − L ( s + n ) z n . Then f = τ s ( a L ) f and uniqueness of power series expansions imply that for all n ∈ N either a n = 0 or e − L ( s + n ) = 1 . The latter property is equivalent to s ∈ − n + 2 πiL Z . Thus, since f is a -eigenfunction of τ s ( a L ) and hence f = 0 it follows that s ∈− n + πiL Z for some n ∈ N , and a m = 0 for m ∈ N , m = n , and f ( z ) = a n z n with a n = 0 . One can even show that a n ∈ R . Conversely, f ( z ) = z n is obviously a holomorphic -eigenfunction for τ s ( a L ) for all s ∈ − n + πiL Z . (cid:3) The following lemma allows us to transfer the result of Proposition 3.1 to anyhyperbolic element. Recall from Section 2.1 that for any hyperbolic element h ∈ PSL ( R ) its x ± ( h ) denote the attracting and repelling fixed points are denoted by x ± ( h ) . Lemma 3.2.
Let s ∈ C . Suppose that g, h ∈ PSL ( R ) are hyperbolic elements forwhich we find p ∈ PSL ( R ) such that g = php − . Suppose further that f h is a -eigenfunction of τ s ( h ) that is holomorphic at the repelling fixed point x − ( h ) of h .Then f g := τ s ( p ) f h is a -eigenfunction of τ s ( g ) that is holomorphic at x − ( g ) .Proof. The statements on the holomorphy follow directly from Section 2.3. Theeigenfunction property is shown by a straightforward calculation, which we providefor the convenience of the reader. We have τ s ( g ) f g = τ s (cid:0) php − (cid:1) τ s ( p ) f h = τ s ( p ) τ s ( h ) f h = τ s ( p ) f h = f g . This completes the proof. (cid:3)
The following lemma is crucial for the consideration of arbitrary Schottky surfaces.However, for hyperbolic cylinder it facilitates the further argumentation. Through-out let(5) S := (cid:20) − (cid:21) . Lemma 3.3.
Let h ∈ PSL ( R ) be hyperbolic. Then there exists k ∈ PSL ( R ) suchthat khk − = h − .Proof. Since h is hyperbolic, h is conjugate within PSL ( R ) to a diagonal element.Thus, there is L ∈ R and p ∈ G such that php − = a L . From Sa L S = a − L = a − L it follows that p − Sphp − Sp = p − Sa L Sp = p − a − L p = h − . This completes the proof. (cid:3)
The combination of Proposition 3.1 and Lemma 3.2 allows us to fully determinethe -eigenspaces of the transfer operators of hyperbolic cyclinders. This providesa proof of Theorem 1.3, see Theorem 3.4 below. Lemma 3.3 or rather its proofsimplifies the argumentation in the proof of Theorem 3.4. Theorem 3.4.
Let C ℓ be a hyperbolic cylinder and let (Γ , , {D , D − } , { S , S − } ) be a choice of Schottky data for C ℓ . Let p ∈ PSL ( R ) be such that S = pa ℓ p − . ERO IS A RESONANCE OF EVERY SCHOTTKY SURFACE 11
For s ∈ C let L s be the associated transfer operator and let H be the Hilbert space asin Section 2.3. Then L s has a -eigenfunction in H if and only if s ∈ − N + πiℓ Z .For s = − n + πiℓ Z , n ∈ N , the -eigenspace of L s is spanned by the two functions ( τ s ( p ) z n , and (0 , τ s ( pS ) z n ) with S from (5) .Proof. The transfer operator L s for s ∈ C is L s = (cid:18) τ s ( S ) 00 τ s ( S − ) (cid:19) . Thus, the -eigenspace of L s is the direct sum of the -eigenspaces of τ s ( S ) and τ s ( S − ) . One easily checks that the element p ∈ PSL ( R ) with the propertiesas claimed exist. Application of Lemma 3.2 and Proposition 3.1 (in this order)completes the proof. (cid:3) Eigenfunctions for transfer operators of arbitrary Schottky groups.
The following proposition is the key result for constructing explicit -eigenfunctionsof the transfer operators for arbitrary Schottky surfaces. Proposition 3.5.
Let s ∈ C and let h ∈ PSL ( R ) be hyperbolic. Suppose that f is a -eigenfunction of τ s ( h ) that is holomorphic at x − ( h ) . Choose k ∈ PSL ( R ) such that khk − = h − (see Lemma 3.3). Let U ⊆ b C r { x ± ( h ) } be an open set.Then τ s ( h ) f and τ s (cid:0) h − k (cid:1) f extend holomorphically to U . Further (cid:0) τ s ( h ) f − ( − Re s τ s (cid:0) h − (cid:1) τ s ( k ) f (cid:1) | U ≡ ⇔ Im s = 0 . Proof.
By Lemma 3.2 we may restrict without loss of generality to h = a L for L > and k = S = (cid:2) − (cid:3) . Then U is any (non-empty) open subset of C r { } .From Proposition 3.1 it follows that s ∈ − n + πiℓL for some n ∈ N , ℓ ∈ Z , andthe eigenfunction f is (up to scaling) z n . Thus, f extends holomorphically to all of C r { } , and hence τ s ( S ) f does so. For every z ∈ U it follows that (cid:0) τ s ( a L ) f − ( − Re s τ s ( a − L ) τ s ( S ) f (cid:1) | U ( z ) = z n − z n − πiℓL = z n (cid:16) − z − πiℓL (cid:17) . This difference vanishes for all z ∈ U if ℓ = 0 , and at most for finitely many z ∈ U if ℓ = 0 . (cid:3) In Theorem 3.6 below we take advantage of Proposition 3.5 for investigating ifand how the -eigenfunctions of the transfer operators for hyperbolic cylinders canbe inherited, or rather extended, to -eigenfunctions of the transfer operators ofarbitrary Schottky surfaces. Theorem 3.6 is the announced more refined version ofTheorem 1.4. Theorem 3.6.
Let X be a Schottky surface and let (cid:16) Γ , r, (cid:8) D ± j (cid:9) rj =1 , (cid:8) S ± j (cid:9) rj =1 (cid:17) be a choice of Schottky data for X . For s ∈ C let L s be the associated transferoperator and H be the associated Hilbert space as defined in Section 2.3. Let s ∈− N . (i) Let j ∈ { , . . . , r } and let f j be a -eigenfunction of τ s ( S j ) that is holo-morphic at x − ( S j ) . Fix k j ∈ PSL ( R ) such that k j S j k − j = S − j . Then f = ( f , . . . , f r , f − , . . . , f − r ) with f ± j := 0 for j ∈ { , . . . , r } r { j } and f − j := − ( − s τ s ( k j ) f j defines a -eigenfunction of L s in H . (ii) The geometric multiplicity of the eigenvalue of L s in H is at least − χ ( X ) .Proof. Without loss of generality, we may assume that j = 1 . Let e f := L s f, e f = ( e f , . . . , e f r , e f − , . . . , e f − r ) . Then e f = τ s ( S ) f, e f − = − ( − s τ s (cid:0) S − (cid:1) τ s ( k ) f , e f j = τ s ( S ) f − ( − s τ s (cid:0) S − k (cid:1) f for j ∈ {± , . . . , ± r } .By hypothesis, e f = f . By Lemma 3.2, e f − = f − , and by Proposition 3.5, e f j = 0 for all j ∈ {± , . . . , ± r } . Thus, f is a -eigenfunction of L s .By Proposition 3.1, f exists, is unique up to scaling, and extends holomorphicallyto b C r { x + ( S ) } . Thus, f extends to an element of H .Moreover, by letting j run through { , . . . , r } this construction yields r = 1 − χ ( X ) linear independent -eigenfunctions of L s . Hence the geometric multiplicity of asan eigenvalue of L s in H is at least − χ ( X ) . (cid:3) Proofs of Theorem 1.1-1.2.
As already explained in the Introduction, The-orem 1.1 and 1.2 now follow from Theorem 3.6 and 3.4 in comparison with theknowledge on location and orders of topological zeros as obtained from (1).
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