Semi-classical approach for Anosov diffeomorphisms and Ruelle resonances
aa r X i v : . [ n li n . C D ] S e p Semi- lassi al approa h for Anosov di(cid:27)eomorphismsand Ruelle resonan esFrédéri Faure ∗ , Ni olas Roy † , Johannes Sjöstrand ‡ O tober 25, 2018Abstra tIn this paper, we show that some spe tral properties of Anosov di(cid:27)eomorphisms an be obtained by semi- lassi al analysis. In parti ular the Ruelle resonan es whi hare eigenvalues of the Ruelle transfer operator a ting in suitable anisotropi Sobolevspa es and whi h govern the de ay of dynami al orrelations, an be treated as thequantum resonan es of open quantum systems in the Aguilar-Baslev-Combes theoryor the more re ent Hel(cid:27)er-Sjöstrand phase-spa e theory [20℄.1Contents1 Introdu tion 22 The model of hyperboli map 52.1 Transfer operators . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 62.2 Dynami s on the otangent spa e . . . . . . . . . . . . . . . . . . . . . . . 8 ∗ † Geometri Analysis group, Institut fr Mathematik, Humbold Université, Berlin. roymath.hu-berlin.de ‡ CMLS, E ole Polyte hnique, FR 91128 Palaiseau edex. johannesmath.polyte hnique.fr1preprint: http://arxiv.org/abs/0802.17802000 Mathemati s Subje t Classi(cid:28) ation:37D20 Uniformly hyperboli systems (expanding, Anosov, Ax-iom A, et .) 37C30 Zeta fun tions, (Ruelle-Frobenius) transfer operators, and other fun tional analyti te hniques in dynami al systems 81Q20 Semi- lassi al te hniquesKeywords: Transfer operator, Ruelle resonan es, de ay of orrelations, Semi- lassi al analysis.1 The es ape fun tion and the anisotropi Sobolev spa es 93.1 Constru tion of the es ape fun tion A m and the pseudodi(cid:27)erential operator ˆ A m . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 93.2 The anisotropi Sobolev spa es . . . . . . . . . . . . . . . . . . . . . . . . 144 Spe trum of resonan es 165 Asymptoti expansion for dynami al orrelation fun tions 205.1 The Lebesgue orrelation fun tion . . . . . . . . . . . . . . . . . . . . . . . 215.2 The SRB orrelation fun tion . . . . . . . . . . . . . . . . . . . . . . . . . 246 Mixing of Anosov maps preserving a smooth measure 257 Trun ation and numeri al al ulation of the resonan e spe trum 288 Con lusion and perspe tives 29A Pseudodi(cid:27)erential operators with variable order 30A.1 Preliminary remarks . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30A.1.1 Semi- lassi al analysis . . . . . . . . . . . . . . . . . . . . . . . . . 30A.1.2 Symbols with onstant order . . . . . . . . . . . . . . . . . . . . . 31A.1.3 Pseudodi(cid:27)erential operators on manifold . . . . . . . . . . . . . . . 31A.2 Symbols with variable orders . . . . . . . . . . . . . . . . . . . . . . . . . 33A.2.1 De(cid:28)nition and main basi properties of S m ( x,ξ ) ρ . . . . . . . . . . . . 33A.2.2 Canoni al examples . . . . . . . . . . . . . . . . . . . . . . . . . . 34A.2.3 A tion of di(cid:27)eomorphisms . . . . . . . . . . . . . . . . . . . . . . . 35A.3 PDO with variable order . . . . . . . . . . . . . . . . . . . . . . . . . . . . 37A.3.1 Asymptoti expansions . . . . . . . . . . . . . . . . . . . . . . . . 37A.3.2 Adjoint and omposition . . . . . . . . . . . . . . . . . . . . . . . . 38A.3.3 Egorov's theorem . . . . . . . . . . . . . . . . . . . . . . . . . . . . 39A.3.4 Sobolev ontinuity . . . . . . . . . . . . . . . . . . . . . . . . . . . 39A.4 Non-isotropi ellipti ity . . . . . . . . . . . . . . . . . . . . . . . . . . . . 39A.4.1 Variable order ellipti ity . . . . . . . . . . . . . . . . . . . . . . . . 39A.4.2 Parametrix and invertibility . . . . . . . . . . . . . . . . . . . . . . 41A.4.3 L - ontinuity and quasi- ompa ity . . . . . . . . . . . . . . . . . . 441 Introdu tionAn Anosov di(cid:27)eomorphism f on a ompa t manifold M is hara terized by the fa t thatunder iterations, every traje tory has hyperboli instability, whi h means that two pointsthat are lose to ea h other will be separated exponentially fast under the dynami s inthe future or in the past. As a onsequen e the behavior of individual traje tories looks2ike unpredi table or (cid:16) haoti (cid:17). Instead of looking at individual traje tories, it is thenmore natural to study a set of traje tories, or equivalently the transport of fun tions (ordensities) under the map. One is led to study the so- alled Ruelle transfer operator ˆ F de(cid:28)ned by ˆ F ϕ = ϕ ◦ f with ϕ ∈ C ∞ ( M ) . The spe tral de omposition of this operatorprovides obje ts invariant under the dynami s and therefore informs us on the long timebehavior of the dynami s, su h as ergodi ity, mixing, de ay of orrelations, entral limittheorem ...(see [34, hap. VII℄,[3, 16, 23℄).The main subje t of this paper is the spe tral properties of the transfer operator ˆ F . Theapproa h we propose is based on an elementary but ru ial observation whi h itself relies onthe hypothesis of hyperboli ity: high Fourier modes of ˆ F n ϕ go towards in(cid:28)nity as n → ∞ or n → −∞ . In other words, the variations of the fun tion ϕ evolve towards (cid:28)ner and(cid:28)ner s ales as n → ±∞ , and as a onsequen e the (cid:16)information(cid:17) about the initial fun tion ϕ disappear from the ma ros opi s ale (the observation s ale). This is the me hanismresponsible for haoti behavior, and in parti ular for the de ay of dynami al orrelationfun tions. In the 70's, David Ruelle has initiated a fruitful theory alled thermodynami formalism [35, 12, 36℄ where he studied the transfer operator ˆ F and de(cid:28)ned the Ruelleresonan es whi h govern the exponential de ay of the dynami al orrelation fun tions.This approa h has re ently been improved onsiderably in the works of M. Blank, S.Gouëzel, G. Keller, C. Liverani [7, 17, 27℄ and V. Baladi and M. Tsujii [4, 6℄ (see [6℄ forsome histori al remarks) where the authors demonstrate that the Ruelle resonan es arethe dis rete spe trum of the transfer operator in suitably de(cid:28)ned fun tional spa es.From a mathemati al point of view, the es ape of the fun tion ˆ F n ϕ towards highFourier modes we are interested in, is similar to the es ape of a quantum wave fun tiontowards in(cid:28)nity in spa e o urring in open quantum systems. In su h systems, studiedsin e a long time be ause of their relevan e to spontaneous emission of light in atoms [9℄or radioa tive de ay in nu lei, physi ists and mathemati ians have elaborated on eptsand te hniques. In the 70's, J. Aguilar, E. Balslev, J.M. Combes, B. Simon and othersdeveloped a mathemati al theory for quantum resonan es, whi h has been improved afterby many authors in the 80's, in parti ular B. Hel(cid:27)er and J. Sjöstrand [20, 37℄.The aim of this paper is to show that some results of C. Liverani et al. [7, 17, 27, 28℄, V.Baladi et al. [4, 6℄ on erning the de(cid:28)nition and properties of Ruelle resonan es (cid:28)t perfe tlywell within the semi- lassi al approa h of quantum resonan es in phase spa e developed in[20, 37, 38℄. The results we present are not new, but we want to show the relevan e of thesemi- lassi al analysis to the theory of hyperboli dynami al systems.Semi- lassi al analysis (or equivalently mi rolo al analysis) has been developed for thestudy of partial di(cid:27)erential equations in the regime of small wave-length or equivalently,high Fourier mode regime [30, 10, 13℄. As we explained above, the very de(cid:28)nition of hy-perboli dynami s involves high Fourier modes and this implies that semi- lassi al analysisshould be a (cid:16)natural(cid:17) approa h for its understanding. The idea of relevan e of semi- lassi alanalysis in the ontext of hyperboli dynami s has been presented and used in [15℄ in asimpler framework (i.e. real analyti al maps on the torus). In this paper we present thelater approa h in wider generality.In semi- lassi al analysis we distinguish two kinds of operators, the pseudo-di(cid:27)erential3perators (PDO) and the Fourier integral operators (FIO). To ea h PDO ˆ P is asso iateda fun tion P = σ (cid:16) ˆ P (cid:17) on the otangent bundle T ∗ M , alled its symbol. To ea h FIO ˆ F is asso iated a symple ti map F on T ∗ M . In semi- lassi al analysis, we manipulatethe symbols instead of the operators, and powerful theorems trans ribe properties of thesymbols in terms of properties of the operators (for example spe tral properties).In our ontext, the Ruelle transfer operator ˆ F is a FIO whose asso iated symple ti mapdenoted F : T ∗ M → T ∗ M , is the lift of f − (the inverse of the Anosov di(cid:27)eomorphism).This is presented in Se tion 2.In Se tion 3, we study the dynami s of F . It appears that in T ∗ M , the traje tories of F are non ompa t, ex ept for the maximal ompa t invariant subspa e, the se tion ξ = 0 ,and this is related in an essential way to the dis reteness of the spe trum of the operator ˆ F obtained after. We onstru t an (cid:16)es ape fun tion(cid:17) A m on the otangent spa e T ∗ M ,whi h de reases stri tly along the non- ompa t traje tories of F , in a ontrolled manner.Sin e it de reases in the unstable dire tion and in reases in the stable dire tion, the es apefun tion A m belongs to a lass of symbols with variable order, de(cid:28)ned in Appendix A.We de(cid:28)ned an asso iated invertible PDO denoted by ˆ A m . We also de(cid:28)ne the anisotropi Sobolev spa e asso iated to ˆ A m in the standard manner: H m def = ˆ A − m ( L ( M )) .In Se tion 4, we show in Theorem 2 that the operator ˆ F a ting on the anisotropi Sobolev spa e H m has a dis rete spe trum outside an a disk of radius ε m (whi h anbe made arbitrary small). The dis rete spe trum does not depend on the hoi e of A m and de(cid:28)nes the Ruelle resonan es. This is the main result of this paper. This theorem hasalready been obtained by various authors with di(cid:27)erent degrees of generalities [17, 27, 4, 28℄,but the proof we present here is di(cid:27)erent as it uses in a simple way three major Theoremsof semi- lassi al analysis: the (cid:16)Composition Theorem for PDO(cid:17), the (cid:16)Egorov's Theorem fortransport(cid:17) and the (cid:16) L ontinuity Theorem(cid:17).As an appli ation of this approa h, we derive expressions for dynami al orrelationfun tions in Se tion 5. In Se tion 6, we propose a new proof for a theorem of D. Anosovwhi h states that an Anosov di(cid:27)eomorphism preserving a smooth measure is mixing. InSe tion 7, we show that a semi- lassi al trun ation of the operator ˆ F gives the Ruelleresonan e spe trum. This latter result is useful for numeri al omputations.In appendix A, we provide a self- ontained presentation of semi- lassi al results adaptedfor this arti le.The ase of uniformly expanding maps an be onsidered with a similar approa h.However it would be mostly simpli(cid:28)ed by the fa t that the es ape fun tion A m wouldhave onstant order m , and the asso iated Sobolev spa e H m are usual (non anisotropi )Sobolev spa es.A knowledgment: We gratefully a knowledge Mady Smets and (cid:16)Le foyer d'humanismede Peyresq(cid:17) for their ni e hospitality during a workshop where major part of this workhas been made. FF a knowledges support by (cid:16)Agen e Nationale de la Re her he(cid:17) underthe grant JC05_52556. We thank the (cid:16)Classi al and quantum resonan es team(cid:17) NaliniAnantharaman, Viviane Baladi, Yves Colin de Verdière, Colin Guillarmou, Lu Hillairet,4rédéri Naud, Stéphane Nonnenma her and Dominique Spehner for dis ussions related tothis work.2 The model of hyperboli mapLet M be a smooth ompa t onne ted manifold. Let f : M → M be a C ∞ Anosovdi(cid:27)eomorphism. We re all the de(cid:28)nition:De(cid:28)nition 1. (see [23℄ page 263) A di(cid:27)eomorphism f : M → M is Anosov (or uniformlyhyperboli ) if there exists a Riemannian metri g , an f -invariant orthogonal de omposi-tion of T M : T M = E u ⊕ E s (1)and < θ < , su h that for any x ∈ M | D x f ( v s ) | g ≤ θ | v s | g , ∀ v s ∈ E s ( x ) (2) (cid:12)(cid:12) D x f − ( v u ) (cid:12)(cid:12) g ≤ θ | v u | g , ∀ v u ∈ E u ( x ) . This means that E s is the stable foliation and E u the unstable foliation for positive time.Remarks:1. Standard examples are hyperboli automorphisms of the torus T n as well as their C small perturbations, thanks to the stru tural stability theorem (see [23℄ page 266).The problem of lassifying manifolds that admit Anosov di(cid:27)eomorphisms turned outto be very di(cid:30) ult. The only known examples are infranil manifolds (whi h ontainthe torus ase) and it is onje tured that they are the only ones [19, p. 16℄. Here isa simple example of Anosov di(cid:27)eomorphism on ( x, y ) ∈ T = R / Z : f : (cid:18) xy (cid:19) (cid:18) x ′ y ′ (cid:19) = (cid:18) (cid:19) (cid:18) xy (cid:19) + (cid:18) ε π sin (2 π (2 x + y )) (cid:19) (3)with ε small enough2. The Ruelle resonan es of this map are depi ted on (cid:28)gure 6page 30.2. The metri g ( x ) ( alled the Lyapounov metri ) and the distributions E u ( x ) , E s ( x ) ⊂ T x M are in general only Hölder ontinuous with respe t to x ∈ M (See [23℄ hap.19). For the purpose of semi- lassi al analysis, one needs a smooth metri in order2This example preserves area dx ∧ dy . 5o onstru t suitable symbols. In this paper, we will assume that M is endowed witha smooth Riemannian metri g satisfying c g ≤ g ≤ c.g (4)uniformly on M , with ≤ c < θ − . (5)With θ ∗ def = c θ , this implies that < θ ∗ < and for any v s ∈ E s ( x ) one has estimatessimilar to Eq.(2), but with the metri g : | D x f ( v s ) | g ≤ c | D x f ( v s ) | g ≤ cθ | v s | g ≤ c θ | v s | g = θ ∗ | v s | g (6)Similarly for v u ∈ E u ( x ) , | D x f − ( v u ) | g ≤ θ ∗ | v u | g . Unless spe i(cid:28)ed, we will alwayswork with this metri g , whi h an be obtained from g by smoothing3.2.1 Transfer operatorsWe denote by dx = dµ Leb an arbitrary4 smooth density normalized by µ Leb ( M ) = 1 . Letus de(cid:28)ne the bounded operator ˆ F on L ( M ) by5 ˆ F ϕ def = ϕ ◦ f, ϕ ∈ L ( M ) (7) alled the Ruelle transfer operator (or Koopman operator).Let us emphasize that in general f does not preserve any smooth measure, but if f preserves the Lebesgue measure µ Leb then ˆ F is unitary in L ( M ) .Let us remark that the L − adjoint operator ˆ F ∗ is given by (cid:16) ˆ F ∗ ϕ (cid:17) ( y ) = (cid:0) ϕ ◦ f − (cid:1) ( y ) (cid:12)(cid:12) D f − ( y ) f (cid:12)(cid:12) − with ψ, ϕ ∈ C ∞ ( M ) . The adjoint operator ˆ F ∗ also alled the Perron-Frobenius oper-ator is usually onsidered sin e it transports densities [27℄. Our main result, Corollary 1page 19, on erns the spe trum of both ˆ F and ˆ F ∗ .3If however the metri g is already given, one an always ful(cid:28)ll Eq.(5) by taking some positive power f n , n ∈ N , of the Anosov map.4We have hosen a density dx in order to de(cid:28)ne L ( M ) . However this hoi e does not play any rolefor the prin ipal results of this paper.5It would have been more natural to onsider ˆ F ϕ def = ϕ ◦ f − instead, but our hoi e here follows thepaper [15℄ where we onsidered expanding maps whi h are not invertible.6ther types of transfer operators. In the ontext of thermodynami formalism ofdynami al systems introdu ed by D. Ruelle et al. (see [43℄ hap. 4, [24℄ hap. 6), a moregeneral lass of transfer operators than Eq.(7) is onsidered and de(cid:28)ned as follow. Let V ∈ C ∞ ( M ) be a smooth (real or omplex) valued fun tion alled the potential and let ˆ F V : C ∞ ( M ) → C ∞ ( M ) be de(cid:28)ned by (cid:16) ˆ F V ϕ (cid:17) ( x ) := ϕ ( f ( x )) e V ( x ) (8)Compared to the simplest ase V = 0 , given in Eq.(7), the new term e V is a pseu-dodi(cid:27)erential operator of order (as de(cid:28)ned in the subsequent se tions and in AppendixA). Consequently the anoni al map F : T ∗ M → T ∗ M asso iated to ˆ F V , Eq.(11), is un- hanged. This implies that our main results, Theorem 1 page 17, Corollary 1 page 19 andtheir proof, are the same6 for this new operator ˆ F V .There is even a slightly more general lass of transfer operators a ting on se tions of linebundles7, for whi h our results work as well. Sin e these transfer operators have interesting onne tions with quantum haos and geometri quantization (see [14℄) we mention them.A general de(cid:28)nition pro eeds as follow.De(cid:28)nition 2. Let L → M be a smooth omplex line bundle over M . A transfer op-erator ˆ F asso iated to the smooth di(cid:27)eomorphism f : M → M is a linear map a tingon smooth se tions, ˆ F : C ∞ ( M ; L ) → C ∞ ( M ; L ) , su h that for any smooth fun tion ψ ∈ C ∞ ( M ) and any smooth se tion s ∈ C ∞ ( M ; L ) one has (cid:16) ˆ F ( ψs ) (cid:17) = ( ψ ◦ f ) . (cid:16) ˆ F s (cid:17) (9)One also requires that for any x ∈ M and s ∈ C ∞ ( M ; L ) , ( s ◦ f ) ( x ) = 0 ⇒ (cid:16) ˆ F s (cid:17) ( x ) = 0 (10)This de(cid:28)nition of transfer operator generalizes Eq.(8), sin e in the ase where L is atrivial line bundle, se tions are identi(cid:28)ed with omplex fun tions thanks to a global nonvanishing se tion r ∈ C ∞ ( M ; L ) : any global se tion s ∈ C ∞ ( M ; L ) an be written s = ϕr with ϕ ∈ C ∞ ( M ) . Let e V ∈ C ∞ ( M ) be de(cid:28)ned by ˆ F ( r ) = e V r whi h is possible from(10). Then (9) gives ˆ F ( ϕr ) = ( ϕ ◦ f ) e V r whi h is equivalent to (8).In this paper we will only onsider the simpler expression Eq.(7).6Of ourse the spe trum of ˆ F V depends on V and the value is in general no more an eigenvalue of ˆ F V . Corollary 2 and 3 page 23 are spe i(cid:28) to ˆ F V =0 .7This works also for ve tor bundles. 7 T ∗ x ′ MT ∗ x M x ′ = f − ( x ) f − x M E ∗ u E ∗ s ξ F ( ξ ) Figure 1: Dynami s of F de(cid:28)ned in Eq.(11), on the otangent spa e T ∗ M .2.2 Dynami s on the otangent spa eIn order to study the spe trum of the operator ˆ F using semi- lassi al analysis later on, weneed to onsider the dynami s indu ed by f in the otangent bundle F : T ∗ M → T ∗ M ,namely the lift of f − . See Figure 1. For any x ∈ M , let x ′ = f − ( x ) , and de(cid:28)ne F : T ∗ x M → T ∗ x ′ Mξ ( D x ′ f ) t ξ (11)In semi- lassi al analysis, the map F is pre isely the anoni al map asso iated to theoperator ˆ F de(cid:28)ned in Eq.(7). This appears in Egorov's Theorem 9. It is the lift of f − rather than f be ause the support of ˆ F ϕ = ϕ ◦ f is the support of ϕ transported by f − .Noti e that the zero se tion ξ = 0 is a ompa t set, invariant by the dynami s and its omplement ontains only unbounded traje tories. This observation is at the origin of themethod whi h leads to the quasi- ompa ity result obtained in the main Theorem 1.Let T ∗ M = E ∗ s ⊕ E ∗ u be the de omposition dual to Eq.(1), i.e. E ∗ s ( E u ) = 0 and E ∗ u ( E s ) =0 .Lemma 1. The de omposition T ∗ M = E ∗ s ⊕ E ∗ u is invariant by F and: | F ( ξ s ) | ≤ θ ∗ | ξ s | ∀ ξ s ∈ E ∗ s | F − ( ξ u ) | ≤ θ ∗ | ξ u | ∀ ξ u ∈ E ∗ u . (12)with θ ∗ = c θ, < θ ∗ < , with c from Eq. (4).8roof. The distribution E ∗ s is invariant by F be ause for any ξ s ∈ E ∗ s , v u ∈ E u , one has F ( ξ s ) ( v u ) = (cid:0) ( Df ) t ξ s (cid:1) ( v u ) = ξ s ( Df ( v u )) = 0 sin e E u is invariant by Df . The same holdsfor E ∗ u . On the other hand, one gets easily onvin ed that Eq. (4) implies the same inequalitiesfor the metri on the dual spa e. Eq.(2) implies that on the dual spa e, for any ξ s ∈ E ∗ s ( x ) , | F ( ξ s ) | g ≤ θ | ξ s | g . Then for any ξ s ∈ E ∗ s ( x ) one has | F ( ξ s ) | g ≤ c | F ( ξ s ) | g ≤ cθ | ξ s | g ≤ c θ | ξ s | g and similarly for (cid:12)(cid:12) F − ( ξ u ) (cid:12)(cid:12) g ≤ c θ | ξ u | , ∀ ξ u ∈ E ∗ u .3 The es ape fun tion and the anisotropi Sobolev spa es3.1 Constru tion of the es ape fun tion A m and the pseudodi(cid:27)er-ential operator ˆ A m In this se tion, we onstru t a fun tion A m on the otangent spa e whi h de reases alongall the unbounded traje tories of F pi tured in Figure 1. It is alled an es ape fun tion.In order to apply semi- lassi al theorems later on, we make sure that A m is a suitablesymbol. This will allow us to onstru t a pseudodi(cid:27)erential operator ˆ A m from the symbol A m . It turns out that an es ape fun tion A m ( x, ξ ) suitable for our purposes must have anorder m in h ξ i whi h depends itself of the dire tion ξ/ | ξ | . This gives rise to general lassesof symbols S m ( x,ξ ) ρ and PDO's Ψ m ( x,ξ ) ρ with variable order m ( x, ξ ) . Their de(cid:28)nitions andmain properties are summarized in Appendix A.Lemma 2. Let u < < s . There exists a order fun tion m ( x, ξ ) ∈ S taking values in [ u, s ] , with the following properties. For any (cid:28)xed x and | ξ | > , m ( x, ξ ) depends onlyon the dire tion ˜ ξ = ξ/ | ξ | of the otangent ve tor. Moreover m ( x, ξ ) = s (resp. u ) in avi inity of the stable dire tion E ∗ s ( x ) (resp. unstable dire tion E ∗ u ( x ) ). See (cid:28)gure 2. Thefun tion m ( x, ξ ) de reases with respe t to the map F : ∃ R > , ∀ | ξ | ≥ R, ( m ◦ F ) ( x, ξ ) − m ( x, ξ ) ≤ , (13)De(cid:28)ne A m ( x, ξ ) def = h ξ i m ( x,ξ ) , h ξ i = q | ξ | . (14)whi h belongs to the lass S m ( x,ξ ) ρ , for any ≤ ρ < . The main property of the symbol A m is: ∃ R > , ∀ | ξ | ≥ R, ( A m ◦ F ) ( x, ξ ) A m ( x, ξ ) ≤ e − ca < (15)with a = min ( − u, s ) and c > independent of the hoi e of u, s .9emarks:ˆ Eq.(15) means that the fun tion A m de reases stri tly and uniformly along the tra-je tories of F in the otangent spa e. We all A m an es ape fun tion.ˆ The onstan y of m in the vi inity of the stable/unstable dire tion allows us tohave a smooth order fun tion m despite the foliations E ∗ s ( x ) , E ∗ u ( x ) have only Hölderregularity.ˆ Inspe tion of the proof shows that c an be hosen arbitrary lose to log (cid:16) θ ∗ (cid:17) .The real symbol A m an be quantized into a pseudodi(cid:27)erential operator ˆ A m of variableorder m ( x, ξ ) , a ording to Eq.(57). Then Corollary 4 and Example 1 tell us that we anmodify the symbol A m at a subleading order (i.e. S m ( x,ξ ) − (2 ρ − ρ ) su h that the operator an be assumed to be formally self-adjoint and invertible on C ∞ ( M ) .Proof of Lemma 2.The fun tion m . Sin e the lifted map F de(cid:28)ned in Eq.(11) is linear in ξ , it de(cid:28)nesa map ˜ F on the osphere bundle S ∗ M = ( T ∗ M \ { } ) / R + , namely the spa e of dire tions ˜ ξ := ξ/ | ξ | , whi h is a ompa t spa e. See Figure 2. The image of E ∗ u , E ∗ s ⊂ T ∗ M by the proje tion T ∗ M \ { } → S ∗ M are denoted respe tively ˜ E ∗ u , ˜ E ∗ s ⊂ S ∗ M . Eq.(12) implies that ˜ E ∗ u is auniform attra tor for ˜ F , and ˜ E ∗ s is a uniform repeller, i.e. ˜ F n (cid:16) ˜ ξ (cid:17) onverges to ˜ E ∗ u (respe t. ˜ E ∗ s ) when n → + ∞ (respe t. n → −∞ ).Let u < < s . Let m ∈ C ∞ ( S ∗ M ; [ u, s ]) with m = s > in a neighborhood ˜ N s of ˜ E ∗ s and m = u < in a neighborhood ˜ N u of ˜ E ∗ u . We also assume that ( ˜ ξ ∈ ˜ N s ⇒ ˜ F − (cid:16) ˜ ξ (cid:17) ∈ ˜ N s ) and ( ˜ ξ ∈ ˜ N u ⇒ ˜ F (cid:16) ˜ ξ (cid:17) ∈ ˜ N u ) (16)See Figure 3.Let N ∈ N and de(cid:28)ne ˜ m ∈ C ∞ ( S ∗ M ; [ u, s ]) by ˜ m := 12 N N − X n = − N m ◦ ˜ F n (17)Then ˜ m ◦ ˜ F − ˜ m = 12 N (cid:16) m ◦ ˜ F N − m ◦ ˜ F − N (cid:17) (18)We will show now that ∀ ˜ ξ ∈ S ∗ M ˜ m (cid:16) ˜ F (cid:16) ˜ ξ (cid:17)(cid:17) − ˜ m (cid:16) ˜ ξ (cid:17) ≤ . (19)10 ∗ ME ∗ u FE ∗ s ˜ ξ ˜ Fξ Figure 2: The map F on the otangent spa e T ∗ M and the indu ed map ˜ F on the ospherebundle S ∗ M . ˜ F ˜ F − ˜ N s ˜ N u ˜ F − N ˜ F N ˜ E ∗ u ˜ E ∗ s ˜ ξN s N u Figure 3: The horizontal axis is a s hemati pi ture of S ∗ M and this shows the onstru tionand properties of the sets N s and N u . 11et N s := S ∗ M \ ˜ F − N (cid:16) ˜ N u (cid:17) and N u := S ∗ M \ ˜ F N (cid:16) ˜ N s (cid:17) . Then m (cid:16) ˜ F N (cid:16) ˜ ξ (cid:17)(cid:17) = u for ˜ ξ / ∈ N s , and similarly m (cid:16) ˜ F − N (cid:16) ˜ ξ (cid:17)(cid:17) = s for ˜ ξ / ∈ N u .For N large enough one has N s ⊂ ˜ N s , N u ⊂ ˜ N u and N s ∩ N u = ∅ . Therefore,ˆ if ˜ ξ ∈ N s then ˜ ξ / ∈ N u and m (cid:16) ˜ F − N (cid:16) ˜ ξ (cid:17)(cid:17) = s ≥ m (cid:16) ˜ F N (cid:16) ˜ ξ (cid:17)(cid:17) and (18) gives ˜ m (cid:16) ˜ F (cid:16) ˜ ξ (cid:17)(cid:17) − ˜ m (cid:16) ˜ ξ (cid:17) ≤ .ˆ Similarly if ˜ ξ ∈ N u then ˜ ξ / ∈ N s and m (cid:16) ˜ F N (cid:16) ˜ ξ (cid:17)(cid:17) = u ≤ m (cid:16) ˜ F − N (cid:16) ˜ ξ (cid:17)(cid:17) thus ˜ m (cid:16) ˜ F (cid:16) ˜ ξ (cid:17)(cid:17) − ˜ m (cid:16) ˜ ξ (cid:17) ≤ .ˆ Finally if ˜ ξ / ∈ ( N u ∪ N s ) then m (cid:16) ˜ F N (cid:16) ˜ ξ (cid:17)(cid:17) − m (cid:16) ˜ F − N (cid:16) ˜ ξ (cid:17)(cid:17) = ( u − s ) < andtherefore ∀ ˜ ξ / ∈ ( N u ∪ N s ) ˜ m (cid:16) ˜ F (cid:16) ˜ ξ (cid:17)(cid:17) − ˜ m (cid:16) ˜ ξ (cid:17) = 12 N ( u − s ) < (20)We have shown Eq.(19).We onstru t a smooth fun tion m on T ∗ M satisfying m ( x, ξ ) = ˜ m (cid:16) ˜ ξ (cid:17) , if | ξ | > , = 0 if | ξ | < / Then (19) implies Eq.(13).From Eq.(16) one dedu es that the set S (cid:16) ˜ ξ (cid:17) := n n ∈ Z / ˜ F n (cid:16) ˜ ξ (cid:17) / ∈ ( N s ∪ N u ) o is onne ted. Moreover the ardinal of this set is uniformly bounded: ∃N ∈ N ∀ ˜ ξ ∈ S ∗ M ♯ S (cid:16) ˜ ξ (cid:17) ≤ N From this we dedu e thatˆ if ˜ ξ ∈ N s then ˜ F N (cid:16) ˜ ξ (cid:17) / ∈ ˜ N u but also ˜ F n (cid:16) ˜ ξ (cid:17) / ∈ ˜ N u for n ≤ N and even ˜ F n (cid:16) ˜ ξ (cid:17) ∈ ˜ N s for n ≤ N − N . Thus (17) gives ˜ m (cid:16) ˜ F (cid:16) ˜ ξ (cid:17)(cid:17) ≥ (cid:18) − N + 12 N (cid:19) s + N + 12 N u ≥ s (21)where the last inequality holds for N large enough.ˆ If ˜ ξ ∈ N u one shows similarly that ˜ m (cid:16) ˜ ξ (cid:17) ≤ (cid:18) − N N (cid:19) u + N N ≤ u (22)where the last inequality holds for N large enough.12he symbol A m . Let A m ( x, ξ ) := h ξ i m ( x,ξ ) with h ξ i = q | ξ | . A m belongs to the lass S m ( x,ξ ) ρ , for any ≤ ρ < from Lemma 6.We will show now the uniform es ape estimate Eq.(15).For | ξ | large enough one has log A m ( x, ξ ) = ˜ m (cid:16) ˜ ξ (cid:17) ln h ξ i , log ( A m ◦ F ) ( x, ξ ) = ˜ m (cid:16) ˜ F (cid:16) ˜ ξ (cid:17)(cid:17) ln h F ( ξ ) i ˆ If ˜ ξ / ∈ ( N s ∪ N u ) then (20) gives ˜ m (cid:16) ˜ F (cid:16) ˜ ξ (cid:17)(cid:17) = ˜ m (cid:16) ˜ ξ (cid:17) + N ( u − s ) . One also has log h F ( ξ ) i = log h ξ i + O (1) . Therefore ˜ m (cid:16) ˜ F (cid:16) ˜ ξ (cid:17)(cid:17) ln h F ( ξ ) i − ˜ m (cid:16) ˜ ξ (cid:17) ln h ξ i = (cid:18) ˜ m (cid:16) ˜ ξ (cid:17) + 12 N ( u − s ) (cid:19) (log h ξ i + O (1)) − ˜ m (cid:16) ˜ ξ (cid:17) ln h ξ i = ˜ m (cid:16) ˜ ξ (cid:17) O (1) + 12 N ( u − s ) log h ξ i≤ − c min ( s, − u ) with c > and if | ξ | is large enough.ˆ If ˜ ξ ∈ N s (neighborhood of the stable dire tion) then | F ( ξ ) | ≤ C ′ | ξ | , with C ′ > ,so < ln h F ( ξ ) i ≤ ln h ξ i − ln C , with C > ( lose to C ′ ). And using (21),(19) ˜ m (cid:16) ˜ F (cid:16) ˜ ξ (cid:17)(cid:17) ln h F ( ξ ) i ≤ ˜ m (cid:16) ˜ F (cid:16) ˜ ξ (cid:17)(cid:17) (ln h ξ i − ln C ) ≤ ˜ m (cid:16) ˜ ξ (cid:17) ln h ξ i − s C ˆ If ˜ ξ ∈ N u (neighborhood of the unstable dire tion) then | F ( ξ ) | ≥ C ′ | ξ | , with C ′ > ,so ln h F ( ξ ) i ≥ ln h ξ i + ln C with C > ( lose to C ′ ). And using (21),(19), ˜ m (cid:16) ˜ F (cid:16) ˜ ξ (cid:17)(cid:17) ln h F ( ξ ) i ≤ ˜ m (cid:16) ˜ ξ (cid:17) (ln h ξ i + ln C ) ≤ ˜ m (cid:16) ˜ ξ (cid:17) ln h ξ i + 12 u ln C In on lusion, for any x and | ξ | large enough, there exists c > independent of u, s su hthat log ( A m ◦ F ) ( x, ξ ) ≤ log A m ( x, ξ ) − c min ( s, − u ) We have obtained the uniform es ape estimate Eq.(15) and (cid:28)nished the proof of Lemma2. 13.2 The anisotropi Sobolev spa esA parti ular feature of the self-adjoint and invertible PDO ˆ A m ∈ Ψ m ( x,ξ ) ρ introdu ed aboveis that its symbol A m has a non-isotropi behavior with respe t to ξ ∈ T ∗ x M . It is aPDO with maximum order a = min ( | u | , s ) , but with variable order m ( x, ξ ) ∈ [ u, s ] , with u < < s . For large | ξ | , the symbol A m in reases in the stable dire tion ξ ∈ E ∗ s ( x ) as A m ( ξ ) ∼ | ξ | s and de reases in the unstable dire tion ξ ∈ E ∗ u ( x ) as A m ( ξ ) ∼ / | ξ | | u | . Inthis se tion we onsider a slightly more general spa e of fun tions m and do not requireany relation with the dynami s of F : we just assume that m is in C ∞ ( T ∗ M ) and is afun tion of (cid:16) x, ξ | ξ | (cid:17) for | ξ | large enough. Therefore m ∈ S is an order fun tion.We de(cid:28)ne the anisotropi Sobolev spa e to be the spa e of distributions (in ludedin D ′ ( M ) ): H m def = ˆ A − m (cid:0) L ( M ) (cid:1) (23)Remarks:ˆ This de(cid:28)nition is very similar to the standard de(cid:28)nition of Sobolev spa es on R d ([39℄p.271): H s def = Op ( h ξ i s ) − (cid:0) L (cid:0) R d (cid:1)(cid:1) ex ept for anisotropy with respe t to ξ . Equivalent de(cid:28)nitions of anisotropi Sobolevspa es have been given in [25℄ and also by V. Baladi et M. Tsujii in [6℄ for the spe i(cid:28) purpose of hyperboli dynami s.ˆ Noti e that ϕ ∈ H m ⇔ ˆ A m ϕ ∈ L ( M ) , so roughly speaking, in the ase of thefun tion m de(cid:28)ned in Lemma 2, it means that the Fourier transform ˆ ϕ ( ξ ) performedin a vi inity of x ∈ M , in reases less than | ξ s | − s − d/ for ξ s ∈ E ∗ s ( x ) and less than | ξ u | − u − d/ for ξ u ∈ E ∗ u ( x ) , with d = dim ( M ) . We an say that if u < < s , then ϕ is regular in the stable dire tion and irregular in the unstable dire tion.Some simple properties of the anisotropi Sobolev spa es H m :1. H m is a Hilbert spa e with the s alar produ t ( ϕ , ϕ ) H m def = (cid:16) ˆ A m ϕ , ˆ A m ϕ (cid:17) L ( M ) , ϕ , ϕ ∈ H m and the map ˆ A m : ( H m , ( ., . ) H m ) → (cid:0) L ( M ) , ( ., . ) L (cid:1) (24)is unitary.2. On L ( M ) , Dom (cid:16) ˆ A m (cid:17) = H m ∩ L ( M ) and Dom (cid:16) ˆ A − m (cid:17) = H − m ∩ L ( M ) .14. There are embedding relations as for usual Sobolev spa es. First H max( m ) ⊂ H m ⊂ H min( m ) (25)If m ′ ≥ m then H m ′ ⊂ H m (26)and H m ′ is dense in H m .4. If ϕ ∈ H m and g ∈ C ∞ ( M ) then gϕ ∈ H m (27)and moreover, the map ϕ → gϕ is ontinuous H m → H m .5. Let H − m def = ˆ A m (cid:0) L ( M ) (cid:1) . The spa es H m and H − m are dual in the following sense: if ψ ∈ H m , ϕ ∈ H − m , wenote ψ ( ϕ ) = ϕ (cid:0) ψ (cid:1) = ( ψ, ϕ ) H m × H − m def = (cid:16) ˆ A m ψ, ˆ A − m ϕ (cid:17) L ( M ) . (28)Then (cid:12)(cid:12) ( ψ, ϕ ) H m × H − m (cid:12)(cid:12) ≤ k ψ k H m k ϕ k H − m . (29)6. If ψ ∈ H m ∩ L ( M ) and ϕ ∈ H − m ∩ L ( M ) then ( ψ, ϕ ) H m × H − m = ( ψ, ϕ ) L ( M ) (30)Sin e the dual bra ket oin ides with the L s alar produ t, we will drop the indi esin the sequel of the paper, and write: ( ψ, ϕ ) def = ( ψ, ϕ ) H m ,H − m
7. If ψ ∈ H m , ϕ ∈ H − m and g ∈ C ∞ , then: ( gψ, ϕ ) = ( ψ, gϕ ) (31)Proof. The proofs of properties 1 to 4 follow dire tly from those of ˆ A m .5. | ( ψ, ϕ ) | = (cid:12)(cid:12)(cid:12)(cid:12)(cid:16) ˆ A m ψ, ˆ A − m ϕ (cid:17) L ( M ) (cid:12)(cid:12)(cid:12)(cid:12) ≤ (cid:13)(cid:13)(cid:13) ˆ A m ψ (cid:13)(cid:13)(cid:13) L (cid:13)(cid:13)(cid:13) ˆ A − m ϕ (cid:13)(cid:13)(cid:13) L = k ψ k H m k ϕ k H − m .6. ( ψ, ϕ ) = (cid:16) ˆ A m ψ, ˆ A − m ϕ (cid:17) L ( M ) = ( ψ, ϕ ) L ( M ) by self adjointness of ˆ A m .7. Let M g denotes multipli ation g ∈ C ∞ . The operator ˆ B g = ˆ A m M g ˆ A − m is bounded in L ( M ) . Moreover, one has ˆ B ∗ g = ˆ A − m M g ˆ A m sin e ˆ A m is self-adjoint. We dedu e that ( gψ, ϕ ) = (cid:16)(cid:16) ˆ A m M g ˆ A − m (cid:17) ˆ A m ψ, ˆ A − m ϕ (cid:17) L = (cid:16) ˆ A m ψ, (cid:16) ˆ A − m M g ˆ A m (cid:17) ˆ A − m ϕ (cid:17) L = ( ψ, gϕ ) . H − s v ,j v ,j H m H H − m H s C ∞ w ,j w ,j Figure 4: S hemati representation of the anisotropi Sobolev spa es H m and their embed-ding relations Eq.(25) with the usual Sobolev spa es H s , s ∈ R . The eigen-distributions v i,j , w i,j of the operator ˆ F whi h appear in Corollary 1 are also represented.4 Spe trum of resonan esWe give now the main result of this paper. Its proof relies on semi- lassi al analysis and isinspired by the study of resonan es in open quantum systems [20, 21℄. It is also inspiredfrom a previous work [15℄ performed within a simple and illuminating model, namelyanalyti al hyperboli map on the torus. In some sense it shows a lose analogy betweenRuelle resonan es and quantum resonan es. The essential point of this approa h is to (cid:28)ndthe dis rete spe trum of resonan es of the operator ˆ F in the Sobolev spa e of distributions H m , thanks to a onjuga y by the es ape operator ˆ A m de(cid:28)ned in Se tion 3.1. First observethat the operator ˆ F de(cid:28)ned in Eq.(7) extends by duality to the distribution spa e D ′ ( M ) by ˆ F ( α ) ( ϕ ) = α (cid:16) ˆ F ∗ ( ϕ ) (cid:17) where α ∈ D ′ ( M ) , ϕ ∈ C ∞ ( M ) and ˆ F ∗ is the L − adjoint operator. One he ks easilythat for ψ, ϕ ∈ L ( M ) , ˆ F ∗ is given by (cid:16) ˆ F ∗ ϕ (cid:17) ( y ) = ( ϕ ◦ f − ) ( y ) (cid:12)(cid:12) D f − ( y ) f (cid:12)(cid:12) − .16heorem 1. Let m be a fun tion whi h satis(cid:28)es the hypothesis of Lemma 2. ˆ F leavesthe anisotropi Sobolev spa e H m globally invariant. The operator ˆ F : H m → H m is a bounded operator and an be written ˆ F = ˆ r m + ˆ k m (32)where ˆ k m is a ompa t operator and k ˆ r m k ≤ ε m = (cid:13)(cid:13)(cid:13) ˆ F (cid:13)(cid:13)(cid:13) L e − ca with onstants c, a > de(cid:28)ned in Lemma 2. Consequently, the essential spe tral radius is smaller than ε m , whi hmeans that ˆ F has a dis rete spe trum λ i outside the ir le of radius ε m .Remarks :ˆ (cid:13)(cid:13)(cid:13) ˆ F (cid:13)(cid:13)(cid:13) L depends on the hoi e of the density dx . We ould have (cid:13)(cid:13)(cid:13) ˆ F (cid:13)(cid:13)(cid:13) L loser to byanother hoi e of dx .ˆ Noti e that is an eigenvalue of ˆ F , with onstant eigenfun tion. We will see inCorollary 2 that the spe tral radius of ˆ F is one, i.e. that there are no eigenvaluesoutside the unit ir le.ˆ For future purpose, let ε > and O ε denotes the set of order fun tions m whi hsatisfy the hypothesis of Lemma 2, and su h that ε m = (cid:13)(cid:13)(cid:13) ˆ F (cid:13)(cid:13)(cid:13) L e − ca < ε : O ε def = n m / ε m = (cid:13)(cid:13)(cid:13) ˆ F (cid:13)(cid:13)(cid:13) L e − ca < ε o (33)The set O ε is non empty.Proof. We use the unitary map between H m and L ( M ) given in Eq.(23), and onsider ˆ Q m def =ˆ A m ˆ F ˆ A − m : L ( M ) → L ( M ) , de(cid:28)ned on a dense domain, and whi h is unitary equivalent to ˆ F : H m → H m : L ( M ) ˆ Q m → L ( M ) ↓ ˆ A − m (cid:9) ↓ ˆ A − m H m ˆ F → H m Instead of working with ˆ Q m dire tly, it is more onvenient to onsider ˆ P m def = ˆ F − ˆ Q m = (cid:16) ˆ F − ˆ A m ˆ F (cid:17) ˆ A − m .
17t follows from Egorov's Theorem 9, that the produ t ˆ F − ˆ A m ˆ F is a PDO in Ψ m ◦ F ( x,ξ ) ρ whosesymbol is A m ◦ F modulo subleading terms in S m ◦ F ( x,ξ ) − (2 ρ − ρ . On the other hand, the ompositionTheorem 8 for PDO tells that ˆ P m is a PDO in Ψ m ◦ F ( x,ξ ) − m ( x,ξ ) ρ whose symbol is P m = A m ◦ FA m modulo subleading orre tions in S m ◦ F ( x,ξ ) − m ( x,ξ ) − (2 ρ − ρ . From the onstru tion of the es apefun tion A m , Eq.(13) insures that ˆ P m ∈ Ψ ρ . On the other hand Eq.(15) gives lim sup P m ≤ e − a.c . This allows us to apply the Lemma 14 of L - ontinuity and obtain that for any ε > , ˆ P m de omposes as ˆ P m = ˆ p ε + ˆ k ε with ˆ k ε ∈ Ψ −∞ a smoothing operator and k ˆ p ε k ≤ e − a.c + ε . Finally, we multiply on the left by ˆ F to obtain ˆ Q m = ˆ F ˆ p ε + ˆ F ˆ k ε . The se ond term is smoothing, hen e ompa t, while the (cid:28)rst one has an operator norm boundedby ( e − ac + ε ) (cid:13)(cid:13)(cid:13) ˆ F (cid:13)(cid:13)(cid:13) = Ce − ac , with any C > (cid:13)(cid:13)(cid:13) ˆ F (cid:13)(cid:13)(cid:13) and the hoi e ε = e − ac (cid:18) C − k ˆ F kk ˆ F k (cid:19) . We haveshown the laimed spe tral results for ˆ Q m : L ( M ) → L ( M ) and therefore for ˆ F : H m → H m .18orollary 1. Let ε > and let m ∈ O ε be an order fun tion as de(cid:28)ned in Eq.(33). If wedenote by π the spe tral proje tor asso iated to ˆ F : H m → H m outside the disk of radius ε , and ˆ K def = ˆ π ˆ F , ˆ R def = (1 − ˆ π ) ˆ F , then we have a spe tral de omposition ˆ F = ˆ K + ˆ R, ˆ K ˆ R = ˆ R ˆ K = 0 (34)and1. The spe tral radius of ˆ R is smaller than ε .2. ˆ K has (cid:28)nite rank. Its spe trum has generalized eigenvalues λ i ( ounting multipli ity) alled the Ruelle resonan es, with ε < | λ i | . The general Jordan de omposition of ˆ K an be written ˆ K = X i ≥ , | λ i | >ε λ i d i X j =1 v i,j ⊗ w i,j + d i − X j =1 v i,j ⊗ w i,j +1 ! (35)with d i the dimension of the Jordan blo k asso iated to the eigenvalue λ i , with v i,j ∈ H m , w i,j ∈ H − m ( w i,j is viewed as a linear form on H m with the duality Eq.(28)).They satisfy w i,j ( v k,l ) = δ ik δ jl .3. The distributions v i,j , w i,j and the orresponding eigenvalues λ i do not depend onthe hoi e of m , but are intrinsi to the operator ˆ F : v i,j ∈ \ m ∈O| λi | H m , w i,j ∈ \ m ∈O| λi | H − m . (36)In other words, v i,j are smooth in every dire tion ex ept in the unstable dire tionwhi h ontains their wave front ([40℄ p.27): W F ( v i,j ) ⊂ E ∗ u . Similarly w i,j are smooth ex ept in the stable dire tion whi h ontains their wavefront: W F ( w i,j ) ⊂ E ∗ s . The resolvent (cid:16) z − ˆ F (cid:17) − has a meromorphi extension from C ∞ ( M ) to D ′ ( M ) ,whose poles are the λ i .Proof. Points 1 and 2 are immediate onsequen es of Theorem 1. The proje tor π an be obtained19rom an integral of the resolvent ˆ R ( z ) = (cid:16) z − ˆ F (cid:17) − : H m → H m on a ir ular ontour of radius ε . We will prove now point 3 namely that the spe trum and eigen-distribution do not depend on m . Let ε > , and m, m ′ ∈ O ε (de(cid:28)ned in Eq.(33)), and suppose (cid:28)rst that m ′ ≥ m . From Eq.(26),one has H m ′ ⊂ H m . Let ˆ F m (resp. ˆ F m ′ ) denotes the restri tion of ˆ F to the distribution spa e H m (resp. H m ′ ). From Theorem , both ˆ F m and ˆ F m ′ are bounded operators and have essentialspe trum radius less than ε . For | z | large enough, the resolvents of ˆ F m and ˆ F m ′ are equal on H m ′ be ause one an write ˆ R m ′ ( z ) = (cid:16) z − ˆ F m ′ (cid:17) − = 1 z ∞ X n =1 ˆ F m ′ z ! n ! = ˆ R m ( z ) sin e ˆ F m ′ = ˆ F m on H m ′ and the sum is onvergent. By meromorphi ontinuation, the resolventsalso oin ide for | z | > ε . By expli it ontour integral of the resolvent on a ir le of radius ε , onededu es that the orresponding (cid:28)nite rank operators ˆ K m ′ and ˆ K m are equal on H m ′ . But sin e H m ′ is dense in H m , one dedu es that there are no eigenspa es of ˆ K m outside H m ′ . Thereforethe eigen-distributions v i,j belongs to H m ′ .Now let m, m ′′ ∈ O ε be any two order fun tions (with no mutual in lusion). From the expli it onstru tion given in the proof of Lemma 2, one an (cid:28)nd m ′ ∈ O ε , su h that m ′ ≥ m and m ′ ≥ m ′′ .Then the above argument shows that v i,j ∈ H m ′ ⊂ (cid:16) H m ∩ H m ′′ (cid:17) .Similar (but dual) arguments for the operator ˆ F ∗ : H − m → H − m show that its eigenve tors w i,j are in (cid:16) H − m ∩ H − m ′′ (cid:17) . We have obtained Eq.(36).Sin e C ∞ ⊂ H m for any ε > , and any m ∈ O ε , and H m ⊂ D ′ ( M ) , we dedu e that (cid:16) z − ˆ F (cid:17) − : C ∞ ( M ) → D ′ ( M ) admits a meromorphi extension on C \ { } .5 Asymptoti expansion for dynami al orrelation fun -tionsOne usually obtains mu h information on ˆ F and the dynami al system f through the studyof dynami al orrelation fun tions.De(cid:28)nition 3. For ψ , ψ ∈ L ( M ) and n ∈ Z , de(cid:28)ne the Lebesgue dynami al orre-lation fun tion C Lebψ ,ψ ( n ) def = (cid:16) ψ , ˆ F n ψ (cid:17) = Z ψ ( x ) ψ ( f n ( x )) dµ Leb (37)20e(cid:28)nition 4. The di(cid:27)eomorphism f is alled Lebesgue-mixing if there exists an in-variant measure8 µ srb alled the Sinai-Ruelle-Bowen (SRB) measure su h that forany ψ , ψ ∈ C ∞ ( M ) , C Lebψ ,ψ ( n ) −→ n →∞ (cid:18)Z ψ dµ Leb (cid:19) (cid:18)Z ψ dµ srb (cid:19) (38)De(cid:28)nition 5. For ψ , ψ ∈ L ( M ) and n ∈ Z , de(cid:28)ne the SRB dynami al orrelationfun tion C srbψ ,ψ ( n ) def = Z ψ ( x ) ψ ( f n ( x )) dµ srb (39)We show in Theorem 3 that Lebesgue-mixing implies SRB-mixing, i.e. C srbψ ,ψ ( n ) −→ n →∞ (cid:18)Z ψ dµ srb (cid:19) (cid:18)Z ψ dµ srb (cid:19) (40)whi h is also the usual de(cid:28)nition of mixing.Let us mention the following onje ture ([23℄ p. 575, foot-note (2), or [32℄ p. 7.)Conje ture 1. A smooth Anosov di(cid:27)eomorphism f : M → M on a onne ted ompa tmanifold M is Lebesgue-mixing.In the parti ular ase where f preserves a smooth measure dx (so ˆ F is unitary on L ( M ) ), this has been proved by Anosov in his PhD-thesis [1℄ (see [8℄ Theorem 6.3.1).In Se tion 6 we provide a di(cid:27)erent proof entirely based on the semi- lassi al approa hdeveloped in this paper.We will assume Lebesgue-mixing in Theorem 3.Theorems 2 and 3 below have been obtained before with various degrees of generality.5.1 The Lebesgue orrelation fun tionThe Ruelle resonan es λ i and asso iated distributions v i,j , w i,j have been de(cid:28)ned in Corol-lary 1. 21heorem 2. For any ψ , ψ ∈ C ∞ ( M ) , ε > su h that ε = | λ i | , ∀ i , and n ≥ , one has C Lebψ ,ψ ( n ) = X i ≥ , | λ i | >ǫ min ( n,d i − X k =0 C kn λ n − ki d i − k X j =1 v i,j (cid:0) ψ (cid:1) w i,j + k ( ψ ) + k ψ k H m k ψ k H − m O ε ( ε n ) . (41)with any m ∈ O ε (de(cid:28)ned in Eq.(33)) and C kn := n !( n − k )! k ! .Remarks:ˆ More generally Eq.(41) still holds for ψ ∈ H m and ψ ∈ H − m , with m ∈ O ε .ˆ The right hand side of Eq.(41) is ompli ated by the possible presen e of (cid:16)Jordanblo ks(cid:17). In the ase where the spe trum λ i is simple ( λ i = λ j ) it reads more simply C Lebψ ,ψ ( n ) = X i ≥ , | λ i | >ǫ λ ni v i (cid:0) ψ (cid:1) w i ( ψ ) + O ε ( ε n ) . Proof. Theorem 2 is dedu ed from Corollary 1. For any ε > , let m ∈ O ε . For any n ≥ wehave ˆ F n = ˆ K n + ˆ R n and (cid:13)(cid:13)(cid:13) ˆ R n (cid:13)(cid:13)(cid:13) H m = O ε ( ε n ) . If ψ ∈ H m , ψ ∈ H − m then we use Eq.(29) towrite C Lebψ ,ψ ( n ) = (cid:16) ψ , ˆ F n ψ (cid:17) (42) = (cid:16) ψ , ˆ K n ψ (cid:17) + (cid:16) ψ , ˆ R na ψ (cid:17) = (cid:16) ψ , ˆ K n ψ (cid:17) + k ψ k H m k ψ k H − m O ε ( ε n ) Using the Jordan Blo k de omposition of ˆ K , Eq.(35), and Eq.(28), we have (cid:16) ψ , ˆ K n ψ (cid:17) = X i ≥ , | λ i | >ε min ( n,d i − X k =0 C kn λ n − ki d i − k X j =1 ( ψ , v i,j ) w i,j + k ( ψ ) , (43) = X i ≥ , | λ i | >ε min ( n,d i − X k =0 C kn λ n − ki d i − k X j =1 v i,j (cid:0) ψ (cid:1) w i,j + k ( ψ ) (44)We have obtained Eq.(41). 22orollary 2. For any i ≥ , | λ i | ≤ . Therefore the spe tral radius of the operator ˆ F : H m → H m is one. If an eigenvalue is on the unit ir le, | λ i | = 1 , then d i = 1 , i.e. ithas no Jordan blo k.Proof. Sin e ˆ F ϕ = ϕ ◦ f , it is lear that λ = 1 is an eigenvalue for the onstant fun tion. Forall n , and ψ , ψ ∈ C ∞ ( M ) , one has (cid:16) ψ , ˆ F n ψ (cid:17) = R ψ ( x ) ψ ( f n ( x )) dµ Leb hen e (cid:12)(cid:12)(cid:12) C Lebψ ,ψ ( n ) (cid:12)(cid:12)(cid:12) = (cid:12)(cid:12)(cid:12)(cid:16) ψ , ˆ F n ψ (cid:17)(cid:12)(cid:12)(cid:12) ≤ | ψ | C | ψ | C Vol ( M ) (45)is bounded uniformly with respe t to n .Suppose that λ i > . Sin e C ∞ ( M ) is dense in H m and H − m , there exists ψ , ψ ∈ C ∞ ( M ) su h that v i, (cid:0) ψ (cid:1) = 0 and w i, ( ψ ) = 0 . Then λ ni v i, (cid:0) ψ (cid:1) w i, ( ψ ) would diverge for n → ∞ ,and Eq.(41) implies that C Lebψ ,ψ ( n ) would diverge also, in ontradi tion with Eq.(45).Similarly, suppose that | λ i | = 1 , but d i ≥ . There exists ψ , ψ ∈ C ∞ ( M ) su h that v i,j (cid:0) ψ (cid:1) = 0 and w i,j ( ψ ) = 0 . Then the term k = d i − in Eq.(41) whi h ontains C kn divergesas n d i − for n → ∞ . Eq.(41) implies that C Lebψ ,ψ ( n ) would diverge also, in ontradi tion withEq.(45).Corollary 3. The following two propositions are equivalent:1. f is Lebesgue-mixing.2. λ = 1 is simple, v = Leb is the Lebesgue measure and w = µ srb the SRB measure.The other eigenvalues satisfy | λ i | < , i ≥ .Therefore: C Lebψ ,ψ ( n ) −→ n →∞ v (cid:0) ψ (cid:1) w ( ψ ) Remarks:ˆ It turns out that the SRB orrelation fun tion C srbψ ,ψ ( n ) admits an asymptoti ex-pansion similar to Eq.(41), see Theorem 3 below.ˆ Without the Lebesgue-mixing assumption and if Conje ture 1 is wrong, there maybe a (cid:28)nite number of eigenvalues on the unit ir le.23roof. The Lebesgue-mixing assumption implies that C Lebψ ,ψ ( n ) onverges for n → ∞ . The onstant fun tion v = 1 is obviously an eigenfun tion of ˆ F with eigenvalue λ = 1 . There areno other eigenvalues on the unit ir le otherwise from Eq.(41), C Lebψ ,ψ ( n ) would not onverge for n → ∞ . We obtain C Lebψ ,ψ ( n ) = (cid:16) ψ , ˆ F n ψ (cid:17) → n →∞ v (cid:0) ψ (cid:1) w ( ψ ) with w = µ srb from De(cid:28)nition4. But Eq.(45) also implies that | w ( ψ ) | ≤ C | ψ | C . Therefore w is distribution of order ,hen e de(cid:28)nes a measure.5.2 The SRB orrelation fun tionWe have shown above that µ srb = w ∈ H − m for any m ∈ O ε and ε < . Noti e that fromEq.(27), we have w ψ ∈ H − m for any ψ ∈ C ∞ ( M ) . Sin e v i,j ∈ H m then Eq.(28) impliesthat v i,j ( w ψ ) = ( v ij , w ψ ) makes sense.Theorem 3. Assume that f is Lebesgue-mixing. If ψ , ψ ∈ C ∞ ( M ) then the asymptoti behavior of the SRB orrelation fun tion Eq.(40) is given by: C srbψ ,ψ ( n ) = w (cid:0) ψ (cid:1) w ( ψ ) (46) + X i ≥ , | λ i | >ǫ min ( n,d i − X k =0 C kn λ n − ki d i − k X j =1 v i,j (cid:0) w ψ (cid:1) w i,j ( ψ ) + O ε ( ε n ) . In parti ular f is SRB-mixing : C srbψ ,ψ ( n ) −→ n →∞ w (cid:0) ψ (cid:1) w ( ψ ) = (cid:18)Z ψ dµ srb (cid:19) (cid:18)Z ψ dµ srb (cid:19) and the onvergen e is exponentially fast.Proof. Using Eq.(28) and Eq.(31), we start with an equivalent expression for the SRB orrelationfun tion: C srbψ ,ψ ( n ) = Z ψ ( x ) ψ ( f n ( x )) dµ srb = (cid:16) w , ψ ˆ F n ψ (cid:17) = (cid:16) w ψ , ˆ F n ψ (cid:17) Then as in Eq.(42), we use the de omposition Eq.(35) and dedu e Eq.(46), sin e v (cid:0) w ψ (cid:1) w ( ψ ) = w (cid:0) ψ (cid:1) w ( ψ ) . 24 Mixing of Anosov maps preserving a smooth measureIn the parti ular ase where f preserves a smooth measure dx , ergodi ity of f and thereforemixing, has been proved by Anosov in his PhD thesis [1℄ (see [8℄ Theorem 6.3.1). In thisse tion we provide a di(cid:27)erent proof entirely based on the semi- lassi al approa h developedin this paper.Theorem 4. Suppose that f preserves a smooth measure dx . Then on the unit ir le,there is no Ruelle resonan e, ex ept with multipli ity one (equivalently f is Lebesgue-mixing from Corollary 3).Proof. of Theorem 4. From Corollary 1 and Corollary 2 an eigenvalue λ = e iθ on the unit ir lewould have no Jordan Blo k and would orrespond to an eigen-ve tor u ∈ H m , ˆ F u = e iθ u . Lemma4 and Lemma 3 below imply that u is a onstant fun tion and λ = 1 .The following Lemma ontains the global aspe t of the problem.Lemma 3. If 9 u ∈ C ( M ) and ˆ F u = λu , with | λ | = 1 then u is a onstant fun tion,and λ = 1 .Proof. Let us assume that u ∈ C ( M ) , with ˆ F u = λu , | λ | = 1 . Let u n := ˆ F n u = u ◦ f n . Onehas u n = λ n u , therefore | du n | ∞ = | du | ∞ < ∞ is bounded uniformly with respe t to n , sin e M is ompa t. On the other hand ( du n ) f − n ( x ) = ( Df n ) t f − n x ( du ) x . Suppose that there exists x ∈ M su h that du x = 0 . If du x / ∈ E ∗ s ( x ) (stable dire tion) then (cid:12)(cid:12)(cid:12) ( du n ) f − n ( x ) (cid:12)(cid:12)(cid:12) diverge when n → + ∞ . If du x ∈ E ∗ s ( x ) then (cid:12)(cid:12)(cid:12) ( du n ) f − n ( x ) (cid:12)(cid:12)(cid:12) diverge when n → −∞ . This ontradi ts | du n | ∞ < ∞ , therefore du = 0 . M is onne ted therefore u is onstant.Lemma 4. Assume there exists a positive smooth density dx on M whi h is invariantunder the map f . Let u ∈ H m , ˆ F u = e iθ u . Then u ∈ C ∞ ( M ) .25 nstable directionstable direction | ξ u | = (1 + δ )(1 + | ξ s | ) ξ u ξ s | ξ u | = a (1 + | ξ s | ) B = 1 B ◦ F − = 1 D = 0 Figure 5: A distribution u ∈ H m is regular in the stable dire tion. If ˆ F u = e iθ u , the ideaof the proof of Lemma 48 is to propagate this regularity under the map F towards theunstable dire tion. For that purpose, we propagate the symbol B and establish in (56)that u is semi- lassi ally negligible in a zone where D = 0 .Proof. We shall make use of some h -pseudodi(cid:27)erential al ulus10. Assume there exists a positivesmooth density dx on M whi h is invariant under the map f . Then ˆ F : L ( M, dx ) → L ( M, dx ) is unitary. The idea of the proof is to use Corollary 1 whi h states that u is C ∞ in every dire tionex ept the unstable dire tion, and use the unitary of ˆ F , to dedu e that u is C ∞ also in theunstable dire tion (by propagation).It is easy to see that there exist symbols ≤ B ( x, ξ ) , C ( x, ξ ) ∈ S ( T ∗ M ) , su h that B + C , (47)and B ( x, ξ ) = 1 on the set k ξ u k ≤ (1 − δ )(1 + k ξ s k ) (48)with support in k ξ u k ≤ (1 + δ )(1 + k ξ s k ) , (49)where we hoose δ > su(cid:30) iently small. See Figure 5.Then B ◦ F − ∈ S is equal to 1 on a set k ξ u k ≤ a (1 + k ξ s k ) (50)and has its support in a set k ξ u k ≤ b (1 + k ξ s k ) , (51)10In parti ular ξ is quantized into the operator hD x = − ih∂/∂x while for ordinary PDO, ξ is quantizedinto D x = − i∂/∂x . 26here < a < b are independent of δ when δ > is small enough. Now we an onstru t orresponding h -pseudodi(cid:27)erential operators ˆ B, ˆ C su h that B + ˆ C + K, (52)where K is negligible in the sense that K = O ( h N ) : H − N → H N , ∀ N ∈ N , (53)and su h that the symbol of ˆ B is equal to B modulo hS − , and modulo h ∞ S −∞ it is equalto 1 on the set (48) and has its support in the set (49). It follows from Egorov Theorem that ˆ F ˆ B ˆ F − = ˆ F ˆ B ˆ F ∗ has the orresponding properties with the sets (48), (49) repla ed by (50), (51).We an (cid:28)nd a self-adjoint h -pseudodi(cid:27)erential operator ˆ D with symbol of lass S , su h that ( ˆ F ˆ B ˆ F ∗ ) − ˆ B = ˆ D + L, (54)where L is negligible as in (53). In fa t, in the region (48) we an take ˆ D = 0 and when we furtherapproa h the unstable dire tions we (cid:28)rst have ˆ F ˆ B ˆ F ∗ = 1 mi ro-lo ally, so that the left hand sidein (54) is ≡ − ˆ B ≡ ˆ C , so that we an take ˆ D = ˆ C . Even loser to the unstable dire tions, weget outside the support of ˆ B and we an take ˆ D = ˆ F ˆ B ˆ F ∗ . Now, let u ∈ H m be as in the proposition and write ˆ Bu = e iθ ˆ B ˆ F − u. Thanks to the properties of ˆ B and m , this quantity belongs to L , and using the unitarity of ˆ F ,we get k ˆ Bu k = k ˆ F ˆ B ˆ F − u k . Combining this with (54), we get k ˆ F ˆ B ˆ F − u k − k ˆ Bu k = k ˆ Du k + ( Lu | u ) , (55)and sin e L is negligible, k ˆ Du k = O ( h ∞ ) . (56)Sin e ˆ D is semi- lassi ally ellipti in the region (1 + δ )(1 + k ξ s k ) ≤ k ξ u k ≤ a (1 + k ξ s k ) , we see that u is mi ro-lo ally O ( h ∞ ) in the region (repla ing ξ → hξ ) (1 + δ )( 1 h + k ξ s k ) ≤ k ξ u k ≤ a ( 1 h + k ξ s k ) , and letting h → , we see that u has no wave-front set in a oni al neighborhood of E ∗ u . Sin e wealready know that WF( u ) ⊂ E ∗ u , we on lude that u ∈ C ∞ , and this ends the proof of Lemma 4.27 Trun ation and numeri al al ulation of the resonan espe trumLet χ : R + → R + be a C ∞ fun tion su h that χ ( x ) = 1 , if x ≤ , and χ ( x ) = 0 , if x ≥ . For r > , let the fun tion χ r on T ∗ M be de(cid:28)ned by χ r ( x, ξ ) = χ ( | ξ | /r ) . Let thetrun ation operator be: ˆ χ r def = Op ( χ r ) Noti e that ˆ χ r is a smoothing operator whi h trun ates large omponents in ξ . In [15℄se tion 2.1.4 and referen es therein, we interpret ˆ χ r as a (cid:16)noisy operator(cid:17) with a noise ofamplitude /r .Theorem 5. (cid:16) ˆ F ˆ χ r (cid:17) is a smoothing operator. For any ε > the spe trum of (cid:16) ˆ F ˆ χ r (cid:17) in L ( M ) outside the disk of radius ε , onverges for r → ∞ , towards the spe trum ofRuelle resonan es ( λ i ) i , ounting multipli ities. The eigenspa es onverge towards theeigen-distributions.Remarks:1. Theorem 5 gives a pra ti al way to ompute numeri ally the resonan e spe trum:one expresses the operator ˆ F in a dis rete basis of L ( M ) , trun ates it smoothly(a ording to the operator ˆ χ r ), and diagonalizes the resulting matrix numeri ally.See Figure 6.2. Theorem 5 gives also a simple way to establish a relation between Ruelle resonan es λ i and the periodi points of the map f , via dynami al zeta fun tions, see e.g. [27, 5℄.On one hand the Atiyah-Bott (cid:28)xed point formula ([2℄ orollary 5.4 p.393), gives forany n ≥ , Tr (cid:16)(cid:16) ˆ F ˆ χ r (cid:17) n (cid:17) −→ r →∞ X x ∈ F ix ( f n ) | det (1 − D x f n ) | and on the other hand, the zeros of the dynami al zeta fun tion d ( z ) = exp − X n ≥ z n n Tr (cid:16)(cid:16) ˆ F ˆ χ r (cid:17) n (cid:17)! = det (cid:16) − z (cid:16) ˆ F ˆ χ r (cid:17)(cid:17) onverge towards (1 /λ i ) i . 28roof. Let ε > , and m ∈ O ε/ . From Eq.(32), the operator ˆ F : H m → H m an be written ˆ F = ˆ r + ˆ k , with k ˆ r k ≤ ε/ , and ˆ k ompa t. In H m , the operator ˆ χ r −→ r →∞ Id onverges strongly,and k ˆ χ r k ≤ C r −→ r →∞ , in parti ular k ˆ χ r k ≤ for r large enough. For | z | > ε write ˆ F − z = ˆ r − z + ˆ k = (ˆ r − z ) (cid:16) r − z ) − ˆ k (cid:17)(cid:16) ˆ F − z (cid:17) − = (cid:16) r − z ) − ˆ k (cid:17) − (ˆ r − z ) − when the (cid:28)rst fa tor on the right is well de(cid:28)ned. Similarly (cid:16) ˆ F ˆ χ r − z (cid:17) − = (cid:16) r ˆ χ r − z ) − ˆ k ˆ χ r (cid:17) − (ˆ r ˆ χ r − z ) − . Let z ∈ C \ σ (cid:16) ˆ F (cid:17) and | z | > ε . Then (ˆ r ˆ χ r − z ) − is well de(cid:28)ned sin e k ˆ r ˆ χ r k ≤ ε . Moreover (ˆ r ˆ χ r − z ) − −→ r →∞ (ˆ r − z ) − strongly and (ˆ r ˆ χ r − z ) − ˆ k ˆ χ r −→ r →∞ (ˆ r − z ) − ˆ k in norm sin e ˆ k is om-pa t. Therefore (cid:16) ˆ F ˆ χ r − z (cid:17) − −→ r →∞ (cid:16) ˆ F − z (cid:17) − onverges strongly. Now the (cid:28)nite rank operator ˆ K in the spe tral de omposition Eq.(34) an be obtained by a ontour integral of the resolvent: ˆ π = 12 πi I | z | = ε (cid:16) z − ˆ F (cid:17) − dz, ˆ K = ˆ π ˆ F Similarly the spe trum of (cid:16) ˆ F ˆ χ r (cid:17) outside the disk of radius ε is the spe trum of the (cid:28)nite rankoperator ˆ K r given by: ˆ π r def = 12 πi I | z | = ε (cid:16) z − ˆ F ˆ χ r (cid:17) − dz, ˆ K r = ˆ π r (cid:16) ˆ F ˆ χ r (cid:17) . The operator ˆ K r −→ r →∞ ˆ K onverges strongly and therefore in norm sin e it has (cid:28)nite rank (seeTh. 9.19 p.98 in [21℄). We dedu e Theorem 5.8 Con lusion and perspe tivesIn this paper we have proposed a semi- lassi al approa h for spe tral properties of Anosovdynami al systems. We dis uss now some possible perspe tives for this work.First, we have treated the ase of hyperboli di(cid:27)eomorphisms f . The ase of expansivemaps whi h are not invertible is more simple, and the method we propose works equallywell, but need some adaptations. For example the transfer operator ˆ F has the samede(cid:28)nition Eq.(7), but the asso iated anoni al map F on T ∗ M is now multivalued (itsgraph on T ∗ M × T ∗ M is well de(cid:28)ned).The ase of partially hyperboli systems (and in parti ular hyperboli (cid:29)ows) where thereis a neutral dire tion is very interesting, and there are important re ent results on erning29igure 6: Ruelle resonan es λ i obtained numeri ally for the model Eq.(3), with ε = 0 . .their spe tral properties [11, 26℄. For the same reasons explained in the introdu tion, wethink that a semi- lassi al approa h is natural and hopefully fruitful for these systems too.Finally let us mention the open question mentioned in the onje ture 1. One an wonderif a semi- lassi al approa h similar to Se tion 6 ould help towards the resolution of this.A Pseudodi(cid:27)erential operators with variable orderA.1 Preliminary remarksA.1.1 Semi- lassi al analysisIn this appendix, we provide a self- ontained series of analyti tools for studying pseudod-i(cid:27)erential operators with slightly more general lasses of symbols than usual ones, namelysymbols with variable order. All the results we give ome from the standard semi- lassi alanalysis but for our spe ial symbol lasses, and are given mainly without proof. We referto [31, 18, 40℄ for the standard results in semi- lassi al analysis. Note that the idea ofusing symbols with variable order is not new. See for example [29, 41, 25℄.Semi- lassi al analysis is a ri h theory whi h gives a sense to pseudodi(cid:27)erential opera-tors ˆ P of the form ϕ → ˆ P ( ϕ ) ( x ) = Z R d e iξ ( x − y ) P ( x, ξ ) ϕ ( y ) dydξ (57)where P ( x, ξ ) is a smooth fun tion alled the symbol of ˆ P and belonging to some appro-priate lass of fun tions satisfying ertain regularity onditions at in(cid:28)nity. These lasses30ield to a powerful symbol al ulus, i.e., a tool for extra ting information in an asymp-toti way about the operator ˆ P by means of its symbol. The main results are basi ally thefollowing :ˆ Composition of PDO's. Given two PDO's ˆ P and ˆ Q , the produ t ˆ A = ˆ P ˆ Q is alsoa PDO and its symbol is given to leading order in ξ by the produ t of the symbols A = P Q .ˆ Sobolev ontinuity. First de(cid:28)ned on C ∞ ( R n ) , pseudodi(cid:27)erential operators are shownto be ontinuous between ertain Sobolev spa es. Moreover, the orresponding op-erator norm is estimated by some norms of the derivatives of the symbol.ˆ Ellipti ity, parametrix. A ondition alled ellipti ity imposed on a symbol is enoughto insure that the orresponding operator is invertible up to a regularizing operator.The (cid:16)almost-inverse(cid:17) is alled a parametrix.A.1.2 Symbols with onstant orderThe typi al lass of symbols one onsiders is the set S mρ,δ ⊂ C ∞ ( R n ) of smooth fun tions P ( x, ξ ) whi h satisfy the following estimates. For any ompa t subset K ⊂ R n and anymulti-index α, β ∈ N n , there is a onstant C K,α,β su h that (cid:12)(cid:12) ∂ αξ ∂ βx P ( x, ξ ) (cid:12)(cid:12) ≤ C K,α,β h ξ i m − ρ | α | + δ | β | (58)for any ( x, ξ ) ∈ K × R n . Here h ξ i means q | ξ | and we use the standard multi-indi esnotation ∂ αx f = ∂ αi f∂x αii ... ∂ αn f∂x αnn . The number m ∈ R is alled the ( onstant) order of thesymbol. These lasses of symbols, introdu ed by Hörmander [22℄, are quite general andallow one nevertheless to develop a symbol al ulus, provided the onstants ρ, δ ful(cid:28)ll ertain onditions. The orresponding lass of PDO's (Formula 57) is denoted by Ψ mρ,δ .The operator ˆ P , denoted sometimes also by P ( x, D ) , is the (left) quantization of P .Nevertheless, as explained in the introdu tion, we need to onsider symbols with anorder m whi h is no longer onstant but depends on the variables ( x, ξ ) . It turns out thatthese lasses are ontained in some Hörmander lasses (with order equal to lim sup m ) butone needs to keep tra k of the fa t that the order is variable in order to develop a moregeneral notion of ellipti ity, for symbols whi h would not be ellipti in the usual sense. Thiswill be explained in the rest of this appendix whi h is devoted to the symbol al ulus forsymbols with variable order. But before, we need one additional remark about the theoryof PDO's on manifolds.A.1.3 Pseudodi(cid:27)erential operators on manifoldThe usual way of de(cid:28)ning PDO's on a para- ompa t Hausdor(cid:27) manifold M is to makea partition of the unity of M and use in ea h hart the semi- lassi al analysis on R n .De(cid:28)ned in this way, the symbol of an operator depends unfortunately on the harts. But31t turns out that, provided < ρ ≤ and ρ + δ ≥ , there is an element of the quotientspa e S mρ,δ /S m − ( ρ − δ ) ρ,δ alled the prin ipal symbol whi h is well-de(cid:28)ned independently ofthe harts. Any member of this equivalen e lass is a fun tion on the otangent bundle T ∗ M , also alled a prin ipal symbol. For te hni al reasons it is ommon and onvenientto assume ρ = 1 − δ and ρ > , and to denote S mρ = S mρ,δ . We will follow from now on this onvention.It is well-known that many results of semi- lassi al analysis on a manifold are given interms of prin ipal symbols. On the other hand, when we onsider symbols with a non- onstant order fun tion m ( x, ξ ) ∈ T ∗ M , we need to manipulate arefully the on ept ofprin ipal symbol, sin e there are two di(cid:27)erent notions depending on whether we onsiderour symbol to have a variable order m ( x, ξ ) or a onstant order equal to lim sup m .To avoid possibly onfusing onsiderations about prin ipal symbols, we will use a very onvenient quantization s heme, developed in [33℄, whi h works on Riemannian manifolds ( X, g ) and provides a notion of total symbol. It is de(cid:28)ned as follows. First, we (cid:28)x a ut-o(cid:27) fun tion χ ∈ C ∞ ( T M, [0 , whi h equals on a neighborhood of the zero se tion T M and is supported in a neighborhood W ⊂ T M of T M in whi h the exponential mapde(cid:28)nes a di(cid:27)eomorphism onto an open neighborhood of the diagonal in M × M . Then, forany u ∈ C ∞ ( M ) we de(cid:28)ne u χ ∈ C ∞ ( T M ) its semi- lassi al lift on T M by u χ ( x, v ) = (cid:26) χ ( v ) u (exp x v ) for ( x, v ) ∈ W else. . Finally, for any symbol p ∈ C ∞ ( T ∗ M ) one de(cid:28)nes the operator ˆ P by u → ˆ P ( u ) ( x ) = Z T ∗ x X p ( x, ξ ) f u χ ( x, ξ ) dξ (59)where e f denotes the Fourier transform of a fun tion f ( x, v ) ∈ C ∞ ( T M ) with respe t tothe v variable, i.e., e f ( x, ξ ) = 1(2 π ) n Z T x X e − i h ξ | v i f ( x, v ) dv. It is shown in [33℄ that this onstru tion gives rise to a notion of total symbol ( alledthere normal symbol) well-de(cid:28)ned independently of the ut-o(cid:27) fun tion χ up to anelement of S −∞ ρ . The lasses of symbols are de(cid:28)ned in the usual way : Fix m ∈ R and ρ > . Then, a fun tion p ∈ C ∞ ( T ∗ M ) belongs to the lass S mρ if in any trivialization ( x, ξ ) : T ∗ M | U → R n , for any ompa t K ⊂ U and any multi-indi es α, β ∈ N n , there isa onstant C K,α,β su h that (cid:12)(cid:12) ∂ αξ ∂ βx p ( x, ξ ) (cid:12)(cid:12) ≤ C K,α,β h ξ i m − ρ | α | +(1 − ρ ) | β | on T ∗ M | U . Now the fun tion h ξ i = q | ξ | is de(cid:28)ned in term of the norm | ξ | = g x ( ξ, ξ ) with the s alar produ t on T ∗ M denoted by the same letter g .32.2 Symbols with variable ordersAs usual, la onstante positive C , qui est le (cid:28)dèle ompagnon de l'analyste, pourra varierd'une formule à l'autre11.A.2.1 De(cid:28)nition and main basi properties of S m ( x,ξ ) ρ As explained before, we want to develop a symboli al ulus for PDO's whose symbol hasan order m whi h depends on the point ( x, ξ ) ∈ T ∗ M . We (cid:28)rst need to explain whi hfun tions are a eptable as order fun tions.De(cid:28)nition 6. An order fun tion m ( x, ξ ) is an element of S r , for some < r < whi h is also bounded at in(cid:28)nity in ξ , i.e., sup x,ξ ∈ T ∗ M | m ( x, ξ ) | < ∞ . We now de(cid:28)ne the lass of symbols of variable order exa tly in the same way as symbolswith onstant order. We will use the slightly abusive notation S m ( x,ξ ) ρ to emphasize thefa t that m ( x, ξ ) is not onstant.De(cid:28)nition 7. Let m ( x, ξ ) ∈ S r be an order fun tion and < ρ < . A fun tion p ∈ C ∞ ( T ∗ M ) belongs to the lass S m ( x,ξ ) ρ if in any trivialization ( x, ξ ) : T ∗ M | U → R n ,for any ompa t K ⊂ U and any multi-indi es α, β ∈ N n , there is a onstant C K,α,β su hthat (cid:12)(cid:12) ∂ αξ ∂ βx p ( x, ξ ) (cid:12)(cid:12) ≤ C K,α,β h ξ i m ( x,ξ ) − ρ | α | +(1 − ρ ) | β | for any ( x, ξ ) ∈ T ∗ M | U .As with the usual al ulus, for any symbol p ∈ S m ( x,ξ ) ρ and any multi-indi es α, β wehave ∂ αξ ∂ βx p ∈ S m ( x,ξ ) − ρ | α | +(1 − ρ ) | β | ρ . This means that ∂ αξ ∂ βx p has an order fun tion given by m ( x, ξ ) − ρ | α | +(1 − ρ ) | β | . Similarly,for any two symbols p ∈ S m ( x,ξ ) ρ and q ∈ S m ′ ( x,ξ ) ρ ′ , the point-wise produ t p ( x, ξ ) q ( x, ξ ) belongs to S m ( x,ξ )+ m ′ ( x,ξ ) ρ ′′ with ρ ′′ = min ( ρ, ρ ′ ) .Another simple but important property of variable order symbols, is that they belongin fa t to some Hörmander lass. This follows from the following fa t.11A. Unterberger [41℄ 33emma 5. For any two non- onstant order m ( x, ξ ) and m ′ ( x, ξ ) satisfying m ( x, ξ ) ≤ m ′ ( x, ξ ) on T ∗ M and any ρ ≥ ρ ′ , the following holds S m ( x,ξ ) ρ ⊂ S m ′ ( x,ξ ) ρ ′ . In parti ular, one has S m ( x,ξ ) ρ ⊂ S sup mρ ′ and S m ( x,ξ ) ρ ⊂ S ε +lim sup mρ ′ for any ε > . Proof. Let p ∈ S m ( x,ξ ) ρ be a symbol. For any α, β ∈ N n , any ompa t K ⊂ M , the (cid:28)rst in lusion omes simply from (cid:12)(cid:12)(cid:12) ∂ αξ ∂ βx p ( x, ξ ) (cid:12)(cid:12)(cid:12) ≤ C h ξ i m ( x,ξ ) − ρ | α | +(1 − ρ ) | β | ≤ C h ξ i m ′ ( x,ξ ) − ρ ′ | α | +(1 − ρ ′ ) | β | . On the other hand, for large enough h ξ i , m is bounded by lim sup m + ε for any ε > . Thisprovides the se ond estimate.This property implies in parti ular that we an use the same lass of residual symbolsas for Hörmander symbols, namely S −∞ := \ m< S mρ whi h is independent of ρ . For onvenien e, we also introdu e the lass of symbols of anyorder S ∞ ρ := [ m> S mρ . A.2.2 Canoni al examplesWe show now that the natural andidate h ξ i m ( x,ξ ) is indeed a suitable symbol.Lemma 6. Let m ∈ S ρ be an order fun tion. The smooth fun tion p ( x, ξ ) = h ξ i m ( x,ξ ) belongs to S m ( x,ξ ) ρ − ε for any ε > . 34roof. We will prove that for any α, β , we have ∂ αξ ∂ βx p ( x, ξ ) = q ( x, ξ ) h ξ i m ( x,ξ ) with q ∈ S ( − ρ + ε ) | α | +(1 − ρ + ε ) | β | ρ (60)for any ε > . First of all, this is true for (cid:28)rst order derivatives. Indeed, for | β | = 1 we omputethe derivative : ∂ βx p ( x, ξ ) = (cid:16) ∂ βx (ln ( h ξ i ) m ( x, ξ )) (cid:17) h ξ i m ( x,ξ ) =: q ( x, ξ ) h ξ i m ( x,ξ ) . The appearan e of logarithmi terms is a tually the worse that an happen when di(cid:27)erentiating asymbol with variable order, but it is easily ontrolled. First, ln ( h ξ i ) is bounded by h ξ i ε for ε > arbitrarily small. Moreover, the logarithm disappears as soon as we take at least one derivativein x or in ξ . Namely, whenever ( α, β ) = (0 , one has ∂ αξ ∂ βx (ln ( h ξ i )) ∈ S −| α | . This shows that ln ( h ξ i ) ∈ S ε for any ε > . This means that ln ( h ξ i ) m ( x, ξ ) ∈ S ε .S ρ = S ερ . Then the derivativeyields to q ( x, ξ ) ∈ S ε +1 − ρρ . Similarly, for | α | = 1 one shows that ∂ αξ p ( x, ξ ) = q ( x, ξ ) h ξ i m ( x,ξ ) with q ∈ S ε − ρρ for any ε > . Let us prove by iteration that Equation (60) holds in general. Suppose itholds for all α, β satisfying | α + β | ≤ N for some N ∈ N . Then, any ( α ′ , β ′ ) with | α + β | = N + 1 has the form ( α + a, β + b ) with | a + b | = 1 , i.e., either ( | a | , | b | ) = (1 , or ( | a | , | b | ) = (0 , . Inthe (cid:28)rst ase, we want to ompute ∂ aξ ∂ αξ ∂ βx p ( x, ξ ) = (cid:0) ∂ aξ q ( x, ξ ) (cid:1) h ξ i m ( x,ξ ) + q ( x, ξ ) ∂ aξ (cid:16) h ξ i m ( x,ξ ) (cid:17) . By assumption, the (cid:28)rst term is in S ( − ρ + ε ) | α | +(1 − ρ + ε ) | β |− ρρ . h ξ i m ( x,ξ ) and the se ond one is in S ( − ρ + ε ) | α | +(1 − ρ + ε ) | β | ρ .S − ρ + ερ . h ξ i m ( x,ξ ) . Together, this provides h ξ i − m ( x,ξ ) ∂ aξ ∂ αξ ∂ βx p ( x, ξ ) ∈ S ( − ρ + ε ) | α + a | +(1 − ρ + ε ) | β | ρ . Similarly, we would (cid:28)nd h ξ i m ( x,ξ ) ∂ bx ∂ αξ ∂ βx p ( x, ξ ) ∈ S ( − ρ + ε ) | α | +(1 − ρ + ε ) | β + b | ρ for | b | = 1 . This proves Formula (60) for any α, β satisfying | α + β | ≤ N + 1 and the formula forall α, β is proved by indu tion. Then, we dedu e that (cid:12)(cid:12)(cid:12) ∂ αξ ∂ βx p ( x, ξ ) (cid:12)(cid:12)(cid:12) ≤ h ξ i m ( x,ξ ) − ( ρ − ε ) | α | +(1 − ρ + ε ) | β | and thus p ( x, ξ ) ∈ S m ( x,ξ ) ρ − ε for any ε > .A.2.3 A tion of di(cid:27)eomorphismsFor any di(cid:27)eomorphism φ : M → M , we denote by φ ∗ : T ∗ M → T ∗ M the lift on the otangent bundle de(cid:28)ned by φ ∗ ( x, ξ ) = (cid:16) φ ( x ) , (cid:0) ( D x φ ) − (cid:1) t (cid:17) .35emma 7. Let p ∈ S m ( x,ξ ) ρ be a symbol with non- onstant order m ( x, ξ ) and φ : M → M a di(cid:27)eomorphism. Then, the omposition p ◦ φ ∗ belongs to S m ◦ φ ∗ ρ .Proof. Set a = p ◦ φ ∗ . For simpli ity, we will write ξ a ◦ φ ∗ for the ξ a omponent of φ ∗ ( x, ξ ) . Wewill prove by iteration that for any order fun tion m ( x, ξ ) , any symbol p ∈ S mρ and any α, β ∈ N d ,one has ∂ αξ ∂ βx ( p ◦ φ ∗ ) = q α,β ◦ φ ∗ where q α,β ∈ S m − ρ | α | +(1 − ρ ) | β | ρ , (61)where we write m = m ( x, ξ ) for shortness. First of all, this is trivially true for α = β = 0 . Then,we suppose it is true for all α, β ∈ N d , with | α + β | ≤ N . If we ompute the derivative ∂ α + aξ ∂ βx of p ◦ φ ∗ with | a | = 1 , we obtain simply ∂ α + aξ ∂ βx ( p ◦ φ ∗ ) = ∂ aξ ( q α,β ◦ φ ∗ )= X | a ′ | =1 ∂ a ′ ξ ( q α,β ) ◦ φ ∗ .∂ aξ (cid:16) ξ a ′ ◦ φ ∗ (cid:17) . The terms ∂ a ′ ξ ( q α,β ) live in S m − ρ | α + a | +(1 − ρ ) | β | ρ by assumption whereas the se ond term in theprodu t belongs to S . This means that ∂ α + aξ ∂ βx ( p ◦ φ ∗ ) = q α + a,β ◦ φ ∗ where the symbol q α + a,β = X | a ′ | =1 ∂ a ′ ξ ( q α,β ) .∂ aξ (cid:16) ξ a ′ ◦ φ ∗ (cid:17) ◦ φ − ∗ belongs to S m − ρ | α + a | +(1 − ρ ) | β | ρ . We onsider now the derivative ∂ αξ ∂ β + bx with | b | = 1 . The ompu-tation is slightly more ompli ated, sin e there is an x -dependen e in the ξ omponent of φ ∗ ( x, ξ ) .Namely, ∂ αξ ∂ β + bx ( p ◦ φ ∗ ) = X | b ′ | =1 ∂ b ′ x ( q α,β ) ◦ φ ∗ .∂ bx (cid:16) φ b ′ ( x ) (cid:17) + X | a ′ | =1 ∂ a ′ ξ ( q α,β ) ◦ φ ∗ .∂ bx (cid:16) ξ a ′ ◦ φ ∗ (cid:17) . In the (cid:28)rst sum, the terms ∂ b ′ x ( q α,β ) belong to S m − ρ | α | +(1 − ρ ) | β + b | ρ and ∂ bx (cid:16) φ b ′ ( x ) (cid:17) is in S . On theother hand, in the se ond sum we have ∂ a ′ ξ ( q α,β ) ∈ S m − ρ | α + a | +(1 − ρ ) | β | ρ but the se ond one in theprodu t is in S . This means that the se ond sum brings a power (1 − ρ ) of ξ . All together, weobtain ∂ αξ ∂ β + bx ( p ◦ φ ∗ ) = q α,β + b ◦ φ ∗ with q α,β + b ∈ S m − ρ | α + a | +(1 − ρ ) | β | +1 − ρρ . This proves thereforeby indu tion Formula (61) for all α, β ∈ N d .Now, the lemma follows easily form this formula. Indeed, for any α, β ∈ N d one has (cid:12)(cid:12)(cid:12) ∂ αξ ∂ βx ( p ◦ φ ∗ ) (cid:12)(cid:12)(cid:12) ≤ C h ξ ◦ φ ∗ i m ◦ φ ∗ − ρ | α | +(1 − ρ ) | β | .
36n the other hand, sin e both D x φ and ( D x φ ) − are uniformly bounded on M , it follows thatthere is a onstant C > su h that C h ξ i ≤ h ξ ◦ φ ∗ i ≤ C h ξ i . This implies that (cid:12)(cid:12)(cid:12) ∂ αξ ∂ βx ( p ◦ φ ∗ ) (cid:12)(cid:12)(cid:12) ≤ C h ξ i m ◦ φ ∗ − ρ | α | +(1 − ρ ) | β | . A.3 PDO with variable orderGiven an order fun tion m ( x, ξ ) and a symbol p ( x, ξ ) in the lass S m ( x,ξ ) ρ , Formula (59)provides an operator from C ∞ ( M ) to C ∞ ( M ) , whi h is a tually ontinuous. By duality, itis also ontinuous from E ′ to D ′ . The lass Ψ m ( x,ξ ) ρ of PDO's is then the set of operators ofthe form (59) modulo a smoothing operator, i.e. an operator whi h sends E ′ into C ∞ ( M ) ontinuously. We denote by Ψ −∞ = T n> Ψ − nρ the lass of smoothing operators and alsothe lass Ψ ∞ ρ = S n> Ψ nρ of all PDOs of type ρ whi h ontains Ψ m ( x,ξ ) ρ for any variableorder m ( x, ξ ) . Noti e that given an operator ˆ p ∈ Ψ ∞ ρ , its symbol is well-de(cid:28)ned only upto an element in S −∞ ρ .We now review the most important properties of PDO's. The proofs for non- onstantorder symbols follow in most ases the line of the proofs for usual symbols (see for example[18, 40℄) and are omitted for shortness of this paper. In all the sequel, the parameter ρ isalways supposed to satisfy ρ > . As well, in order to avoid any dis ussion about properlysupported operators, we assume from now on M to be ompa t, sin e it will be the asefor the appli ation of these tools to Ruelle-Polli ott resonan es.A.3.1 Asymptoti expansionsSemi- lassi al analysis is naturally an asymptoti theory. In order to prove the basi theorems about omposition, Egorov, ellipti ity or fun tional al ulus, one needs to givea sense to formal series like P p j , where { p j } j ∈ N is a sequen e of symbols with de reasingorders. Su h a series is most of the time divergent, but it is possible to (cid:28)nd a symbol p whi h is asymptoti ally equivalent to the series. This is an adaptation of an old result byBorel. 37heorem 6. Let p j ∈ S m j ( x,ξ ) ρ be a sequen e of symbols with variable order m j ∈ S r , with ρ < r ≤ , satisfying m j ↓ −∞ , in the sense that, for all j ∈ N sup x,ξ m j ( x, ξ ) → −∞ and m j +1 ( x, ξ ) ≤ m j ( x, ξ ) . Then, there exists a symbol p ∈ S m ( x,ξ ) ρ su h that for all N ≥ p − N − X j =0 p j ∈ S m N ( x,ξ ) ρ . The symbol p is unique modulo a residual symbol, i.e., an element of S −∞ .The proof of this theorem is a straightforward adaptation of the proof for usual symbols,whi h an be found for example in [30, II, 3℄ or [18℄. This fa t implies automati ally the orresponding result for asymptoti sums of PDO's. Namely, if ˆ p j ∈ Ψ m j ( x,ξ ) ρ is sequen eof PDO with de reasing orders, then there exist an operator ˆ p ∈ Ψ m ( x,ξ ) ρ whi h satis(cid:28)es ˆ p − N − X j =0 ˆ p j ∈ Ψ m N ( x,ξ ) ρ for all N ≥ .A.3.2 Adjoint and ompositionTheorem 7. Let ˆ p ∈ Ψ m ( x,ξ ) ρ be a PDO with non- onstant order symbol p ∈ S m ( x,ξ ) ρ . Thenthe adjoint ˆ p ∗ is itself a PDO in Ψ m ( x,ξ ) ρ and its symbol p ∗ ∈ S m ( x,ξ ) ρ satis(cid:28)es p ∗ ( x, ξ ) − p ( x, ξ ) ∈ S m ( x,ξ ) − (2 ρ − ρ , where denotes the omplex onjugate.Here, the adjoint means the formal L -adjoint de(cid:28)ned on the same domain C ∞ ( M ) .38heorem 8. Let ˆ p ∈ Ψ m ( x,ξ ) ρ and ˆ q ∈ Ψ m ′ ( x,ξ ) ρ be two PDO's with non- onstant order m ( x, ξ ) and m ′ ( x, ξ ) . Then the produ t ˆ a := ˆ p ˆ q is a PDO in Ψ m ( x,ξ )+ m ′ ( x,ξ ) ρ and itssymbol a ( x, ξ ) satis(cid:28)es a ( x, ξ ) − p ( x, ξ ) q ( x, ξ ) ∈ S m ( x,ξ )+ m ′ ( x,ξ ) − (2 ρ − ρ . A.3.3 Egorov's theoremEgorov's Theorem des ribes how PDO's transform under onjugation with a Fourier Inte-gral Operator. We will nevertheless avoid talking about general FIO's and restri t ourselvesto the simplest ase, namely the omposition by a di(cid:27)eomorphism on M , whi h is su(cid:30) ientfor our purposes. See [40, p.24℄Theorem 9. Let ˆ p ∈ Ψ m ( x,ξ ) ρ be a PDO with non- onstant order m ( x, ξ ) and f : M → M a di(cid:27)eomorphism. Denote by ˆ F the pull-ba k operator ˆ F ( u ) = u ◦ f and by F : T ∗ M → T ∗ M the lift of f − to the otangent bundle de(cid:28)ned by F ( x, ξ ) = (cid:0) f − ( x ) , ( D x f ) t (cid:1) . Then,the onjugation ˆ a := ˆ F − ˆ p ˆ F belongs to Ψ m ◦ F ( x,ξ ) ρ and its symbol a ( x, ξ ) satis(cid:28)es a ( x, ξ ) − p ◦ F ( x, ξ ) ∈ S m ◦ F ( x,ξ ) − (2 ρ − ρ . A.3.4 Sobolev ontinuityIt is a well-known fa t that on a ompa t manifold, a PDO of onstant order m extendsto a ontinuous operator H s → H s − m for all s . Thanks to Lemma 5, a PDO with variableorder m ( x, ξ ) extends to a ontinuous operators H s → H s − m + with m + = lim sup m ( x, ξ ) .On the other hand the embeddings H s ֒ → H s ′ for s ′ < s are ompa t. In parti ular,smoothing operators are ompa t in any Sobolev spa e.A.4 Non-isotropi ellipti ityA.4.1 Variable order ellipti ityWe know from the standard theory of PDO's that an operator ˆ p ∈ Ψ mρ is invertible modulo Ψ −∞ with an (cid:16)inverse(cid:17) in Ψ − mρ as soon as its symbol p satis(cid:28)es an ellipti ity ondition.We now show that the lassi al de(cid:28)nition of ellipti ity extends in a natural way to symbolswith variable order m ( x, ξ ) . This leads to a more general notion of ellipti ity whi h isproved to be equivalent to the existen e of a parametrix Ψ − m ( x,ξ ) ρ .39e(cid:28)nition 8. A symbol a non- onstant order p ∈ S m ( x,ξ ) ρ is alled ellipti if there is a C > su h that | p ( x, ξ ) | ≥ C h ξ i m ( x,ξ ) whenever h ξ i ≥ C . An operator ˆ p is ellipti if itssymbol is ellipti .One an easily he k that the following statement is equivalent to ellipti ity.Lemma 8. A symbol p ∈ S m ( x,ξ ) ρ is ellipti if and only if there exists a symbol q ∈ S − m ( x,ξ ) ρ su h that p ( x, ξ ) q ( x, ξ ) − ∈ S −∞ ρ . Example 1. For any order fun tion m ( x, ξ ) the symbol p ( x, ξ ) = h ξ i m ( x,ξ ) is ellipti andone an hoose q = h ξ i − m ( x,ξ ) on the whole of T ∗ M .For usual symbols, it is well-known that ellipti ity is a phenomenon of the prin ipalsymbol. This is also true for variable order symbols, in the following sense.Lemma 9. Let p ∈ S m ( x,ξ ) ρ be an ellipti symbol. Then any other symbol q ∈ S m ( x,ξ ) ρ satisfying p − q ∈ S m ( x,ξ ) − ερ for some ε > is ellipti as-well.Proof. Suppose p = q + s with s ∈ S m ( x,ξ ) − ερ . Ellipti ity of p means that for large enough h ξ i onehas | p ( x, ξ ) | ≥ C h ξ i m ( x,ξ ) . On the other hand, for large h ξ i one has also | s ( x, ξ ) | ≤ C h ξ i m ( x,ξ ) − ε .Therefore | q ( x, ξ ) | ≥ C h ξ i m ( x,ξ ) (cid:0) − C ′ h ξ i − ε (cid:1) ≥ C ′′ h ξ i m ( x,ξ ) for large h ξ i , hen e q is ellipti .Noti e that this notion of ellipti ity is more general than the usual one. Indeed, thesymbol h ξ i m ( x,ξ ) with an order whi h takes its values, say between − and +1 is not ellipti in the usual sense. Indeed, the symbol h ξ i m ( x,ξ ) with an order taking its values, say between-1 and +1 is not ellipti in the usual sense, when we view it as a symbol with onstantorder sup m . 40.4.2 Parametrix and invertibilityThe main point in onsidering this notion of non-isotropi ellipti ity is of ourse that it isequivalent to the existen e of a parametrix, as explained in Theorem 10 below.De(cid:28)nition 9. Let ˆ p ∈ Ψ m ( x,ξ ) ρ be any PDO. A parametrix of ˆ p is a PDO ˆ q ∈ Ψ − m ( x,ξ ) ρ su h that ˆ p ˆ q − I ∈ Ψ −∞ and ˆ q ˆ p − I ∈ Ψ −∞ . Theorem 10. An operator ˆ p ∈ Ψ m ( x,ξ ) ρ admits a parametrix if and only if its symbol p isellipti .For this reason, we will say equally that the symbol p or the operator ˆ p is ellipti .The onstru tion is standard. We just he k that it works as-well in the variable order ontext. Assume p ∈ S m ( x,ξ ) ρ is ellipti . Lemma 8 implies that there is a q ∈ S − m ( x,ξ ) ρ su hthat ˆ p ˆ q = I − ˆ r where ˆ r ∈ Ψ − (2 ρ − ρ be ause of Theorem 8. This implies ˆ r j ∈ Ψ − j (2 ρ − ρ forall j ∈ N and it follows that ˆ q N = ˆ q (cid:0) I + ˆ r + ... + ˆ r N (cid:1) satis(cid:28)es ˆ p ˆ q N − − I ∈ Ψ − N (2 ρ − ρ . On the other hand, thanks to the re-summation Theorem 6 we an (cid:28)nd a ˆ q R ∈ Ψ − m ( x,ξ ) ρ satisfying ˆ q R − ˆ q N − ∈ Ψ − m ( x,ξ ) − N (2 ρ − ρ for all N ∈ N . Therefore we have ˆ p ˆ q R − I ∈ Ψ − N (2 ρ − ρ for all N ∈ N , hen e ˆ q R is a right parametrix for ˆ p . Similarly, we an onstru t a leftparametrix ˆ q L ∼ ( I + ˆ s + ˆ s + ... ) ˆ q with ˆ s ∈ Ψ − (2 ρ − ρ given by ˆ p ˆ q = I − ˆ s . Finally, thefa t that ˆ q L − ˆ q R ∈ Ψ −∞ ρ omes from the observation that both ˆ q L ˆ p ˆ q R − ˆ q R and ˆ q L ˆ p ˆ q R − ˆ q L are smoothing. Therefore, say ˆ q L is a (both sided) parametrix for ˆ p .Proof.The existen e of a parametrix has many interesting onsequen es, su h as those listedbelow. First of all, it is well-known that a standard ellipti PDO (with onstant order m )is Fredholm H s → H s − m for any s . For PDO's with variable order, a slightly weaker resultholds.Lemma 10. Let ˆ p ∈ Ψ m ( x,ξ ) ρ be ellipti . Then the kernel of the operator ˆ p : H s → H s − m + with m + = lim sup m ( x, ξ ) is (cid:28)nite dimensional and ontained in C ∞ ( M ) .41roof. The key point in this proof is the fa t that for any smoothing operator ˆ r ∈ Ψ −∞ , theoperator I + ˆ r : H s → H s is Fredholm for any s and its kernel is ontained in C ∞ ( M ) . See forexample [31, h. 7℄ for a proof for s = 0 whi h extends straightforwardly to the ase s = 0 .Now, the ellipti ity ˆ p implies the existen e of a left parametrix ˆ q ∈ Ψ − m ( x,ξ ) ρ , i.e. ˆ q ˆ p = I +ˆ r with ˆ r ∈ Ψ −∞ . This operator extends to ˆ q : H s − m + → H s − m + − m − where m − = lim sup ( − m ( x, ξ )) .It follows that ker ˆ p is ontained in the kernel of I + ˆ r : H s → H s ⊂ H s − m + − m − whi h is (cid:28)nite dimensional and itself ontained in C ∞ ( M ) .This lemma has the onsequen e that we an make ˆ p invertible by adding a smoothingoperator, as shown in Lemma 12. This is useful in pra ti e, sin e one often needs to onstru t PDO's whose symbol satis(cid:28)es ertain properties whi h are not modi(cid:28)ed by addinga residual term. One needs (cid:28)rst a preliminary result.Lemma 11. Let ˆ p ∈ Ψ m ( x,ξ ) ρ be an ellipti and formally self-adjoint operator. Then ˆ p viewed as an unbounded operator on L admits a self-adjoint extension.Proof. The formal self-adjointness of ˆ p implies a ording to Theorem 7 that its symbol satis(cid:28)es p − Re ( p ) ∈ S m ( x,ξ ) − (2 ρ − ρ , hen e | Re ( p ( x, ξ )) | ≥ c. | p ( x, ξ ) | , with c > , for large enough | ξ | . We an thus suppose that Re ( p ( x, ξ )) > for large enough | ξ | (if Re ( p ) has the opposite sign, then the following argumentapplies to − ˆ p ). We an therefore apply Lemma 13 of the next se tion to show that ˆ p = ˆ b ∗ ˆ b − ˆ K with ˆ b ∈ Ψ m ( x,ξ ) ρ and ˆ K ∈ Ψ −∞ . Sin e ˆ K is bounded in L , it follows that ˆ p is bounded frombelow in L : (ˆ pu, u ) ≥ (cid:12)(cid:12)(cid:12) ˆ bu (cid:12)(cid:12)(cid:12) − (cid:12)(cid:12)(cid:12) ˆ K (cid:12)(cid:12)(cid:12) | u | . This allows us to onstru t the Friedri hs extension of ˆ p , whi h is self-adjoint on a domain in L (see e.g. [42, p. 317℄).Lemma 12. Let ˆ p ∈ Ψ m ( x,ξ ) ρ be an ellipti and formally self-adjoint operator. Then thereexists a formally self-adjoint and smoothing operator ˆ r ∈ Ψ −∞ ρ su h that ˆ p + ˆ r is invertible C ∞ ( M ) → C ∞ ( M ) with inverse in Ψ − m ( x,ξ ) ρ .42roof. This is also a standard onstru tion. First, Lemma 10 tells us that ˆ p : H m + → L with m + = lim sup m ( x, ξ ) has a (cid:28)nite dimensional kernel ontained in C ∞ ( M ) . But this impliesthat viewed as an unbounded operator on L with domain H m + , the operator ˆ p has also a (cid:28)nitedimensional kernel ontained in C ∞ ( M ) . On the other hand, ˆ p has a self-adjoint extension on L thanks to Lemma 11. Denote by D the domain of this extension. This leads to the orthogonalde omposition L = im (ˆ p ) ⊥ ⊕ ker (ˆ p ) and the restri tion ˆ p : im (ˆ p ) ∩ D → im (ˆ p ) is thus invertible.Therefore, the operator ˆ P given in matrix form by ˆ P := ˆ p | im (ˆ p ) ∩D I ! is invertible. We (cid:28)rst remark that ˆ P is related to ˆ p by ˆ P = ˆ p (1 − π ) + π with π : L → ker ˆ p the orthogonal L -proje tion. Sin e ker p is a (cid:28)nite dimensional subspa eof C ∞ ( M ) , the proje tion π is a smoothing operator. It follows that ˆ r := ˆ P − ˆ p = (1 − ˆ p ) π is self-adjoint and smoothing. In parti ular ˆ P ∈ Ψ m ( x,ξ ) ρ is a PDO and de(cid:28)nes an inje tivemap C ∞ ( M ) → C ∞ ( M ) . On the other hand, the existen e of a parametrix ˆ Q ∈ Ψ − m ( x,ξ ) ρ for ˆ p , and thus for ˆ P , implies that ˆ P is also surje tive C ∞ ( M ) → C ∞ ( M ) . Finally, denote by ˆ P − : C ∞ ( M ) → C ∞ ( M ) the inverse of ˆ P whi h is ontinuous by the open mapping Theorem.One has ˆ Q = ˆ Q ˆ P ˆ P − = ˆ P − + ˆ r ˆ P − with ˆ r smoothing. The last term is also smoothing sin e ˆ P − is ontinuous and ˆ r smoothing. Thisimplies that ˆ P − is itself a PDO in Ψ − m ( x,ξ ) ρ .Colle ting the results of this se tion, we obtain the following orollary.Corollary 4. For any real ellipti symbol q ∈ S m ( x,ξ ) ρ , there is an operator ˆ p ∈ Ψ m ( x,ξ ) ρ satisfying ˆ p − ˆ q ∈ Ψ m ( x,ξ ) − (2 ρ − ρ , whi h is formally self-adjoint and invertible C ∞ ( M ) → C ∞ ( M ) .Proof. Let ˆ q ∈ Ψ m ( x,ξ ) ρ be the quantized of the symbol q and take the real part ˆ a := (ˆ q + ˆ q ∗ ) . It isself-adjoint and a ording to Theorem 7, its symbol satis(cid:28)es a ( x, ξ ) = q ( x, ξ ) mod S m ( x,ξ ) − (2 ρ − ρ .Then, Lemma 9 implies that the ellipti ity of q is not destroyed by a modi(cid:28) ation of order m ( x, ξ ) − (2 ρ − . Therefore a ∈ S m ( x,ξ ) ρ is ellipti as-well. Finally, Theorem 10 shows that thisimplies the existen e of a parametrix for ˆ a . Consequently, one an (cid:28)nd a self-adjoint PDO ˆ p = ˆ a mod Ψ −∞ whi h is invertible C ∞ ( M ) → C ∞ ( M ) and with inverse in Ψ − m ( x,ξ ) ρ (see Lemma 12).43.4.3 L - ontinuity and quasi- ompa ityThe next result, due originally to Hörmander, is very useful. It tells us that one an takethe (cid:16)square root(cid:17) of a positive ellipti operator.Lemma 13. Let p ∈ S m ( x,ξ ) ρ be an ellipti symbol satisfying p − Re ( p ) ∈ S m ( x,ξ ) − ερ forsome ε > and Re ( p ( x, ξ )) > for h ξ i ≥ c . Then, there exists ˆ b ∈ Ψ m ( x,ξ ) ρ su h that ˆ p − ˆ b ∗ ˆ b ∈ Ψ −∞ . Proof. Thanks to Lemma 9, the real part Re ( p ) is ellipti and satis(cid:28)es therefore Re ( p ) ≥ C h ξ i m ( x,ξ ) for large h ξ i . This implies that we an ertainly (cid:28)nd a smooth b ( x, ξ ) whi h o-in ides with p Re ( p ) for large h ξ i , i.e., outside from a ompa t set. Straightforward omputationsshow that b ∈ S m ( x,ξ ) ρ . On the other hand we have | b | = Re ( p ) mod S −∞ . Then, the symboli al ulus gives ˆ b ∗ ˆ b = \ Re ( p ) mod Ψ m ( x,ξ ) − (2 ρ − ρ = b p mod Ψ m ( x,ξ ) − ερ , where we have assumed without loss of generality that ε ≤ ρ − . The rest of the pro edure is astandard iterative onstru tion whi h shows that for any N , there are ˆ b j ∈ Ψ m ( x,ξ ) − jερ , j : 1 ..N ,satisfying (cid:16) ˆ b ∗ + ... + ˆ b ∗ N (cid:17) (cid:16) ˆ b + ... + ˆ b N (cid:17) = b p mod Ψ m ( x,ξ ) − ( N +1) ερ . Then, a Borel resummation (Theorem 6) yields the result.The last but not the least result of this appendix is standard, sin e it on erns symbolsof ( onstant) order . It is usually a way to prove L - ontinuity of PDO's, but it yieldsalso a way to show that a PDO's with a (cid:16)small(cid:17) symbol is quasi- ompa t, whi h is theproperty we use in the ontext of Ruelle-Polli ott resonan es.Lemma 14. Let p ∈ S ρ be a symbol and denote L = lim sup ( x,ξ ) ∈ T ∗ M | p ( x, ξ ) | . Then, for any ε > there is a de omposition ˆ p = ˆ p ε + ˆ K ε with ˆ K ε ∈ Ψ −∞ and k ˆ p ε k ≤ L + ε . 44roof. The (cid:28)rst remark is that for any ε > the operator (cid:0) L + ε (cid:1) I − ˆ p ∗ ˆ p =: ˆ q is self-adjoint andin the lass S ρ , whi h means q − R ( q ) ∈ S − (2 ρ − ρ . On the other hand, q = (cid:0) L + ε (cid:1) − | p | modulo Ψ − (2 ρ − ρ , whi h is positive for large ξ . Therefore we an apply Lemma 13 and obtain ˆ b ∈ Ψ ρ su hthat ˆ q = ˆ b ∗ ˆ b − ˆ K with K ∈ Ψ −∞ . Then, for any u ∈ L ( M ) one has k ˆ p ( u ) k = (ˆ p ∗ ˆ p ( u ) , u )= (cid:0) L + ε (cid:1) k u k − (cid:16) ˆ b ∗ ˆ b ( u ) , u (cid:17) + (cid:16) ˆ K ( u ) , u (cid:17) = (cid:0) L + ε (cid:1) k u k − (cid:13)(cid:13)(cid:13) ˆ b ( u ) (cid:13)(cid:13)(cid:13) + (cid:16) ˆ K ( u ) , u (cid:17) . From this follows the upper bound k ˆ p ( u ) k ≤ (cid:0) L + ε (cid:1) k u k + (cid:16) ˆ K ( u ) , u (cid:17) . (62)The next step is to introdu e the spe tral proje tor π λ of the Lapla ian − ∆ on ( −∞ , λ ] for largeenough λ , whi h will be hosen later depending on ε in a suitable way. Noti e that this proje tionis smoothing. Then, we de ompose ˆ p = ˆ p ε + ˆ r ε := ˆ p (1 − π λ ) + ˆ pπ λ . It follows (cid:28)rst that ˆ r ε is smoothing. On the other hand, the upper bound (62) yields k ˆ p ε ( u ) k ≤ (cid:0) L + ε (cid:1) k (1 − π λ ) u k + (cid:16) ˆ K (1 − π λ ) ( u ) , (1 − π λ ) ( u ) (cid:17) ≤ (cid:0) L + ε (cid:1) k u k + (cid:13)(cid:13)(cid:13) ˆ K (1 − π λ ) (cid:13)(cid:13)(cid:13) k u k where we have used k − π λ k ≤ and the Cau hy-S hwarz inequality. Finally, we show that we an make (cid:13)(cid:13)(cid:13) ˆ K (1 − π λ ) (cid:13)(cid:13)(cid:13) arbitrarily small. Sin e ˆ K is smoothing, it is ontinuous H s → L for any N > . In parti ular, we an de ompose ˆ K (1 − π λ ) = ˆ K (1 − ∆) N (1 − ∆) − N (1 − π λ ) and (cid:13)(cid:13)(cid:13) ˆ K (1 − ∆) N (cid:13)(cid:13)(cid:13) L ≤ C N . On the other hand, the spe tral theorem yields (cid:13)(cid:13)(cid:13) (1 − ∆) − N (1 − π λ ) (cid:13)(cid:13)(cid:13) ≤ λ − N . Therefore we have showed that (cid:13)(cid:13)(cid:13) ˆ K (1 − π λ ) (cid:13)(cid:13)(cid:13) ≤ C N λ N whi h an be made arbitrarily small,by taking λ = ε for example. This proves k ˆ p ε ( u ) k ≤ (cid:0) L + 2 ε (cid:1) k u k .Referen es[1℄ D. Anosov. Geodesi (cid:29)ows on ompa t riemannian manifolds of negative urvature.Pro eedings of the Steklov Mathemati al Institute, 90:1:1(cid:21)235, 1967.[2℄ M. F. Atiyah and R. Bott. A Lefs hetz (cid:28)xed point formula for ellipti omplexes. I.Ann. of Math. (2), 86:374(cid:21)407, 1967. 453℄ V. Baladi. Positive transfer operators and de ay of orrelations. Singapore: WorldS ienti(cid:28) , 2000.[4℄ V. Baladi. Anisotropi Sobolev spa es and dynami al transfer operators: C ∞∞