DENSITY AND LOCATION OF RESONANCES FOR CONVEX CO-COMPACT HYPERBOLIC SURFACES ´ ERIC ´ FRED NAUD

Abstract. Let X = Γ\H2 be a convex co-compact hyperbolic surface and let δ be the Hausdorff dimension of the limit set. Let ∆X be the hyperbolic Laplacian. We show that the density of resonances of the Laplacian ∆X in rectangles {σ ≤ Re(s) ≤ δ, |Im(s)| ≤ T } is less than O(T 1+τ (σ) ) in the limit T → ∞, where τ (σ) < δ as long as σ > 2δ . This improves the previous fractal Weyl upper bound of Zworski [33] and goes in the direction of a conjecture stated in [16].

1. Introduction and results In this work, we will focus on the distribution of resonances of the Laplacian for a class of hyperbolic Riemann surfaces of infinite volume. The spectral theory of these objects can be viewed as a relevant picture for more realistic physical models such as obstacle or potential scattering, but is also interesting in itself because of its connection with counting problems and number theory (see for example the recent works of Bourgain-Gamburd-Sarnak [6], or Bourgain-Kontorovich [7]). Resonances replace the missing eigenvalues in non-compact situations and their precise distribution and localization in the complex plane is still a widely open subject. Let us be more specific. Let H2 denote the usual hyperbolic plane with its standard metric with curvature −1, and let X = Γ\H2 be a convex co-compact hyperbolic surface (see next paragraph for details). Here Γ is a discrete group of isometries of H2 , convex co-compact (that is to say finitely generated, without non trivial parabolic elements or elliptic elements). In this paper we will assume that Γ is non-elementary which is equivalent to say that X is not a hyperbolic cylinder. The limit set Λ(Γ) is commonly defined as Λ(Γ) := Γ.z ∩ ∂H2 , where Γ.z denotes the orbit of z ∈ H2 under the action of Γ. This limit set actually does not depend on the choice of z and is a Cantor set in the above setting. We will denote by δ(Γ) the Hausdorff dimension of Λ(Γ). Let ∆X denote the hyperbolic Laplacian on the surface X. The spectral Key words and phrases. Laplacian on hyperbolic surfaces, Resonances, Selberg’s Zeta function, Ruelle Transfer operators, Topological Pressure. 1

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theory on L2 (X) has been described by Lax and Phillips [17] :[1/4, +∞) is the continuous spectrum and has no embedded eigenvalues. The rest of the spectrum is made of a (possibly empty) finite set of eigenvalues, starting at δ(1−δ). The fact that the bottom of the spectrum is related to the dimension δ was first pointed out by Patterson [24] for convex co-compact groups. This result was later extented for geometrically finite groups by Sullivan [30, 31]. By the preceding description of the spectrum, the resolvent RX (s) = (∆X − s(1 − s))−1 : L2 (X) → L2 (X), is therefore well defined and analytic on the half-plane {Re(s) > 21 } except at a possible finite set of poles corresponding to the finite point spectrum. Resonances are then defined as poles of the meromorphic continuation of RX (s) : C0∞ (X) → C ∞ (X) to the whole complex plane. The set of poles is denoted by RX , and this contains possible eigenvalues and genuine resonances. This continuation is usually performed via the analytic Fredholm theorem after the construction of an adequate parametrix. The first result of this kind in the more general setting of asymptotically hyperbolic manifolds is due to Mazzeo and Melrose [20]. A more precise parametrix for hyperbolic surfaces was constructed by Guillop´e and Zworski [12, 13]. Note that by the above construction and choice of spectral parameter s = σ + it, the resonance set (including possible eigenvalues) RX is included in the half-plane {Re(s) ≤ δ}. If δ > 12 , then the set of genuine resonances (RX minus eigenvalues) is included in the half-plane {Re(s) < 21 }. One of the basic problems of the theory is to locate the resonances with the largest real part, which play a key role in various asymptotic problems, including hyperbolic lattice point counting and wave asymptotics. Another central and related question is the existence of a fractal Weyl law when counting resonances in strips. More precisely, in the papers by Zworski and Guillop´e-Lin-Zworski, [15, 33] the following is proved. For all σ ≤ δ set N (σ, T ) := #{z ∈ RX : σ ≤ Re(z) ≤ δ and 0 ≤ Im(z) ≤ T }, where counting is done with multiplicities. Then for all σ ≤ δ, one can find Cσ > 0 such that for all T ≥ 1, one has (1)

N (σ, T ) ≤ Cσ T 1+δ .

The first upper bound of this type involving a ”fractal” dimension is due to Sj¨ostrand [28] for potential scattering, see also [29]. In [15], the authors give some numerical evidence that the above estimate may be optimal provided σ is negative enough, but no rigorous results so far have confirmed this conjecture, and the best existing lower bound is a sublinear omega estimate: see [14]. On the other hand, it is conjectured

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in [16] that for all  > 0, there are only finitely many resonances in the half-plane   δ Re(s) ≥ +  , 2 hence suggesting that one has to take σ ≤ δ/2 to be able to prove a lower bound of the type N (σ, T ) ≥ Bσ T 1+δ for large T . In this paper, we show the following upper bound which provides a more precise picture of the distribution of resonances. Theorem 1.1. Let Γ be a non-elementary convex co-compact subgroup of PSL2 (R). Then for all σ ≥ 2δ , one can find τ (σ) ≥ 0, Cσ > 0 such that for any T ≥ 1, N (σ, T ) ≤ Cσ T 1+τ (σ) , where τ ( 2δ ) = δ, τ (σ) < δ for all σ > 2δ . Moreover, one can find σ0 > 2δ such that the map σ 7→ τ (σ) is real-analytic, strictly convex and strictly decreasing on [ 2δ , σ0 ]. In particular, its derivative at σ = 2δ satisfies τ 0 ( 2δ ) < 0. The main novelty of that result is this τ (σ) < δ for all σ > 2δ which shows that the upper bound (1) cannot be sharp unless σ ≤ 2δ . An explicit expression for τ (σ) is provided in the last section (see formula (14)), involving the topological pressure of the Bowen-Series map. Theorem 1.1 is relevant for σ close enough to the critical value 2δ since we know from a previous work of the author [21] that there always exists a spectral gap i.e. one can find  > 0 such that RX ∩ {Re(s) ≥ δ − } = {δ}. Unfortunately, this gap  is hardly explicit. Similarly if δ > 12 , then this spectral gap is already given by Lax-Phillips theory 1, but the above theorem is still non-trivial even in that case since 2δ < 12 . An analog of our model Γ\H2 is obstacle scattering outside several convex bodies, where resonances of the Dirichlet laplacian ∆D are defined by the meromorphic continuation of (∆D − λ2 )−1 defined on the upper half-plane, to C. They have been numerically investigated in [19] and experimentally in [4] via microwave scattering. Both papers show that the density of resonances ”peaks” for Im(λ) close to −γcl /2, where γcl is the classical escape rate. The escape rate γcl measures the exponential rate at which particles escape to infinity under the action of the classical flow. In our setting, with the above choice of spectral parameter s, this corresponds exactly to the line Re(s) = 2δ . We refer the reader to [22] for a comprehensive survey on questions related to 1We

recall that RX has at most finitely many points with Re(s) ≥ 21 , the largest one being the simple eigenvalue at s = δ.

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fractal Weyl laws and spectral gaps for various open (chaotic) quantum systems. The only similar result we are aware of so far in the rigorous mathematical literature is the spectral deviations obtained by N. Anantharaman [1] for the eigenvalues of the damped wave equation on compact negatively curved manifolds. The techniques we rely on to prove Theorem 1.1 are rather different although they both share some ergodic theory and thermodynamical formalism. We would like to point out that Theorem 1.1 has a natural interpretation in terms of Selberg’s zeta function. Let us denote by P the set of primitive closed geodesics on X = Γ\H2 and given C ∈ P we denote by l(C) its length. Selberg’s zeta function ZΓ (s) is usually defined through the infinite product (convergent for Re(s) > δ ) Y
ZΓ (s) := 1 − e−(s+k)l(C) . (C,k)∈P×N

It is known since the work of Patterson-Perry [25] that this function extends analytically to C and the non-trivial 2 zeros of ZΓ (s) are the resonances. Therefore Theorem 1.1 can be read as a statement on zeros of Selberg’s zeta function, and this is actually the starting point of our proof. From a number theoretic point of view, Theorem 1.1 is the analog of a result of Selberg on the density of zeros of the Riemann zeta function ζ(s) in the critical strip { 21 < Re(s) < 1}. Using the same notations, Selberg’s result (see Titchmarsh [32], chapter 9) is (uniformly for 21 ≤ σ ≤ 1)   1 1− 41 (σ− ) 2 log T . N (σ, T ) = O T The analogy stops here: although we do use some complex analytic tools which are reminiscent of number theory, the main part of our argument is very different since there is no available representation of ZΓ (s) involving Dirichlet series and arithmetic sums. Let us make a few comments on the organization of the paper. In the next section we recall the transfer operator approach for Selberg’s zeta function. In section §3 we prove the necessary a priori bounds to control the growth of a (modified ) Selberg zeta function in strips, slightly generalizing the method of [15]. In §4 we use a classical lemma of Littlewood to relate the counting function for resonances to a ”mean square estimate”. The idea of §6, which is the core of that paper, is to exhibit some cancellations in the quadratic sums to beat the pointwise bound of [15] leading to (1). To achieve this goal some lower bounds on the derivatives of ”off-diagonal phases” are required and proved in 2”trivial”

or topological zeros of ZΓ (s) are located at s = −n, n ∈ N0 , with mutiplicities 2n + 1, and correspond with multiplicities to the resonances of H2 . See Theorem 10.1 in [5] for a precise factorization of ZΓ (s).

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§5. The technique we use in §6 bears some similarity with the work of Dolgopyat [11, 21] and can be viewed as an ”averaged” Dolgopyat estimate. We would like to mention that the ideas presented here should extend without major difficulties to higher dimensional Schottky groups; actually only §5 needs to be significantly modified. Acknowledgements I would like to thank both referees for valuable remarks that definitely helped to improve the readability of the paper. I wish to thank Colin Guillarmou, St´ephane Nonnenmacher and Maciej Zworski for their reading and comments. I also thank D. Jakobson for our collaboration which motivated this paper.

2. Transfer operator and Selberg’s zeta function We use the notations of §1. Let H2 denote the Poincar´e upper halfplane H2 = {x + iy ∈ C : y > 0} endowed with its standard metric of constant curvature −1 dx2 + dy 2 ds2 = . y2 The group of (positive) isometries of H2 is naturally isomorphic to PSL2 (R) through the action of 2 × 2 matrices viewed as M¨obius transforms az + b z 7→ , ad − bc = 1. cz + d Below we recall the definition of Fuchsian Schottky groups which will be used to uniformize convex co-compact surfaces. A Fuchsian Schottky group is a discrete subgroup of PSL2 (R) built as follows. Let D1 , . . . , Dp , Dp+1 , . . . , D2p be 2p Euclidean open discs in C orthogonal to the line R ' ∂H2 . We assume that for all i 6= j, Di ∩ Dj = ∅. Let γ1 , . . . , γp ∈ PSL2 (R) be p isometries such that for all i = 1, . . . , p, we have b \ Dp+i , γi (Di ) = C b := C ∪ {∞} stands for the Riemann sphere. where C D3

D1

γ1

D2

D4

γ2

A Schottky group with two generators 3 3Here

γ1 (∂D1 ) = ∂D3 and γ2 (∂D2 ) = ∂D4 . The small discs inside are obtained after ”iteration”, for example D3 contains from left to right γ1 (D2 ), γ1 (D4 ) and γ1 (D3 ).

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The discrete group Γ generated by γ1 , . . . , γp and their inverses is called a classical Fuchsian Schottky group. If p > 1 then Γ is said to be non elementary. It is always a free, geometrically finite, discrete group and if in addition we require that for all i 6= j, Di ∩ Dj = ∅, then the quotient Riemann surface X = Γ\H2 is a convex co-compact surface, i.e. a geometrically finite surface with no cusps. The converse is true: up to an isometry, all convex cocompact hyperbolic surfaces can be uniformized by a group as above, see [9]. Of course, the choice of the γi is not unique. We recall that for Fuchsian Schottky groups as defined above the limit set Λ(Γ) satisfies Λ(Γ) ⊂ ∪2p i=1 Di , in particular it is a compact subset of C. Let Γ ⊂ PSL2 (R) be a Fuchsian Schottky group as defined earlier: Γ = hγ1 , . . . , γp ; γ1−1 , . . . , γp−1 i, b \ Dp+i . We also set for i = 1, . . . , p, γp+i := γ −1 . For where γi (Di ) = C i all j = 1, . . . , 2p, let H 2 (Dj ) denote the Bergman space of holomorphic functions defined by ( ) Z H 2 (Dj ) := f : Dj → C : f holomorphic and |f |2 dm < +∞ , Dj

here m stands for the usual Lebesgue measure on C. Each function space H 2 (Dj ) is a Hilbert space when endowed with the obvious norm. We set 2p M H 2 := H 2 (Dj ). j=1

The Ruelle transfer operator is a bounded linear operator Ls : H 2 → H 2 defined by (z ∈ Di , s ∈ C ) (Ls (f ))i (z) :=

X

(γj0 (z))s fj+p (γj (z)),

j6=i

with the notation f = (f1 , . . . , f2p ), and j + p is understood mod 2p. We have to say a few words about the complex powers here: we have γj0 (Di ) ⊂ C \ R− for i 6= j so γj0 (z)s is understood as 0

γj0 (z)s := esL(γj (z)) ,

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where L is the complex logarithm on C\R− which coincides on R+ \{0} with the usual logarithm, that is to say Z z dζ (2) L(z) = . ζ 1 This operator Ls acts as a compact, trace class operator on H 2 and we refer the reader to [5, 15] for a proof (Lemma 15.6 in [5]) . The Fredholm determinant associated to this family is the Selberg zeta function defined above: for all s ∈ C, we have ZΓ (s) = det(I − Ls ). For a proof of this fact, see [5] Theorem 15.8. This remarkable formula that dates back to the work of Pollicott [26] in the co-compact case is the starting point of our analysis. Unfortunately, it is not enough to have the trace class mapping property on H 2 and we will need 4 to use as in [15] a family of Hilbert spaces H 2 (h) depending on a small scale parameter 0 < h ≤ h0 which will allow us to work closer to the limit set Λ(Γ). We recall the construction taken from [15], see also [5] §15.4. Let 0 < h and set Λ(h) := Λ(Γ) + (−h, +h), then for all h small enough, Λ(h) is a bounded subset of R whose connected components have length at most Ch where C > 0 is independent of h, see [5] Lemma 15.12. Let {I` (h), ` = 1, . . . , N (h)} denote these connected components. The existence of a finite Patterson-Sullivan measure µ supported on Λ(Γ) plus Sullivan’s Shadow Lemma (see [5], chapter 14) show that (see also [5] P. 312, first displayed equation) X µ(I` (h)) = µ(Λ(h)), A−1 hδ N (h) ≤ `

for some uniform
A > 0, hence the number N (h) of connected components is O h−δ . Given 1 ≤ ` ≤ N (h), let D` (h) be the unique euclidean open disc in C orthogonal to R such that D` (h) ∩ R = I` (h). Now set N (h) 2

H (h) :=

M

H 2 (D` (h)).

`=1

We will see in the next section that for n ≥ n0 (with n0 independent of h and s) that the iterates Lsn = Ls ◦ . . . ◦ Ls act as compact trace class operators on H 2 (h). Moreover, the Fredholm determinant (n)

ZΓ (s) := det(I − Lsn ) 4The

main reason that forces us to perform this ”small scale analysis” is the exponential blow-up of phases in the complex domain as |Im(s)| → ∞, see §3, estimate (11).

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has among its zeros all zeros of the original Selberg zeta function ZΓ (s) and we will count resonances using this determinant instead of the original zeta function. The goal of the next section is to provide a proof of the following fact, which is a slight modification of the argument of Guillop´e-Lin-Zworski in [15]. Proposition 2.1. Fix σ0 ≤ δ. There exists C > 0, n0 ∈ N0 and R > 0 such that for all σ0 ≤ Re(s) ≤ δ and n ≥ n0 , we have for |Im(s)| ≥ R, (n)

log |ZΓ (s)| ≤ C|Im(s)|δ enP (σ0 ) , where P (σ) is the topological pressure at σ. We refer the reader to the next paragraph for a definition of topological pressure. 3. Basic pointwise estimates The goal of this section is to prove the previous Proposition. We first introduce some notations. We recall that γ1 , . . . , γp are generators of the Schottky group Γ, as defined in the previous section. Considering a finite sequence α with α = (α1 , . . . , αn ) ∈ {1, . . . , 2p}n , we set γα := γα1 ◦ . . . ◦ γαn . We then denote by Wn the set of admissible sequences of length n by Wn := {α ∈ {1, . . . , 2p}n : ∀ i = 1, . . . , n − 1, αi+1 6= αi + p mod 2p}. We point out that α ∈ Wn if and only if γα is a reduced word in the free group Γ. For all j = 1, . . . , 2p, we define Wnj by Wnj := {α ∈ Wn : αn 6= j}. If α ∈ Wnj , then γα maps Dj into Dα1 +p . Given the above notations and f ∈ H 2 , we have for all z ∈ Dj and n ∈ N, X Lsn (f )(z) = (γα0 (z))s f (γα (z)). α∈Wnj

We will need throughout the paper some distortion estimates for these maps γα . More precisely we have for all j = 1, . . . , 2p, • (Uniform hyperbolicity). One can find C > 0 and 0 < θ < θ < 1 such that for all n, j and α ∈ Wnj , for all z ∈ Dj we have n

C −1 θ ≤ |γα0 (z)| ≤ Cθn . • (Bounded distortion). There exists M1 > 0 such that for all n, j and all α ∈ Wnj , 00 γ sup α0 ≤ M1 . γ D j

α

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The second estimate, called bounded distortion, will be used constantly throughout the paper. In particular it implies that for all z1 , z2 ∈ Dj , for all α ∈ Wnj , we have (3)

e−|z1 −z2 |M1 ≤

|γα0 (z1 )| ≤ e|z1 −z2 |M1 . |γα0 (z2 )|

These estimates are rather standard facts in the classical ergodic theory of uniformly expanding Markov maps. For the sake of completeness, we provide some details and references on their proofs. To prove the first estimate, it is enough to show that one can find m ≥ 1 such that for all j and all α ∈ Wmj , we have for all z ∈ Dj , |γα0 (z)| < 1. This is usually shown using an argument on isometric circles, see for example [5], Proposition 15.4. See also [8], Lemma 3. Setting η :=

sup j z∈Dj ,α∈Wm

|γα0 (z)| < 1,

Writing n = mq + r with r < m, we have for all α ∈ Wnj and z ∈ Dj ,
n |γα0 (z)| ≤ Cη q ≤ C η 1/m , where C is a constant independent of n, j, z. A similar idea gives the lower bound. To prove the second estimate, write for all z ∈ Dj and α ∈ Wnj , γα00 d {L(γα0 (z))} (z) = 0 γα dz  n  00 X γαk = ◦ γαk+1 ...αn (z) × γα0 k+1 ...αn (z), 0 γ αk k=1 which by the first estimate is bounded in absolute value by e C

n X

θn−k ≤ M1 .

k=1

Another critical tool in our analysis is the Topological pressure and Bowen’s formula. Let Ij := Dj ∩ R. The Bowen-Series map T : ∪2p i=1 Ii → R ∪ {∞} is defined by T (x) = γi (x) if x ∈ Ii . The non-wandering set of this map is exactly the limit set 5 Λ(Γ) of the group: Λ(Γ) =

+∞ \

T −n (∪2p i=1 Ii ).

n=1 5This

identity follows from the orbit equivalence, Lemma 15.3 in [5].

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The limit set is T -invariant and given a continuous map ϕ : Λ(Γ) → R, the topological pressure P (ϕ) can be defined through the variational formula:   Z P (ϕ) = sup hµ (T ) + ϕdµ , µ

Λ

where the supremum is taken over all T -invariant probability measures on Λ, and hµ (T ) stands for the measure-theoretic entropy. A celebrated result of Bowen [8] says that the map σ 7→ P (−σ log T 0 ) is convex 6, strictly decreasing and vanishes exactly at σ = δ(Γ), the Hausdorff dimension of the limit set. An alternative way to compute the topological pressure is to look at weighted sums on periodic orbits i.e. we have !1/n X (n) (4) eP (ϕ) = lim eϕ (x) , n→+∞

T n x=x

with the notation ϕ(n) (x) = ϕ(x)+ϕ(T x)+. . .+ϕ(T n−1 x). In particular, we have the identity !1/n X P (−σ log T 0 ) n 0 −σ (5) e = lim [(T (x)) ] . n→+∞

T n x=x

Equivalence between the two formulas for the pressure is a basic fact of ergodic theory, see for example [23], chapters 4 & 5. For simplicity, we will use the notation P (σ) in place of P (−σ log T 0 ). Bowen’s formula now reads σ < δ ⇒ P (σ) > 0, σ > δ ⇒ P (σ) < 0.

(6)

We will seldom use the two previous formulas in our analysis but will rather use the following upper bound. Lemma 3.1. For all σ0 , M in R with 0 ≤ σ0 < M , one can find C0 > 0 such that for all n large enough and M ≥ σ ≥ σ0 , we have   2p X X  sup(γα0 )σ  ≤ C0 enP (σ0 ) . (7) j=1

α∈Wnj

Ij

Proof. Assume that n is taken large enough so that for all α, sup γα0 < 1. Ij

For all σ ≥ σ0 and x ∈ Ij , we have by bounded distortion property (3) X X sup(γα0 )σ ≤ C M (γα0 (x))σ0 = C M Lσn0 (1)(x). α∈Wnj 6Convexity

Ij

α∈Wnj

follows obviously from the variational formula above.

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The Ruelle-Perron-Frobenius Theorem ([2], Theorem 1.5 or [23], Theorem 2.2) applied to Lσ0 : C 1 (I) → C 1 (I), with I = ∪j Ij , says that this operator is quasi-compact 7, has spectral radius eP (σ0 ) and eP (σ0 ) is the only eigenvalue on the circle {|z| = eP (σ0 ) }. Moreover, eP (σ0 ) is a simple eigenvalue whose spectral projection is given by Z f 7→ P(f ) := ϕ0 f dµ0 , I

where µ0 is a probability measure and ϕ0 is a positive C 1 density on I. Using Holomorphic functional calculus, we can therefore decompose Lσn0 = enP (σ0 ) P + N n , where N has spectral radius at most θ0 eP (σ0 ) for some θ0 < 1. The spectral radius formula gives in particular 1/n

lim kN n kC 1 ≤ θ0 eP (σ0 ) .

n→∞

It is now clear that we have X n sup |γα0 |σ ≤ C0 sup ϕ0 enP (σ0 ) + C1 θe0 enP (σ0 ) , α∈Wnj

Ij

for some constants C0 , C1 and θ0 < θe0 < 1 and the proof is done.  We can now show that, for n sufficiently large, we have a well defined operator Lsn : H 2 (h) → H 2 (h). In the sequel, we assume that h > 0 is small enough so that Λ(h) ⊂ I, otherwise things do not make sense. Set Ej (h) := {1 ≤ ` ≤ N (h) : D` (h) ⊂ Dj }. Given α ∈ Wnj and ` ∈ Ej (h), we have by contraction for n large enough, γα (D` (h)) ⊂ D`0 (h) for some `0 ∈ {1, . . . , N (h)}. Actually we need a quantitative estimate which is as follows. Lemma 3.2. There exists n0 such that for all n ≥ n0 , for all α ∈ Wnj and all ` ∈ Ej (h), there exists an index `0 such that γα (D` (h)) ⊂ D`0 (h) with dist(γα (D` (h)), ∂D`0 (h)) ≥ 21 h. 7It

means that outside a disc of radius r0 < eP (σ0 ) , the spectrum is made of isolated eigenvalues of finite multiplicity, see [2] chapter 1, for precise references.

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Proof. Since we are dealing with discs orthogonal to R, it is enough to work with the real intervals I` (h). Let α ∈ Wnj and let ` ∈ Ej (h). Pick x ∈ I` (h) ∩ Λ(Γ). Because Λ(Γ) is Γ-invariant, γα (x) is in some I`0 (h) ∩ Λ(Γ), with dist(γα (x), ∂I`0 (h)) ≥ h. Given an arbitrary y ∈ I` (h), we have for some C > 0 uniform in h, |γα (x) − γα (y)| ≤ Cθn h, therefore we get dist(γα (x), ∂I`0 (h)) ≥ h − Cθn h ≥ 21 h, provided n is large enough, say n ≥ n0 , with n0 independent of h.  For all n ≥ n0 as above, and all h > 0 small enough, the transfer operator acts as a compact trace class operator Lsn : H 2 (h) → H 2 (h), which follows from the preceding Lemma 3.2, see for example [5], Lemma 15.6. To estimate the pointwise growth of the Fredholm determinant (n) ZΓ (s) = det(I − Lsn ) we will use some singular values estimates on the space H 2 (h) with h = |Im(s)|−1 . Notice that this zeta function does not depend on h. Indeed, provided Re(s) > δ, we have the formula ! ∞ X 1 (8) det(I − Lsn ) = exp − Tr(Lsnk ) , k k=1 and the trace can be explicitly computed (see [5], Lemma 15.7): X (T p )0 (x)−s , Tr(Lsp ) = p )0 (x))−1 1 − ((T p x∈Λ:T x=x the above sums over periodic orbits of the Bowen-Series map are clearly independent of the choice of function space, hence of h which will be adjusted to optimize our estimates. First we need to recall some facts on singular values of compact operators and our basic reference is the book of Simon [27]. If H is a complex Hilbert space and T : H → H is a compact operator, the singular values µk (T ) are the eigenvalues √ (ordered decreasingly) of the self-adjoint positive operator T ∗ T . If we have ∞ X µk (T ) < +∞, k=1

then T is said to be trace class. If T is trace class, then the eigenvalue sequence |λ1 (T )| ≥ |λ2 (T )| ≥ . . . ≥ |λk (T )|

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is summable and the trace Tr(T ) is defined by Tr(T ) =

∞ X

λk (T ),

k=1

while the determinant det(I + T ) is given by det(I + T ) :=

∞ Y

(1 + λk (T )) .

k=1

Weyl’s inequalities show that we have ∞ ∞ Y Y (1 + |λk (T )|) ≤ (1 + µk (T )) . k=1

k=1

In this paper we will use the following remark: Lemma 3.3. Let T : H → H be a trace class operator and (e` )`∈N a Hilbert basis of H, then we have ∞ X log | det(I + T )| ≤ kT (e` )k. `=0

Proof. By Weyl’s inequality, we write ∞ ∞ X X √ log | det(I + T )| ≤ log(1 + µk (T )) ≤ µk (T ) = Tr( T ∗ T ). k=0

k=0

But the classical Lidskii theorem says that ∞ ∞ X X √ √ √ ∗ ∗ h T T e` , e` i ≤ k T ∗ T e` k, Tr( T T ) = `=0

`=0

√ by the Cauchy-Schwarz inequality, but we have k T ∗ T e` k = kT e` k.  We can now give a proof of Proposition 2.1. First we need an explicit orthonormal basis of H 2 (h). Setting for ` ∈ {1, . . . , N (h)}, D` (h) := D(c` , r` ) with c` ∈ R and h ≤ r` < Ch, we denote by e`k the function in H 2 (h) defined for z ∈ Dj (h) by ( 0 if j 6= ` e`k (z) = q k+1 1  z−cj k if j = `. π rj rj It is straightworfard to check that (e`k )k∈N,1≤`≤N (h) is an orthonormal basis of H 2 (h). We now use Lemma 3.3 and write (n) log |ZΓ (s)|

≤

(h) ∞ N X X k=0 `=1

kLsn (e`k )kH 2 (h) .

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Using the Cauchy-Schwarz inequality, we get (N (h) = O(h−δ )) 1/2  N (h) ∞ X δ X (n)  (9) log |ZΓ (s)| ≤ Ch− 2 kLsn (e`k )k2H 2 (h)  . k=0

`=1

On the other hand we have N (h)

X

kLsn (e`k )k2H 2 (h)

=

X

|Lsn (e`k )|2 dm,

D`0 (h) `=1

j=1 `0 ∈Ej (h)

`=1

N (h)

2p X X Z

where m is the usual Lebesgue measure on C. Given `0 ∈ Ej (h), we can write Z N (h) X X Z (k) n ` 2 (10) |Ls (ek )| dm ≤ |(γα0 )s ||(γβ0 )s |Fα,β dm, D`0 (h) `=1

D`0 (h)

α,β∈Wnj

where N (h) (k) Fα,β (z)

:=

X

|e`k ◦ γα (z)||e`k ◦ γβ (z)|.

`=1

We now need to prove the following remark. Lemma 3.4. Setting Ωj (h) := ∪`∈Ej (h) D` (h), we have for all α, β ∈ Wnj , (k) sup Fα,β ≤ Ch−2 ρk , Ωj (h)

with C and 0 < ρ < 1 uniform. (k)

Proof. First remark that given z ∈ Ωj (h), we have either Fα,β (z) = 0 or (k) Fα,β (z) = |e`k0 ◦ γα (z)||e`k0 ◦ γβ (z)|, for some `0 ∈ {1, . . . , N (h)}. Then combine Lemma 3.2 with the explicit formula for e`k0 to obtain the result.  Here comes the main point of this section (and is the key idea in the proof of [15]). For all z ∈ D`0 (h), we write 0

0

0

γα0 (z)s = esL(γα (x)) es{L(γα (z))−L(γα (x))} , where x is chosen in I`0 (h). Notice that L(γα0 (x)) is real, so using the bounded distortion property, we get for some C > 0, (11)

Re(s) C|Im(s)|h

|γα0 (z)s | ≤ (γα0 (x))

e

.

It is now clear that if we set |Im(s)| = h−1 we get (we recall that σ = Re(s) ≤ M ) !Re(s) sup |(γα0 (z))s | ≤ CM z∈D`0 (h)

sup γα0 (x)

x∈Ij

,

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15

and this is what justifies the choice of the space H 2 (h). Therefore we have obtained (C changes from line to line) 2  N (h) 2p X X X sup |γα0 |σ  kLsn (e`k )k2H 2 (h) ≤ Ch−δ ρk  j=1 α∈Wnj

`=1

Ij

≤ Ch−δ ρk e2nP (σ0 ) . The proof is done by inserting the above estimate in formula (9) and summing over k.  4. Applying Littlewood’s Lemma We now show how to reduce the proof of Theorem 1.1 to a mean square estimate on transfer operators. To this end we apply a result of Littlewood borrowed from Titchmarsh [32]. More precisely, we prove the following. Define M (σ, T ) by M (σ, T ) := #{s ∈ RX : σ ≤ Re(s) ≤ δ and T /2 ≤ Im(s) ≤ T }. Proposition 4.1. Fix σ0 < δ, then one can find ν0 > 0, T0 > 0 such that for all σ0 < σ < δ, there exists Cσ , Cσ0 > 0 such that for all T ≥ T0 and n(T ) = [ν log T ] with 0 < ν ≤ ν0 , Z T (n(T )) M (σ, T ) ≤ Cσ log |ZΓ (σ0 + it)|dt + Cσ0 T. T /2

Proof. We start by recalling a version of Littlewood’s Lemma which suits our needs. Let σ0 < 1 and T > 0. Let f be a function which is holomorphic on a neighborhood of the rectangle RT,σ0 := [σ0 , 2] + i[T /2, T ], and assume that f does not vanish on the segment 2+i[T /2, T ]. Denote by ZT,σ0 the set of zeros of f on RT,σ0 . Then we have the formula (obtained by taking imaginary parts in [32], formula (9.9.1)) Z T Z T X 2π (Re(z) − σ0 ) = log |f (σ0 + it)|dt − log |f (2 + it)|dt T /2

z∈ZT,σ0

Z

2

T /2

Z

2

Arg(f (σ + iT ))dσ −

+ σ0

Arg(f (σ + iT /2))dσ. σ0

The function Arg(f ) is defined as follows. Let Ω be a bounded simply connected neighborhood of RT,σ0 on which f is holomorphic. By removing from Ω a finite number of horizontal segments Sj , j = 1, . . . , m, one obtains a simply connected domain
Ω \ ∪m S , j j=1 on which f does not vanish, hence one can define a complex logarithm log f . Arg(f ) is then defined as the imaginary part of log f . To extend

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Arg(f ) one can take upper limits (see Titchmarsh [32], section 9.9, for more details). The resulting function Arg(f ) is well defined on RT,σ0 \Z, locally integrable and discontinuous on a finite union of segments. Fix σ0 < σ < δ < 2. Then since RX is a subset of the set of zeros of (n) (n) ZΓ (s), applying the above formula to ZΓ (s) we get Z T (n(T )) log |ZΓ (σ0 + it)|dt 2π(σ − σ0 )M (σ, T ) ≤ T /2

Z

T

−

(n(T )) log |ZΓ (2

 + it)|dt + O

 +O

sup σ0 ≤σ≤2

T /2

sup σ0 ≤σ≤2

(n(T )) |Arg(ZΓ (σ

(n(T )) |Arg(ZΓ (σ

 + iT ))|

 + iT /2))| .

We need to prove the following. Lemma 4.2. For all n large enough, we have for all t, (n)

|Re(ZΓ (2 + it))| ≥ 21 . Proof. First remark that for Re(s) > δ formula (8) gives ! ! ∞ ∞ X X 1 1 Re(Tr(Lsnk )) cos Im(Tr(Lsnk )) , Re(det(I−Lsn )) = exp − k k k=1 k=1 and the trace formula combined with the pressure formula (5) shows that for all  > 0 and p large, |Tr(Lsp )| ≤ Cep(P (Re(s))+) . We therefore obtain

(n) |Re(ZΓ (2 + it))| ≥ exp −Cen(P (2)+) | cos Cen(P (2)+) |. (n)

Since P (2) < 0 by (6), |Re(ZΓ (2 + it))| ≥ 12 as long as n is taken large enough .  This lower bound shows that Z T T (n(T )) − log |ZΓ (2 + it)|dt ≤ log(2). 2 T /2 (n)

It remains to control Arg(ZΓ (s)) and we will use another classical result of Titchmarsh essentially based on Jensen’s formula, see [32], 9.4. Lemma 4.3. Fix 2 > σ0 > 0 and let f be a holomorphic function on the half-plane {Re(s) ≥ 0}. Suppose that |Re(f (2 + it))| ≥ m > 0 for all t ∈ R and assume that f (s) = f (s) for all s. For all t ≥ 0 and σ ≥ 0, set Aσ,t := sup |f (σ 0 + it0 )|. σ≤σ 0 ≤2 0≤t0 ≤t

DENSITY AND LOCALIZATION OF RESONANCES

17

Then if T is not the imaginary part of a zero of f (s), for all σ ≥ σ0 > 0, |Arg(f (σ + iT ))| ≤ Cσ0 (log A0,T +2 − log m) + 3π/2, where Cσ0 does not depend on T . To finish the proof of Proposition 4.1, we use the pointwise estimate of Proposition 2.1 which combined with Titchmarsh’s Lemma 4.3 and the lower bound of Lemma 4.2 gives
(n(T )) sup |Arg(ZΓ (σ + iT ))| = O T δ+νP (σ0 ) = O(T ), σ0 ≤σ≤2

as long as we choose ν ≤

1−δ . P (σ0 )

The proof is done, provided that there (n(T ))

are no zeros on {Im(s) = T } and {Im(s) = T /2}. If ZΓ (s) vanishes e on these lines, we simply replace T by some T < T ≤ T + 1 so that (n(T )) ZΓ (s) does not vanish on {Im(s) = Te} and T /2 by some T 0 with (n(T )) T /2−1 ≤ T 0 < T /2 so that ZΓ (s) does not vanish on {Im(s) = T 0 }. We then write M (σ, T ) ≤ #{s ∈ RX : Re(s) ≥ σ and T 0 ≤ Im(s) ≤ Te} Z

Te

≤C T0

Z

T

≤C

(n(T ))

log |ZΓ

(n(T ))

log |ZΓ

(σ0 + it)|dt + O(T )

(σ0 + it)|dt + CT + O(T ),

T /2

by the pointwise estimate of Proposition 2.1 and assuming ν small enough.  We now state the main estimate which is the core argument of the paper. Let ϕ0 ∈ C0∞ (R) be a smooth compactly supported function R such that Supp(ϕ0 ) = [−1, +1], ϕ0 > 0 on (−1, +1) and ϕ0 (x)dx = 1. We define a probability measure µT on R by the formula   Z Z 1 +∞ t − 2T ϕ0 f (t)dt. f dµT := T −∞ T R For simplicity, we will use below the notation k.k(h) in place of k.kH 2 (h) . Proposition 4.4. There exist ν > 0, 0 < ρ < 1 and C > 0 such that for all σ > 2δ one can find ε(σ) > 0 such that for all T ≥ 1 and all k∈N N (h) Z X (n(T )) kLσ+it (e`k )k2(h) dµT (t) ≤ Cρk T δ−ε(σ) , `=1

R

where we have taken n(T ) := [ν log T ], h = T −1 .

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The proof of this proposition is postponed to §6 and occupies the rest of the paper. Let us show how the combination of Proposition 4.4 and Proposition 4.1 implies Theorem 1.1. By Proposition 4.1, we have for δ < σ0 < σ, 2   Z 5T 2 5 (n(T )) M σ, T ≤ C log |ZΓ (σ0 + it)|dt + O(T ), 5 2 T 4 and we use Lemma 3.3 to write Z 5T (h) Z X NX 2 (n(T )) log |ZΓ (σ0 + it)|dt ≤ 5 T 4

k∈N `=1

5 T 2 5 T 4

(n(T ))

kLσ0 +it (e`k )k(h) dt.

Because we have 5 T 2

Z

5 T 4

−1

 ≤

(n(T ))

kLσ0 +it (e`k )k(h) dt

inf [−3/4,1/2]

ϕ0

Z T

(n(T ))

kLσ0 +it (e`k )k(h) dµT (t),

R

we obtain Z 5T (h) Z X NX 2 (n(T )) (n(T )) kLσ0 +it (e`k )k(h) dµT (t). log |ZΓ (σ0 + it)|dt ≤ CT 5 T 4

k∈N `=1

R

P By R applying Cauchy-Schwarz inequality twice (with respect to ` and dµT ), the above quantity is seen to be less than  1/2 N (h) Z X δ X (n(T ))  CT h− 2 kLσ0 +it (e`k )k2(h) dµT (t) ≤ CT 1+δ−ε(σ0 )/2 . k∈N

`=1

R

We have therefore obtained  o n 1 1+max δ− ε(σ0 ),0 2 . M (σ, T ) = O T To get an estimate on the counting function N (σ, T ), we just write   N0 X T N (σ, T ) ≤ M σ, k + O(1) 2 k=0 T 2N0

≤ 1. Since we have for T large,   n o N0 X 1 T 1+max δ− ε(σ0 ),0 2 M σ, k ≤ CT , 2 k=0

where N0 is such that

with C independent of T , the proof is done. The dependence of ε on σ0 will be discussed at the end of §6.

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19

5. A key lower bound Given α, β ∈ Wnj , we set for all x ∈ Ij = Dj ∩ R, Φα,β (x) := log γα0 (x) − log γβ0 (x). The goal of this section is to prove the following fact which will be a key result in the next section on mean square estimates. Proposition 5.1. There exists η0 > 0 and 1 > θ > 0 such that for all n ≥ 1 and all j = 1, . . . , 2p, we have for all α 6= β ∈ Wnj , n

inf |Φ0α,β (x)| ≥ η0 θ .

x∈Ij

The proof of this proposition will follow from the next Lemma which is of geometric nature. Lemma 5.2. Let Γ be a Schottky group as above, then there exists a constant CΓ > 0 such that for all   a b ∈ Γ \ {Id}, |c| ≥ CΓ . γ= c d Proof. Let Γ 6= Id be an element of Γ, and assume that   a b , γ= c d with c = 0. By looking at γ −1 one can assume that |a| ≥ 1. In addition, since Γ has no non-trivial parabolic element, we have actually |a| > 1. The action of γ on the Riemann sphere is therefore of the type γ(z) = a2 z + ab. Now pick an element of Λ(Γ) which is different from the fixed point of γ. Its orbit under the action of (γ n )n∈N goes to infinity as n → +∞ which contradicts the fact that Λ is γ-invariant and compact (recall that for the type of Schottky group we use, ∞ 6∈ Λ(Γ)). Therefore c 6= 0. Remark that by definition of Γ, we have γ −1 (∞) = −d/c ∈ ∪2p j=1 Dj . Consequently, one can find a constant Q1√such that for all γ 6= Id, |d/c| ≤ Q1 . Notice also that we have (i = −1) 1 Im(γ(i)) = 2 . c + d2 Since the limit set Λ(Γ) is compact, the orbit Γ.i has to be bounded in C for the usual euclidean distance. Hence there exists Q2 such that for all γ ∈ Γ, 1 ≤ Q2 . 2 c + d2 As a consequence we get 1 ≤ c2 (Q2 + Q2 Q21 ),

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and the proof is done.  We can now complete the proof of Proposition 5.1. Writing aα x + b α , with aα dα − bα cα = 1, γα (x) = cα x + dα we have |cβ dα − cα dβ | |Φ0α,β (x)| = 2 = 2|cβ dα − cα dβ |(γα0 (x))1/2 (γβ0 (x))1/2 . |cβ x + dβ ||cα x + dα | We can now remark that since Γ is a free group, α 6= β implies γα ◦γβ−1 6= Id, and we have the formula      aα b α dβ −bβ ∗ ∗ −1 γα ◦ γβ = = . cα d α −cβ aβ cα d β − d α cβ ∗ We can therefore apply the above Lemma and the proof ends by using bounded distortion and the lower bound for the derivatives. A comment on the meaning of this Lemma. It is very similar to ”nonintegrability” conditions used in all variants of Dolgopyat’s method [11]. Indeed, the phases Φα,β = log γα0 − log γβ0 can be understood in terms of the ”temporal distance function” for which lower bounds are required. See for example [3, 21] for non-integrability conditions which are the closest to what we use. We refer the reader to the paper of Liverani [18] for a comprehensive survey on the temporal distance function in the context of contact Anosov Flows. Although we have fully used the group structure in the above proof, we believe that a similar control could be obtained in greater generality i.e. for axiom A contact flows. 6. Mean square estimates The goal of this final section is to prove Proposition 4.4. We recall that in the sequel we take h = T −1 , n(T ) = [ν log T ], 0 < ν << 1. We first start by writing N (h) Z 2p Z X X (n(T )) ` 2 kLσ+it (ek )k(h) dµT (t) = `=1

R

j=1

Ωj (h)

Z NX (h)

(n(T ))

|Lσ+it (e`k )|2 dµT dm.

R `=1

For all z ∈ Dj and α, β ∈ Wnj we use the notation Φα,β (z) := L(γα0 (z)) − L(γβ0 (z)), where L is given by (2). This notation coincides on the real axis with the one introduced in the previous section. Remark that since Φα,β is real on the real axis and because of bounded distortion (3), we have on any disc D` (h), (12)

sup |eiT Φα,β (z) | = O(1), z∈D` (h)

DENSITY AND LOCALIZATION OF RESONANCES

21

uniformly on T and `. We refer to the end of §3 where the same idea is used and explained in detail. For all z ∈ Ωj (h), we have by a change of variable Z NX (h) (n(T )) |Lσ+it (e`k )(z)|2 dµT = R `=1

X

(k)

(γα0 (z))σ (γβ0 (z))σ Gα,β (z)c ϕ0 (−T Φα,β (z))e2iT Φα,β (z) ,

α,β∈Wnj

where ϕ c0 is the usual Fourier transform of ϕ0 defined by Z +∞ ϕ0 (x)e−ixξ dx, ϕ c0 (ξ) := −∞

and

(k) Gα,β

is the following sum (see also Lemma 3.4): N (h) (k) Gα,β (z)

:=

X

e`k ◦ γα (z)e`k ◦ γβ (z),

`=1

and according to the notations of §3, we have clearly (k)

(k)

|Gα,β (z)| ≤ Fα,β (z). The goal now is to gain some decay as T goes to +∞ by using the key observation of Lemma 5.1. We split the above sum into two contributions, the ”diagonal” one plus the ”off-diagonal” one: X (k) (γα0 (z))σ (γβ0 (z))σ Gα,β (z)c ϕ0 (−T Φα,β (z))e2iT Φα,β (z) α,β∈Wnj

= Sdiag + Sof f diag , where we have set X Sdiag := |γα0 (z)|2σ G(k) ϕ0 (−T Φα,α (z))ei2T Φα,α (z) , α,α (z)c α∈Wnj

X

Sof f diag :=

(k)

ϕ0 (−T Φα,β (z))e2iT Φα,β (z) . (γα0 (z))σ (γβ0 (z))σ Gα,β (z)c

α6=β∈Wnj

We mention that the phases e2iT Φα,β will not be used to obtain cancellations in the sequel, all decay will come from the fourier transform ϕ c0 (−T Φα,β (z)). We first deal with the diagonal contribution, which by bounded distortion and the pressure estimate (together with Lemma 3.4) gives |Sdiag | ≤ Cen(T )P (2σ) h−2 ρk , and therefore Z Ωj (h)

|Sdiag |dm ≤ Ch−δ en(T )P (2σ) ρk .

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We clearly have a gain as long as P (2σ) < 0, which by the identity (6) amounts to say that σ > 2δ . To deal with the off-diagonal sum, we will use the Paley-Wiener-Schwartz estimate 8 for ξ ∈ C and all q ≥ 0, |c ϕ0 (ξ)| ≤ Cq

(13)

e|Im(ξ)| . (1 + |ξ|)q

This implies that for all q, we have
|c ϕ0 (−T Φα,β (z))| = Oq (1 + |T Φα,β (z)|)−q . The trouble comes from Φα,β (z) which may be vanishing on some part of Ωj (h) even if α 6= β. We prove the following Lemma. Lemma 6.1. Fix an  > 0 and 0 < η < 1 such that η +  < 1. There exists ν small enough (recall that n(T ) = [ν log T ]) such that for all α, β ∈ Wnj with α 6= β, one can split the set Ej (h) = Ej[ t Ej] so that: • We have #Ej[ = O(hη−1 ), Ej[ is the ”bad” set of indices. • For all ` ∈ Ej] and z ∈ D` (h), |Φα,β (z)| ≥ Chη+ , Ej] is the ”good” set of indices where a lower bound is available. Proof. By Proposition 5.1, for all α 6= β, the function x 7→ Φα,β (x) is strictly monotonic on the interval Ij , and its derivative is uniformly n bounded from below by C1 θ . Two cases can occur: • Either Φα,β (x) vanishes at some point x0 ∈ Ij and we set Jα,β = [x0 − hη , x0 + hη ], where 0 < η < 1 will be adjusted later on. Then for all x ∈ Ij \ Jα,β we have n

|Φα,β (x)| ≥ C1 hη θ . • The map x 7→ Φα,β (x) does not vanish on Ij := (aj , bj ). According to the sign of Φα,β , we set Jα,β = [aj , aj + hη ] or Jα,β = [bj − hη , bj ]. In both cases we have for x ∈ Ij \ Jα,β , n

|Φα,β (x)| ≥ C1 hη θ . Now define Ej] by Ej] = {` ∈ Ej (h) : D` (h) ∩ Jα,β = ∅}. Given ` ∈ Ej] , we have by bounded distortion for all z ∈ D` (h), n

|Φα,β (z)| ≥ C1 hη θ − C2 h ≥ C3 hη+ , provided η +  < 1 and ν| log θ| ≤ . It remains to count Ej[ := Ej (h) \ Ej] . 8which

is simply obtained by repeated integration by parts in our case

DENSITY AND LOCALIZATION OF RESONANCES

23

Except for possibly two of them, ` ∈ Ej[ implies R∩D` (h) ⊂ Jα,β , hence by volume comparison (#Ej[ − 2)Ch ≤ |Jα,β | = hη . Therefore #Ej[ = O(hη−1 ).  Going back to Sof f diag , we have using Lemma 3.4, estimate (12) and (13) Z Z X dm(z) 0 σ k −2 0 σ |Sof f diag |dm ≤ Cq ρ h kγα k∞ kγβ k∞ . q (1 + |T Φ α,β (z)|) Ω (h) Ωj (h) j α6=β We then write Z Z Z dm(z) dm(z) dm(z) = + , q q q Ωj (h) (1 + |T Φα,β (z)|) Ω[j (h) (1 + |T Φα,β (z)|) Ω]j (h) (1 + |T Φα,β (z)|) with the notations Ω[j (h) :=

[

D` (h) and Ω]j (h) :=

[

D` (h).

`∈Ej]

`∈Ej[

We recall to the reader that #Ej[ = O(hη−1 ) and #Ej] = O(h−δ ). Applying the above Lemma and taking  > 0 small enough so that η +  < 1, we get with T = h−1 and q large enough Z

dm(z) 2+η−1 2−δ+q(1−(η+)) = O h + O h . q Ωj (h) (1 + |T Φα,β (z)|)
= O h2+η−1 . Therefore, Z

|Sof f diag |dm ≤ Cρk e2nP (σ) hη−1 .

Ωj (h)

Adding all of our estimates, we have reached N (h) Z X
(n(T )) kLσ+it (e`k )k2(h) dµT (t) ≤ Cρk T δ+νP (2σ) + T 1−η+2νP (σ) . `=1

R

Here, we have the freedom to choose η as close to 1 as we want and ν as small as desired. Therefore, for all , one can find ν() such that for all n(T ) = [ν log T ] with 0 < ν ≤ ν(), and for all σ ≥ 2δ , N (h) Z

X `=1

R


(n(T )) kLσ+it (e`k )k2(h) dµT (t) ≤ Cρk T δ+νP (2σ) + T  .

24

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The proof of Proposition 4.4 is done. We can now say a few more words on the function τ (σ) of the main Theorem 1.1: using the above choice of ν() then for all σ > σ0 ≥ 2δ , we can take nν o τ (σ) = δ + max P (2σ0 ), ( − δ)/2 . 2 so for example taking σ0 = 21 (σ + 2δ ) gives o n ν δ δ (14) τ (σ) = max δ + P (σ + 2 ), 2 + /2 . 2 We have not attempted to optimize the choice of constants at all. The function τ chosen above inherits all classical properties of the topological pressure ([23], chapter 4). The map σ 7→ P (σ) is convex, strictly decreasing and real analytic. In addition one can show that in our setup, it is strictly convex. We detail briefly the argument. We recall that T : Λ → Λ is the Bowen-Series map associated to Γ. By a standard formula (see [23]), we have 2 Z  Z 1 00 (n) P (σ) = lim g − n gdµσ dµσ n→∞ n Λ where g = log |T 0 | and g (n) := g + g ◦ T + . . . + g ◦ T n−1 and µσ is the equilibrium measure of −σg. From this formula one can infer that P 00 (σ) > 0 for all σ if and only if g is not cohomologous to a constant. That is to say one cannot find a constant K ∈ R such that for all n ∈ N and all periodic points w with T n w = w we have g (n) (w) = nK. Since there is a one to one correspondence between the length of closed geodesics on X = Γ\H2 and the set {g (n) (w) : T n w = w, n ∈ N}, P 00 (σ) = 0 would imply that the length spectrum is contained in a lattice. This cannot occur since it is known that every non elementary Fuchsian group produces a non-lattice length spectrum [10]. References [1] N. Anantharaman. Spectral deviations for the damped wave equation. Geom. Funct. Anal., 20(3):593–626, 2010. [2] V. Baladi. Positive transfer operators and decay of correlations, volume 16 of Advanced series in Nonlinear dynamics. World Scientific, Singapore, 2000. [3] V. Baladi and B. Vall´ee. Euclidean algorithm are gaussian. J. Number Theory, 110 (2005), no 2, 331-386. [4] S. Barkofen, T. Weich, A. Potzuweit, H. -J. St¨ockmann, U. Kuhl, and M. Zworski. Experimental observation of spectral gap in microwave n-disk systems. Arxiv preprint, 2012. [5] D. Borthwick. Spectral theory of infinite-area hyperbolic surfaces, volume 256 of Progress in Mathematics. Birkh¨auser Boston Inc., Boston, MA, 2007. [6] J. Bourgain, A. Gamburd, and P. Sarnak. Generalization of Selberg’s 3/16 theorem and affine sieve. Acta Math. 207 (2011), no 2, 255-290.

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