Micro-local analysis of contact Anosov flows and band structure of the Ruelle spectrum
MMicro-local analysis of contact Anosov flows and bandstructure of the Ruelle spectrum
Frédéric Faure
Univ. Grenoble Alpes, CNRS, Institut Fourier, F-38000 Grenoble, [email protected]
Masato Tsujii
Department of Mathematics, Kyushu University,Moto-oka 744, Nishi-ku, Fukuoka, 819-0395, [email protected]
Abstract
We develop a geometrical micro-local analysis of contact Anosov flow, such as geodesicflow on negatively curved manifold. We use the method of wave-packet transform discussedin [17] and observe that the transfer operator is well approximated (in the high frequencylimit) by a “quantization” of an induced transfer operator acting on sections of somevector bundle on the trapped set. This gives a few important consequences: The discreteeigenvalues of the generator of transfer operators, called Ruelle spectrum, are structuredinto vertical bands. If the right-most band is isolated from the others, most of the Ruellespectrum in it concentrate along a line parallel to the imaginary axis and, further, thedensity satisfies a Weyl law as the imaginary part tend to infinity. Some of these resultswere announced in [14]. Remark . On this pdf file, you can click on the colored words, they contain an hyper-linkto wikipedia or other multimedia contents. See Appendix A for some convention of notationsused in this paper. a r X i v : . [ m a t h . D S ] F e b ontents X X and pull back operator e tX . . . . . . . . . . . . . . . . . . . . . 142.2 More general pull back operator e tX F with generator X F . . . . . . . . . . . . . 152.3 Anosov vector field . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 152.4 Contact Anosov flow . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16 X g on T ∗ M . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 163.2 Wave-packet transform T . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 173.3 Lifted flow ˜ φ t : T ∗ M → T ∗ M and its generator ˜ X . . . . . . . . . . . . . . . . . 173.4 Propagation of singularities . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19 ˜ φ t on the cotangent space T ∗ M Σ . . . . . . . . . . . . . . . . . . . . . . . . . . . . 25 W . . . . . . . . . . . . . . . . 265.2 Anisotropic spaces H W ( M ; F ) . . . . . . . . . . . . . . . . . . . . . . . . . . . . 265.3 Decay of norm outside the trapped set . . . . . . . . . . . . . . . . . . . . . . . 275.4 Discrete Ruelle spectrum in anisotropic Sobolev spaces H W ( M ; F ) . . . . . . . 27 Σ N → Σ . . . . . . . . . . . . . . . . . . . . . . . 286.2 Equivalent family of operators . . . . . . . . . . . . . . . . . . . . . . . . . . . . 306.3 Description of the operator e tX near the trapped set Σ . . . . . . . . . . . . . . 306.4 Factorization formula for ˜Op Σ (cid:16) d ˜ φ t (cid:17) . . . . . . . . . . . . . . . . . . . . . . . . 326.5 Transfer operators on N s . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32 B.1 Transfer operator . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 57B.2 More general pull back operators e tX F . . . . . . . . . . . . . . . . . . . . . . . . 57 C Bargmann transform and Metaplectic operators 58
C.1 Weyl Heisenberg group . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 58C.2 Bargmann Transform on a Euclidean vector space ( E, g ) . . . . . . . . . . . . . 59C.3 Linear map φ : ( E , g ) → ( E , g ) . . . . . . . . . . . . . . . . . . . . . . . . . . 62C.4 Compatible triple g, Ω , J . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 64C.5 Bergman projector on a symplectic vector space ( F, Ω , g ) . . . . . . . . . . . . . 65C.6 The metaplectic decomposition of F ⊕ F ∗ . . . . . . . . . . . . . . . . . . . . . 65C.7 Metaplectic decomposition of a linear symplectic map Φ : ( F , Ω ) → ( F , Ω ) . . 67C.8 Some useful decomposition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 69C.9 Taylor operators T k on S ( E ) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 69C.10 Analysis on T ( E ⊕ E ∗ ) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 70 D Linear expanding maps 73
D.1 Anisotropic Sobolev space H W ( E ) . . . . . . . . . . . . . . . . . . . . . . . . . 73D.2 Result . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 74D.3 Discrete Ruelle spectrum in a simple toy model . . . . . . . . . . . . . . . . . . 753 Introduction
Contact Anosov flow:
In this paper we consider a smooth contact Anosov vector field X on a closed contact manifold ( M, A ) with contact one form A . The vector field X isconsidered as a derivation X : C ∞ ( M ) → C ∞ ( M ) and generates the pull-back operators bythe flow φ t : M → M , i.e. e tX u = u ◦ φ t with u ∈ C ∞ ( M ) and t ∈ R . More generally, if F is avector bundle over M (for example the differential forms F = Λ • ( M ) ) we will consider the Liederivative X F acting on smooth sections C ∞ ( M ; F ) of F over the vector field X , generatorof the pull back operator e tX F . As a particular and important example, if ( N , g N ) is a closedRiemannian manifold with negative curvature, the geodesic flow X is a contact Anosov vectorfield on the unit cotangent bundle M = T ∗ N , where the contact one form A = pdq is theLiouville one-form. Question of long time behavior:
Anosov flow is a typical model of chaotic dynamicswith sensitivity to initial conditions. A typical question in dynamical systems theory that isaddressed in this paper is to describe the long time action of the flow generated by X onsmooth sections, i.e. for any given smooth sections u, v ∈ C ∞ ( M ; F ) , describe the correlation (cid:104) v | e tX F u (cid:105) L ( M ; F ) for t → ∞ ? In other words we are looking for a “good description” of the pullback operator e tX F for t → ∞ .In the case of a trivial bundle F = C , it is known from Sinaï [30] that contact Anosovflows have mixing property, i.e. for any smooth functions u, v ∈ C ∞ ( M ; C ) , we have (cid:104) v | e tX u (cid:105) L → t → + ∞ (cid:104) v | (cid:105) L (cid:104) Vol( M ) | u (cid:105) L where L = L ( M, dm ) with the invariant measure dm (2.12) on M . This mixing property dominates the long time behavior and says that the op-erator e tX converges (in the weak sense) to the rank one operator Π = (cid:104) Vol( M ) | . (cid:105) L . In thispaper we will be interested to find a description of the operator e tX in the limit of large t , upto an error that decays as e − Λ t with an arbitrary large rate Λ > . Discrete Ruelle spectrum and anisotropic Sobolev space:
Although the space L ( M ; F ) may be natural to consider in order to describe the operator e tX F , this is not what we will do.We will instead deform L ( M ; F ) and consider a family of other spaces H W ( M ; F ) , definedfrom a weight function W on T ∗ M , in which the essential spectrum of the generator X F ispushed in the direction Re ( z ) → −∞ as far as we want, revealing some intrinsic discretespectrum called Ruelle resonances . See Figure 1.2. This fact has been first obtained byButterley and Liverani in [4].This discrete spectrum governs the long time behavior of e tX F but only few is knownabout it for a general Anosov flow (or Axiom A flow). The function space H W ( M ; F ) iscalled anisotropic Sobolev space [22][2][12, 13] because it is a Hilbert space of distributionalsections that contains smooth sections C ∞ ( M ; F ) and its order depends on the stable/unstabledirections of the dynamics.In the special case of a contact Anosov vector field X , much more can be said about theRuelle spectrum of X F at least about the effective action of the operator e tX F . This is thesubject of this paper, and for this, we will use a function space H W ( M ; F ) defined in a previouspaper [17] and designed for that purpose. For every k ∈ N , we introduce the finite rank vector bundle over M : F k := | det E s | − / ⊗ Pol k ( E s ) ⊗ F → M (1.1) See Definition (2.4). Ruelle spectrum can be calculated explicitly in only exceptional cases. Appendix D.3 presents the Ruellespectrum of a very simple toy model on R . See [14, Prop. 4.1] for the Ruelle spectrum of the geodesic flow ona compact hyperbolic surface. E s ⊂ T M is the stable directions of the Anosov flow, | det E s | / is the half densities bundleand | det E s | − / = (cid:16) | det E s | / (cid:17) ∗ its dual, Pol k ( E s ) is the bundle of homogeneous polynomials ofdegree k . Then we consider X F k : S (cid:91) ( M ; F k ) → S (cid:91) ( M ; F k ) being the Lie derivative of sectionsof F k induced from the initial derivation X F and the generated group e tX F k : S (cid:91) ( M ; F k ) →S (cid:91) ( M ; F k ) , t ∈ R , that we call the classical dynamics . We introduce the related quantities γ ± k := lim t →±∞ log (cid:13)(cid:13) e tX F k (cid:13)(cid:13) /tL ∞ . (1.2) Remark . We can give obvious (but rough) estimates for γ ± k as follows. If < λ min ≤ λ max denote the minimal and maximal Lyapounov exponents of the Anosov flow φ t , we have − (cid:18) d k (cid:19) λ max + c F ≤ γ − k ≤ γ + k ≤ − (cid:18) d k (cid:19) λ min + C F , (1.3)where dim M = 2 d + 1 and c F ≤ C F depend on X F (we have c F = C F = 0 when X F = X isthe vector field itself with F = C ).We consider the symplectic manifold Σ := R ∗ A = { ω A ( m ) , ω ∈ R ∗ , m ∈ M } being the symplectization of the contact manifold ( M, A ) . This manifold Σ , as a sub-manifold of T ∗ M , is also the trapped set (or non wandering set) of the induced dynamics ˜ φ t = ( dφ t ) ∗ on T ∗ M and therefore invariant. In order to define the quantization of the classical dynamics e tX F k : S (cid:91) ( M ; F k ) → S (cid:91) ( M ; F k ) , we first need to consider the pull back bundle F k → Σ on thesymplectic manifold Σ by π : Σ → M . This also extends the classical dynamics as e tX F k : S (cid:91) (Σ; F k ) → S (cid:91) (Σ; F k ) (1.4)over the Hamiltonian flow ˜ φ t = ( dφ t ) ∗ : Σ → Σ . We denote T k : S ( E s ) → Pol k ( E s ) ⊂ S (cid:48) ( E s ) the Taylor projector and extend it as a bundle operator T k ∈ L ( F , F (cid:48) ) with bundles F := | det E s | − / ⊗ S ( E s ) ⊗ F and F (cid:48) := | det E s | − / ⊗ S (cid:48) ( E s ) ⊗ F . For any given K ∈ N , we denote T [0 ,K ] := (cid:76) Kk =0 T k . For any t ∈ R , we have a well defined map e tX F : S (cid:91) (Σ; F ) → S (cid:91) (Σ; F ) .The quantization of e tX F T [0 ,K ] gives a quantum evolution operator for t ∈ R , Op σ ,σ (cid:0) e tX F T [0 ,K ] (cid:1) : C ∞ ( M ; F ) → C ∞ ( M ; F ) . This quantization depends on parameters σ , σ > that measures the distance to Σ ⊂ T ∗ M outside of which we truncate. See Figure 1.1. This quantization relies on metaplectic quan-tization on the tangent bundle T Σ and will be shown to be equivalent to Weyl quantizationin some simple case. The semi-classical parameter is the frequency ω along the flow direc-tion (equivalently the parameter along Σ ) and the semi-classical limit is | ω | → ∞ (usuallydenoted (cid:126) = 1 / | ω | → ).The next theorem shows how the pull-back operator e tX F is well approximated by theoperator Op σ ,σ (cid:0) e tX F T [0 ,K ] (cid:1) . Theorem 1.2. “Emergence of quantum dynamics” . For any K ∈ N and (cid:15) > , wecan choose an anisotropic Sobolev space H W ( M ; F ) , such that for any t > , we can take σ > σ > large enough and then the cut-off frequency ω > large enough such that e tX F = Op σ ,σ (cid:0) e tX F T [0 ,K ] (cid:1) + R ω + O (cid:16) e t ( γ + K +1 + (cid:15) ) (cid:17) , (1.5) where R ω is a finite rank operator. The proof of Theorem 1.2 is given in Section 7.4. The bundle E s → M is Hölder continuous but smooth along the flow and E s directions. See Notation 6.11for S (cid:91) ( M ; F k ) .
5n (1.5) the notation O ( ∗ ) is for an operator whose norm in H W ( M ; F ) is bounded by aconstant multiple of ∗ , independent on t . From (1.3) we have γ + k → k →∞ −∞ , hence the term O (cid:16) e t ( γ + K +1 + (cid:15) ) (cid:17) represents a family of operators that decay very fast for t → + ∞ in operatornorm in the anisotropic Sobolev space H W ( M ; F ) . Remark . One interpretation of Theorem 1.2 is that for large time t (cid:29) , an effective quan-tum dynamics emerges from the contact Anosov dynamics . This quantum dynamicsis the quantization of the Hamiltonian dynamics that takes place on Σ .To show (1.5) we will use microlocal analysis directly on T ∗ M following the approach pro-posed in [17], using a metric g on T ∗ M compatible with the symplectic form, that measuresthe size of wave-packets or coherents states in accordance with the uncertainty principle. Few comments about the result (1.5) • As explained in the introduction, in the case of a trivial bundle F = C , contact Anosovflow have mixing property e tX → t → + ∞ (cid:104) Vol( M ) | . (cid:105) L . Although this mixing propertydominates the long time behavior, the limit operator here (cid:104) Vol( M ) | . (cid:105) L , belongs to thefinite rank operator R ω in (1.5) that we do not describe in this paper. We insteadconsider terms that are exponentially small compared to it, but that belongs to an infinitedimensional effective space.• In (1.1), the vector bundle F → M is arbitrary but notice that the special choice of thebundle F = | det E s | / is specially interesting because it gives F = (6 . | det E s | − / ⊗ Pol ( E s ) ⊗ | det E s | / = C (1.6)i.e. the trivial bundle over Σ . The paper [16] is devoted to that case. Notice that F = | det E s | / is not a smooth bundle but only Hölder continuous and technically weneed to consider an extension to a Grassmanian bundle. Eq.(1.2) gives γ ± = 0 and γ +1 < . For that case and taking K = 0 , Eq.(1.5) shows that for t (cid:29) , the operator e tX F is well described by the dominant term Op σ (cid:0) e tX F T (cid:1) that is the “quantization” ofthe dynamics X itself. From Theorem 1.4 below, for ω = Im ( z ) → ∞ , the eigenvalues of X F accumulate on the imaginary axis Re ( z ) = γ ± = 0 with density given by the Weyllaw, separated by a uniform spectral gap γ +1 < . See Figure 1.2.• In the special case of the geodesic flow on a surface of constant and negative curvature,i.e. N = Γ \ SL ( R ) / SO ( R ) , giving M = Γ \ SL , with Γ being a co-compact subgroup of SL R , then (1.5) is exact and the dominant quantum operator Op σ (cid:0) e tX F T (cid:1) is conjugated(hence with same spectrum) to the wave operator exp (cid:16) ± it (cid:113) ∆ − (cid:17) on S ( N ) , [14, 7].This operator is indeed considered in physics as the Schrödinger operator, giving thequantum description of a free particle.For the case of geodesic flow on a non-constant negatively curved Riemannian manifold ( N , g ) , let us observe that the symplectic phase space Σ is (symplectically) isomorphicto a double cover of the cotangent bundle T ∗ N \ { } and that the principal symbol of thegenerator of the leading quantum operator Op σ (cid:0) e tX F (cid:1) in (1.5) is the frequency ω , equalto the principal symbol of √ ∆ on T ∗ N under this isomorphism, where ∆ = d † d is theLaplace Beltrami operator on S ( N ) . However we do not expect that the spectra of bothoperators coincide in general. We will investigate this question in a future work.• In semi-classical analysis, the Egorov Theorem [33, p.26] or the WKB approximation [3,p.11] show that classical Hamiltonian dynamics emerges in the high frequency limit (andfinite time) of quantum dynamics. This is how in physics, geometrical optics is derived6rom electromagnetic waves, Newtonian (and Hamilton) mechanics is derived from quan-tum waves mechanics of Schrödinger, etc. Inspired from these physical phenomena thisis how quantization has been defined in mathematics, for example ordinary quantization[33, p.2] that defines a pseudo-differential operator on S ( R n ) from a Hamiltonian function(symbol) on R n , or geometric quantization in a more geometric setting [36]. In this paperwe have exhibited the converse and maybe unexpected phenomena: how quantummechanics emerges from the classical mechanics when this later is chaotic. Fromthe mathematical point of view, an interesting consequence is that it furnishes a natu-ral quantization of a given classical dynamics among all possible quantizations. Thisnatural quantization has indeed the preferable properties (that characterize it) that thesemi-classical Van-Vleck formula or semi-classical Trace formula are asymptotically exact,i.e. they have error terms that decay exponentially fast with t → ∞ at large but fixed ω .See [15, Section 1.5,1.6,1.7] for more discussions. The proof of the emerging quantum dynamics in (1.5) can be summarized by the followingmechanisms that will be explained in details in this paper and illustrated on Figure 1.1.1. In the limit of high frequencies, evolution of functions by the pull-back operator e tX F iswell described on the cotangent bundle T ∗ M with the induced flow ˜ φ t := ( dφ t ) ∗ , t ∈ R .This is because e tX F is a Fourier integral operator (i.e. has micro-local property (3.20)).2. For an Anosov contact flow, the dynamics induced on the cotangent bundle T ∗ M is a“scattering dynamics” on the trapped set Σ ⊂ T ∗ M and Σ is symplectic and normallyhyperbolic. See Figure 1.1. In terms of dynamics, the subset Σ is the non wandering setfor the flow ˜ φ t on T ∗ M and the orbits on the outside of Σ go to infinity either as t → + ∞ or t → −∞ . As a consequence, for large time | t | (cid:29) , the outer part of the trapped set Σ has a negligible contribution, because information (measured by our adapted operatornorm) escapes away. This will be given in Theorem 5.2. So only the dynamics on Σ playsa role for our purpose. But due to the uncertainty principle in T ∗ M , we still have toconsider a vicinity of Σ in T ∗ M . The uncertainty is precisely measured by a metric g on T ∗ M compatible with the symplectic structure. We will hence consider a vicinity of Σ ofa given size σ (measured by g ). It is important to remark that we will take σ large, butthe vicinity of size σ at frequency ω projected down on M has size (cid:16) σω − / that goesto zero as ω goes to infinity. This will allow us to use the linearization of the dynamicsas a local approximation.3. In a vicinity of size σ of the trapped set Σ (i.e. the non wandering set) that matters, thereis a micro-local decoupling between the directions tangent to Σ and those (symplectically)normal to Σ , represented by a normal vector bundle denoted N → Σ . The dynamics onthe normal direction N is hyperbolic and responsible for the emergence of polynomialfunctions along the stable direction N s ≡ E s . We introduce approximate projectors Op σ ( T k ) that restricts functions to the symplectic trapped set Σ and that are polynomialvalued along N s with degree k . This projector plays a similar role as the Bergmanprojector (or Szegö projector) in geometric quantization. What remains for large time, isan effective Hilbert space of quantum waves that live on the trapped set Σ , valued in thevector bundle F k (6.22).The very simple toy model that is useful to have in mind is given in Section D.3. It shows inparticular the emergence of polynomials Pol k ( E s ) as in (6.22). low X N s ( ρ ) m E s σT ∗ MM E u Trapped set Σ ρ = ω A ( m ) Size (cid:16) σω − / N u ( ρ ) N ( ρ ) Figure 1.1: The dynamics induced on T ∗ M scatters on the trapped set Σ ⊂ T ∗ M definedin (4.9). Σ is a symplectic sub-manifold of T ∗ M and at every point ρ = ω A ( m ) ∈ Σ , thesymplectic-normal bundle N ( ρ ) = N u ( ρ ) ⊕ N s ( ρ ) (a symplectic linear subspace of T ρ T ∗ M )splits into unstable/stable subspaces, see (6.2). The main geometrical object consideredin this paper is this fibration N s → Σ → M . Beware that for a geodesic flow on ( N , g ) , thisfibration sequence continues with M = T ∗ N → N . γ +2 γ − γ +1 γ − γ +0 B ω ε Re( z ) H Spectrum that controlsemerging behavior b o und e d r e s o l v e n t r (cid:29) discrete spectrumIntrinsic bounded resolventbounded resolvent ˇ γ − λ max r + C (cid:48) Essential spectrum ω = Im( z ) − λ min r + C B B Figure 1.2: The dots represent the intrinsic Ruelle discrete spectrum of the Lie derivative X F ina Sobolev space H W ( M ; F ) . From [17], the essential spectrum is in a vertical band that can bemoved arbitrarily far on the left by changing the weight W , and reveals this intrinsic discretespectrum. The right most eigenvalues in the first band B dominate the emerging behavior of e tX F for t (cid:29) . 8here are many consequences of the effective description (1.5) of the dynamics. In thispaper we describe few of them that are illustrated on Figure 1.2.The generator of the classical dynamics (1.4) is X F k and its spectrum in L (Σ; F k ) iscontained in the vertical band B k := (cid:2) γ − k , γ + k (cid:3) × i R with γ ± k defined in (1.2). We alwayshave γ − k +1 < γ − k and γ + k +1 < γ + k but this does not guaranty that the bands are separatedby gaps i.e. that γ + k +1 < γ − k , B k +1 ∩ B k = ∅ . We will follow general ideas from classical-quantum correspondence principles for quantization in the case where the classical symbol isan operator valued function on a symplectic manifold Σ (this situation is also present in physicsfor elastic waves, electromagnetic waves, Dirac equation etc), we derive the following resultsfor the operator X F . In the semi-classical limit, i.e. for ω = Im ( z ) → ±∞ on the spectral plane z ∈ C , Theorem 1.4below shows that the spectrum of X F is discrete and asymptotically contained in the verticalbands (cid:83) k ∈ N B k and the resolvent is uniformly bounded in the gaps. This is illustrated on Figure1.2. Theorem 1.4. “Band structure of the Ruelle spectrum”.
For any (cid:15) > , C > ,there exists C (cid:15) > , ω (cid:15) > such that the Ruelle eigenvalues σ ( X F ) are contained inthe following spectral domain that consists of a union of a “low frequency horizontalband” and “vertical bands” : ( σ ( X F ) ∩ { Re ( z ) > − C } ) ⊂ {| Im ( z ) | ≤ ω (cid:15) } ∪ (cid:32) (cid:91) k ∈ N (cid:8) Re ( z ) ∈ (cid:2) γ − k − (cid:15), γ + k + (cid:15) (cid:3)(cid:9)(cid:33) , (1.7) and the resolvent operator is uniformly bounded in the “gaps”: (cid:13)(cid:13) ( z − X ) − (cid:13)(cid:13) H W ≤ C (cid:15) , ∀ z ∈ C s.t. | Im ( z ) | > ω (cid:15) , Re ( z ) ∈ (cid:2) γ + k +1 + (cid:15), γ − k − (cid:15) (cid:3) . (1.8)The proof is given in Section 7.3.From a general theorem in semi-group theory [8, p.276, p.307], Theorem 1.4 that concernsthe generator X F is equivalent to the following result that concerns the semi-group (cid:0) e tX F (cid:1) t ≥ and illustrated on Figure 1.3. Theorem 1.5. ” Ring spectrum”.
Let k ∈ N . For any t > , the op-erator e tX F : H W ( M ; F ) → H W ( M ; F ) has discrete spectrum on the ring (cid:110) z ∈ C , e tγ + k +1 < | z | < e tγ − k (cid:111) (if non empty, i.e. if γ + k +1 < γ − k ) and on | z | > e tγ +0 .Remark . An immediate consequence of Theorem 1.5, if γ + k +1 < γ − k for example, is thepossibility to perform a contour integral of the resolvent of e tX F on the circle of radius r = e t ( γ + k +1 + (cid:15) ) with γ + k +1 + (cid:15) < γ − k , (cid:15) > so that there is no eigenvalue on this circle. UsingCauchy integral and holomorphic functional calculus, we get a spectral projector Π [0 ,k ] = Id − πi (cid:117) (cid:0) z − e tX F (cid:1) − dz on the first k external bands and one can approximate e tX F for t > bythe spectral restriction e tX F Π [0 ,k ] with the following estimate: (cid:13)(cid:13) e tX F − e tX F Π [0 ,k ] (cid:13)(cid:13) H W ≤ C (cid:15) e t ( γ + k +1 + (cid:15) ) , where e tX F Π [0 ,k ] is compared to a quantum evolution operator in this paper. This operator canbe used to describe with great accuracy the decay of correlations (cid:104) v | e tX F u (cid:105) as t → ∞ .9 tγ +0 e tγ − e tγ +1 e tγ − Im( z ) Re( z ) Figure 1.3: Band spectrum of e tX F in H W ( M ; F ) for t > , from Theorem 1.5. For k ∈ N , and γ ± k defined in (1.2), the operator e tX F has discrete spectrum in the annulus e tγ + k +1 < | z | < e tγ − k (if γ + k +1 < γ − k ) called “spectral gap”. It has also discrete spectrum on | z | > e tγ +0 and possiblyessential spectrum elsewhere. Compare with Figure 1.2. If B k ∩ (cid:16)(cid:83) k (cid:48) (cid:54) = k B k (cid:48) (cid:17) = ∅ , i.e. the band B k is isolated, Theorem 1.7 below shows that the densityof discrete eigenvalues of X F in B k in the limit ω → ∞ approaches to rank ( F k ) Vol ( M ) ω d (2 π ) d +1 with rank ( F k ) = (1 . (cid:18) k + d − d − (cid:19) rank ( F ) . (1.9)where dim M = 2 d + 1 . It is analogous to the usual Weyl law. Theorem 1.7. “ Weyl law for isolated bands ”. Let k ∈ N . For any (cid:15) > , such that γ + k +1 < γ − k − (cid:15) and γ + k + (cid:15) < γ − k − (this last condition is for k ≥ only) lim δ → + ∞ lim sup ω →±∞ | ω | d δ (cid:93) (cid:8) σ ( X F ) ∩ (cid:0)(cid:2) γ − k − (cid:15), γ + k + (cid:15) (cid:3) × i [ ω, ω + δ ] (cid:1)(cid:9) = rank ( F k ) Vol ( M )(2 π ) d +1 (1.10)The proof is obtained in Section 8. Remark . The result holds true for a group of isolated bands k ∈ [ k , k ] with k ≤ k ,assuming γ + k +1 < γ − k and γ + k < γ − k − . For the case k = 0 we only have to assume γ + k +1 < γ − k . We define in (9.1) the maximal and minimal exponents ˇ γ − k ≤ ˇ γ + k of the bundle maps e tX F k with respect to the contact volume on Σ , satisfying (cid:2) ˇ γ − k , ˇ γ + k (cid:3) ⊂ (cid:2) γ − k , γ + k (cid:3) . The following theoremin addition to Theorem 1.7 shows that most of eigenvalues in band B k belong to a narrowerband Re ( z ) ∈ (cid:2) ˇ γ − k , ˇ γ + k (cid:3) . 10 heorem 1.9. “Ergodic concentration of the spectrum” . Let k ∈ N . For any (cid:15) > small enough, δ > and ω → + ∞ , we have lim δ → + ∞ lim sup ω →±∞ | ω | d δ (cid:93) (cid:8) σ ( X F ) ∩ (cid:0)(cid:2) γ − k − (cid:15), γ + k + (cid:15) (cid:3) \ (cid:2) ˇ γ − k − (cid:15), ˇ γ + k + (cid:15) (cid:3)(cid:1) × i [ ω, ω + δ ] (cid:9) = 0 . (1.11)The proof is given in Section 9.For example, in the special case of band k = 0 and rank ( F ) = 1 we have rank ( F ) = (1 . ,and from ergodicity of X , most of eigenvalues accumulate on the vertical line ˇ γ − = ˇ γ +0 = 1Vol ( M ) (cid:90) M Ddm being the space average of the “damping function” D ∈ C ( M ; R ) defined by D ( m ) := V ( m ) + div X /E s ( m ) with the potential function V given in (2.5). The band structure of the Ruelle spectrum and Weyl law described above reflect in fact adeeper geometric origin that can be explained in terms of geometric quantization. Let us saymore about this.
Wave-packet transform T : In paper [15, Section 4.1.5], see also (3.5), we introduce awave-packet transform T : C ∞ ( M ; F ) → S ( T ∗ M ; F ) , (1.12)satisfying T † T = Id L ( M ; F ) and that is used for micro-local analysis on the whole cotangentspace T ∗ M . For example, using a characteristic function χ Σ ,σ : T ∗ M → [0 , for the neigh-borhood of Σ ⊂ T ∗ M at distance σ > measured with a specific metric g , we define theoperator ˆ χ σ = (5 . T † χ Σ ,σ T : C ∞ ( M ; F ) → C ∞ ( M ; F ) that “restricts functions to their micro-local components” near Σ .Theorem 5.2 will show that the norm decays exponentially fast outside the trapped set Σ with an arbitrary exponential rate, i.e. we have (cid:13)(cid:13) e tX F (Id − ˆ χ σ ) (cid:13)(cid:13) H W ( M ) ≤ Ce − t Λ for t ≥ , withan arbitrary rate Λ > . This implies that for our study, we can consider only the componentof the dynamics e tX ˆ χ σ near the trapped set. Quantization:
For a map a ∈ S (cid:91) (Σ; End ( F )) and time t ∈ R , we will define a quantumoperator in Definition 7.1 Op σ ,σ (cid:0) e tX F a (cid:1) : C ∞ ( M ; F ) → C ∞ ( M ; F ) . In (6.7), we define an equivalence relation ≈ t between two families of Fourier integral op-erators that depend on σ > (or σ , σ > ). This equivalence means that for any t ∈ R , thedifference of the Schwartz kernel of the operators on phase space and on the graph of ˜ φ t , van-ishes for large σ (or σ large and then σ large depending on t , that we write as σ (cid:29) σ (cid:29) )and for high frequencies | ω | → ∞ .In Theorem 7.2 we will obtain Theorem 1.10. “Approximation of the dynamics by quantum operator”.
Forany t ∈ R , and σ (cid:29) σ (cid:29) large enough depending on t , we have e tX F ˆ χ σ ≈ t Op σ ,σ (cid:0) e tX F (cid:1) ˆ χ σ . (1.13)11n Theorem 7.7 we will obtain Theorem 1.11. “Composition formula”.
For slowly varying symbols a, b ∈S (cid:91) (cid:0) Σ; End (cid:0) F [0 ,K ] (cid:1)(cid:1) and any t, t (cid:48) ∈ R , σ , σ , σ , σ > we have Op σ ,σ (cid:0) e tX F a (cid:1) Op σ ,σ (cid:16) e t (cid:48) X F b (cid:17) ≈ t + t (cid:48) Op σ ,σ (cid:16) e tX F ae t (cid:48) X F b (cid:17) . (1.14)In particular, since for any k, k (cid:48) ∈ N we have [ T k , T k (cid:48) ] = δ k = k (cid:48) T k and (cid:2) T k , e tX F (cid:3) = 0 , we willget in corollary 7.8 that Op σ (cid:0) e tX F T k (cid:1) Op σ (cid:16) e t (cid:48) X F T k (cid:48) (cid:17) ≈ t + t (cid:48) δ k = k (cid:48) Op σ (cid:16) e ( t + t (cid:48) ) X F T k (cid:17) . (1.15)In Theorem 7.4 we will also get some boundness theorems that reflect Definition 1.2 at thequantum level, such as Theorem 1.12. “Boundness” . For any k ∈ N , ∀ (cid:15) > , ∃ C k,(cid:15) > , ∀ t ≥ , ∀ σ , σ > , ∃ ω σ , ω σ > , (cid:13)(cid:13) Op σ ,σ (cid:0) e tX F T k (cid:1)(cid:13)(cid:13) H W ( M ) ≤ C k,(cid:15) e t ( γ + k + (cid:15) ) , (1.16) (cid:13)(cid:13) Op σ ,σ (cid:0) e − tX F T k (cid:1)(cid:13)(cid:13) H W ( M ) ≤ C k,(cid:15) e − t ( γ − k − (cid:15) ) . (1.17)The band spectrum of the Ruelle spectrum in Theorem 1.4 and Theorem 1.5 is a directmanifestation of these boundness estimates together with the algebraic structure of (1.15). If u ∈ C ( M ; E u ) , s ∈ C ( M ; E s ) are respectively Hölder continuous sections of the unsta-ble/stable bundles of the Anosov dynamics, we naturally associate to them the following“operator-valued symbols” (see Section 10 for a more detailed Definition) ι s : C (Σ; F k ) → C (Σ; F k − ) (1.18)by point-wise tensor-contraction with the polynomial in (1.1) and u ∨ : C (Σ; F k ) → C (Σ; F k +1 ) (1.19)by point-wise symmetric tensor product. We will see in Lemma 10.2 that they satisfy thepoint-wise Weyl algebra (also called symplectic Clifford algebra) [ ι s , u ∨ ] = ω ( d A ) ( s, u ) Id F . (1.20)where d A is the symplectic form on E u ⊕ E s . Theorem 1.13. “Quantized Weyl algebra”.
Quantization of the symbols (1.18) and(1.19) gives approximate horocycle operators Op σ ( ι s ) , Op σ ( u ∨ ) : C ∞ ( M ; F ) → C ∞ ( M ; F ) that satisfy the approximate Weyl algebra, for σ (cid:29) σ (cid:29) , ˆ χ σ [Op σ ( ι s ) , Op σ ( u ∨ )] ˆ χ σ ≈ ˆ χ σ Op σ (( d A ) ( s, u )) X ˆ χ σ (1.21) and for any t ∈ R and σ (cid:29) σ (cid:29) , ˆ χ σ e tX F Op σ ( ι s ) ˆ χ σ ≈ t ˆ χ σ Op σ ( ι dφ t s ) e tX F ˆ χ σ (1.22) ˆ χ σ e tX F Op σ ( u ∨ ) ˆ χ σ ≈ t ˆ χ σ Op σ (cid:0)(cid:0) dφ t u (cid:1) ∨ (cid:1) e tX F ˆ χ σ (1.23)12 roof. We deduce (1.21) from (1.20), using composition of operators (7.8) and notice that ω isthe principal symbol of X . We deduce Eq. (1.22) from Egorov formula (7.12).Theorem 1.13 can be considered as a generalization of the sl ( R ) algebra [ U, S ] = X thatcomes up with the geodesic flow on M = Γ \ SL ( R ) , and more generally the algebra of so ( n, ,see [18, 7, 24]. We can call Op σ ( ι s ) , Op σ ( u ∨ ) the “approximate horocycle operators ”.They respectively map band B k to B k ± . We present here some related works specifically on the topics of band structure of the Ruellespectrum for contact Anosov dynamics and the emergence of an effective quantum dynamicsas expressed in Eq.(1.5) using micro-local analysis. This paper is a continuation along thefollowing series of papers.• In [9], the case of a contact U (1) -extension of the Arnold’s cat map (cid:18) (cid:19) on T is treated. Band structure of the Ruelle spectrum and emergence of the quantum catmap dynamics has been shown. This model is also called “prequantum cat map” since itsconstruction follows the prequantization procedure of Kostant, Souriau, Kirillov that isequivalent to a contact U (1) -extension of the Hamiltonian dynamics. Remark . The symplectic form Ω on T is not exact so we need to consider a nontrivial line bundle L → T with curvature Ω , called prequantum line bundle. On theopposite, in the model of contact Anosov flow considered in this paper the symplecticform Ω = dθ on Σ ⊂ T ∗ M is exact, θ being the Liouville one-form (3.12), hence theprequantum line bundle L → Σ is trivial and we have decided to ignore it although itspresence manifests all along the paper, for example in the definition of the twist operator(3.24). In a more correct description, we could add L in the tensor product F ⊗ L → Σ in (6.22).• In [15], the case of a U (1) -extension of an arbitrary Anosov map φ : M → M is con-sidered (also called prequantum Anosov map). Band structure of the Ruelle spectrumand emergence of the quantum dynamics has been shown. The models in [9, 15] can beconsidered as toy models (or simplified models) for the contact Anosov flow consideredhere because they are analogous (but not equivalent) to a time one contact Anosov flowmap.• The paper [14] announced the band spectrum for contact Anosov flows, (less precise)Weyl law and the method presented in this paper.• In the paper [16], one has considered the interesting choice of the bundle F = | det E s | / ,giving the trivial bundle (1.6). This model is treated in relation with the “semi-classicalzeta function” that generalizes the Selberg zeta function to non constant curvature. Tech-niques presented in papers [16, 15] are enough to show the band structure for contactAnosov flows, however they are less geometrical than those in the present paper and there-fore less easy to use. Technically, the case of the Hölder continuous bundle F = | det E s | / is more tricky than the case a smooth bundle F and needs to consider an extension to aGrassmanian bundle.C. Guillarmou and M. Cekic in [23] prove the first band for contact Anosov flows in dimension3 using horocycle operators. For the special case of hyperbolic manifolds, endowed with analgebraic structure, band spectrum of Anosov dynamics has been studied in [7] and [24].In different situations, band structure and Weyl law for the spectrum of resonances hasbeen studied in [32],[31] for convex obstacles and by Semyon Dyatlov in [6] for regular normallyhyperbolic trapped sets. 13 .5 Organization of the paper In Section 2 we recall the definition of contact Anosov vector field X , the Lie derivative X F and its spectral properties, i.e. the discrete Ruelle spectrum. The subsequent Sections presentthe techniques that we use and the proofs of the main Theorems presented in Section 1.3.In Section 3 we present the micro-local analysis of a general non vanishing vector field X ona manifold M . The strategy is to lift the analysis of the evolution operator e tX on the cotangentbundle T ∗ M using the wave-packet transform. In the high frequency limit (semi-classical limit),the linearization of the flow gives a good description and the analysis takes place on T T ∗ M .Specifically for contact Anosov flow, in Section 4, we recall the main properties of the inducedHamiltonian flow ˜ φ t on T ∗ M with the definition of the trapped set Σ ⊂ T ∗ M (also called thesymplectization of ( M, A ) ). In Section 5 we recall from [17] that the properties of the pullback operator in the semi-classical limit ω → ∞ come from the analysis of ˜ φ t in a parabolicneighborhood of the trapped set Σ . This remark leads us in Section 6 to study the differentialof the lifted flow d ˜ φ t on T Σ T ∗ M . Since Σ is symplectic and normally hyperbolic, we denote N → Σ the normal symplectic vector bundle and N s the stable sub-bundle. There appear thevector bundles F k = | det N s | − / ⊗ Pol k ( N s ) ⊗ F → Σ , for every k ∈ N , that play a major role.The interpretation and consequences of the previous analysis are given in Section 7, wherewe get the emergence of a quantum dynamics on the vector bundles F k and band structure ofthe spectrum. Section 8 gives the proof of the Weyl law and Section 9 explains the concentrationin narrower bands. Section 10 gives Definitions and proofs for approximate horocycle operators.Appendix A gives some convention of notations used in this paper. Appendix B containsadditional comments about flows. In the previous analysis of the differential d ˜ φ t on T Σ T ∗ M ,we use Bargmann transform, i.e. wave-packet transform on an Euclidean vector space withmetaplectic operators Op (cid:16) d ˜ φ t (cid:17) obtained by quantization of the linear symplectic maps d ˜ φ t .All the details of this is given in Section C of the appendix. Also we use results for linearexpanding maps given in Section D. Acknowledgement.
F. Faure acknowledges Claude Gignoux, Colin Guillarmou, Victor Maucout,Malik Mezzadri, Stéphane Nonnenmacher for their support, for interesting and motivatingdiscussions. F. Faure acknowledges M.S.R.I. and organizers of the micro-local semester 2020where a part of this work has been developed. M. Tsujii acknowledges partially supported byJSPS KAKENHI Grant Number 15H03627 and 22340035 during this work. X X and pull back operator e tX Let M be a C ∞ closed connected manifold. Let X ∈ C ∞ ( M ; T M ) be a C ∞ vector field on M , considered as a first order differential operator acting on smooth functions C ∞ ( M ) , i.e. inlocal coordinates y = ( y , y , . . . , y dim M ) , X ≡ dim M (cid:88) j =1 X j ( y ) ∂∂y j . For t ∈ R , let φ t : M → M (2.1)be the C ∞ flow defined by d ( u ◦ φ t ) dt = Xu , ∀ u ∈ C ∞ ( M ) , i.e. e tX u := u ◦ φ t = (cid:0) φ t (cid:1) ◦ u, (2.2)where ( φ t ) ◦ = e tX denotes the pull back operator acting on functions (following notations inAppendix A). See Appendix B.1 for additional remarks about the transfer operator.14 .2 More general pull back operator e tX F with generator X F It will be interesting to consider the following more general situation. Let π : F → M be asmooth complex vector bundle of finite rank over M . Let X F a derivation over X , i.e. X F is a first order differential operator acting on sections of FX F : C ∞ ( M ; F ) → C ∞ ( M ; F ) (2.3)satisfying the Leibniz rule X F ( f u ) = X ( f ) u + f X F ( u ) , ∀ f ∈ C ∞ ( M ) , ∀ u ∈ C ∞ ( M ; F ) . (2.4) Example 2.1. If F is the bundle of differential forms F = Λ • ( T M ) (or a more general tensorfield). The Lie derivative X F (obtained from the the differential dφ t ) satisfies (2.4). Example 2.2.
A simple but useful example is the trivial rank one bundle F = M × C . Let V ∈ C ∞ ( M ; C ) called the Gibbs potential function. The operator X F = X + V (2.5)satisfies (2.4), where V is seen as the multiplication operator by the function V . By integrationwe get, with u ∈ C ∞ ( M ) , e tX F u = e (cid:82) t V ◦ φ s ds (cid:124) (cid:123)(cid:122) (cid:125) amplitude u ◦ φ t (cid:124) (cid:123)(cid:122) (cid:125) transport , (2.6)More remarks about general pull back operators e tX F are given in Section B.2. We make the hypothesis that X is an Anosov vector field on M . This means that for everypoint m ∈ M the tangent vector X ( m ) is non zero and we have a continuous splitting of thetangent space T m M = E u ( m ) ⊕ E s ( m ) ⊕ E ( m ) (cid:124) (cid:123)(cid:122) (cid:125) R X ( m ) (2.7)that is invariant under the action of the differential dφ t and there exist λ min > , C > and asmooth metric g M on M such that ∀ t ≥ , ∀ m ∈ M, (cid:13)(cid:13)(cid:13) dφ − t/E u ( m ) (cid:13)(cid:13)(cid:13) g M ≤ Ce − λ min t , (cid:13)(cid:13) dφ t/E s ( m ) (cid:13)(cid:13) g M ≤ Ce − λ min t . (2.8)See Figure 2.1. The linear subspace E u ( m ) , E s ( m ) ⊂ T m M are called the unstable/stablespaces and the one dimensional space E ( m ) := R X ( m ) is called the neutral direction or flowdirection. In general the maps m → E u ( m ) , m → E s ( m ) and m → E u ( m ) ⊕ E s ( m ) are onlyHölder continuous with some Hölder exponent respectively [26]: β u , β s , β ∈ ]0 , , β := min ( β u , β s ) . (2.9)Let A ∈ C ( M ; T ∗ M ) be the continuous one form on M called Anosov one form definedfor every m ∈ M by the conditions A ( m ) ( X ( m )) = 1 and Ker ( A ( m )) = E u ( m ) ⊕ E s ( m ) . (2.10)From this definition, A is preserved by the flow φ t and the map m → A ( m ) is Höldercontinuous with exponent β . Notice that A is the unique continuous one form preserved by the flow with the normalization condition A ( X ) = 1 . table unstableVector field Flow E s ( m ) E u ( m ) E s E u Xφ t ( m ) m E E ( m ) X M
Figure 2.1: Anosov flow φ t generated by a vector field X on a compact manifold M . In addition we assume in this paper that the Anosov vector field X is contact , i.e. that theAnosov one form A , (2.10), is a smooth contact one form on M . This means that thedistribution E u ( m ) ⊕ E s ( m ) is smooth w.r.t. m ∈ M (hence with exponent β = 1 ) and that E u ( m ) ⊕ E s ( m ) endowed with the two form d A ( m ) is a linear symplectic space for every m ∈ M . Then dim M = 2 d + 1 is odd with d := dim E u ( m ) = dim E s ( m ) . Remark . the vector field X is the Reeb vector field of A , i.e. X is determined from A by A ( X ) = 1 and ( d A ) ( X, . ) = 0 . (2.11)We will write dm := 1 d ! A ∧ ( d A ) ∧ d (2.12)for the corresponding smooth and non-degenerate volume form on M invariant by the flow φ t . X In this Section we consider a general smooth and non vanishing vector field X (not necessarilyAnosov) on a smooth closed manifold M . We review some results given in [17]. We definethe wave packet transform T : S ( M ) → S ( T ∗ M ) that maps distributions on M to functionson T ∗ M and satisfies T † T = Id . Then we recall the theorem of propagation of singularities,saying that the Schwartz kernel of the lifted pull back operator T e tX T † , with t ∈ R , decays veryfast outside the graph of the flow ˜ φ t := ( dφ t ) ∗ : T ∗ M → T ∗ M acting on the cotangent space T ∗ M . Moreover we give an approximate description of this Schwartz kernel in the vicinity ofthe graph of ˜ φ t . g on T ∗ M Let us suppose that for j = 1 , . . . J , κ j : m ∈ U j → y = ( x, z ) ∈ R dx × R z are flow boxcoordinate charts where U j are open subset on M and ( dκ j ) ( X ) = ∂∂z . On each chart, let usdenote η = ( ξ, ω ) ∈ R d × R the dual coordinates associated to ( x, z ) . Since A is invariant by the flow, for any u , u ∈ E s ( m ) we have ∀ t ∈ R , ( d A ) ( u , u ) = (cid:0) d (cid:0) ( φ t ) ∗ A (cid:1)(cid:1) ( u , u ) = ( d A ) ( dφ t ( u ) , dφ t ( u )) → t → + ∞ hence d A /E s = 0 meaning that E s ( m ) is isotropic.Similarly E u ( m ) is isotropic hence E u ( m ) , E s ( m ) are Lagrangian and d = dim E u ( m ) = dim E s ( m ) . From Cartan formula and invariance of A : L X A = ι X d A + dι X A and ι X A = 1 give ( d A ) ( X, . ) = 0 . g on T ∗ M introduced in [17, section4.1.2] with parameters α ⊥ = and α (cid:107) = 0 . Let δ > , (3.1)and define the function, for η ∈ R d +1 , δ ⊥ ( η ) := min (cid:110) δ , | η | − / (cid:111) . (3.2)In coordinates, we define the metric g on T ∗ R d +1 = R d +1) y,η at point (cid:37) = ( y, η ) ∈ R d +1) by g ( (cid:37) ) := (cid:18) dxδ ⊥ ( η ) (cid:19) + (cid:0) δ ⊥ ( η ) dξ (cid:1) + (cid:18) dzδ (cid:19) + ( δ dω ) . (3.3)This gives a metric g j := ( dκ j ) ∗ g on T ∗ U j . It is shown in [17, Lemma 4.8] that on eachintersection U j ∩ U j (cid:48) , the metric g j and g j (cid:48) are relatively bounded by each over uniformly in η .This gives a (class of equivalent) metric g on T ∗ M . We will denote dist g ( ρ (cid:48) , ρ ) the distancebetween two points ρ, ρ (cid:48) ∈ T ∗ M according to a metric g in this class. It is shown in [17, Lemma4.8] that the metric g is geodesically complete. T As in [17, def. 4.22], for every coordinate neighborhood U j ⊂ M with j ∈ { , . . . J } and ρ ∈ T ∗ U j we define a wave packet Φ j,ρ ∈ C ∞ ( M ) and the “wave packet transform” T : (cid:40) C ∞ ( M ; C ) → S (cid:0) T ∗ M ; C J (cid:1) u → (cid:0) (cid:104) Φ j,ρ | u (cid:105) L ( M ) (cid:1) ρ ∈ T ∗ M,j ∈{ ...J } . (3.5)that satisfies the ” resolution of identity on C ∞ ( M ) ” [17, prop. 4.23] Id /C ∞ ( M ) = T † T (3.6)where T † : L (cid:16) T ∗ M ⊗ C J ; dρ (2 π ) d +1 (cid:17) → L ( M ; dm ) is the L -adjoint of T and with respect tothe canonical measure dρ ≡ dydη on T ∗ M . Remark . Below for simplicity, we will ignore ⊗ C J in the notation though keeping it in mind. ˜ φ t : T ∗ M → T ∗ M and its generator ˜ X In this section we define and describe the flow ˜ φ t = ( dφ t ) ∗ : T ∗ M → T ∗ M induced from theflow φ t . It will appear in Theorem 3.10 below, that for the analysis of e tX , this flow ˜ φ t on T ∗ M plays a major role. Notation . We will consider ρ ∈ T ∗ M and for the projection π : T ∗ M → M we will set m = π ( ρ ) ∈ M (see Figure 4.1). We refer to [17, def. 4.21] for the precise expression of a wave packet Φ j,ρ . Here it is enough to say that ona chart U j , for a given (cid:37) = ( y, η ) ∈ T ∗ R n +1 and in local coordinates y (cid:48) ∈ R n +1 , then for large | η | (cid:29) , function Φ j,ρ is equivalent to a Gaussian wave packet in “vertical Gauge”: Φ j,(cid:37) ( y (cid:48) ) ∼ a (cid:37) χ ( y (cid:48) − y ) exp (cid:16) iη. ( y (cid:48) − y ) − (cid:107) ( y (cid:48) , y ) (cid:107) g (cid:37) (cid:17) (3.4)with y (cid:48) = ( x (cid:48) , z (cid:48) ) ∈ R n +1 and where χ ∈ C ∞ (cid:0) R n +1 (cid:1) is some cut-off function with χ ≡ near the origin and a (cid:37) > is such that (cid:107) Φ j,(cid:37) (cid:107) L ( R n +1 ) = 1 and (cid:107) ( y (cid:48) , y ) (cid:107) g (cid:37) = (cid:12)(cid:12)(cid:12)(cid:12) ( z (cid:48) − z ) δ (cid:12)(cid:12)(cid:12)(cid:12) + (cid:12)(cid:12)(cid:12)(cid:12) ( x (cid:48) − x ) δ ⊥ ( η ) (cid:12)(cid:12)(cid:12)(cid:12) is obtained from the metric (3.3). .3.1 Frequency function ω We define the frequency function ω ∈ C ∞ ( T ∗ M ; R ) by ∀ ρ ∈ T ∗ M, ω ( ρ ) := ρ ( X ) . (3.7)that measures the oscillations along the flow direction. Remark . In terms of micro-local analysis, the function ω is the principal symbol of theoperator − iX , see [13, footnote page 332.]. ˜ φ t : T ∗ M → T ∗ M Let us denote the differential of the flow map φ t : M → M by dφ t : T m M → T φ t ( m ) M. By duality we get a map ˜ φ t := (cid:0) dφ t (cid:1) ∗ : T ∗ m M → T ∗ φ − t ( m ) M. (3.8)defined by (cid:104) v | (cid:0) dφ t (cid:1) ∗ ( ρ ) (cid:105) T m M = (cid:104) dφ t v | ρ (cid:105) T ∗ φt ( m ) M (3.9)for any ρ ∈ T ∗ φ t ( m ) M , v ∈ T m M with m = π ( v ) . Notice that ˜ φ t is a lift of the inverse map φ − t map, i.e. π (cid:16) ˜ φ t ( ρ ) (cid:17) = φ − t ( π ( ρ )) . (3.10) θ and the symplectic two form Ω Let θ be the canonical Liouville one form on the vector bundle π : T ∗ M → M [28, p.90]: atany point ρ ∈ T ∗ M , θ ρ is defined by ∀ V ∈ T ρ ( T ∗ M ) , θ ρ ( V ) = ρ ( dπ ( V )) . (3.11)Using the local coordinates ( y j ) j =1 ,... d +1 on M and dual coordinates ( η j ) j on T ∗ M , we have θ := d +1 (cid:88) j =1 η j dy j . (3.12)Let Ω be the canonical symplectic form: Ω = dθ = d +1 (cid:88) j =1 dη j ∧ dy j . (3.13)It gives a bundle isomorphism ˇΩ : T T ∗ M → T ∗ T ∗ M defined by ∀ V ∈ T T ∗ M, ˇΩ ( V ) := Ω ( V, . ) ∈ T ∗ T ∗ M. ˜ X on T ∗ M Let ˜ X be the vector field on T ∗ M that generates the flow ˜ φ t (3.8). Notice from (3.10) that ( dπ ) (cid:16) ˜ X (cid:17) = − X. (3.14)18 emma 3.4. [20, thm 2.124p.112, in coordinates],[29, prop. 1.21 p.15]. The Lie deriva-tive of θ by the vector field ˜ X vanishes: L ˜ X θ = 0 , (3.15) and ˜ X is determined by ˜ X = ˇΩ − ( d ω ) , (3.16) i.e. ˜ X is the Hamiltonian vector field on ( T ∗ M, Ω) , with Hamiltonian function ω ,Eq.(3.7).Remark . As a consequence of (3.16) we have ˜ X ( ω ) = Ω (cid:16) ˜ X, ˜ X (cid:17) = 0 , i.e. for every ω ∈ R ,the frequency level Σ ω := ω − ( ω ) = { ρ ∈ T ∗ M, X ( ρ ) = ω } ⊂ T ∗ M (3.17)that is a affine sub-bundle of T ∗ M is preserved by ˜ φ t . Remark . For ω (cid:54) = 0 , the one form − ω θ is a contact one form on the frequency level ω − ( ω ) and ˜ X is its Reeb vector field. Remark . From now on we will often use the knowledge about Bargmann transform that issummarized in Section C of the appendix and we ask the readers to check the notation andtheir knowledge. We will often refer to this Section C.We first introduce a definition that will be used to express that some operator R is “undercontrol” or “negligible” in our analysis. It will means that the Schwartz kernel of T R T ∗ decaysvery fast outside the graph of ˜ φ t defined in (3.8) and that on the graph, the Schwartz kernel isbounded by some given function a ( ρ ) . Definition 3.8.
Let R : S ( M ) → S (cid:48) ( M ) be an operator and t ∈ R . Let a ∈ C ( T ∗ M ; R + ) be a positive function on T ∗ M . We say that R ∈ Ψ ˜ φ t ,g ( a ) if for any N > , there exists a constant C N,t > such that for any ρ, ρ (cid:48) ∈ T ∗ M (cid:12)(cid:12) (cid:104) δ ρ (cid:48) |T R T † δ ρ (cid:105) L ( T ∗ M ) (cid:12)(cid:12) ≤ C N,t (cid:68) dist g (cid:16) ρ (cid:48) , ˜ φ t ( ρ ) (cid:17)(cid:69) − N a ( ρ ) (3.18)For simplicity, we will denote Ψ ˜ φ t ( a ) = Ψ ˜ φ t ,g ( a ) if we use the metric g defined in (3.3). Remark . The following useful properties follow from Shur test. If R ∈ Ψ ˜ φ t ( a ) and (cid:107) a (cid:107) L ∞ ( T ∗ M ) < ∞ , then (cid:107) R (cid:107) L ( M ) ≤ C t (cid:107) a (cid:107) L ∞ ( T ∗ M ) . For σ > , we will use later in Section 7 the re-scaledmetric g/σ . Notice that dist g/σ (cid:16) ρ (cid:48) , ˜ φ t ( ρ ) (cid:17) = σ − dist g (cid:16) ρ (cid:48) , ˜ φ t ( ρ ) (cid:17) and in the right hand sideof (3.18) this corresponds to consider that the Schwartz kernel decays very fast at distancegreater than σ . If R ∈ Ψ ˜ φ t ,g/σ ( a ) and if (cid:107) a (cid:107) L ∞ ( T ∗ M ) < ∞ , then Shur’s test gives (cid:107) R (cid:107) L ( M ) ≤ C t,σ (cid:107) a (cid:107) L ∞ ( T ∗ M ) . (3.19)19 .4.1 Result for the wave front set The next theorem that is proved in [17, Lemma 4.38] expresses that the Schwartz kernel of T e tX T † decays very fast outside the graph of ˜ φ t := ( dφ t ) ∗ : T ∗ M → T ∗ M . Theorem 3.10 ( Propagation of singularities ) . [17, Lemma 4.38]. For each t ∈ R , e tX ∈ Ψ ˜ φ t (1) . (3.20) Remark . This result is similar to the description of evolution of the wave-front set inmicro-local analysis as in [33, Prop. 9.5, page 29.]
Theorem 3.10 says nothing for the value of the Schwartz kernel (cid:104) δ ρ (cid:48) |T e tX T † δ ρ (cid:105) L ( T ∗ M ) if ρ (cid:48) and ˜ φ t ( ρ ) are in bounded distance from each other, i.e. near the graph of ˜ φ t . The next theorem3.14 below will completement this lack of information. It will give an approximate expressionfor the Schwartz kernel of T e tX T † in the vicinity of the graph of ˜ φ t . This description will beuseful later to get Theorem 6.8.For ρ ∈ T ∗ M , with | ρ | (cid:29) , we will use the fact that the metric g can be approximated bya Euclidean metric in a vicinity of ρ . Consequently the wave-packet transform T : C ∞ ( M ) →S ( T ∗ M ) can be approximated by a Bargmann transform B constructed from local Bargmantransforms on the tangent bundle T ρ T ∗ M in the vicinity of the zero section. We first need tointroduce some operators.Below we view S ( T T ∗ M ) as the set of functions u : ∈ ρ ∈ T ∗ M → u ρ ∈ S ( T ρ T ∗ M ) , i.e. wehave a natural identification S ( T T ∗ M ) ≡ S ( ρ ∈ T ∗ M ; S ( T ρ T ∗ M )) . (3.21) The operator (cid:103) exp ◦ . ( T ∗ M, g ) is geodesically complete. Let us write exp : T T ∗ M → T ∗ M (3.22)for the exponential map associated to the metric g on T ∗ M . This map induces a twisted pullback operator (cid:103) exp ◦ : C ∞ ( T ∗ M ) → C ∞ ( T T ∗ M ) (3.23)defined as follows. For u ∈ C ∞ ( T ∗ M ) , v ∈ T ρ T ∗ M with ρ ∈ T ∗ M , (cid:0) (cid:103) exp ◦ u (cid:1) ( v ) := e − iρ ( dπ ( v )) u (exp ( v )) (3.24) Remark . The importance of the phase in (3.24) is clear from the lemma C.32 that considersthe linear case. This phase reflects the existence of a (trivial) complex line bundle L over T ∗ M with connection given by the canonical Liouville one form with respect to a global trivialization.This line bundle is usually called the prequantum line bundle. As said before, in this paperwe can ignore this line bundle since it is trivial and work with respect to a global section of it,giving phases like in (3.24). See Remark 1.14. Truncation operator χ σ . Let σ > . The operator χ σ truncates functions in a ball ofradius σ centered on the zero section χ σ : C ∞ ( T T ∗ M ) → S (cid:48) ( T T ∗ M ) u ( v ) → (cid:40) u ( v ) if (cid:107) v (cid:107) g ρ ≤ σ otherwise. (3.25)where v ∈ T ρ T ∗ M and ρ ∈ T ∗ M . 20 he horizontal vector bundle H . For ρ ∈ T ∗ M , there is the vertical linear subspace V ( ρ ) := dπ − (0) ⊂ T ρ T ∗ M that is isomorphic to T ∗ π ( ρ ) M and we define the horizontal space H ( ρ ) := ( V ( ρ )) ⊥ g by g ρ -orthogonality, i.e. we have T T ∗ M = V ⊥ g ⊕ H. We have that V ( ρ ) and H ( ρ ) are Ω -Lagrangian subspace of T ρ T ∗ M and we have an isomor-phism that is Ω -symplectic and g -isometric (see Lemma C.15), T T ∗ M = H ⊕ V → H ⊕ H ∗ . (3.26) The operator B . Let us denote the fiber-wise Bargmann transform in vertical gauge de-fined in (C.8) for the metric g ρ , as follows: for every ρ ∈ T ∗ M , this is the operator B ρ : S ( H ( ρ )) → S ( H ( ρ ) ⊕ H ( ρ ) ∗ ) = (3 . S ( T ρ T ∗ M ) , so that for u ∈ S ( H ( ρ )) , ( x, ξ ) ∈ H ( ρ ) ⊕ H ( ρ ) ∗ , we let ( B ρ u ) ( x, ξ ) := (cid:104) ϕ x,ξ | u (cid:105) L ( H ⊕ H ∗ ) with the Gaussian wave packet with unit norm ϕ x,ξ ( y ) = π − n/ e iξ · ( y − x ) e − (cid:107) y − x (cid:107) g , n = dim H ( ρ ) = 2 d + 1 . The collection of these operators B := ( B ρ ) ρ ∈ T ∗ M gives an operator B : S ( H ) → S ( T T ∗ M ) . (3.27)The L -adjoint operator B † : S ( T T ∗ M ) → S ( H ) is given by (cid:0) B † ρ v (cid:1) ( y ) := (cid:90) v ( x, ξ ) ϕ x,ξ ( y ) dxdξ (2 π ) n (3.28) The operator ( dφ tH ) ◦ . At time t , for every m ∈ M , the flow φ t : M → M has a differential dφ t ( m ) : T m M → T φ t ( m ) M . Let π : T ∗ M → M be the projection and dπ : T T ∗ M → T M itsdifferential. For every ρ ∈ T ∗ M , the differential ( dπ ) ( ρ ) restricted to the vector space H ( ρ ) gives an isomorphism: ( dπ ) ( ρ ) : H ( ρ ) → T π ( ρ ) M. Using the induced flow ˜ φ t : T ∗ M → T ∗ M defined in (3.8), we get an isomorphism (cid:0) dφ tH (cid:1) ( ρ ) := (( dπ ) ( ρ )) − ◦ (cid:0) dφ t ( π ( ρ )) (cid:1) ◦ ( dπ ) ( ρ ) : H ( ρ ) → H (cid:16) ˜ φ − t ( ρ ) (cid:17) and a pull back operator (cid:0) dφ tH (cid:1) ◦ : C ∞ ( H ) → C ∞ ( H ) . Since the contact volume (2.12) is preserved, we have (cid:12)(cid:12) det (cid:0) dφ tH (cid:1)(cid:12)(cid:12) = 1 . (3.29) The operator r . Let r : S ( T T ∗ M ) → S ( T ∗ M ) be the map that restricts u ∈ S ( T T ∗ M ) to its value at the zero section ( r u ) ( ρ ) := u ( ρ, . Remark . We may also denote r / = r . Referring to the structure Fiber/Base for thebundle T T ∗ M → T ∗ M , the notation r / means here that the value is taken at in the fiber ofthe bundle. Later we will meet the notation r / Σ meaning that the value is taken on Σ on the basis of the bundle. 21ecall the map ˜ φ t : T ∗ M → T ∗ M defined in (3.8). Its differential d ˜ φ t : T T ∗ M → T T ∗ M gives a push-forward operator (cid:16) d ˜ φ t (cid:17) −◦ : S ( T T ∗ M ) → S ( T T ∗ M ) . For later analysis it will bepreferable to deal with the operator (cid:16) d ˜ φ t (cid:17) −◦ instead of the operator ( dφ tH ) ◦ . We can relatethese two operators as follows. As in (C.16) we define first the “metaplectic correction” Υ (cid:16) d ˜ φ t (cid:17) := (cid:32) det (cid:32) (cid:32) Id + (cid:18)(cid:16) d ˜ φ t (cid:17) − (cid:19) † (cid:16) d ˜ φ t (cid:17) − (cid:33)(cid:33)(cid:33) / . As in Definition C.31, we define the fiber-wise operator ˜Op (cid:16) d ˜ φ t (cid:17) = ( C. (cid:16) Υ (cid:16) d ˜ φ t (cid:17)(cid:17) / P (cid:16) d ˜ φ t (cid:17) −◦ P = ( C. (cid:12)(cid:12) det (cid:0) dφ tH (cid:1)(cid:12)(cid:12) / B (cid:0) dφ tH (cid:1) ◦ B † (3.30) = (3 . B (cid:0) dφ tH (cid:1) ◦ B † . (3.31) : S ( T T ∗ M ) → S ( T T ∗ M ) . (3.32) In the next theorem we use and compose the operators defined in the previous section with T in (3.5), and obtain a good approximation of the pull back operator e tX . Theorem 3.14 ( “More precise expression for the propagation of singularities” ) . For any t ∈ R , and σ > , we have e tX = (cid:0) T † r (cid:1) ˜Op (cid:16) d ˜ φ t (cid:17) (cid:0) χ σ (cid:103) exp ◦ T (cid:1) + R t,σ (3.33) with a remainder operator R t,σ that satisfies, ∀ N > , ∀ ≤ γ < , ∃ C N > , R t,σ ∈ Ψ ˜ φ t (cid:0)(cid:0) δ ⊥ ( ρ ) (cid:1) γ σ + C N σ − N (cid:1) (3.34) with δ ⊥ ( ρ ) defined in (3.2).Remark . In practice, in order to use Theorem 3.14, we first take any t , some N > , thencutoff σ (cid:29) large enough, so that C t,N σ − N (cid:28) . Then we take frequencies | ρ | (cid:29) largeenough to get also C t (cid:0) δ ⊥ ( ρ ) (cid:1) γ σ (cid:28) in the right hand side of (3.34). Using the Shur test, thisguaranties that the remainder operator R t,σ restricted to high frequencies is arbitrary small inoperator norm. Proof.
Let ˜ R t,σ := T e tX T † − r ˜Op (cid:16) d ˜ φ t (cid:17) (cid:0) χ σ (cid:103) exp ◦ (cid:1) . We will first show the following estimate. ∀ t ∈ R , ∃ C t > , ∀ σ > , ∀ ρ, ρ (cid:48) ∈ T ∗ M with dist g (cid:16) ρ (cid:48) , ˜ φ t ( ρ ) (cid:17) ≤ σ , (cid:12)(cid:12)(cid:12) (cid:104) δ ρ (cid:48) | ˜ R t,σ δ ρ (cid:105) (cid:12)(cid:12)(cid:12) ≤ C t (cid:0) δ ⊥ ( ρ ) (cid:1) γ σ + C t,N σ − N (3.35)with δ ⊥ ( ρ ) = min (cid:110) δ , | ρ | − / (cid:111) given in (3.2) and any ≤ γ < .In case of a linear map A : E → E and Euclidean metric g on E ⊕ E ∗ , (3.35) is shown inLemma C.32, where the cut-off χ σ introduces the error term C t,N σ − N , ∀ N . The non-linearitybut temperate property of the metric g is related to the distortion function ∆ ( ρ ) = δ ⊥ ( ρ ) givenin [17, Lemma 4.13] and that gives an error term ∆ ( ρ ) γ σ for any γ < , hence C t (cid:0) δ ⊥ ( ρ ) (cid:1) γ σ .From (3.3), the cutoff at distance σ measured by the metric g corresponds to a distance22 (cid:0) σδ ⊥ ( ρ ) (cid:1) on M . Assume (cid:0) δ ⊥ ( ρ ) (cid:1) γ σ < C . Since we neglect non linearity of the map φ t ,we get from Taylor expansion at second order, an error term O t (cid:0) σδ ⊥ ( ρ ) (cid:1) . This error termis bounded by the previous one so we don’t need to write it. Otherwise, the left hand side of(3.35) is bounded by a constant C t so the claim (3.35) is always true.Due to χ σ , then (cid:104) δ ρ (cid:48) | r ˜Op (cid:16) d ˜ φ t (cid:17) (cid:0) χ σ (cid:103) exp ◦ (cid:1) δ ρ (cid:105) vanishes if dist g (cid:16) ρ (cid:48) , ˜ φ t ( ρ ) (cid:17) > C t σ with C t independent on ρ, ρ (cid:48) . Hence, with Theorem 3.10 and (3.35) we have that (cid:12)(cid:12)(cid:12) (cid:104) δ ρ (cid:48) | ˜ R t,σ δ ρ (cid:105) L ( T ∗ M ) (cid:12)(cid:12)(cid:12) ≤ C N,t (cid:68) dist g (cid:16) ρ (cid:48) , ˜ φ t ( ρ ) (cid:17)(cid:69) − N (cid:0)(cid:0) δ ⊥ ( ρ ) (cid:1) γ σ + C N σ − N (cid:1) . The same inequality holds with ˜ R t,σ replaced by T R t,σ T † . From Definition (3.18) we deduce(3.34). ˜ φ t on the cotangent space T ∗ M In the following we suppose that X is a smooth contact Anosov vector field on a closedmanifold M as defined and described in Section 2.4. The dual decomposition of (2.7) is T ∗ m M = E ∗ u ( m ) ⊕ E ∗ s ( m ) ⊕ E ∗ ( m ) (4.1)with E ∗ u := ( E s ⊕ E ) ⊥ = { ρ ∈ T ∗ M, Ker ρ ⊃ E s ⊕ E } , (4.2) E ∗ s := ( E u ⊕ E ) ⊥ = { ρ ∈ T ∗ M, Ker ρ ⊃ E u ⊕ E } , (4.3) E ∗ := ( E u ⊕ E s ) ⊥ = { ρ ∈ T ∗ M, Ker ρ ⊃ E u ⊕ E s } (4.4) = (2 . R A = { ω A ( m ) , m ∈ M, ω ∈ R } . (4.5)Hence dim E ∗ u ( m ) = dim E ∗ s ( m ) = d, dim E ∗ ( m ) = 1 . From (2.9), the map m ∈ M → E ∗ u ( m ) is β u -Hölder continuous and m ∈ M → E ∗ s ( m ) is β s -Hölder continuous. However the maps m ∈ M → E ∗ ( m ) = R A ( m ) m ∈ M → ( E ∗ u ⊕ E ∗ s ) ( m ) = E ⊥ ( m ) = Ker ( X ) ( m ) are smooth. A cotangent vector ρ ∈ T ∗ M is decomposed accordingly to the dual decomposition(4.1) as ρ = ρ u + ρ s + ρ (4.6)with components ρ u ∈ E ∗ u , ρ s ∈ E ∗ s , ρ = ω A ( m ) ∈ E ∗ with the frequency ω = (3 . ω ( ρ ) ∈ R .By duality, the hyperbolicity assumption (2.8) gives that the components (4.6) of ρ ( t ) =˜ φ t ( ρ ) satisfy ∃ C > , ∀ ρ ∈ T ∗ M, ∀ t ≥ , (cid:107) ρ u ( t ) (cid:107) g M ≥ C e λ min t (cid:107) ρ u (0) (cid:107) g M , (cid:107) ρ s ( t ) (cid:107) g M ≤ Ce − λ min t (cid:107) ρ s (0) (cid:107) g M , ω ( t ) = ω (0) . (4.7)See figure 4.1. 23 E u E s ω A ( m ) A ( m ) E ∗ ( m ) Trapped set T ∗ m M T ∗ φ t ( m ) M φ t ( m ) φ t M u exp( tX ) u ρ ( t )˜ XE ∗ u ( m ) E ∗ s ( m ) ρ ˜ φ t m Σ ω ( m ) Figure 4.1: The Anosov flow φ t = exp ( tX ) on M induces a Hamiltonian flow ˜ φ t = exp (cid:16) t ˜ X (cid:17) incotangent space T ∗ M . The magenta lines on M represent “internal oscillations” of a function u . These oscillations correspond to a cotangent vector ρ ∈ T ∗ φ t ( m ) M . Transported by the flowthe oscillations of e tX u = u ◦ φ t increase and the wave front of these oscillations become parallelto E s ⊕ E equivalently ρ ( t ) = ˜ φ t ( ρ ) converges to the direction of E ∗ u ⊂ T ∗ M and remainsin the frequency level Σ ω := ω − ( ω ) . The trapped set of the lifted flow ˜ φ t is the rank onevector bundle E ∗ = R A (green line) where A is the Anosov one form. From Lemma 4.2, Σ := E ∗ \ { } is a symplectic sub-manifold of T ∗ M , with dimΣ = 2 ( d + 1) .24 .2 The symplectic trapped set Σ From (4.7) we deduce the following Lemma.
Lemma 4.1.
The set E ∗ = R A defined in (4.4) is the trapped set (or non wanderingset ) of the flow ˜ φ t in the sense that E ∗ = (cid:110) ρ ∈ T ∗ M | ∃K ⊂ T ∗ M compact, s.t. ˜ φ t ( ρ ) ∈ K , ∀ t ∈ R (cid:111) (4.8) = (4 . { ω A ( m ) ∈ T ∗ M, m ∈ M, ω ∈ R } . From (4.6), the trapped set E ∗ is characterized by ρ u = 0 , ρ s = 0 and from (4.7), E ∗ is transversely hyperbolic . The following lemma shows that
Σ := E ∗ \ { } is a symplectic manifold, called the symplec-tization of M [5, Section 11.2][1, Appendix 4.]. We will denote π : T ∗ M → M the projectionand ( π ◦ dm ) the form dm (2.12) pulled back on T ∗ M . Lemma 4.2.
The set
Σ := E ∗ \ { } := { ω A ( m ) ∈ T ∗ M | m ∈ M, ω ∈ R \ { }} (4.9) is a smooth symplectic sub-manifold of T ∗ M with dimΣ = 2 ( d + 1) . The inducedvolume form on Σ is given by d(cid:37) := 1( d + 1)! ( dθ ) ∧ ( d +1) = ω d ( dω ) ∧ ( π ◦ dm ) (4.10) Proof.
For m ∈ M , ω ∈ R \ { } , at point ω A ( m ) ∈ Σ ⊂ T ∗ M we have θ = (3 . ω ( π ∗ A ) (4.11)where π ∗ is the pull back map on forms. Hence dθ = d ( ω ( π ∗ A )) = dω ∧ ( π ∗ A ) + ω ( π ∗ d A ) giving the following volume form on E ∗ d(cid:37) : = 1( d + 1)! ( dθ ) ∧ ( d +1) = 1( d + 1)! ( d + 1) ω d · dω ∧ (cid:16) ( π ∗ A ) ∧ ( π ∗ d A ) ∧ d (cid:17) = (2 . ω d ( dω ) ∧ π ∗ dm which does not vanish since ω (cid:54) = 0 . In Lemma 4.1 we have observed that the trapped set E ∗ is transversely hyperbolic. In thissection, we recall that from this important property, and following [13] or [17, Def 5.8], wedefine a weight function or escape function W : T ∗ M → R + that decays outside the trappedset (i.e. that is a Lyapounov function for the flow ˜ φ t ). Then we define the anisotropic Sobolevspace H W ( M ) and obtain that the norm of the pull back operator e tX restricted to the outsideof E ∗ decays exponentially fast with an arbitrarily large rate. This is the important propertythat will able us to restrict the analysis to a neighborhood of the trapped set E ∗ in lattersections. 25 .1 Definition and properties of the escape function W We will use the notation (cid:104) x (cid:105) := max (1 , | x | ) , x ∈ R . (5.1)As in [17, Def 5.8], we consider the positive valued function W ∈ C ( T ∗ M ; R + , ∗ ) called weightfunction defined as follows. Let R > , γ ∈ [1 − β, and h > that we can choose and willlet be small. For ρ ∈ T ∗ M , with stable/unstable components ρ s ∈ E ∗ s , ρ u ∈ E ∗ u , we define W ( ρ ) := (cid:68) h γ ( ρ ) (cid:107) ρ s (cid:107) g (cid:69) R (cid:68) h γ ( ρ ) (cid:107) ρ u (cid:107) g (cid:69) R (5.2)with h γ ( ρ ) := h (cid:68) (cid:107) ρ u + ρ s (cid:107) g (cid:69) − γ (5.3)This function W has a few properties given in [17, Thm 5.9]. The first property is that W decays exponentially with respect to the flow ˜ φ t (defined in (3.8)) outside a fixed distance ofthe trapped set E ∗ : ∃ C > , ∀ t ≥ , ∃ C t > , ∀ ρ ∈ T ∗ M, W (cid:16) ˜ φ t ( ρ ) (cid:17) W ( ρ ) ≤ (cid:40) CCe − ( λr ) t if dist g ( ρ, E ∗ ) > C t (5.4)with r = R (1 − γ ) > , the order of W . This property is a consequence of the definitionof W in (5.2) and the hyperbolicity properties of the flow (4.7). A second property called h γ -temperate property is that ∃ C > , ∃ N > , ∀ ρ, ρ (cid:48) ∈ T ∗ M, W ( ρ (cid:48) ) W ( ρ ) ≤ C (cid:104) h γ ( ρ ) dist g ( ρ (cid:48) , ρ ) (cid:105) N , and means that the function W is relatively bounded for distances less than dist g ( ρ (cid:48) , ρ ) ≤ h γ ( ρ ) − (that may be large when h γ ( ρ ) (cid:28) , far from the trapped set) and has polynomialgrowth at larger distances. Another property called slow variation property is that ∃ µ > , ∃ C > , ∃ N > , ∀ h > , ∀ ρ, ρ (cid:48) ∈ T ∗ M, W ( ρ (cid:48) ) W ( ρ ) ≤ C h µ (cid:104) dist g ( ρ (cid:48) , ρ ) (cid:105) N , (5.5)where h enters in W through (5.3). H W ( M ; F ) H W ( M ; F ) Let us summarize the definition of the space H W ( M ; F ) that we use in this paper. For moredetails, we refer to the paper [17]. For simplicity we will forget the vector bundle F , that playsno role for this construction, i.e. we first consider only a trivial bundle F = M × C .For any u ∈ C ∞ ( M ) we define its H W ( M ) -norm (cid:107) u (cid:107) H W ( M ) := (cid:107) W T u (cid:107) L ( T ∗ M ) where W is used as a multiplication operator on S ( T ∗ M ) . Let H W ( M ) be the Hilbert spaceobtained by completion of C ∞ ( M ) with respect to this norm and called anisotropic Sobolevspace : H W ( M ) := { u ∈ C ∞ ( M ) } (cid:107) . (cid:107) H W . (5.6)In other words W T : H W ( M ) → L ( T ∗ M ) is an isometry.26 emark . The function W in (5.2) satisfies ∃ C > , ∀ ρ ∈ T ∗ M, C − (cid:68) | ρ | g M (cid:69) −| r | ≤ W ( ρ ) ≤ C (cid:68) | ρ | g M (cid:69) | r | . (5.7)with r = R − γ ) > . (5.8)Consequently H r ( M ) ⊂ H W ( M ) ⊂ H − r ( M ) . (5.9)where H r ( M ) := Op (cid:18)(cid:68) | ξ | g M (cid:69) − r (cid:19) ( L ( M )) is the standard Sobolev space of order r . Wedefine similarly H W ( M ; F ) by completion of the space of smooth sections C ∞ ( M ; F ) . Let σ > that will be chosen large enough later. Define the following characteristic function χ Σ ,σ : T ∗ M → R + , such that for every ρ ∈ T ∗ Mχ Σ ,σ ( ρ ) := (cid:40) if (cid:107) ρ − ω A ( π ( ρ )) (cid:107) g ρ ≤ σ if (cid:107) ρ − ω A ( π ( ρ )) (cid:107) g ρ ≥ σ . (5.10)with ω = ω ( ρ ) . Let ˆ χ σ := Op ( χ Σ ,σ ) := T † χ Σ ,σ T : C ∞ ( M ) → C ∞ ( M ) (5.11)be the corresponding PDO operator, as defined in [17, def. 4.26] that extracts components nearthe trapped set. Theorem 5.2. [17, Lemma 5.14]. For any Λ > , we can choose R large enough in (5.2)as a parameter of the function W , such that ∃ C > , ∀ t ≥ , ∃ σ t , ∀ σ > σ t we have (cid:13)(cid:13) e tX (Id − ˆ χ σ ) (cid:13)(cid:13) H W ( M ) ≤ Ce − Λ t , (5.12) (cid:13)(cid:13) (Id − ˆ χ σ ) e tX (cid:13)(cid:13) H W ( M ) ≤ Ce − Λ t . (5.13) Remark . Consider the decomposition e tX = e tX (Id − ˆ χ σ ) + e tX ˆ χ σ . Theorem 5.2 means thatthe first component e tX (Id − ˆ χ σ ) decays exponentially fast as O (cid:0) e − Λ t (cid:1) with an arbitrarily large Λ (cid:29) . We deduce that in order to describe the dominant effect of the operator e tX and Ruelleresonances on the spectral domain Re ( z ) > − Λ , one has to study the second component e tX ˆ χ σ namely the dynamics e tX restricted to a neighborhood of the trapped set Σ = E ∗ \ { } . Then,from Theorem 3.14, we only need to study the operator ˜Op (cid:16) d ˜ φ t (cid:17) restricted to the trapped set Σ , i.e. the linearized dynamics on the trapped set. This is the subject of the next section. H W ( M ; F ) In paper [17], Theorem 5.2 is used to prove the following theorem.27 heorem 5.4. “Group property and discrete spectrum” [17] . For a general Anosovvector field X on M , the family of operators e tX F : H W ( M ; F ) → H W ( M ; F ) , t ∈ R , form a strongly continuous group and the generator X F has discrete spectrum denoted σ ( X F ) , on C − rλ min + (cid:15) ≤ Re ( z ) ≤ C (cid:48) with C, C (cid:48) independent of the parameter R (Notthat r is defined from R by (5.8)). This spectrum is intrinsic and called future Ruellespectrum in [17]. See Figure 1.2.
Remark . The discrete spectrum of X F (and eigenspace) are said “intrinsic” because they donot depend on H W ( M ; F ) . However more refined spectral properties as the norm of the resol-vent (cid:13)(cid:13) ( z − X F ) − (cid:13)(cid:13) H W ( M ) depend on the choice of the space H W ( M ; F ) and play an importantrole in this paper. Σ Theorem 5.2 and Remark 5.3 above show that the action of the pull back operator e tX is welldescribed by its micro-local restriction to a vicinity of the trapped set Σ ⊂ T ∗ M . The purposeof this section and the next one is to give a accurate description of this restriction. Since thepull back operator e tX is closely related to the differential d ˜ φ t of the induced flow on T ∗ M (as we have seen in Theorem 3.14 of propagation of singularities), we will first describe in thissection the geometry of T ∗ M in the vicinity of the trapped set and the action of d ˜ φ t on thisvicinity. N → Σ The following definition comes from Lemma 4.2, i.e. the fact that
Σ := E ∗ \ { } is symplectic.We will denote the vector bundle of tangent spaces over Σ by T Σ T ∗ M := { T ρ ( T ∗ M ) | ρ ∈ Σ } . efinition 6.1. Since Σ is symplectic, we can define K := T Σ , N := ( T Σ) ⊥ Ω (6.1)being the tangent bundle of Σ and its symplectic orthogonal in T Σ T ∗ M , so that T Σ T ∗ M = K ⊥ Ω ⊕ N (6.2)with dim K = 2 d + 2 and dim N = 2 d . Furthermore, each component decomposes as K = ( K u ⊕ K s ) ⊥ Ω ⊕ K , N = N u ⊕ N s , (6.3) dim K u = dim K s = dim N u = dim N s = d, dim K = 2 , with K u/s := K ∩ dπ − (cid:0) E u/s (cid:1) , K = E n ⊕ E ∗ n , N u/s := N ∩ dπ − (cid:0) E u/s (cid:1) , (6.4) E n := R ˜ X, E ∗ n := K ∩ dπ − ( { } ) , dim K u = dim K s = dim N u = dim N s = d, dim K = 2 , dim E n = dim E ∗ n = 1 . Each component K u , K s , E n , E ∗ n , N u , N s is a Ω -isotropic vector space, invariant under thedynamics d ˜ φ t and the indices u, s, n denotes respectively instability, stability or neutrality.See Figure 1.1. Remark . • Notice that due to (3.10) that reverses order of time, N u with is stable for d ˜ φ t andrespectively for N s that is unstable.• The symplectic normal bundle N → Σ is smooth, but the isotropic sub-bundle N u → Σ is only β u -Hölder continuous.• Beware that N ( ρ ) (cid:42) T ∗ m M since T ∗ m M is Lagrangian whereas N ( ρ ) is symplectic, i.e. N ( ρ ) is transverse to the fibers of T ∗ M .By definition K ⊥ Ω ⊕ N are orthogonal for the symplectic form Ω . But a priori they are notorthogonal for the metric g given in (3.3). Since this last property would be useful later, wewill modify the metric g according to the next lemma to get g -orthogonality as well. Lemma 6.3.
On the trapped set Σ , we construct a modified metric ˜ g from the existingmetric g as follows. Let ˜ g /N = g /N , ˜ g /K = g /K and assume K ⊥ ˜ g ⊕ N . Then1. ˜ g is compatible with Ω .2. The decomposition (6.2) is orthogonal with respect to the metric ˜ g (and Ω as before): T Σ T ∗ M = K ⊥ Ω , ˜ g ⊕ N (6.5)
3. The metric ˜ g is equivalent to the metric g , uniformly with respect to ρ ∈ Σ for | ω | ≥ .Proof. By construction we have claim 1,2. For the proof of Claim 3 let us remark that forthe metric g on Σ given in (3.3), the subspaces K, N are uniformly apart form each over with29espect to ρ ∈ Σ , i.e. ∃ < C < , ∀ ρ ∈ Σ , ∀ u ∈ K ρ , ∀ v ∈ N ρ , | g ρ ( u, v ) | < C (cid:107) u (cid:107) g ρ (cid:107) v (cid:107) g ρ . (6.6)Indeed, let ρ = ω A ( m ) ∈ Σ . From (3.2) we have δ ⊥ ( ρ ) (cid:16) | ρ | − / (cid:16) | ω | − / , hence from (3.3),we have that a scaling of the coordinates by ω / gives a metric on T ρ T ∗ M that does not dependon ω (up to a vanishing term as | ω | → ∞ ). The same is true for the symplectic form on T ρ T ∗ M .This gives the bound (6.6) with C < independent on ω . We will often use the notation ≈ t (in particular ≈ if t = 0 ) defined as follows. For this, we useDefinition 3.8 above. Definition 6.4.
Let t ∈ R be given. For two family of operators A = ( A σ,σ ) σ,σ , B =( B σ,σ ) σ,σ depending on parameters σ > , and sometimes on σ > also, we will write A ≈ t B (6.7)if A σ,σ = B σ,σ + r σ,σ with a remainder operator r σ,σ , such that ∀ N > , ∀ σ > σ > , r σ,σ ∈ Ψ ˜ φ t ,g/σ ( a N,σ,σ ) , (6.8)with, for ρ ∈ T ∗ M , a N,σ,σ ( ρ ) = C N,σ,σ ( ω ( ρ )) (cid:104) dist g ( ρ, Σ) − σ (cid:105) − N , (6.9) lim σ →∞ lim σ →∞ lim | ω |→∞ C N,σ,σ ( ω ) = 0 . (6.10) Notation . Let us remark that we have C N,σ,σ ( ω ) arbitrarily small, by taking first thedistance σ (cid:29) large enough, then σ large enough and finally frequency | ω | (cid:29) large enough.We will sometimes use the notation ∀ σ (cid:29) σ (cid:29) , ∀ | ω | (cid:29) , (6.11)to mean that a condition holds if we take σ, σ , ω in such a way. Remark . In fact, we will have the more precise bound (but we will not need), for examplein (6.18), C N,σ,σ ( ω ) ≤ C N (cid:16) σ − N + ( σ − σ ) − N + | ω | − β/ σ + ( ω − ω σ ) − N (cid:17) , (6.12)where < β ≤ is the Hölder exponent of the distributions E u , E s . Eq.(6.12) implies (6.10). e tX near the trapped set Σ We first define some operators. Looking at Figure 1.1 may help. We have defined the operator (cid:103) exp ◦ : S ( T ∗ M ) → S ( T T ∗ M ) in (3.23). We denote r / Σ : S ( T T ∗ M ) → S ( T Σ T ∗ M ) therestriction operator to the base space Σ ⊂ T ∗ M , see Remark 3.13. We also denote ˜Op Σ (cid:16) d ˜ φ t (cid:17) : S ( T Σ T ∗ M ) → S ( T Σ T ∗ M ) the operator ˜Op (cid:16) d ˜ φ t (cid:17) defined in (3.30), restricted to S ( T Σ T ∗ M ) . This is possible since Σ ininvariant by ˜ φ t and ˜Op (cid:16) d ˜ φ t (cid:17) is fiber-wise. We denote r N = r N/ : S ( T Σ T ∗ M ) → S ( N ) therestriction operator to the sub-bundle N ⊂ T Σ T ∗ M . We have that the exponential map exp N : N → T ∗ M (6.13)30s a diffeomorphism from the neighborhood of the zero section to its image. Since the metric g on T ∗ M is moderate [17, Lemma 4.13 page 27], this implies that the size σ of this neighborhoodcan be taken as large as we need, if we consider frequencies | ω | > ω σ with ω σ large enough thatdepends on σ . To express and use this, we introduce the following cut-off function in frequency χ | ω |≥ ω σ : N → [0 , defined as follows. For ω σ > , and v ∈ N , χ | ω |≥ ω σ ( v ) = (cid:40) if | ω ( v ) | ≥ ω σ otherwiseThen for σ > , we use the operator χ σ that truncates in fibers, defined in (3.25). Hence, forany σ > we can take ω σ > large enough so that the following twisted push forward operatoris well defined (cid:94) (cid:0) exp − N (cid:1) ◦ χ | ω |≥ ω σ χ σ : S ( N ) → S ( T ∗ M ) as the inverse expression of (3.24). Namely, for w ∈ S ( N ) , ρ ∈ T ∗ M , let v = exp − N ( ρ ) ∈ N ρ , ρ ∈ Σ , (cid:18) (cid:94) (cid:0) exp − N (cid:1) ◦ w (cid:19) ( ρ ) := e iρ ( dπ ( v )) w ( v ) . (6.14)We combine these previous operators together with T : C ∞ ( M ) → S ( T ∗ M ) in (3.5), anddefine Definition 6.7.
We define the operators ˇ T σ := r / Σ χ | ω |≥ ω σ χ σ (cid:0) (cid:103) exp ◦ (cid:1) T : C ∞ ( M ) → S ( T Σ T ∗ M ) (6.15) ˇ T ∆ σ := T † (cid:94) (cid:0) exp − N (cid:1) ◦ r N : S ( T Σ T ∗ M ) → C ∞ ( M ) . (6.16)The next theorem shows that with these operators we get a good approximation of theoperator e tX in a neighborhood of the trapped set Σ in terms of the differential d ˜ φ t on Σ . Wewill use the notation ≈ t defined in (6.7) and the operator ˆ χ σ defined in (5.11). Theorem 6.8.
For any fixed t ∈ R , we have for σ (cid:29) σ , e tX ˆ χ σ ≈ t ˇ T ∆ σ ˜Op Σ (cid:16) d ˜ φ t (cid:17) ˇ T σ ˆ χ σ (6.17) Proof.
From Theorem 3.14 we have e tX ˆ χ σ = (3 . (cid:0) T † r (cid:1) ˜Op (cid:16) d ˜ φ t (cid:17) (cid:0) χ σ (cid:103) exp ◦ T (cid:1) ˆ χ σ + R with R satisfying, for any N > and N (cid:48) > , R ∈ Ψ ˜ φ t (cid:16)(cid:16)(cid:0) δ ⊥ ( ρ ) (cid:1) γ σ + C N (cid:48) σ − N (cid:48) + C N ( σ − σ ) − N (cid:17) C N (cid:104) dist g ( ρ, Σ) − σ (cid:105) − N (cid:17) , (6.18)where the factor C N (cid:104) dist g ( ρ, Σ) − σ (cid:105) − N comes from the cut-off χ Σ ,σ and the term C N ( σ − σ ) − N comes from the truncation operator ˆ χ σ in front of ˇ T σ . Recall that every constant depends on t from Definition 3.8. On the trapped set we have (cid:0) δ ⊥ ( ρ ) (cid:1) γ (cid:16) (3 . min (cid:16) δ γ , | ω | − γ/ (cid:17) . We continueusing Lemma C.27, e tX ˆ χ σ = ( C. T † (cid:94) (cid:0) exp − N (cid:1) ◦ r N χ σ ˜Op Σ (cid:16) d ˜ φ t (cid:17) r / Σ χ σ (cid:103) exp ◦ T ˆ χ σ + R = ˇ T ∆ σ ˜Op Σ (cid:16) d ˜ φ t (cid:17) ˇ T σ ˆ χ σ + R. We take > γ > β so that (cid:104) ω (cid:105) − γ/ ≤ (cid:104) ω (cid:105) − β/ and we have R ∈ Ψ ˜ φ t ,g/σ ( a N,σ ,σ ) with a N,σ ,σ defined in (6.9). This gives (6.17). 31 .4 Factorization formula for ˜Op Σ (cid:16) d ˜ φ t (cid:17) Now that we have obtained expression (6.17), the objective is the study of the operator ˜Op Σ (cid:16) d ˜ φ t (cid:17) : S ( T Σ T ∗ M ) → S ( T Σ T ∗ M ) restricted to the trapped set Σ (recall remark 5.3). The decomposition (6.5) is orthogonal forboth the symplectic form and the modified metric ˜ g , and preserved by the differential of theflow, i.e. (cid:16) d ˜ φ t (cid:17) / Σ ≡ (cid:18) d ˜ φ tK d ˜ φ tN (cid:19) . Remark . Eq. (6.5) implies that on the trapped set Σ we have a natural identification S ( T Σ T ∗ M ) = S ( K ) ⊗ Σ S ( N ) . (6.19)where S ( K ) ⊗ Σ S ( N ) is a notation for the space of sections S (Σ; S ( K ) ⊗ S ( N )) , i.e. thetensor product is fiber-wise over Σ and not global. Theorem 6.10. “Factorization formula” . On the trapped set Σ , with respect to theidentification (6.19) we have ˜Op Σ (cid:16) d ˜ φ t (cid:17) = ˜Op (cid:16) d ˜ φ tK (cid:17) ⊗ Σ ˜Op (cid:16) d ˜ φ tN (cid:17) . (6.20)Formula (6.20) is pointwise on the vector space T ρ T ∗ M for ρ ∈ Σ and is shown in LemmaC.22. N s Now we consider specifically the second component ˜Op (cid:16) d ˜ φ tN (cid:17) in the right hand side of (6.20).We have seen in (6.3) that the normal bundle N = N u ⊕ N s is symplectic and that the sub-bundle N s is Lagrangian. We will now focus on this Lagrangian bundle N s → Σ of unstabledirections. Notation . The bundle N s → Σ defined in (6.4) is Hölder continuous with exponent β . Wedefine S (cid:91) ( N s ) := S (cid:91) (Σ; S ( N s )) := (cid:110) f ∈ C β (Σ; S ( N s )) | f is C ∞ along the orbit of ˜ φ t and sup ω ( ρ ) ≥ ω f ( ρ ) = O (cid:0) (cid:104) ω (cid:105) −∞ (cid:1)(cid:41) i.e. the set of Hölder continuous function on Σ valued in S ( N s ( ρ )) for each ρ ∈ Σ , smooth inthe neutral direction K , and that decay very fast on Σ as ω → ∞ . This notation will alsoconcerns later other space of sections over Σ , denoted similarly S (cid:91) ( N s ⊕ N ∗ s ) , S (cid:91) (Σ; F ) etc. F → Σ We recall that definition and basic properties of Bargmann transform are given in Section C ofthe appendix. We will use them in this section. The bundle N s has been defined in (6.4). In(C.8) we define the Bargman transform B N s : S (cid:91) ( N s ) → S (cid:91) ( N s ⊕ N ∗ s ) = ( C. S (cid:91) (cid:0) N s ⊕ N ⊥ g s (cid:1) = S (cid:91) ( N ) . S N s ,N u : N → N be the symplectic map given in (C.42) such that S N s ,N u ( N s ) = N s , S N s ,N u (cid:0) N ⊥ g s (cid:1) = N u . Then ˜Op ( S N s ,N u ) : S (cid:91) ( N ) → S (cid:91) ( N ) is defined as in (C.31) and we put B N s ,N u := ˜Op ( S N s ,N u ) B N s : S (cid:91) ( N s ) → S (cid:91) ( N ) . (6.21) Definition 6.12.
Let F := | det N s | − / ⊗ S ( N s ) (6.22)be a Hölder continuous bundle over Σ where | det N s | − / is the dual of the half densitybundle. Following Notation 6.11 we define the space of sections of FS (cid:91) (Σ; F ) := (cid:40) f ∈ C β (Σ; F ) | f is C ∞ along the orbit of ˜ φ t and sup ω ( ρ ) ≥ ω f ( ρ ) = O (cid:0) (cid:104) ω (cid:105) −∞ (cid:1)(cid:41) and denote X F : S (cid:91) (Σ; F ) → S (cid:91) (Σ; F ) (6.23)the Lie derivative of sections of F , i.e. generator of the group of operators, ∀ t ∈ R , e tX F := (cid:12)(cid:12)(cid:12) det (cid:16) d ˜ φ tN s (cid:17)(cid:12)(cid:12)(cid:12) − / (cid:16) d ˜ φ tN s (cid:17) −◦ : S (cid:91) (Σ; F ) → S (cid:91) (Σ; F ) . (6.24) Remark . The linear map d ˜ φ tN s is expanding on N s , hence (cid:12)(cid:12)(cid:12) det (cid:16) d ˜ φ tN s (cid:17)(cid:12)(cid:12)(cid:12) > . Lemma 6.14.
We have ˜Op (cid:16) d ˜ φ tN (cid:17) = B N s ,N u e tX F B † N s ,N u . (6.25) Proof.
From Lemma C.26 (with
Φ = d ˜ φ tN and φ = (cid:16) d ˜ φ tN s (cid:17) − ) we have ˜Op (cid:16) d ˜ φ tN (cid:17) = (cid:12)(cid:12)(cid:12) det (cid:16) d ˜ φ tN s (cid:17)(cid:12)(cid:12)(cid:12) − / B N s ,N u (cid:16) d ˜ φ tN s (cid:17) −◦ B † N s ,N u : S (cid:91) ( N ) → S (cid:91) ( N ) (6.26) = (6 . B N s ,N u e tX F B † N s ,N u . (6.27) ˇOp (cid:0) e tX F a (cid:1) We will consider a ∈ S (cid:91) (Σ; End ( S ( N s ))) called a symbol that denotes (as in Notation 6.11) amap a : ρ ∈ Σ → a ( ρ ) ∈ End ( S ( N s )) . For generally we will consider a ∈ S (cid:91) (Σ; L ( S ( N s ) , S (cid:48) ( N s ))) .Then from (6.21), B N s ,N u a B † N s ,N u ∈ L ( S (cid:91) ( N ) , S (cid:48) (cid:91) ( N )) . Definition 6.15.
For a ∈ S (cid:91) (Σ; End ( S ( N s ))) (or a ∈ S (cid:91) (Σ; L ( S ( N s ) , S (cid:48) ( N s ))) ) wedefine ˇOp ( a ) := (cid:102) Op (cid:0) Id S ( K ) (cid:1) ⊗ Σ (cid:16) B N s ,N u a B † N s ,N u (cid:17) : S (cid:91) ( T Σ T ∗ M ) → S (cid:48) (cid:91) ( T Σ T ∗ M ) , and for any t ∈ R , ˇOp (cid:0) e tX F a (cid:1) := (cid:102) Op (cid:16) d ˜ φ tK (cid:17) ⊗ Σ (cid:16) B N s ,N u e tX F a B † N s ,N u (cid:17) : S (cid:91) ( T Σ T ∗ M ) → S (cid:48) (cid:91) ( T Σ T ∗ M ) , (6.28)33rom (6.20) and (6.25) we get the following lemma. Lemma 6.16.
For any t ∈ R , we have ˇOp (cid:0) e tX F (cid:1) = ˜Op Σ (cid:16) d ˜ φ t (cid:17) . (6.29) Remark . Later we will use that for any t, t (cid:48) ∈ R , a, b ∈ S (cid:91) (Σ; End ( S ( N s ))) , ˇOp (cid:0) e tX F a (cid:1) ˇOp (cid:16) e t (cid:48) X F b (cid:17) = ˇOp (cid:16) e tX F ae t (cid:48) X F b (cid:17) . (6.30) T k Since d ˜ φ tN s : N s → N s is linear, the push-forward operator (cid:16) d ˜ φ tN s (cid:17) −◦ keeps invariant the spaceof homogeneous polynomial functions of order k ∈ N denoted Pol k ( N s ) ⊂ S (cid:48) ( N s ) , for any k .To express this property, we consider the finite rank “Taylor projectors”, defined from (C.39),denoted T k : S (cid:91) ( N s ) → Pol k ( N s ) ⊂ S (cid:48) (cid:91) ( N s ) (6.31)and T [0 ,K ] := K (cid:88) k =0 T k , T ≥ ( K +1) = Id − T [0 ,K ] , (6.32)satisfying [ T k , T k (cid:48) ] = δ k = k (cid:48) T k and (cid:2) e tX F , T k (cid:3) = 0 . From the last relation, it is interesting to denote F k := | det N s | − / ⊗ Pol k ( N s ) (6.33)that is a finite rank bundle over Σ . Then the image of T k is S (cid:91) (Σ; F k ) . We denote X F k thegenerator X F in (6.23) restricted to the space of sections S (cid:91) (Σ; F k ) . We also denote F [0 ,K ] := K (cid:77) k =0 F k , (6.34)the direct sum bundle with generator X F [0 ,K ] . X F k in L (Σ; F k ) The operators (cid:0) e tX F k (cid:1) t ∈ R in (6.24) form a group of point-wise linear bundle operators over ˜ φ t : Σ → Σ , consequently L norm and L p norms for p ∈ [1 , ∞ ] are equal (cid:13)(cid:13) e tX F k (cid:13)(cid:13) L = (cid:13)(cid:13) e tX F k (cid:13)(cid:13) L p . (6.35) Definition 6.18.
For any k ∈ N , let γ ± k := lim t →±∞ log (cid:13)(cid:13) e tX F k (cid:13)(cid:13) /tL ∞ (6.36)34 emma 6.19. We have the estimates ∀ (cid:15) > , ∃ C (cid:15) > , ∀ t ≥ , (cid:13)(cid:13) e tX F k (cid:13)(cid:13) L ≤ C (cid:15) e t ( γ + k + (cid:15) ) , (cid:13)(cid:13) e − tX F k (cid:13)(cid:13) L ≤ C (cid:15) e − t ( γ − k − (cid:15) ) . (6.37) Equivalently ∀ (cid:15) > , ∃ C (cid:15) > , ∀ z ∈ C , Re ( z ) > γ + k + (cid:15) or Re ( z ) < γ − k − (cid:15) implies (cid:13)(cid:13) ( z − X F k ) − (cid:13)(cid:13) L ≤ C (cid:15) . (6.38) Consequently the spectrum of the operator X F k in L (Σ; F k ) is contained in the verticalband B k := (cid:8) Re ( z ) ∈ (cid:2) γ − k , γ + k (cid:3)(cid:9) . (6.39) Remark . In fact the spectrum of X F k is made by essential spectrum in this band B k . Proof. (6.36) implies (6.37) that implies (6.38) and (6.39) by writing the convergent expressionfor
Re ( z ) > γ + k + (cid:15) : ( z − X F k ) − = (cid:90) ∞ e t ( X F k − z ) dt that gives (cid:13)(cid:13) ( z − X F k ) − (cid:13)(cid:13) ≤ (cid:90) ∞ e − Re( z ) t (cid:13)(cid:13) e tX F k (cid:13)(cid:13) dt ≤ (cid:90) ∞ e − Re( z ) t C (cid:15) (cid:48) e t ( γ + k + (cid:15) (cid:48) ) dt ≤ C (cid:15) if (cid:15) (cid:48) < (cid:15) .We have equivalent results for the family of operators (cid:0) ˇOp (cid:0) e tX F T k (cid:1)(cid:1) t ∈ R in S (cid:91) (cid:0) Σ; Im (cid:0) ˇOp ( T k ) (cid:1)(cid:1) : Lemma 6.21.
For k ∈ N , we have (cid:13)(cid:13) ˇOp (cid:0) e tX F T k (cid:1)(cid:13)(cid:13) L = (cid:13)(cid:13) e tX F k (cid:13)(cid:13) L (6.40) Proof.
From Definition C.24, ˜Op (cid:16) d ˜ φ tK (cid:17) is unitary. Hence (6.28) gives (6.40). H W ( T Σ T ∗ M ) and remainder term Now we will consider the operator ˇOp (cid:0) e tX F T ≥ ( K +1) (cid:1) : S (cid:91) ( T Σ T ∗ M ) → S (cid:48) (cid:91) ( T Σ T ∗ M ) . We definea weight function W : N → R + similar to W : T ∗ M → R + in (5.2), with the same parameters h > , < γ < , R > , as follows. For v ∈ N decomposed as v = v u + v s with v u ∈ N u , v s ∈ N s , let W ( v ) := (cid:68) h γ ( v ) (cid:107) v s (cid:107) g (cid:69) R (cid:68) h γ ( v ) (cid:107) v u (cid:107) g (cid:69) R (6.41)with h γ ( v ) = h (cid:68) (cid:107) v (cid:107) g (cid:69) − γ . The metric g on T ∗ M is moderate [17, Lemma 4.13 page 27]. This implies that under theexponential map (6.13) exp N : N → T ∗ M , the functions W and W are equivalent in any σ > neighborhood of the trapped set, provided we consider frequencies | ω | > ω σ with ω σ large enough. In more precise terms, ∃ C ≥ , ∀ σ > , ∃ ω σ > , ∀ | ω | > ω σ , ∀ v ∈ N with ω (exp N ( v )) = ω and (cid:107) v (cid:107) g ≤ σ we have C W ( v ) ≤ W (exp N ( v )) ≤ C W ( v ) . (6.42)35e extend W : N → R + as a function on T Σ T ∗ M = K ⊕ N by tensoring with function along K and define the Sobolev space H W ( T Σ T ∗ M ) := W − L ( T Σ T ∗ M ) = L ( T Σ T ∗ M ; W ) . Lemma 6.22.
For K ∈ N and R (cid:29) large enough so that K < r − (with r = R (1 − γ ) the order of W defined in (5.8)), we have that ∀ (cid:15) > , ∃ C (cid:15) > , ∀ t ≥ , (cid:13)(cid:13) ˇOp (cid:0) e tX F T ≥ ( K +1) (cid:1) χ | ω |≥ (cid:13)(cid:13) H W ( T Σ T ∗ M ) ≤ C (cid:15) e ( γ + K +1 + (cid:15) ) t . (6.43) Proof.
This is given by Proposition D.3.
Proposition 6.23.
For every k < r − (with r = R (1 − γ ) the order of W defined in(5.8)), there exists C k such that (cid:13)(cid:13) ˇOp ( T k ) (cid:13)(cid:13) H W ( T Σ T ∗ M ) ≤ C k . (6.44) Proof.
This is direct consequence of Lemma D.6.
In this section we will collect the results of the previous sections. For this, we introduce firstthe definition of a symbol and a quantum operator acting on S (cid:91) (Σ; F ) . We show few usefulproperties such as the composition formula. Then we use them to prove the main Theoremsof this paper, namely Theorem 1.4 about the band spectrum, Theorem 7.13 that gives somedecay estimates and Theorem 7.15 that approximate the transfer operator e tX by some quantumoperator. We consider the operators ˇ T σ : C ∞ ( M ) → S (cid:91) ( T Σ T ∗ M ) and ˇ T ∆ σ : S (cid:91) ( T Σ T ∗ M ) → C ∞ ( M ) that defined in (6.15). Recall that parameter σ > is the distance to the trapped set Σ afterwhich there is a cut-off. Recall also that although not explicitly written, there is a cutoff atfrequencies smaller than some parameter ω σ > in ˇ T σ , ˇ T ∆ σ . We will also use the bundle ofoperators ˇOp (cid:0) e tX F a (cid:1) : S (cid:91) ( T Σ T ∗ M ) → S (cid:48) (cid:91) ( T Σ T ∗ M ) from Definition 6.15. Definition 7.1.
For a ∈ S (cid:91) (Σ; End ( F )) called a symbol and for t ∈ R , σ , σ > ,the quantization of the symplectic bundle map e tX F a : S (cid:91) (Σ; F ) → S (cid:91) (Σ; F ) is theoperator Op σ ,σ (cid:0) e tX F a (cid:1) := ˇ T ∆ σ ˇOp (cid:0) e tX F a (cid:1) ˇ T σ : C ∞ ( M ) → C ∞ ( M ) . (7.1)In the case σ = σ , for simplicity we will denote Op σ (cid:0) e tX F a (cid:1) := Op σ ,σ (cid:0) e tX F a (cid:1) . Later we will use Definition (7.1) only with symbols a = T k , a = T [0 ,K ] , a ∈ End (cid:0) F [0 ,K ] (cid:1) , a = T ≥ ( K +1) , or a = Id F . 36 .1.2 Approximation of the dynamics by quantum operators We will use the notation ≈ t defined in Definition 6.4 and notation σ (cid:29) σ (cid:29) defined in (6.11). Theorem 7.2. “Approximation of the dynamics by quantum operator”.
For any t ∈ R , and σ (cid:29) σ (cid:29) , we have e tX ˆ χ σ ≈ t Op σ (cid:0) e tX F (cid:1) ˆ χ σ . (7.2) Proof.
This is from (6.17) from Theorem 6.8 and Definitions (7.1) and (6.29).
The next lemma shows the uniform boundedness of the operators ˇ T σ , ˇ T ∆ σ with respect to σ . Lemma 7.3. ∃ C > , ∀ σ > , ∃ ω σ > , (cid:13)(cid:13) ˇ T σ (cid:13)(cid:13) H W ( M ) →H W ( T Σ T ∗ M ) ≤ C, (cid:13)(cid:13) ˇ T ∆ σ (cid:13)(cid:13) H W ( T Σ T ∗ M ) →H W ( M ) ≤ C. (7.3) Proof.
Recall the expression ˇ T σ = (6 . r / Σ χ | ω |≥ ω σ χ σ (cid:0) (cid:103) exp ◦ (cid:1) T where, from (6.42), the operator r / Σ χ | ω |≥ ω σ χ σ (cid:0) (cid:103) exp ◦ (cid:1) satisfies ∃ C > , ∀ σ > , ∃ ω σ > , (cid:13)(cid:13) r / Σ χ | ω |≥ ω σ χ σ (cid:0) (cid:103) exp ◦ (cid:1)(cid:13)(cid:13) L ( T ∗ M ; W ) →H W ( T Σ T ∗ M ) ≤ C. Since T : H W ( M ) → L ( T ∗ M ; W ) is an isometry, we have (cid:107)T (cid:107) H W ( M ) → L ( T ∗ M ; W ) ≤ and wededuce the first bound in (7.3). Similarly from the expression ˇ T ∆ σ = (6 . T † (cid:94) (cid:0) exp − N (cid:1) ◦ r N χ | ω |≥ ω σ χ σ ,we deduce the second claim. Theorem 7.4.
For any K ∈ N , we have ∀ (cid:15) > , ∃ C K,(cid:15) > , ∀ t ≥ , ∀ σ , σ > , ∃ ω σ , ω σ > , (cid:13)(cid:13) Op σ ,σ (cid:0) e tX F T ≥ ( K +1) (cid:1)(cid:13)(cid:13) H W ( M ) ≤ C K,(cid:15) e t ( γ + K +1 + (cid:15) ) , (7.4) and (cid:13)(cid:13) Op σ ,σ (cid:0) e tX F T [0 ,K ] (cid:1)(cid:13)(cid:13) H W ( M ) ≤ C K,(cid:15) e t ( γ +0 + (cid:15) ) . (7.5) For negative time ∀ (cid:15) > , ∃ C K,(cid:15) > , ∀ t ≤ , ∀ σ , σ > , ∃ ω σ , ω σ > , (cid:13)(cid:13) Op σ ,σ (cid:0) e tX F T [0 ,K ] (cid:1)(cid:13)(cid:13) H W ( M ) ≤ C K,(cid:15) e t ( γ − K − (cid:15) ) . (7.6) Proof.
This is from Lemma 7.3, Lemma 6.21 and Lemma 6.19.
Remark . In particular, taking t = 0 in (7.5), we get some uniform boundedness for theoperators Op σ ,σ ( T k ) as follows. For any k ∈ N , ∃ C k > , ∀ σ , σ > , ∃ ω σ , ω σ > , (cid:13)(cid:13) Op σ ,σ ( T k ) (cid:13)(cid:13) H W ( M ) ≤ (6 . , . C k . (7.7)37 .2 Composition formula We first introduce the following definition concerning symbols a that enter in quantum operators(7.1). Recall F [0 ,K ] defined in (6.34). Definition 7.6.
For K ∈ N , a symbol a ∈ S (cid:91) (cid:0) Σ; End (cid:0) F [0 ,K ] (cid:1)(cid:1) is slowly varying withrespect to the metric g on Σ if there exists h ∈ C ( R ∗ ; R + ) that satisfies h ( ω ) → as ω → ∞ and N ≥ such that ∀ ρ, ρ (cid:48) ∈ Σ , (cid:107) a ( ρ (cid:48) ) − a ( ρ ) (cid:107) H W ≤ h ( ω ( ρ )) (cid:104) dist g ( ρ (cid:48) , ρ ) (cid:105) N . .For example for any k ≤ K , T k ∈ S (cid:91) (cid:0) Σ; End (cid:0) F [0 ,K ] (cid:1)(cid:1) is slowly varying because the bundle F [0 ,K ] → Σ is Hölder continuous with some Hölder exponent < β ≤ , giving h ( ω ) = O (cid:0) ω − β/ (cid:1) .The next composition formula are a consequence of the exact relation (6.30) after quanti-zation. Theorem 7.7. “Composition formula”.
For slowly varying symbols a, b ∈S (cid:91) (cid:0) Σ; End (cid:0) F [0 ,K ] (cid:1)(cid:1) and any t, t (cid:48) ∈ R , we have for any σ , σ , σ , σ (cid:29) , Op σ ,σ (cid:0) e tX F a (cid:1) Op σ ,σ (cid:16) e t (cid:48) X F b (cid:17) ≈ t + t (cid:48) Op σ ,σ (cid:16) e tX F ae t (cid:48) X F b (cid:17) . (7.8) For (cid:28) σ (cid:28) σ , σ or (cid:28) σ (cid:28) σ , σ depending on t , we have Op σ ,σ (cid:0) e tX F (cid:1) Op σ ,σ (cid:16) e t (cid:48) X F (cid:17) ≈ t + t (cid:48) Op σ ,σ (cid:16) e tX F e t (cid:48) X F (cid:17) . (7.9) For (cid:28) σ (cid:28) σ , σ and σ (cid:29) we have Op σ ,σ (cid:0) e tX F a (cid:1) Op σ ,σ (cid:16) e t (cid:48) X F (cid:17) ≈ t + t (cid:48) Op σ ,σ (cid:16) e tX F ae t (cid:48) X F (cid:17) . (7.10) For (cid:28) σ (cid:28) σ , σ and σ (cid:29) we have Op σ ,σ (cid:0) e tX F (cid:1) Op σ ,σ (cid:16) e t (cid:48) X F b (cid:17) ≈ t + t (cid:48) Op σ ,σ (cid:16) e tX F e t (cid:48) X F b (cid:17) . (7.11) Proof.
On Euclidean space model without cutoff (i.e. σ = ∞ ), the relations in Theorem 7.7are exact. The corrections and error terms have different origin:1. The truncation at size σ , σ .2. The non-linearity but temperate and moderate property of the metric g from [17, Lemma4.13], as already discussed in the proof of Theorem 3.14.3. The slow variation of the symbols, from Definition 7.6.These errors terms are contained in the definition of the equivalence ≈ t + t (cid:48) .38 orollary 7.8. As particular cases of (7.8), we have • Egorov formula: for a slowly varying symbol a ∈ S (cid:91) (cid:0) Σ; End (cid:0) F [0 ,K ] (cid:1)(cid:1) , and t ∈ R , Op σ ,σ (cid:0) e tX F (cid:1) Op σ ,σ ( a ) ≈ t Op σ ,σ (cid:0) e t ad X F a (cid:1) Op σ ,σ (cid:0) e tX F (cid:1) . (7.12) where (cid:28) σ , σ (cid:28) σ , σ and with the natural action e t ad X F a = e t [ X F ,. ] a = e tX F ae − tX F . • Approximate projectors : for any k, k (cid:48) ∈ N , Op σ ( T k ) Op σ ( T k (cid:48) ) ≈ δ k = k (cid:48) Op σ ( T k ) . (7.13)• “Decomposition of the dynamics in components” : for any k, k (cid:48) ∈ N , Op σ (cid:0) e tX F T k (cid:1) Op σ (cid:16) e t (cid:48) X F T k (cid:48) (cid:17) ≈ t + t (cid:48) δ k = k (cid:48) Op σ (cid:16) e ( t + t (cid:48) ) X F T k (cid:17) . (7.14) Remark . The algebraic structure of (7.14) will manifest itself later in the Ruelle spectrumwith the band structure.
Proof.
We write Op σ ,σ (cid:0) e tX F (cid:1) Op σ ,σ ( a ) ≈ t (7 . Op σ ,σ (cid:0) e tX F a (cid:1) =Op σ ,σ (cid:0) e t ad X F ae tX F (cid:1) ≈ t (7 . Op σ ,σ (cid:0) e t ad X F a (cid:1) Op σ ,σ (cid:0) e tX F (cid:1) where (cid:28) σ , σ (cid:28) σ , σ . This gives (7.12). Eq.(7.13) and Eq.(7.14) come from (7.8). For the proof, we follow the strategy presented in the proof of [17, Lemma 5.16]. Let K ∈ N , (cid:15) (cid:48) > be fixed and consider z ∈ C in the spectral plane such that γ + K +1 + (cid:15) (cid:48) < Re ( z ) < γ − K − (cid:15) (cid:48) . (7.15)We put ω = Im ( z ) . Let δ > . Following the same notations as in [17, Section 5.3.3], we willconsider the frequency intervals J ω := [ ω − , ω + 1] , J (cid:48) ω,δ := [ ω − δ, ω + δ ] , so that J ω ⊂ J (cid:48) ω,δ . We consider the following partition of the cotangent bundle T ∗ M = Ω ∪ Ω ∪ Ω (7.16)with Ω ,σ := (cid:8) ρ ∈ T ∗ M | dist g ( ρ, Σ) ≤ σ and ω ( ρ ) ∈ J (cid:48) ω,δ (cid:9) , Ω ,σ := (cid:8) ρ ∈ T ∗ M | dist g ( ρ, Σ) > σ and ω ( ρ ) ∈ J (cid:48) ω,δ (cid:9) , Ω := (cid:8) ρ ∈ T ∗ M | ω ( ρ ) / ∈ J (cid:48) ω,δ (cid:9) . For j = 0 , , , we define χ Ω j : T ∗ M → [0 , as being the characteristic function of Ω j and asin (5.11), we set ˆ χ Ω j := T † χ Ω j T , Id H W ( M ) = (7 . ˆ χ Ω ,σ + ˆ χ Ω ,σ + ˆ χ Ω . We will now construct the resolvent of X at z from contributions of each domain Ω j with j = 0 , , . Ω ,σ If we take σ sufficiently large relative to σ , we have ˆ χ Ω ,σ ≈ Op σ (Id) ˆ χ Ω ,σ . We have Op σ (cid:0) T ≥ ( K +1) (cid:1) + Op σ (cid:0) T [0 ,K ] (cid:1) = (6 . Op σ (Id) . Hence, using notation (6.11), ∀ (cid:15) (cid:48)(cid:48) > , ∀ σ (cid:29) σ (cid:29) , ∀ ω (cid:29) , (cid:13)(cid:13) ˆ χ Ω ,σ − (cid:0) Op σ (cid:0) T ≥ ( K +1) (cid:1) + Op σ (cid:0) T [0 ,K ] (cid:1)(cid:1) ˆ χ Ω ,σ (cid:13)(cid:13) H W ( M ) ≤ (cid:15) (cid:48)(cid:48) . Key estimates:Lemma 7.10.
We have ∀ (cid:15) > , ∃ C (cid:15) > , ∀ t ≥ , ∃ σ t > , ∀ σ > σ t , ∃ ω σ > , (cid:13)(cid:13) e − tX Op σ (cid:0) T [0 ,K ] (cid:1)(cid:13)(cid:13) H W ( M ) ≤ C (cid:15) e − t ( γ − K − (cid:15) ) (7.17) and (cid:13)(cid:13) e tX Op σ (cid:0) T ≥ ( K +1) (cid:1)(cid:13)(cid:13) H W ( M ) ≤ C (cid:15) e t ( γ + K +1 + (cid:15) ) . (7.18) Proof.
Let t ≥ . We write for σ (cid:29) σ (cid:29) σ , e − tX Op σ (cid:0) T [0 ,K ] (cid:1) ≈ t ˆ χ σ e − tX ˆ χ σ Op σ (cid:0) T [0 ,K ] (cid:1) ≈ t (7 . ˆ χ σ Op σ (cid:0) e − tX F (cid:1) ˆ χ σ Op σ (cid:0) T [0 ,K ] (cid:1) (7.19) ≈ t ˆ χ σ Op σ (cid:0) e − tX F (cid:1) Op σ (cid:0) T [0 ,K ] (cid:1) ≈ t (7 . ˆ χ σ Op σ (cid:48) ,σ (cid:0) e − tX F T [0 ,K ] (cid:1) , (7.20)with σ (cid:48) that depends on σ and t . Then (7.5) gives (7.17).From Theorem 7.2 and Theorem 7.7 we have for σ (cid:29) σ , e tX Op σ (cid:0) T ≥ ( K +1) (cid:1) = ˆ χ σ e tX Op σ (cid:0) T ≥ ( K +1) (cid:1) + (1 − ˆ χ σ ) e tX Op σ (cid:0) T ≥ ( K +1) (cid:1) . (7.21)For the second term of (7.21), from Theorem 5.2, we have for Λ large enough, (cid:13)(cid:13) (1 − ˆ χ σ ) e tX (cid:13)(cid:13) H W ( M ) ≤ (5 . Ce − Λ t ≤ C (cid:15) e t ( γ + K +1 + (cid:15) ) . For the first term of (7.21), we have for σ (cid:29) σ (cid:29) σ (cid:29) σ depending on t , ˆ χ σ e tX Op σ (cid:0) T ≥ ( K +1) (cid:1) ≈ t ˆ χ σ ˆ χ σ e tX ˆ χ σ Op σ (cid:0) T ≥ ( K +1) (cid:1) ≈ t (7 . ˆ χ σ ˆ χ σ Op σ (cid:0) e tX F (cid:1) ˆ χ σ Op σ (cid:0) T ≥ ( K +1) (cid:1) ≈ t ˆ χ σ Op σ (cid:0) e tX F (cid:1) Op σ (cid:0) T ≥ ( K +1) (cid:1) = ˆ χ σ Op σ (cid:0) e tX F (cid:1) Op σ (Id) − ˆ χ σ Op σ (cid:0) e tX F (cid:1) Op σ (cid:0) T [0 ,K ] (cid:1) ≈ t (7 . , . ˆ χ σ Op σ (cid:0) e tX F (cid:1) − ˆ χ σ Op σ (cid:0) e tX F T [0 ,K ] (cid:1) = ˆ χ σ Op σ (cid:0) e tX F T ≥ ( K +1) (cid:1) . Then (7.4) gives (7.18).
Remark . In Lemma 7.10, due to Theorem 7.7, we need to take σ large enough dependingon t . This was not needed in Theorem 7.4. 40 pproximate resolvent R (0)in ( z ) and R (0)out ( z ) : For T ≥ we define R (0)in ( z ) := − (cid:90) − T e − tz e tX Op σ (cid:0) T [0 ,K ] (cid:1) ˆ χ Ω ,σ dt. (7.22)Then ( z − X ) R (0)in ( z ) = (cid:2) e − tz e tX (cid:3) − T Op σ (cid:0) T [0 ,K ] (cid:1) ˆ χ Ω ,σ = Op σ (cid:0) T [0 ,K ] (cid:1) ˆ χ Ω ,σ − e T z e − T X Op σ (cid:0) T [0 ,K ] (cid:1) ˆ χ Ω ,σ . Hence if we take (cid:15) > so that < (cid:15) < (cid:15) (cid:48) , we have (cid:13)(cid:13)(cid:13) R (0)in ( z ) (cid:13)(cid:13)(cid:13) H W ( M ) ≤ (7 . , . (cid:90) T e t Re( z ) C (cid:15) e − t ( γ − K − (cid:15) ) dt ≤ (7 . C (cid:15) (cid:90) T e − ( (cid:15) (cid:48) − (cid:15) ) t dt ≤ C (cid:15) (cid:15) (cid:48) − (cid:15) , and (cid:13)(cid:13)(cid:13) ( z − X ) R (0)in ( z ) − Op σ (cid:0) T [0 ,K ] (cid:1) ˆ χ Ω ,σ (cid:13)(cid:13)(cid:13) H W ( M ) ≤ (7 . e T Re( z ) C (cid:15) e − T ( γ − K − (cid:15) ) ≤ (7 . C (cid:15) e − ( (cid:15) (cid:48) − (cid:15) ) T . Similarly we define R (0)out ( z ) := (cid:90) T e − tz e tX Op σ (cid:0) T ≥ ( K +1) (cid:1) ˆ χ Ω ,σ dt (7.23)and get (cid:13)(cid:13)(cid:13) R (0)out ( z ) (cid:13)(cid:13)(cid:13) H W ( M ) ≤ (7 . , . C (cid:15) (cid:15) (cid:48) − (cid:15) , and (cid:13)(cid:13)(cid:13) ( z − X ) R (0)out ( z ) − Op σ (cid:0) T ≥ ( K +1) (cid:1) ˆ χ Ω ,σ (cid:13)(cid:13)(cid:13) H W ( M ) ≤ (7 . , . C (cid:15) e − ( (cid:15) (cid:48) − (cid:15) ) T . Ω ,σ The next lemma is similar to Theorem 5.2. Let Λ ≥ − γ + K +1 . Key estimate:
From [17, (5.62) p.73], we have that ∃ C > , ∀ t ≥ , ∃ σ t , ∀ σ > σ t (cid:13)(cid:13) e tX ˆ χ Ω ,σ (cid:13)(cid:13) H W ( M ) ≤ Ce − Λ t ≤ Ce tγ + K +1 . (7.24) Approximate resolvent:
For T ≥ we define R (1) ( z ) := (cid:90) T e − tz e tX ˆ χ Ω ,σ dt (7.25)and similarly to above we get (cid:13)(cid:13) R (1) ( z ) (cid:13)(cid:13) H W ( M ) ≤ (7 . , . C(cid:15) (cid:48) , and (cid:13)(cid:13) ( z − X ) R (1) ( z ) − Op σ (cid:0) T ≥ ( K +1) (cid:1) ˆ χ Ω ,σ (cid:13)(cid:13) H W ( M ) ≤ (7 . , . Ce − (cid:15) (cid:48) T . .3.3 Contribution on Ω We define R (2) ( z ) := (cid:90) ρ ∈ Ω z − i ω ( ρ ) Π ( ρ ) dρ (2 π ) d +1 where Π ( ρ ) is the projector on the wave packet Φ ρ that enters in (3.5), see [17, p.34]. We havethe following properties. Lemma 7.12. [17, Lemma 5.18]There exists
C > such that for any ω ∈ R , δ > , (cid:13)(cid:13) R (2) ( z ) (cid:13)(cid:13) H W ( M ) ≤ Cδ , and (cid:13)(cid:13) ( z − X ) R (2) ( z ) − ˆ χ Ω (cid:13)(cid:13) H W ( M ) ≤ Cδ .
The final approximate resolvent is R ( z ) := R (0)in ( z ) + R (0)out ( z ) + R (1) ( z ) + R (2) ( z ) . From previous estimates we have (cid:107) R ( z ) (cid:107) H W ( M ) ≤ C (cid:15) (cid:15) (cid:48) − (cid:15) + C(cid:15) (cid:48) + Cδ , and (cid:107) ( z − X ) R ( z ) − Id (cid:107) H W ( M ) ≤ (cid:15) (cid:48)(cid:48) + 2 C (cid:15) e − ( (cid:15) (cid:48) − (cid:15) ) T + Ce − (cid:15) (cid:48) T + Cδ .
Hence ( z − X ) R ( z ) = Id − r ( z ) with (cid:107) r ( z ) (cid:107) H W ( M ) ≤ (cid:15) (cid:48)(cid:48) + 2 C (cid:15) e − ( (cid:15) (cid:48) − (cid:15) ) T + Ce − (cid:15) (cid:48) T + Cδ < where the last inequality holds if T (cid:29) , | ω | (cid:29) , δ (cid:29) are large enough. This gives that (Id − r ( z )) − is bounded and ( z − X ) ˜ R ( z ) = Id with ˜ R ( z ) = R ( z ) (Id − r ( z )) − . With a similar construction on the left ˜ R l ( z ) ( z − X ) = Id ,we deduce that (cid:13)(cid:13) ( z − X ) − (cid:13)(cid:13) ≤ C (cid:15) is bounded by C (cid:15) > that does not depend on ω . We have shown (1.8) and this implies (1.7).This finishes the proof of Theorem 1.4. Recall that the space H W ( M ) in (5.6) depends on the weight W that itself depends on param-eter R . We have r = R (1 − γ ) . Let K ∈ N and γ + K +1 defined in (6.36). We take the parameter R large enough such that − C − rλ min + (cid:15) < γ + K +1 . Then Theorem 5.4 shows that the generator X in H W ( M ) has discrete Ruelle spectrum on Re ( z ) > γ + K +1 .Let χ : R + → [0 , with χ ( x ) = 1 for x ≤ and χ ( x ) = 0 for x > . Let (cid:36) > be acut-off frequency and χ (cid:36) : T ∗ M → R + given by χ (cid:36) ( ρ ) = χ (cid:16) (cid:107) ρ (cid:107) gM (cid:36) (cid:17) for ρ ∈ T ∗ M . Using theapproximate projectors Op σ ( T k ) we define ˆ χ (low) (cid:36) := T † χ (cid:36) T , (7.26)42 χ (in) (cid:36),σ,K := Op σ (cid:0) T [0 ,K ] (cid:1) (cid:0) Id − ˆ χ (low) (cid:36) (cid:1) , (7.27) ˆ χ (out) (cid:36),σ,K := (cid:0) Id − Op σ (cid:0) T [0 ,K ] (cid:1)(cid:1) (cid:0) Id − ˆ χ (low) (cid:36) (cid:1) , (7.28)so that Id H ( M ) = ˆ χ (in) (cid:36),σ,K + ˆ χ (out) (cid:36),σ,K + ˆ χ (low) (cid:36) . (7.29)Observe that ˆ χ (low) (cid:36) : S (cid:48) ( M ) → S ( M ) is a smoothing operator hence compact. In the previ-ous decomposition (7.29), the important operators for us are ˆ χ (in) (cid:36),σ,K , ˆ χ (out) (cid:36),σ,K that removes lowfrequencies and respectively select the bands with indices k ≤ K and k ≥ K + 1 . Theorem 7.13. “Some decay estimates”.
For any K ∈ N , we choose H W ( M ; F ) withparameter R (cid:29) large enough. For negative time we have ∀ (cid:15) > , ∃ C (cid:15) > , ∀ t ≤ , ∃ σ t > , ∀ σ > σ t , ∃ ω σ > , ∀ (cid:36) > , (cid:13)(cid:13)(cid:13) e tX F ˆ χ (in) (cid:36),σ,K (cid:13)(cid:13)(cid:13) H W ( M ; F ) ≤ C (cid:15) e t ( γ − K − (cid:15) ) . (7.30) For positive time we have ∀ (cid:15) > , ∃ C (cid:15) > , ∀ t ≥ , ∃ σ t > , ∀ σ > σ t , ∃ ω σ > , ∀ (cid:36) > , (cid:13)(cid:13)(cid:13) e tX F ˆ χ (out) (cid:36),σ,K (cid:13)(cid:13)(cid:13) H W ( M ; F ) ≤ C (cid:15) e t ( γ + K +1 + (cid:15) ) . (7.31) Remark . In fact, (7.31) holds true if we take “ K = − ”, i.e. we have ∀ (cid:15) > , ∃ C (cid:15) > , ∀ t ≥ , ∃ σ t > , ∀ σ > σ t , ∃ ω σ > , ∀ (cid:36) > , (cid:13)(cid:13) e tX F (cid:0) − ˆ χ (low) (cid:36) (cid:1)(cid:13)(cid:13) H W ( M ; F ) ≤ C (cid:15) e t ( γ +0 + (cid:15) ) . (7.32)This has been already shown in [35], and from this we deduce that the operator e tX F in H W ( M ; F ) has discrete spectrum outside the radius e tγ +0 , see Figure 1.3. Proof.
Fix t ∈ R . We have for σ (cid:29) σ > , e tX = e tX ˆ χ σ + e tX (Id − ˆ χ σ ) ≈ t (7 . Op σ (cid:0) e tX F T [0 ,K ] (cid:1) ˆ χ σ + Op σ (cid:0) e tX F T ≥ ( K +1) (cid:1) ˆ χ σ + e tX (Id − ˆ χ σ ) (7.33) Computation of e tX ˆ χ (out) (cid:36),σ,K : Let us suppose t ≥ . For the first term in (7.33) we have from(7.13), Op σ (cid:0) e tX F T [0 ,K ] (cid:1) ˆ χ σ ˆ χ (out) (cid:36),σ ,K = Op σ (cid:0) e tX F T [0 ,K ] (cid:1) ˆ χ σ (cid:0) Id − Op σ (cid:0) T [0 ,K ] (cid:1)(cid:1) (Id − ˆ χ (cid:36) ) ≈ t . (7.34)For the second term in (7.33), we have ∀ (cid:15) > , ∃ C (cid:15) > , ∀ t ≥ , ∀ σ > , ∃ ω σ > , (cid:13)(cid:13)(cid:13) Op σ (cid:0) e tX F T ≥ ( K +1) (cid:1) ˆ χ σ ˆ χ (out) (cid:36),σ,K (cid:13)(cid:13)(cid:13) H W ( M ) ≤ (7 . C (cid:15) e t ( γ + K +1 + (cid:15) ) . For the last term in (7.33), we take − Λ < γ + K +1 so that (5.12) gives for σ > σ t (cid:29) , (cid:13)(cid:13) e tX (Id − ˆ χ σ ) (cid:13)(cid:13) H W ( M ) ≤ Ce − Λ t ≤ Ce tγ + K +1 . We deduce that ∀ (cid:15) > , ∃ C (cid:15) > , ∀ t ≥ , ∃ σ t > , ∀ σ > σ t , ∃ ω σ > , ∀ (cid:36) > , (cid:13)(cid:13)(cid:13) e tX ˆ χ (out) (cid:36),σ,K (cid:13)(cid:13)(cid:13) H W ( M ) ≤ C (cid:15) e t ( γ + K +1 + (cid:15) ) . (7.35)This is (7.31). 43 omputation of e tX ˆ χ (in) (cid:36),σ,K : We have e tX ˆ χ (in) (cid:36),σ,K = e tX Op σ (cid:0) T [0 ,K ] (cid:1) Op (1 − χ (cid:36) ) . Hence, from(7.17), ∀ (cid:15) > , ∃ C (cid:15) > , ∀ t ≤ , ∃ σ t > , ∀ σ > σ t , ∃ ω σ > , ∀ (cid:36) > , (cid:13)(cid:13)(cid:13) e tX ˆ χ (in) (cid:36),σ,K (cid:13)(cid:13)(cid:13) H W ( M ) ≤ C (cid:15) e t ( γ − K − (cid:15) ) . (7.36)This is (7.30).The following theorem describes the operator e tX F ˆ χ (in) (cid:36),σ,K for large time t (cid:29) in term ofsum of quantum operators Op σ (cid:0) e tX F k T k (cid:1) plus an error term that is small for any t > if wetake the size σ (cid:29) large then σ (cid:29) large and then the cut-off frequency (cid:36) (cid:29) large enough. Theorem 7.15. “Approximation of the dynamics” . For any t ∈ R , e tX F ˆ χ (in) (cid:36),σ,K = Op σ ,σ (cid:0) e tX F k T [0 ,K ] (cid:1) (cid:0) Id − ˆ χ (low) (cid:36) (cid:1) + r t (7.37) with lim σ →∞ lim σ →∞ lim | (cid:36) |→∞ (cid:107) r t (cid:107) H W ( M ) = 0 . Proof.
Let K ∈ N . e tX ˆ χ (in) (cid:36),σ,K = (7 . e tX Op σ (cid:0) T [0 ,K ] (cid:1) (cid:0) Id − ˆ χ (low) (cid:36) (cid:1) ≈ t (7 . Op σ (cid:0) e tX F T [0 ,K ] (cid:1) Op σ (cid:0) T [0 ,K ] (cid:1) (cid:0) Id − ˆ χ (low) (cid:36) (cid:1) = Op σ (cid:0) e tX F T [0 ,K ] (cid:1) (cid:0) Id − ˆ χ (low) (cid:36) (cid:1) This gives Theorem 7.15 by applying Shur’s lemma to the remainder.
Proof of Theorem 1.2.
This comes from decomposition (7.29) and estimate (7.31). The otherterms form a compact operator that can be absorbed in the finite rank operator R ω and theremainder O (cid:16) e t ( γ + K +1 + (cid:15) ) (cid:17) of (1.5). The aim of this section is to prove Theorem 1.7. We do this in three steps.
For ω ∈ R , δ > and k ∈ N , we define an “approximate projector” P for the bands k (cid:48) ∈ (cid:74) , k (cid:75) and frequency interval [ ω − δ, ω + δ ] as follows. We will relate its trace to a symplecticvolume. We denote χ [ ω − δ,ω + δ ] : S (cid:91) (cid:0) Σ; F [0 ,k ] (cid:1) → S (cid:91) (cid:0) Σ; F [0 ,k ] (cid:1) the multiplication operator by thecharacteristic function [ ω − δ,ω + δ ] : Σ → R ρ → (cid:40) if ω ( ρ ) ∈ [ ω − δ, ω + δ ]0 if not. Definition 8.1.
For k ∈ N , ω, δ ∈ R we define P := Op σ (cid:0) χ [ ω − δ,ω + δ ] T [0 ,k ] (cid:1) : C ∞ ( M ) → C ∞ ( M ) (8.1)44he next lemma shows that the trace of P is related to the symplectic volume of the supportof function [ ω − δ,ω + δ ] on the trapped set Σ . Lemma 8.2.
The operator P is trace class in H W ( M ) and ∃ C > , ∀ σ > , ∃ C σ > , ∀ δ > , ∀ ω > δ , (cid:107) P (cid:107) Tr ≤ C σ ω d δ, (cid:107) P (cid:107) H W ≤ C σ , (8.2) (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) Tr ( P ) − rank (cid:0) F [0 ,k ] (cid:1) Vol ( M ) ω d (2 π ) d +1 (2 δ ) (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) ≤ Cω d δ (cid:18) C N σ − N + | ω | − β/ σ d +2) + δω (cid:19) + Cω d (8.3) Remark . For any (cid:15) > we can take σ large enough and then let ω be large enough dependingon σ so that the right hand side of (8.3) is smaller than (cid:15)ω d δ + Cω d . Proof.
We have P = Op σ (cid:0) χ [ ω − δ,ω + δ ] T [0 ,k ] (cid:1) = (7 . ˇ T ∆ σ ˇOp (cid:0) T [0 ,k ] (cid:1) χ [ ω − δ,ω + δ ] ˇ T σ . So ∀ N, ∃ C N > , P ∈ Ψ Id ,g/σ (cid:16) C N (cid:10) dist g/σ ( ρ, Σ) (cid:11) − N (cid:104) dist ( ω ( ρ ) , [ ω − δ, ω + δ ]) (cid:105) − N (cid:17) , (8.4)that shows that the Schwartz kernel of T P T † decreases very fast outside a compact domain K ⊂ T ∗ M of volume O σ (cid:0) ω d δ (cid:1) . This implies that P is trace class and (8.2). We have Tr ( P ) = (8 . Tr (cid:0) ˇ T σ ˇ T ∆ σ ˇOp (cid:0) T [0 ,k ] (cid:1) χ [ ω − δ,ω + δ ] (cid:1) = (cid:90) ρ ∈ Σ [ ω − δ,ω + δ ] ( ρ ) Tr (cid:0) T [0 ,k ] (cid:1) d(cid:37) (2 π ) d +1 + R with d(cid:37) = ω d ( dω ) ∧ ( π ∗ dm ) the symplectic volume on Σ , given in (4.10) and with an error R that is bounded: ∃ C > , ∀ N > , ∃ C N , ∀ σ , | R | ≤ ω d δ (cid:16) C N σ − N + C | ω | − β/ σ (cid:17) σ d +1) + Cω d , where Cω d is due to sharp cut-off in frequency. We have Tr (cid:0) T [0 ,k ] (cid:1) = (6 . dim (cid:0) Pol [0 ,k ] ( N s ) (cid:1) = (6 . rank (cid:0) F [0 ,k ] (cid:1) . This gives Tr ( P ) = rank (cid:0) F [0 ,k ] (cid:1) (2 π ) d +1 Vol ( M ) (cid:18)(cid:90) ω + δω − δ ω d dω (cid:19) + R. For ω ≥ δ , we have (cid:82) ω + δω − δ ω (cid:48) d dω (cid:48) = ω d (2 δ ) (cid:0) O (cid:0) δω (cid:1)(cid:1) . We get (8.3). Definition 8.4.
For z ∈ C , Λ > , let R ( z ) := ( z − ( X − Λ P )) − (8.5) D ( z ) := ( z − X ) − − R ( z ) (8.6) Remark . A similar construction of truncated resolvent R ( z ) has been used in [17, Proof ofProp. 5.15]. Following the argument in [11, Section 3.2.3], (8.4) implies first that P is Hilbert Schmidt. Now let B =Op ( b ) with positive symbol b : ρ ∈ T ∗ M → R + , ∗ that decays fast enough outside K so that B is Hilbert-Schmidtand C := P B − is also Hilbert Schmidt. Then P = CB being product of two Hilbert-Schmidt operators is atrace class operator [21, Lemma 7.2 p.67]. ( z ) ω ω − δω + δ Im ( z ) γ +0 γ − k ˜ R ( ω, δ, δ (cid:48) ) B + ( ω, δ, δ (cid:48) ) δ (cid:48) R ( ω, δ, δ (cid:48) ) (cid:15)B + ( ω, δ, δ (cid:48) ) Γ − Γ + Figure 8.1: Picture in the spectral plane for the rectangular domain ˜ R ( ω, δ, δ (cid:48) ) defined in (8.7)and the sub- domains B ± ( ω, δ, δ (cid:48) ) defined in (8.9).Let δ (cid:48) > and define the rectangle in the spectral plane ˜ R ( ω, δ, δ (cid:48) ) := (cid:2) γ − k − (cid:15), γ +0 + 2 (cid:15) (cid:3) × i [ ω − δ + δ (cid:48) , ω + δ − δ (cid:48) ] (8.7)The next Lemma is similar to [17, Lemma 5.16]. Lemma 8.6.
There exists
C > such that, if δ (cid:48) > in (8.7) and Λ > in (8.5) aresufficiently large, then for any ω ∈ R , δ > δ (cid:48) > , z ∈ ˜ R ( ω, δ, δ (cid:48) ) , (cid:107)R ( z ) (cid:107) H W ( M ) ≤ C. (8.8)Let us consider the following union of two horizontal bands (see Figure 8.1) B ± ( ω, δ, δ (cid:48) ) := (cid:2) γ − k − (cid:15), γ +0 + (cid:15) (cid:3) × i [ ω ± ( δ − δ (cid:48) ) , ω ± ( δ − δ (cid:48) )] , (8.9) B ( ω, δ, δ (cid:48) ) := B + ( ω, δ, δ (cid:48) ) ∪ B − ( ω, δ, δ (cid:48) ) . (8.10)We will need later (in the proof of Lemma 8.11), the following lemma. Lemma 8.7.
The operator D ( z ) is meromorphic w.r.t. z ∈ ˜ R ( ω, δ, δ (cid:48) ) and trace classoutside of its poles. In particular Tr ( D ( z )) ∈ L . Moreover, with z = x + iy , ∃ C > , ∀ δ > , ∀ ω > δ, (cid:90) B ( ω,δ,δ (cid:48) ) | Tr ( D ( z )) | dxdy ≤ Cδ (cid:48) ω d ln δ. (8.11) Proof.
We first show few lemmas.
Lemma 8.8. D ( z ) is trace class outside of its poles and Tr ( D ( z )) = ∂ z ln F ( z ) (8.12) with A ( z ) := Id − R ( z ) Λ P, (8.13) F ( z ) := det ( A ( z )) . (8.14)46 roof. D ( z ) is meromorphic because both ( z − X ) − and R ( z ) are meromorphic, see [17,Lemma 5.16]. Then D ( z ) = (8 . ( z − X ) − − R ( z )= ( z − ( X − Λ P )) − (( z − ( X − Λ P )) − ( z − X )) ( z − X ) − = R ( z ) Λ P ( z − X ) − . (8.15)In this last expression P is trace class. ( z − X ) − and R ( z ) are bounded outside of their poleshence D ( z ) is trace class outside of its poles. We have D ( z ) = (8 . ( z − X ) − − R ( z )= ( z − ( X − Λ P ) + Λ P ) − − R ( z )= (8 . R ( z ) (1 − R ( z ) Λ P ) − − R ( z )= (8 . R ( z ) (cid:0) A ( z ) − − (cid:1) . (8.16)We compute ∂ z A ( z ) = (8 . R ( z ) Λ P = (8 . R ( z ) (1 − A ( z )) = (8 . D ( z ) A ( z ) , (8.17)then ∂ z ln A ( z ) = ( ∂ z A ( z )) A ( z ) − = (8 . D ( z ) , ln F ( z ) = (8 . ln det ( A ( z )) = Tr (ln A ( z )) ,∂ z ln F ( z ) = Tr ( ∂ z ln A ( z )) = Tr ( D ( z )) . We have obtained (8.12).For later use, we also notice R ( z ) = (8 . A ( z ) ( z − X ) − . (8.18) Lemma 8.9.
There exists
C > , such that for every z ∈ B ( ω, δ, δ (cid:48) ) , | F ( z ) | ≤ e C (ln δ ) ω d . (8.19) and for every z ∈ B ( ω, δ, δ (cid:48) ) with Re ( z ) = γ − k − (cid:15) , | F ( z ) | ≥ e − C (ln δ ) ω d . (8.20) Proof.
For z ∈ B − ( ω, δ, δ (cid:48) ) , writing λ j ( z ) , j = 1 , , · · · , J, for the eigenvalues of the trace classoperator R ( z ) Λ P , we have log | F ( z ) | = log | det (Id − R ( z ) Λ P ) | = (cid:88) j log | − λ j ( z ) |≤ (cid:88) j log (1 + | λ j ( z ) | ) ≤ (cid:88) j | λ j ( z ) | ≤ (cid:107)R ( z ) Λ P (cid:107) Tr . (8.21)For the last inequality, see e.g. [21, p.64]. Notice that (cid:107)R ( z ) Λ P (cid:107) Tr ≤ (8 . , . Cδω d but this is notenough to get (8.19). We will improve the last bound as follows. We decompose the frequency47nterval [ ω − δ, ω + δ ] as the union of (cid:98) δ (cid:99) (the integer part of δ ) intervals of bounded length l = δ [ δ ] , so that [ ω − δ, ω + δ ] = (cid:91) w ∈{ , ,..., (cid:98) δ (cid:99)} I w (8.22)with I w := [ ω − δ + ( w − l, ω − δ + wl ] . (8.23)Similarly to (8.1) we define P w := ˇ T ∆ σ ˇOp (cid:0) T [0 ,k ] (cid:1) χ I w ˇ T σ : C ∞ ( M ) → C ∞ ( M ) i.e. an approximate projector on frequency interval of size l and we write P = (cid:88) w P w . (8.24)Similarly to (8.2) we have (cid:107) P w (cid:107) Tr ≤ Cω d . and (cid:107)R ( z ) Λ P w (cid:107) Tr ≤ w Cω d (8.25)uniformly for ≤ w ≤ (cid:98) δ (cid:99) . Hence (cid:107)R ( z ) Λ P (cid:107) Tr ≤ [ δ ] (cid:88) w =1 (cid:107)R ( z ) Λ P w (cid:107) Tr ≤ Cω d ln δ. We get | F ( z ) | ≤ (8 . e (cid:107)R ( z )Λ P (cid:107) Tr ≤ e Cω d ln δ . We have obtained (8.19). To prove the second claim (8.20), we consider an arbitrary z ∈ B ( ω, δ, δ (cid:48) ) with Re ( z ) = γ − k − (cid:15) . From Corollary 1.4 we have (cid:13)(cid:13) ( z − X ) − (cid:13)(cid:13) ≤ C . Then (1 − R ( z ) Λ P ) − = A ( z ) − = (8 . ( z − X ) − R ( z ) − = (8 . (cid:0) z − X ) − Λ P (cid:1) is uniformly bounded from (8.2). Hence | − λ j ( z ) | > c with some c > independent on ω andtherefore we have | log | − λ j ( z ) || < C (cid:48) | λ j ( z ) | for some C (cid:48) > . Thus − log | F ( z ) | = − (cid:88) j log | − λ j ( z ) | ≤ C (cid:48) (cid:88) j | λ j ( z ) | ≤ C (cid:48) (cid:107)R ( z ) Λ P (cid:107) Tr ≤ C (cid:48) ω d ln δ, giving (8.20).Let σ ( X ) denotes the discrete Ruelle spectrum of X . Lemma 8.10.
We have ∃ C > , ∀ ω , ∀ z ∈ B ( ω, δ, δ (cid:48) ) , (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) ∂ z ln F ( z ) − (cid:88) z j ∈ σ ( X ) ∩ B ( ω,δ,δ (cid:48) ) z − z j (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) ≤ C (ln δ ) ω d (8.26) where the sum over z j ∈ σ ( X ) ∩ B ( ω, δ, δ (cid:48) ) is counted with multiplicity of the Ruelle eigenvalues. roof. This is lemma α in [34, p.56]. We reproduce the proof here. From (8.8), F ( z ) is awell defined holomorphic function in the domains B ( ω, δ, δ (cid:48) ) and from (8.18), the zeroes of F ( z ) coincide up to multiplicities with the poles of ( z − X ) − , i.e. Ruelle eigenvalues z j ∈ σ ( X ) ∩ B ( ω, δ, δ (cid:48) ) . Using Weyl upper bound O (cid:0) ω d (cid:1) on the density in [17, Thm.3.6], note thatthe number of such Ruelle eigenvalues is bounded by Cδ (cid:48) ω d ≤ Cδω d . From (8.20), we can fixsome z ∈ B ( ω, δ, δ (cid:48) ) with Re ( z ) = γ − k − (cid:15) so that | F ( z ) | ≥ (8 . e − C (cid:48) δω d . The function G ( z ) := F ( z ) (cid:81) z j ( z − z j ) F ( z ) (cid:81) z j ( z − z j ) . (8.27) G ( z ) is holomorphic on B ( ω, δ, δ (cid:48) ) and from estimate (8.19) on the boundary of B ( ω, δ, δ (cid:48) ) andmaximum modulus principle we get ∀ z ∈ B ( ω, δ, δ (cid:48) ) , | G ( z ) | ≤ (8 . , . , . e Cω d ln δ . (8.28)Moreover G ( z ) has no zero on on the simply connected domain B ( ω, δ, δ (cid:48) ) hence g ( z ) := ln G ( z ) (8.29)is well defined as a holomorphic function on B ( ω, δ, δ (cid:48) ) with g ( z ) = 0 and satisfies Re g ( z ) ≤ (8 . Cω d ln δ . By Borel-Carathéodory theorem this implies | g ( z ) | < C (cid:48) ω d ln δ on a region slightly larger than B ( ω, δ, δ (cid:48) ) and therefore by Cauchy integral formula | ∂ z g ( z ) | = (8 . , . (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) ∂ z ln F ( z ) − (cid:88) z j z − z j (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) < C (cid:48)(cid:48) ω d ln δ giving the conclusion.Now we finish with the proof of Lemma 8.7. Using Weyl upper bound on the density in [17,Thm.3.6], we have, with z = x + iy , that (cid:88) z j ∈ σ ( X ) ∩ B ( ω,δ,δ (cid:48) ) (cid:90) B ( ω,δ,δ (cid:48) ) | z − z j | dxdy < C (cid:0) δ (cid:48) ω d (cid:1) ln δ (cid:48) . (8.30)Then (cid:90) B ( ω,δ,δ (cid:48) ) | Tr ( D ( z )) | dxdy = (8 . (cid:90) B ( ω,δ,δ (cid:48) ) | ∂ z ln F ( z ) | dxdy ≤ (8 . , . Cδ (cid:48) ω d ln δ + Cδ (cid:48) ω d ln δ (cid:48) ≤ Cδ (cid:48) ω d ln δ. We have obtained (8.11). Here we do not need to give the constants and the proof is simple: g maps B ( ω, δ, δ (cid:48) ) on the half-plane Re ( z ) < Cδω d . We post-compose by a Moebius transformation f which maps this half-plane to the unit diskand maps g ( z ) = 0 to . Then the image of B ( ω, δ, δ (cid:48) ) by f ◦ g is contained in a disk | z | < r < . This impliesthat the image g ( B ( ω, δ, δ (cid:48) )) is contained in a (large) disk | z | < C (cid:48) δω d . .3 Counting resonances by argument principle From Lemma 8.6, the truncated resolvent R ( z ) is bounded on the rectangle ˜ R ( ω, δ, δ (cid:48) ) andhence by the definition (8.6) the poles of z → Tr D ( z ) coincides with the poles of ( z − X ) − ,i.e., the Ruelle resonance. In the following, we count the number of Ruelle resonance in therectangle ˜ R ( ω, δ, δ (cid:48) ) by using a slightly generalized version of the argument principle. Welearned this method from the paper of S. Dyatlov [6, Section 11].In the following, we suppose that δ > , δ (cid:48) > is fixed so that δ (cid:48) < δ and we considerthe limit | ω | → ∞ . We take a C ∞ function f δ,δ (cid:48) : R → [0 , that is f δ,δ (cid:48) ( ω (cid:48) ) = 1 for ω (cid:48) ∈ [ − δ + 2 δ (cid:48) , δ − δ (cid:48) ] and f δ,δ (cid:48) ( ω (cid:48) ) = 0 for | ω (cid:48) | > δ − δ (cid:48) and | ∂ ω (cid:48) f | < Cδ (cid:48)− . Then for ω ∈ R , z ∈ C ,let f ω,δ,δ (cid:48) ( z ) := f δ,δ (cid:48) (Im ( z − iω )) Let us define the rectangle in the spectral plane (see Figure 8.1) R ( ω, δ, δ (cid:48) ) := (cid:2) γ − k − (cid:15), γ +0 + (cid:15) (cid:3) × i [ ω − δ + 2 δ (cid:48) , ω + δ − δ (cid:48) ] . In the following we suppose that the k -th band is separated from ( k + 1) -st band i.e. γ + k +1 < γ − k . Remark that R ( ω, δ, δ (cid:48) ) is a proper subset of ˜ R ( ω, δ, δ (cid:48) ) defined in (8.7). We have that ∀ z ∈ R ( ω, δ, δ (cid:48) ) , f ω,δ,δ (cid:48) ( z ) = 1 and ∀ z ∈ ˜ R ( ω, δ, δ (cid:48) ) , | ∂ z f ω,δ,δ (cid:48) ( z ) | < Cδ (cid:48)− . (8.31)We consider the following paths Γ := Γ − ∪ Γ + on C , parameterized as Γ − : ω ∈ R → z = γ − k − (cid:15) + iω ∈ C , Γ + : ω ∈ R → z = γ +0 + (cid:15) + iω ∈ C . From (8.8) and (1.8), there is no pole of D ( z ) on Γ ∩ supp ( f ω,δ,δ (cid:48) ) supposing that | ω | is suffi-ciently large. We know that D ( z ) is in trace class. Hence the integral (cid:82) Γ + − Γ − f ω,δ,δ (cid:48) ( z ) Tr ( D ( z )) dz is well defined. The next lemma shows that this integral is close to the number of Ruelle reso-nances in the rectangle R ( ω, δ, δ (cid:48) ) . Lemma 8.11.
We have ∃ C > , ∀ δ > , ∀ ω > δ, ∀ < δ (cid:48) < δ , (cid:12)(cid:12)(cid:12)(cid:12)(cid:18) πi (cid:90) Γ + − Γ − f ω,δ,δ (cid:48) ( z ) Tr ( D ( z )) dz (cid:19) − (cid:93) (cid:16) σ ( X ) ∩ ˜ R ( ω, δ, δ (cid:48) ) (cid:17)(cid:12)(cid:12)(cid:12)(cid:12) < Cω d max (ln δ, δ (cid:48) ) (8.32) Proof.
From the definition of f ω,δ,δ (cid:48) above, we have f ω,δ,δ (cid:48) ( z ) = 0 if | Im ( z ) − ω | > δ − δ (cid:48) , so wecan close the contour integral on horizontal segments Im ( z ) = ω ± ( δ − δ (cid:48) ) . The residues of thepoles of D ( z ) inside the contour are the spectral projector onto eigenspaces. Hence Cauchyintegral formula for smooth functions gives πi (cid:90) Γ + − Γ − f ω,δ,δ (cid:48) ( z ) Tr ( D ( z )) dz = (cid:88) z j ∈ σ ( X ) ∩ ˜ R ( ω,δ,δ (cid:48) ) f ω,δ,δ (cid:48) ( z j )+ 1 π (cid:90) ˜ R ( ω,δ,δ (cid:48) ) ( ∂ z f ω,δ,δ (cid:48) ) ( z ) Tr ( D ( z )) dxdy with z = x + iy . We have ∃ C > , ∀ δ > , ∀ δ (cid:48) < δ, ∀ ω > δ , (cid:12)(cid:12)(cid:12)(cid:12)(cid:90) ˜ R ( ω,δ,δ (cid:48) ) ( ∂ z f ω,δ,δ (cid:48) ) ( z ) Tr ( D ( z )) dxdy (cid:12)(cid:12)(cid:12)(cid:12) ≤ δ (cid:48)− (cid:90) B ( ω,δ,δ (cid:48) ) | Tr ( D ( z )) | dxdy ≤ (8 . , . Cδ (cid:48)− (ln δ ) δ (cid:48) ω d = Cω d ln δ. (8.33)50rom properties of f ω,δ,δ (cid:48) and using Weyl upper bound on the density in [17, Thm.3.6] we have (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) (cid:88) z j ∈ σ ( X ) ∩ ˜ R ( ω,δ,δ (cid:48) ) f ω,δ,δ (cid:48) ( z j ) − (cid:93) (cid:110) z j ∈ σ ( X ) ∩ ˜ R ( ω, δ, δ (cid:48) ) (cid:111)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) ≤ Cω d δ (cid:48) . We have obtained (8.32).
The last step is to relate the integral πi (cid:82) Γ + − Γ − f ω,δ,δ (cid:48) ( z ) Tr ( D ( z )) dz in (8.32) to the trace of P in (8.3). We define the operator P := Op σ (cid:0) χ [ ω − δ + δ (cid:48) / ,ω + δ − δ (cid:48) / T [0 ,k ] (cid:1) . For z ∈ R ( ω, δ, δ (cid:48) ) , let d ( z ) = min { Im( z ) − ( ω + δ + δ (cid:48) / , Im( z ) + ( ω − δ − δ (cid:48) / } ≥ δ (cid:48) / . Lemma 8.12. “Approximate expressions of D ( z ) ” . There exists C > such thatfor any T > , ω > δ > δ (cid:48) > and any z ∈ C with Im ( z ) ∈ [ ω − δ + δ (cid:48) , ω + δ − δ (cid:48) ] and Re ( z ) = γ +0 + (cid:15) , we have (cid:13)(cid:13)(cid:13)(cid:13) D ( z ) − (cid:18)(cid:90) T e − tz e t ( X − Λ) dt (cid:19) (Λ P ) (cid:18)(cid:90) T e − tz e tX dt (cid:19)(cid:13)(cid:13)(cid:13)(cid:13) Tr ≤ Cω d (cid:18) e − (cid:15)T + δ (cid:48) ( d ( z )) (cid:19) (8.34) If instead
Re ( z ) = γ − k − (cid:15) , we have (cid:13)(cid:13)(cid:13)(cid:13) D ( z ) − (cid:18)(cid:90) T e − tz e t ( X − Λ) dt (cid:19) (Λ P ) (cid:18)(cid:90) − T e − tz e tX dt (cid:19)(cid:13)(cid:13)(cid:13)(cid:13) Tr ≤ Cω d (cid:18) e − (cid:15)T + δ (cid:48) ( d ( z )) (cid:19) (8.35) Proof.
Let z ∈ C with Im ( z ) ∈ [ ω − δ + δ (cid:48) , ω + δ − δ (cid:48) ] . In the following we let T > besufficiently large. Suppose Re ( z ) = γ +0 + (cid:15) . As in (8.24), we decompose P = (cid:80) w ∈ W P w with W := { , , . . . , (cid:98) δ (cid:99)} . As in (8.22) with interval I w defined in (8.23) we decompose [ ω − δ + δ (cid:48) / , ω + δ − δ (cid:48) /
2] = (cid:91) w ∈ W I w with a subset W ⊂ W . So we can write P = (cid:88) w ∈ W P w . We have D ( z ) = (8 . (cid:88) w ∈ W D w ( z ) with D w ( z ) := Λ R ( z ) P w ( z − X ) − . For w ∈ W , let ˜ D w ( z ) := Λ (cid:18)(cid:90) T e − tz e t ( X − Λ) dt (cid:19) P w (cid:18)(cid:90) T e − tz e tX dt (cid:19) . ( z − X + Λ) ˜ D w ( z ) ( z − X ) = Λ (cid:2) e − tz e t ( X − Λ) (cid:3) T P w (cid:2) e − tz e tX (cid:3) T = Λ P w + Λ r w , with r w = e T ( − z + X − Λ) P w + P w e T ( − z + X ) + e T ( − z + X − Λ) P w e T ( − z + X ) . We deduce that ˜ D w ( z ) = D w ( z ) + Λ R ( z ) r w ( z − X ) − with R ( z ) r w ( z − X ) − = R ( z ) e T ( − z + X − Λ) P w ( z − X ) − + R ( z ) P w ( z − X ) − e T ( − z + X ) + R ( z ) e T ( − z + X − Λ) P w ( z − X ) − e T ( − z + X ) . For the right hand side, we have the following estimates. If w ∈ W then as in (8.25), we have R ( z ) r w ( z − X ) − = O Tr (cid:18) ω d e − (cid:15)T I w , Im ( z )) (cid:19) , where dist ( I w , Im ( z )) is the distance between I w and Im ( z ) . If w ∈ W \ W then D w ( z ) = O Tr (cid:18) ω d d ( z )) (cid:19) . Then (cid:88) w ∈ W (cid:13)(cid:13) R ( z ) r w ( z − X ) − (cid:13)(cid:13) Tr = (cid:88) w ∈ W O (cid:18) ω d e − (cid:15)T I w , Im ( z )) (cid:19) = O (cid:0) ω d e − (cid:15)T (cid:1) , (cid:88) w ∈ W \ W O Tr (cid:18) ω d d ( z )) (cid:19) = O Tr (cid:18) ω d δ (cid:48) ( d ( z )) (cid:19) . Hence (cid:13)(cid:13)(cid:13)(cid:13) D ( z ) − (cid:18)(cid:90) T e − tz e t ( X − Λ) dt (cid:19) (Λ P ) (cid:18)(cid:90) T e − tz e tX dt (cid:19)(cid:13)(cid:13)(cid:13)(cid:13) Tr = O (cid:18) ω d (cid:18) e − (cid:15)T + δ (cid:48) ( d ( z )) (cid:19)(cid:19) . Now we give the key estimate.
Lemma 8.13.
For any arbitrarily small c > , if we take δ large enough, δ (cid:48) > largeenough and δ/δ (cid:48) large enough, we have (cid:12)(cid:12)(cid:12)(cid:12)(cid:18) πi (cid:90) Γ + − Γ − f ω,δ,δ (cid:48) ( z ) Tr ( D ( z )) dz (cid:19) − Tr ( P ) (cid:12)(cid:12)(cid:12)(cid:12) < cω d δ (8.36) Proof.
We use the notation O Tr ( ∗ ) for an operator whose trace norm is bounded by a constantmultiple of ∗ . We first consider the integration on Γ + . From the last lemma, we have (cid:90) Γ + f ω,δ,δ (cid:48) ( z ) Tr ( D ( z )) dz = (cid:90) R f δ,δ (cid:48) ( ω (cid:48) − ω ) Tr (cid:0) D (cid:0) γ +0 + (cid:15) + iω (cid:48) (cid:1)(cid:1) d ( iω (cid:48) )= (8 . i Tr (cid:20)(cid:90) ω (cid:48) ∈ R f δ,δ (cid:48) ( ω (cid:48) − ω ) (cid:18)(cid:90) Tt (cid:48) =0 e − t (cid:48) ( γ +0 + (cid:15) + iω (cid:48) ) e t (cid:48) ( X − Λ) dt (cid:48) (cid:19) (Λ P ) (cid:18)(cid:90) Tt =0 e − t ( γ +0 + (cid:15) + iω (cid:48) ) e tX dt (cid:19)(cid:21) dω (cid:48) (8.37) + O T r (cid:0) ω d (cid:0) δe − (cid:15)T + δ (cid:48) (cid:1)(cid:1) . ˜ f ω,δ,δ (cid:48) ( t ) := (cid:90) ω (cid:48) ∈ R e − t ( γ +0 + (cid:15) + iω (cid:48) ) f δ,δ (cid:48) ( ω (cid:48) − ω ) dω (cid:48) = e − t ( γ +0 + (cid:15) + iω ) ˆ f δ,δ (cid:48) ( t ) , where ˆ f δ,δ (cid:48) ( t ) := (cid:82) R e − itω (cid:48) f δ,δ (cid:48) ( ω (cid:48) ) dω (cid:48) satisfies a fast decay for | t | (cid:29) δ (cid:48)− as follows: ∀ N > , ∃ C N > , ∀ t, (cid:12)(cid:12)(cid:12) ˆ f δ,δ (cid:48) ( t ) (cid:12)(cid:12)(cid:12) ≤ C N (cid:104) δ (cid:48) t (cid:105) − N and (cid:90) R ˆ f δ,δ (cid:48) ( t ) dt = 2 πf δ,δ (cid:48) (0) = 2 π. (8.38)Then, with the change of variables ( t, t (cid:48) ) → ( t, s = t + t (cid:48) ) , we see (cid:90) Γ + f ω,δ,δ (cid:48) ( z ) Tr ( D ( z )) dz = (8 . i (cid:90) T (cid:90) T ˜ f ω,δ,δ (cid:48) ( t + t (cid:48) ) Tr (cid:16) e t (cid:48) ( X − Λ) P Λ e tX (cid:17) dt (cid:48) dt + O T r (cid:0) ω d (cid:0) δe − (cid:15)T + δ (cid:48) (cid:1)(cid:1) = i (cid:90) T (cid:32)(cid:90) min { T,s } t =max { ,s − T } e − s ( γ +0 + (cid:15) + iω ) ˆ f δ,δ (cid:48) ( s ) Tr (cid:0) e ( s − t )( X − Λ) P Λ e tX (cid:1) dt (cid:33) ds + O T r (cid:0) ω d (cid:0) δe − (cid:15)T + δ (cid:48) (cid:1)(cid:1) . From (8.38), by letting δ (cid:48) be large (and ˆ f δ,δ (cid:48) ( t ) be more concentrate to ), we get (cid:12)(cid:12)(cid:12)(cid:12)(cid:90) Γ + f ω,δ,δ (cid:48) ( z ) Tr ( D ( z )) dz (cid:12)(cid:12)(cid:12)(cid:12) = O T r (cid:0) δ (cid:48)− ω d δ (cid:1) + O T r (cid:0) ω d (cid:0) δe − (cid:15)T + δ (cid:48) (cid:1)(cid:1) . Next we consider the integration on Γ − . Similarly to the argument above, we see (cid:90) − Γ − f ω,δ,δ (cid:48) ( z ) Tr ( D ( z )) dz = i (cid:90) f δ,δ (cid:48) ( ω (cid:48) − ω ) Tr (cid:0) D (cid:0) γ − − (cid:15) − iω (cid:48) (cid:1)(cid:1) dω (cid:48) = (8 . i Tr (cid:20)(cid:90) f δ,δ (cid:48) ( ω (cid:48) − ω ) (cid:18)(cid:90) T e − t (cid:48) ( γ − − (cid:15) − iω (cid:48) ) e t (cid:48) ( X − Λ) dt (cid:48) (cid:19) (Λ P ) (cid:18)(cid:90) − T e − t ( γ − − (cid:15) − iω (cid:48) ) e tX dt (cid:19)(cid:21) dω (cid:48) + O Tr (cid:0) ω d (cid:0) δe − (cid:15)T + δ (cid:48) (cid:1)(cid:1) = i (cid:90) Ts = − T (cid:32)(cid:90) min { ,s } t =max {− T,s − T } e − s ( γ − − (cid:15) − iω (cid:48) ) ˆ f δ,δ (cid:48) ( s ) Tr (cid:0) e s ( X − Λ) e − t ( X − Λ) Λ P e tX (cid:1) dt (cid:33) ds + O Tr (cid:0) ω d (cid:0) δe − (cid:15)T + δ (cid:48) (cid:1)(cid:1) =2 πi (cid:90) t = − T Tr (cid:0) e − t ( X − Λ) Λ P e tX (cid:1) dt + O T r (cid:0) δ (cid:48)− ω d δ (cid:1) + O Tr (cid:0) ω d (cid:0) δe − (cid:15)T + δ (cid:48) (cid:1)(cid:1) =2 πi Tr (cid:16)(cid:2) e t Λ P (cid:3) t = − T (cid:17) + O T r (cid:0) δ (cid:48)− ω d δ (cid:1) + O Tr (cid:0) ω d (cid:0) δe − (cid:15)T + δ (cid:48) (cid:1)(cid:1) =2 πi Tr ( P ) + O T r ( e − (cid:15)T ω d ) + O T r (cid:0) δ (cid:48)− ω d δ (cid:1) + O Tr (cid:0) ω d (cid:0) δe − (cid:15)T + δ (cid:48) (cid:1)(cid:1) =2 πi Tr ( P ) + O Tr (cid:0) ω d δ (cid:48) (cid:1) + O Tr ( e − (cid:15)T ω d ) + O Tr (cid:0) δ (cid:48)− ω d δ (cid:1) + O Tr (cid:0) ω d (cid:0) δe − (cid:15)T + δ (cid:48) (cid:1)(cid:1) . Therefore, by letting T large δ (cid:29) δ (cid:48) (cid:29) and then letting ω large, we obtain the requiredestimate. Using the previous estimates we get that 53 (cid:12)(cid:12)(cid:12)(cid:12) (cid:93) (cid:16) σ ( X ) ∩ ˜ R ( ω, δ, δ (cid:48) ) (cid:17) − rank (cid:0) F [0 ,k ] (cid:1) Vol ( M ) ω d (2 π ) d +1 (2 δ ) (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) < (8 . , . , . Cω d max { δ (cid:48) , ln δ } + cω d δ + Cω d δ (cid:18) C N σ − N + | ω | − β/ σ + δω (cid:19)
10 Proof of Theorem 1.13 (horocycle operators)
Suppose s ∈ C β ( M ; E s ) is a Hölder continuous section of the stable bundle over M and u ∈ C β ( M ; E u ) a section of the unstable bundle. Let π : T ∗ M → M be the bundle map.We have seen in (6.4) that for ρ ∈ Σ , the differential dπ gives a linear isomorphism dπ : N ( ρ ) = N s ( ρ ) ⊕ N u ( ρ ) → E u ( m ) ⊕ E s ( m ) with m = π ( ρ ) . By pull back we get sections ˜ s := ( dπ ) − ( s ) ∈ C (Σ; N s ) and ˜ u := ( dπ ) − ( u ) ∈ C (Σ; N u ) . Recall from Section 2.4 that d A is a symplectic form on E u ⊕ E s and from Definition 6.1 that N ( ρ ) = N s ( ρ ) ⊕ N u ( ρ ) isa Ω -symplectic vector space and that Ω (˜ u, ˜ s ) = (4 . ω ( d A ) ( u, s ) , hence ˜ u (so then u ) defines adual vector ˜ u ∗ := Ω (˜ u, . ) = (4 . ω ( d A ) ( u, dπ ( . )) ∈ N ∗ s ( ρ ) . (10.1)For k ∈ N , let p ∈ C (Σ; F k ) a continuous section of F k = (6 . | det N s | − / ⊗ Pol k ( N s ) ⊗ F over Σ where Pol k ( N s ) ≡ Sym k ( N ∗ s ⊗ . . . ⊗ N ∗ s ) is equivalent to the symmetric tensor algebraof degree k . Definition 10.1.
For s ∈ C β ( M ; E s ) we define ι s : (cid:40) C (Σ; F k ) → C (Σ; F k − ) p → ι ˜ s p = p (˜ s, ., . . . , . ) (10.2)as the interior product, i.e. the pointwise (i.e. independently in every fiber) contractionin the first entry of p by ˜ s := ( dπ ) − ( s ) ∈ C (Σ; N s ) . For u ∈ C β ( M ; E u ) we define u ∨ : (cid:40) C (Σ; F k ) → C (Σ; F k +1 ) p → ˜ u ∗ ∨ p = Sym (˜ u ∗ ⊗ p ) as the symmetric point-wise tensor product of p by ˜ u ∗ := ω ( d A ) ( u, dπ ( . )) ∈ C (Σ; N ∗ s ) . Lemma 10.2. “Weyl algebra”.
For s ∈ C β ( M ; E s ) , u ∈ C β ( M ; E u ) and k ∈ N wehave [ kι s , u ∨ ] = ω ( d A ) ( s, u ) Id F : C (Σ; F k ) → C (Σ; F k ) (10.3) where ω is the frequency function (3.7) on Σ and Id is the identity operator in fibers.Proof. This is point wise relation so we consider a point ρ ∈ Σ with frequency ω = (3 . ω ( ρ ) andthe vector space N s ( ρ ) defined in (6.4). We use the isomorphism Pol k ( N s ) ≡ Sym k ( N ∗ s ⊗ . . . ⊗ N ∗ s )
55o derive (10.3) that goes as follows. If ( e i ) i is a basis of N s ( ρ ) , we write ˜ s = (cid:80) di =1 σ i e i ∈ N s ( ρ ) with components σ = ( σ i ) i ∈ R d . For p ∈ Sym k ( N ∗ s ⊗ . . . ⊗ N ∗ s ) , we associate a degree k poly-nomial P ∈ Pol k (cid:0) R d (cid:1) by P ( σ ) := p (˜ s, ˜ s, . . . ˜ s ) . We define e ∗ i ∨ p := Sym ( e ∗ i ⊗ p ) and observethat ( e ∗ i ∨ p ) (˜ s ) = σ i P ( σ ) and ( kι e i p ) (˜ s ) = ( ∂ σ i P ) ( σ ) . Then the “Weyl algebra relation ” onpolynomials [ ∂ σ i , σ j ] = δ i = j Id gives (cid:2) kι e i , e ∗ j ∨ (cid:3) = δ i = j Id hence for ˜ s ∈ N s , ˜ u ∗ ∈ N ∗ s we have [ kι ˜ s , ˜ u ∗ ∨ ] = ˜ u ∗ (˜ s ) Id . With (10.1), we get (10.3).The Weyl algebra relation (10.3) involves operator valued symbols on C (Σ; F k ) so we can“quantize” it and get operators on C ∞ ( M ) . This gives Theorem 1.13. A General notations used in this paper • Dual ∗ . If E is a vector space and E ∗ its dual space, x ∈ E, ξ ∈ E ∗ we denote the dualityby (cid:104) ξ | x (cid:105) ∈ R . If E , E are vector space and A : E → E is a linear map, we will denote A ∗ : E ∗ → E ∗ the induced dual map on dual spaces defined by (cid:104) A ∗ v | u (cid:105) = (cid:104) v | Au (cid:105) , ∀ u ∈ E , v ∈ E ∗ .• Bilinear map.
Let
Ω : E × E → R be a bilinear map on a vector space E . It defines alinear map ˇΩ : E → E ∗ by ˇΩ ( u ) ( . ) = Ω ( u, . ) . – If ˇΩ is invertible ( Ω is said to be non degenerated), it induces a bilinear form on E ∗ denoted Ω − : E ∗ × E ∗ → R and defined by Ω − ( α , α ) := Ω (cid:0) ˇΩ − α , ˇΩ − α (cid:1) = α (cid:0) ˇΩ − α (cid:1) . – The bilinear map
Ω : E × E → R is anti-symmetric iff ˇΩ ∗ = − ˇΩ . (A.1)The bilinear map g : E × E → R is symmetric iff ˇ g ∗ = ˇ g .• Adjoint † . Suppose A : E → E is a linear map and Ω (respect. Ω ) is a nondegenerated bilinear form on E (respect. E ). The Ω -adjoint of A is A † Ω : E → E defined by Ω (cid:0) u, A † Ω v (cid:1) = Ω ( Au, v ) , ∀ u ∈ E , ∀ v ∈ E hence given by A † Ω = ˇΩ A ∗ ˇΩ − In this paper bilinear forms will be either a symplectic form
Ω ( ., . ) or a metric g ( ., . ) ora L -scalar product (cid:104) . | . (cid:105) (complex valued).• Pull-back ◦ . If f : M → N is a smooth map between two manifolds, – We denote f ◦ the pull back operator f ◦ : (cid:40) C ∞ ( N ) → C ∞ ( M ) u → u ◦ f and if f is a diffeomorphism, we denote f −◦ := ( f − ) ◦ the push forward operator,so that f −◦ f ◦ = Id . – We denote df : T M → T N the differential of f that is a linear bundle map anddenote ( df ) ∗ : T ∗ N → T ∗ M its dual.56 More informations on flows
B.1 Transfer operator
Suppose that X is a vector field on M and ( φ t ) ◦ = e tX is the pull back operator definedin (2.2). For any smooth measure dµ on M , the L ( M, dµ ) scalar product is defined by (cid:104) v | u (cid:105) L := (cid:82) M uvdµ with u, v ∈ C ∞ c ( M ) . Then the L -adjoint ( φ t ) ◦† of ( φ t ) ◦ , defined by (cid:104) v | ( φ t ) ◦† u (cid:105) L = (cid:104) ( φ t ) ◦ v | u (cid:105) L , ∀ u, v ∈ C ∞ c ( M ) is given by (cid:0) φ t (cid:1) ◦† u = (cid:12)(cid:12) det (cid:0) dφ − t (cid:1)(cid:12)(cid:12) . (cid:0) φ − t (cid:1) ◦ = e tX † , with X † = − X − div µ X. ( φ t ) ◦† is called the Ruelle-Perron-Frobenius operator or transfer operator. Observe that ( φ t ) ◦† pushes forward probability distributions because (cid:90) M (cid:0) φ t (cid:1) ◦† udµ = (cid:104) | (cid:0) φ t (cid:1) ◦† u (cid:105) L = (cid:104) (cid:0) φ t (cid:1) ◦ (cid:124) (cid:123)(cid:122) (cid:125) | u (cid:105) L = (cid:90) M udµ. In particular the evolution of Dirac measures gives ( φ t ) ◦† δ m = δ φ t ( m ) and is equivalent toevolution of points under the flow map φ t . B.2 More general pull back operators e tX F Suppose that X is a smooth vector field on M and F → M is a vector bundle over M . Supposethat X F : C ∞ ( M ; F ) → C ∞ ( M ; F ) is a derivation over X defined by (2.4). B.2.1 Expression in local frames
For a general vector bundle F of rank r , with respect to a local frame ( e , . . . e r ) , a section u ∈ C ∞ ( M ; F ) is expressed as u ( m ) = (cid:80) rj =1 u j ( m ) e j ( m ) with m ∈ M and components u j ( m ) ∈ C . We introduce a matrix of potential functions V k,j ( m ) ∈ C defined by ( X F e k ) ( m ) = (cid:88) j V k,j ( m ) e j ( m ) . (B.1)Then the operator X F in (2.3) is expressed as ( X F u ) ( m ) = (2 . r (cid:88) j =1 (cid:32) ( Xu j ) ( m ) + r (cid:88) k =1 V k,j ( m ) u k ( m ) (cid:33) e j ( m ) . (B.2)We see that the expression (B.2) generalizes (2.5). B.2.2 Expression of e tX F from a flow ˜ φ tF on F An equivalent way to present the operator X F in (2.4), is to consider a vector field ˜ X F on F ,a lift of X , i.e. ( dπ ) ˜ X F = X , with the projection π : F → M , and such that the flow mapgenerated by ˜ X F ˜ φ tF : F → F i.e. (cid:16) ˜ φ tF (cid:17) ◦ = e t ˜ X F : S ( F ) → S ( F ) , is a smooth linear bundle map over φ t which means that forevery m ∈ M, t ∈ R , ˜ φ tF ( m ) : F ( m ) → F ( φ t ( m )) is a linear map. See Figure B.1. This definesa group of pull back operators with generator X F acting on a section u ∈ C ∞ ( M ; F ) ofthe vector bundle F : e tX F : C ∞ ( M ; F ) → C ∞ ( M ; F ) , t ∈ R . φ t X φ t ( m ) m uF ( m ) ˜ X F e tX F u e tX F ˜ φ tF Figure B.1: pull back operator e tX F by e tX F u := ˜ φ − tF (cid:0) u ◦ φ t (cid:1) . (B.3)We get e tX F ( f u ) = ˜ φ − tF (cid:0)(cid:0) e tX f (cid:1) ( u ◦ φ t ) (cid:1) = (cid:0) e tX f (cid:1) (cid:0) e tX F u (cid:1) and deduce (2.4) by derivation w.r.t. t . C Bargmann transform and Metaplectic operators
In this section we collect results that concerns the Bargmann transform on a vector space E ,the quantization of affine and linear symplectic map, called Heisenberg group and metaplecticoperators. All these definitions and results are well known and important in many fields ofmathematics and physics. They are at the core of micro-local analysis. We present thesedefinitions and results in a form that is adapted to our paper that relies strongly on them.References are [19],[25],[27, appendix 4.4],[15, chap.3, chap.4]. C.1 Weyl Heisenberg group
Let E be a real vector space, n = dim E . For x ∈ E , ξ ∈ E ∗ we denote (cid:104) ξ | x (cid:105) ∈ R the duality.Let dx ∈ | Λ n ( E ) | be a density on E let dξ be the induced density on E ∗ . Let us denote theFourier transform F : (cid:40) S ( E ) → S ( E ∗ ) u → v ( ξ ) = π ) n/ (cid:82) e − i (cid:104) ξ | x (cid:105) u ( x ) dx Its L -adjoint is F † : (cid:40) S ( E ∗ ) → S ( E ) v → u ( x ) = π ) n/ (cid:82) e i (cid:104) ξ | x (cid:105) v ( ξ ) dξ and we have that F † F = Id S ( E ) hence F − = F † .For x ∈ E , ξ ∈ E ∗ we define the translation maps T x : (cid:40) E → Ey → y + x , T ξ : (cid:40) E ∗ → E ∗ ξ (cid:48) → ξ (cid:48) + ξ And denote T −◦ x , T −◦ ξ the push forward operators. We will denote ˆ T x,ξ := T −◦ x (cid:0) F − T −◦ ξ F (cid:1) (C.1)58 emma C.1. “Weyl Heisenberg group”. On S ( E ) , one has (cid:0) F − T −◦ ξ F (cid:1) T −◦ x = e i (cid:104) ξ | x (cid:105) T −◦ x (cid:0) F − T −◦ ξ F (cid:1) . (C.2) ˆ T x,ξ ˆ T x (cid:48) ,ξ (cid:48) = e i (cid:104) ξ | x (cid:48) (cid:105) ˆ T x + x (cid:48) ,ξ + ξ (cid:48) . In particular ˆ T x,ξ is a unitary operator in L ( E, dx ) and ˆ T − x,ξ = ˆ T † x,ξ = ˆ T − ( x,ξ ) e iξx . (C.3) Proof.
Let x ∈ E . Then ( F δ x ) ( ξ (cid:48) ) = 1(2 π ) n/ e − iξ (cid:48) x , (cid:0) T −◦ ξ F δ x (cid:1) ( ξ (cid:48) ) = 1(2 π ) n/ e − i ( ξ (cid:48) − ξ ) x , (cid:0) F − T −◦ ξ F δ x (cid:1) ( x (cid:48) ) = 1(2 π ) n (cid:90) e iξ (cid:48) x e − i ( ξ (cid:48) − ξ ) x dξ (cid:48) = e iξx δ x ( x (cid:48) ) Hence F − T −◦ ξ F δ x = e iξx δ x . Hence T −◦ x (cid:0) F − T −◦ ξ F (cid:1) δ x (cid:48) = e iξx (cid:48) δ x + x (cid:48) , (cid:0) F − T −◦ ξ F (cid:1) T −◦ x δ x (cid:48) = e iξ ( x + x (cid:48) ) δ x + x (cid:48) = e i (cid:104) ξ | x (cid:105) T −◦ x (cid:0) F − T −◦ ξ F (cid:1) δ x (cid:48) . Then ˆ T x,ξ ˆ T x (cid:48) ,ξ (cid:48) = T −◦ x (cid:0) F − T −◦ ξ F (cid:1) T −◦ x (cid:48) (cid:0) F − T −◦ ξ (cid:48) F (cid:1) = e i (cid:104) ξ | x (cid:48) (cid:105) T −◦ x T −◦ x (cid:48) (cid:0) F − T −◦ ξ F (cid:1) (cid:0) F − T −◦ ξ (cid:48) F (cid:1) = e i (cid:104) ξ | x (cid:48) (cid:105) ˆ T x + x (cid:48) ,ξ + ξ (cid:48) . C.2 Bargmann Transform on a Euclidean vector space ( E, g ) Let E be a vector space with an Euclidean metric g . We have the canonical identification T ∗ E = E ⊕ E ∗ , with the canonical symplectic form Ω on E ⊕ E ∗ given by: for ( x , ξ ) , ( x , ξ ) ∈ E ⊕ E ∗ , Ω (( x , ξ ) , ( x , ξ )) = (cid:104) ξ | x (cid:105) − (cid:104) ξ | x (cid:105) . (C.4)We denote dx the density on E induced by g . Definition C.2. A Gaussian wave packet in vertical gauge (V) with parameters ( x, ξ ) ∈ E ⊕ E ∗ , is the function ϕ (V) x,ξ ∈ S ( E ) given by: for y ∈ E , ϕ (V) x,ξ ( y ) := π − n e i (cid:104) ξ | y − x (cid:105) e − (cid:107) y − x (cid:107) g (C.5)Similarly a Gaussian wave packet in radial gauge (R) is ϕ (R) x,ξ ( y ) := e i (cid:104) ξ | x (cid:105) ϕ (V) x,ξ ( y ) (C.6) Remark
C.3 . We have 59 (cid:13)(cid:13)(cid:13) ϕ (V) x,ξ (cid:13)(cid:13)(cid:13) L ( E,dx ) = (cid:13)(cid:13)(cid:13) ϕ (R) x,ξ (cid:13)(cid:13)(cid:13) L ( E,dx ) = 1 with dx the density on E associated to the metric g .• We have ϕ (V) x,ξ = ( C. ˆ T x,ξ ϕ (V)0 , (C.7) Definition C.4.
The
Bargmann transform is B (V) : (cid:40) S ( E ) → S ( E ⊕ E ∗ ) u → (cid:16) ( x, ξ ) → (cid:104) ϕ (V) x,ξ | u (cid:105) L ( E ; dx ) (cid:17) . (C.8) Lemma C.5.
With respect to the density π ) n dxdξ on E ⊕ E ∗ , the L -adjoint operator B † (V) : L (cid:16) E ⊕ E ∗ ; π ) n dxdξ (cid:17) → L ( E ; dx ) is given by (cid:16) B † (V) v (cid:17) ( y ) = (cid:90) E ⊕ E ∗ ϕ (V) x,ξ ( y ) v ( x, ξ ) dxdξ (2 π ) n (C.9) and satisfies the resolution of identity Id L ( E ) = B † (V) B (V) = (cid:90) E ⊕ E ∗ π ( x, ξ ) dxdξ (2 π ) n (C.10) with π ( x, ξ ) := ϕ (V) x,ξ (cid:104) ϕ (V) x,ξ | . (cid:105) that is the rank one orthogonal projector in L ( E ) onto C ϕ (V) x,ξ .Proof. We first check (C.9). For u ∈ S ( E ) and v ∈ S ( E ⊕ E ∗ ) we have (cid:104) u |B † (V) v (cid:105) L = (cid:104)B (V) u | v (cid:105) L = (cid:90) E ⊕ E ∗ (cid:104) ϕ (V) x,ξ | u (cid:105) L v ( x, ξ ) dxdξ (2 π ) n = (cid:90) E u ( y ) (cid:90) E ⊕ E ∗ ϕ (V) x,ξ ( y ) v ( x, ξ ) dxdξ (2 π ) n dy. Now we check that (cid:13)(cid:13) B (V) u (cid:13)(cid:13) = (cid:107) u (cid:107) that is equivalent to (C.10). We have (cid:13)(cid:13) B (V) u (cid:13)(cid:13) = (cid:90) E ⊕ E ∗ (cid:12)(cid:12)(cid:12) (cid:104) ϕ (V) x,ξ | u (cid:105) L (cid:12)(cid:12)(cid:12) dxdξ (2 π ) n = π − n (cid:90) e − i (cid:104) ξ | y − x (cid:105) e − (cid:107) y − x (cid:107) g u ( y ) dye i (cid:104) ξ | y (cid:48) − x (cid:105) e − (cid:107) y (cid:48) − x (cid:107) g u ( y (cid:48) ) dy (cid:48) dxdξ (2 π ) n But (cid:90) e i (cid:104) ξ | y (cid:48) − y (cid:105) dξ = (2 π ) n δ ( y (cid:48) − y ) (cid:90) E e −(cid:107) y (cid:107) dy = π n hence (cid:13)(cid:13) B (V) u (cid:13)(cid:13) = π − n (cid:90) e −(cid:107) y − x (cid:107) g | u ( y ) | dydx = (cid:90) | u ( y ) | dy = (cid:107) u (cid:107) . P (V) := B (V) B † (V) : S ( E ⊕ E ∗ ) → S ( E ⊕ E ∗ ) (C.11)be the orthogonal projector in L ( E ⊕ E ∗ ) onto Im (cid:0) B (V) (cid:1) called the Bergman projector . Byusing ϕ (R) x,ξ instead of ϕ (V) x,ξ , we can define similarly operators in radial gauge B (R) , B † (R) , P (R) := B (R) B † (R) and we get similar relations.The metric g on E induces a canonical metric on E ⊕ E ∗ denoted g := g ⊕ g − . Lemma C.6.
The Schwartz kernel of P (V) , P (R) are given by (cid:104) δ x (cid:48) ,ξ (cid:48) |P (V) δ x,ξ (cid:105) = exp (cid:18) i ξ (cid:48) + ξ ) ( x (cid:48) − x ) − (cid:107) ( x (cid:48) , ξ (cid:48) ) − ( x, ξ ) (cid:107) g ⊕ g − (cid:19) (C.12) (cid:104) δ x (cid:48) ,ξ (cid:48) |P (R) δ x,ξ (cid:105) = exp (cid:18) − i x (cid:48) , ξ (cid:48) ) , ( x, ξ )) − (cid:107) ( x (cid:48) , ξ (cid:48) ) − ( x, ξ ) (cid:107) g ⊕ g − (cid:19) (C.13) Proof.
We have (cid:104) δ x (cid:48) ,ξ (cid:48) |P (V) δ x,ξ (cid:105) = (cid:104) ϕ (V) x (cid:48) ,ξ (cid:48) | ϕ (V) x,ξ (cid:105) = π − n (cid:90) e − i (cid:104) ξ (cid:48) | y − x (cid:48) (cid:105) e − (cid:107) y − x (cid:48) (cid:107) g e i (cid:104) ξ | y − x (cid:105) e − (cid:107) y − x (cid:107) g dy = π − n e i ( (cid:104) ξ (cid:48) | x (cid:48) (cid:105)−(cid:104) ξ | x (cid:105) ) − (cid:107) x (cid:48) (cid:107) g − (cid:107) x (cid:107) g (cid:90) e −(cid:107) y (cid:107) g + by dy with b = ( x (cid:48) + x ) + i ( ξ − ξ (cid:48) ) . We use the Gaussian integral (cid:90) E e − α (cid:107) y (cid:107) + by dy = (cid:16) πα (cid:17) n e α (cid:107) b (cid:107) , (C.14)with α = 1 , giving (cid:104) δ x (cid:48) ,ξ (cid:48) |P (V) δ x,ξ (cid:105) = e i ( (cid:104) ξ (cid:48) | x (cid:48) (cid:105)−(cid:104) ξ | x (cid:105) ) − (cid:107) x (cid:48) (cid:107) − (cid:107) x (cid:107) e ( (cid:107) x (cid:48) + x (cid:107) −(cid:107) ξ − ξ (cid:48) (cid:107) +2 i (cid:104) ξ − ξ (cid:48) | x (cid:48) + x (cid:105) )= e i ( (cid:104) ξ (cid:48) | x (cid:48) (cid:105)−(cid:104) ξ | x (cid:105) + (cid:104) ξ − ξ (cid:48) | x (cid:48) + x (cid:105) ) e − ( (cid:107) x (cid:48) − x (cid:107) + (cid:107) ξ − ξ (cid:48) (cid:107) )= e i (cid:104) ξ (cid:48) + ξ | x (cid:48) − x (cid:105) e − (cid:107) ( x (cid:48) ,ξ (cid:48) ) − ( x,ξ ) (cid:107) g + g − We have (cid:104) δ x (cid:48) ,ξ (cid:48) |P (R) δ x,ξ (cid:105) = e i ( (cid:104) ξ | x (cid:105)−(cid:104) ξ (cid:48) | x (cid:48) (cid:105) ) (cid:104) δ x (cid:48) ,ξ (cid:48) |P (V) δ x,ξ (cid:105) = e i ( −(cid:104) ξ (cid:48) | x (cid:105) + (cid:104) ξ | x (cid:48) (cid:105) ) e − (cid:107) ( x (cid:48) ,ξ (cid:48) ) − ( x,ξ ) (cid:107) g + g − = e − i Ω(( x (cid:48) ,ξ (cid:48) ) , ( x,ξ )) e − (cid:107) ( x (cid:48) ,ξ (cid:48) ) − ( x,ξ ) (cid:107) g + g − Lemma C.7.
Let π : E ⊕ E ∗ → E denotes the projector on the first component, and π ◦ : S ( E ) → S ( E ⊕ E ∗ ) the pull back operator. We have B (V) = π n P (V) π ◦ Proof.
We compute the Schwartz kernel of both sides for y ∈ E, ( x (cid:48) , ξ (cid:48) ) ∈ E ⊕ E ∗ . We have (cid:104) δ x (cid:48) ,ξ (cid:48) |B (V) δ y (cid:105) = ϕ (V) x (cid:48) ,ξ (cid:48) ( y ) and (cid:104) δ x (cid:48) ,ξ (cid:48) |P (V) π ◦ δ y (cid:105) = (cid:90) e i ( ξ (cid:48) + ξ )( x (cid:48) − x ) − ( (cid:107) x (cid:48) − x (cid:107) + (cid:107) ξ − ξ (cid:48) (cid:107) ) δ ( x − y ) dxdξ (2 π ) n = (2 π ) − n e i ξ (cid:48) ( x (cid:48) − y ) − (cid:107) x (cid:48) − y (cid:107) − (cid:107) ξ (cid:48) (cid:107) (cid:90) e i ξ ( x (cid:48) − y )+ ξ.ξ (cid:48) − (cid:107) ξ (cid:107) dξ
61e use the Gaussian integral (C.14) with α = , b = i ( x (cid:48) − y ) + ξ (cid:48) , α (cid:107) b (cid:107) = 14 (cid:16) (cid:107) ξ (cid:48) (cid:107) − (cid:107) x (cid:48) − y (cid:107) (cid:17) + i x (cid:48) − y ) ξ (cid:48) (cid:104) δ x (cid:48) ,ξ (cid:48) |P (V) π ◦ δ y (cid:105) = (2 π ) − n e i ξ (cid:48) ( x (cid:48) − y ) − (cid:107) x (cid:48) − y (cid:107) − (cid:107) ξ (cid:48) (cid:107) (cid:16) πα (cid:17) n e α (cid:107) b (cid:107) = (4 π ) n (2 π ) − n π n π − n e iξ (cid:48) ( x (cid:48) − y ) − (cid:107) x (cid:48) − y (cid:107) = π − n ϕ (V) x (cid:48) ,ξ (cid:48) ( y ) C.3 Linear map φ : ( E , g ) → ( E , g ) Suppose φ : ( E , g ) → ( E , g ) is a linear invertible map between two metric spaces. Let φ ◦ : S ( E ) → S ( E ) be the pull-back operator defined by φ ◦ u := u ◦ φ . Let Φ := φ − ⊕ φ ∗ : E ⊕ E ∗ → E ⊕ E ∗ be the induced map and Φ −◦ v := v ◦ Φ − the push-forward operator. Wehave defined in (C.8), B : S ( E ) → S ( E ⊕ E ∗ ) and B : S ( E ) → S ( E ⊕ E ∗ ) where B j denotes either B j, (V) or B j, ( R ) . Lemma C.8.
Suppose φ : ( E , g ) → ( E , g ) is a linear invertible map between twometric spaces. We have that φ ◦ = Υ ( φ ) B † Φ −◦ B (C.15) with Υ ( φ ) := (cid:18) det (cid:18) (cid:16) Id + (cid:0) φ − (cid:1) † φ − (cid:17)(cid:19)(cid:19) / (C.16) where ( φ − ) † is the metric-adjoint of φ − defined by (cid:104) x | ( φ − ) † x (cid:105) g = (cid:104) φ − x | x (cid:105) g forany x ∈ E , x ∈ E .Remark C.9 . If φ is an isometry then φ † φ = Id and d ( φ ) = 1 . Proof.
We compute the Schwartz kernels. For y (cid:48) ∈ E , z (cid:48) ∈ E , (cid:104) δ z (cid:48) |B † Φ −◦ B δ y (cid:48) (cid:105) = (cid:104)B δ z (cid:48) | Φ −◦ B δ y (cid:48) (cid:105) For y ∈ E , η ∈ E ∗ , ( B δ y (cid:48) ) ( y, η ) = (cid:104) ϕ (1) y,η | δ y (cid:48) (cid:105) = π − n e − i (cid:104) η | y (cid:48) − y (cid:105) e − (cid:107) y (cid:48) − y (cid:107) g For z ∈ E , ξ ∈ E ∗ , (cid:0) Φ −◦ B δ y (cid:48) (cid:1) ( z, ξ ) = π − n e − i (cid:104) φ ∗− ξ | y (cid:48) − φz (cid:105) e − (cid:107) y (cid:48) − φz (cid:107) g ( B δ z (cid:48) ) ( z, ξ ) = π − n e − i (cid:104) ξ | z (cid:48) − z (cid:105) e − (cid:107) z (cid:48) − z (cid:107) g Hence (cid:104)B δ z (cid:48) | Φ −◦ B δ y (cid:48) (cid:105) = π − n (cid:90) e i (cid:104) ξ | z (cid:48) − z (cid:105) e − (cid:107) z (cid:48) − z (cid:107) g e − i (cid:104) φ ∗− ξ | y (cid:48) − φz (cid:105) e − (cid:107) y (cid:48) − φz (cid:107) g dzdξ (2 π ) n We do the symplectic change of variables ( z, ξ ) ∈ E ⊕ E ∗ → ( y, η ) = Φ − ( z, ξ ) = (cid:16) φ ( z ) , φ ∗ − ξ (cid:17) ∈ ( E ⊕ E ∗ ) , i.e. ξ = φ ∗ η , z = φ − y , and get (cid:104) δ z (cid:48) |B † Φ −◦ B δ y (cid:48) (cid:105) = π − n (cid:90) e i (cid:104) φ ∗ η | z (cid:48) − φ − y (cid:105) e − (cid:107) z (cid:48) − φ − y (cid:107) g e − i (cid:104) η | y (cid:48) − y (cid:105) e − (cid:107) y (cid:48) − y (cid:107) g dydη (2 π ) n = π − n (cid:90) e i (cid:104) η | φz (cid:48) − y (cid:48) (cid:105) e − (cid:107) φ − ( φz (cid:48) − y ) (cid:107) g e − (cid:107) y (cid:48) − y (cid:107) g dydη (2 π ) n
62e have (cid:90) e i (cid:104) η | φz (cid:48) − y (cid:48) (cid:105) dη (2 π ) n = δ ( φz (cid:48) − y (cid:48) ) = (cid:104) δ z (cid:48) | φ ◦ δ y (cid:48) (cid:105) hence (cid:104) δ z (cid:48) |B † Φ −◦ B δ y (cid:48) (cid:105) = (cid:104) δ z (cid:48) | φ ◦ δ y (cid:48) (cid:105) π − n (cid:90) e − (cid:107) φ − ( y (cid:48) − y ) (cid:107) g e − (cid:107) y (cid:48) − y (cid:107) g dy = (cid:104) δ z (cid:48) | φ ◦ δ y (cid:48) (cid:105) π − n (cid:90) E e − (cid:107) φ − Y (cid:107) g e − (cid:107) Y (cid:107) g dY = (cid:104) δ z (cid:48) | φ ◦ δ y (cid:48) (cid:105) π − n (cid:90) e − (cid:104) Y | (cid:16) Id+ ( φ − ) † φ − (cid:17) Y (cid:105) dY = (cid:104) δ z (cid:48) | φ ◦ δ y (cid:48) (cid:105) π − n (2 π ) n det (cid:16) Id + ( φ − ) † φ − (cid:17) / = (cid:104) δ z (cid:48) | φ ◦ δ y (cid:48) (cid:105) Υ ( φ ) − Giving (C.15).Later we will use that
Lemma C.10.
With
Υ ( φ ) defined in (C.16), we have Υ (cid:0) φ − (cid:1) = | det φ | Υ ( φ ) (C.17) Proof.
Notice that det (cid:0) φ † φ (cid:1) = | det φ | where the determinant is measured with respect to localdensities. Υ ( φ ) = det (cid:18) (cid:0) Id + φ − † φ − (cid:1)(cid:19) = (cid:0) det (cid:0) φ † φ (cid:1)(cid:1) − det (cid:18) (cid:0) φ † φ + Id (cid:1)(cid:19) = | det φ | − Υ (cid:0) φ − (cid:1) Corollary C.11. If E is endowed with two metrics g , g then Id S ( E ) = Υ g ,g B † g g B g with Υ g ,g = (cid:18) det (cid:18) (cid:0) Id + Id † (cid:1)(cid:19)(cid:19) / = n (cid:89) j =1 (cid:18) I j (cid:19) / (C.18) ( I j ) j ∈ R n are inertial moments of g with respect to g i.e. in suitable coordinates, g = (cid:80) j dy j , g = (cid:80) j I j dy j . Consequently P g ,g := Υ g ,g B g B † g g : S ( E ⊕ E ∗ ) → S ( E ⊕ E ∗ ) is a projector, with Im ( P g ,g ) = Im ( B g ) , Ker ( P g ,g ) = (Im ( B g )) ⊥ .Proof. We apply Lemma C.8 with φ = Id : ( E, g ) → ( E, g ) and Φ = Id . We have (cid:104) u | (cid:0) φ − (cid:1) † v (cid:105) g = (cid:104) φ − u | v (cid:105) g = (cid:104) u | v (cid:105) g In coordinates such that g = (cid:80) j dy j , g = (cid:80) j I j dy j , (Id) † j,k = (cid:0) φ − (cid:1) † j,k = I j δ j = k giving (C.18). 63 .4 Compatible triple g, Ω , J Let us recall the definition of a compatible triple g, Ω , J . Let F be a vector space endowed witha symplectic bilinear form Ω and an Euclidean metric g . Let J := ˇ g − ˇΩ : F → F (C.19)with ˇΩ : F → F ∗ , ˇ g : F → F ∗ defined in Section A. Definition C.12.
We say that Ω , g are compatible structures on F if J := ˇ g − ˇΩ : F → F is an almost complex structure on F , i.e. J = − Id (C.20). Proposition C.13.
This implies that for any u, v ∈ Fg ( u, v ) = Ω ( u, J v ) , Ω ( u, v ) = g ( J u, v ) g ( J u, J v ) = g ( u, v ) , Ω (
J u, J v ) = Ω ( u, v ) . Proof.
We have ˇ g − ˇΩˇ g − ˇΩ = − Id hence ˇ g = − ˇΩˇ g − ˇΩ and g ( u, v ) = g ( v, u ) = (cid:104)− ˇΩˇ g − ˇΩ v | u (cid:105) = (cid:104) ˇ g − ˇΩ v | ˇΩ u (cid:105) = Ω ( u, J v ) g ( J u, v ) = (cid:104) ˇ g ˇ g − ˇΩ u | v (cid:105) = Ω ( u, v ) g ( J u, J v ) = (cid:104) ˇ g ˇ g − ˇΩ u | ˇ g − ˇΩ v (cid:105) = (cid:104)− ˇΩˇ g − ˇΩ u | v (cid:105) = g ( u, v )Ω ( J u, J v ) = (cid:104) ˇΩˇ g − ˇΩ u | ˇ g − ˇΩ v (cid:105) = (cid:104)− ˇΩˇ g − ˇΩˇ g − ˇΩ u | v (cid:105) = Ω ( u, v ) Example C.14.
Let ( E, g ) be a vector space with Euclidean metric g . On E ⊕ E ∗ , the canonicalsymplectic form Ω in (C.4), is compatible with the induced metric g := g ⊕ g − . Lemma C.15.
Let ( F, Ω , g ) be a vector space with a symplectic structure Ω and com-patible metric g . If E ⊂ F is a linear Lagrangian subspace, let E ⊥ g be the orthogonalsubspace with respect to g . Then E ⊥ g = J ( E ) is also Lagrangian and E = J (cid:0) E ⊥ g (cid:1) .Hence we have an isomorphism Ψ that is symplectic (for Ω ) and isometric (for g ) Ψ : F = E ⊕ E ⊥ g → E ⊕ E ∗ (C.21) where E ⊕ E ∗ is endowed with the canonical symplectic form (C.4) and the metric g = g ⊕ g − .Proof. Let v , v ∈ E and u := J ( v ) , u := J ( v ) . Then g ( u , v ) = (def g ) Ω ( u , J v ) = Ω ( J v , J v ) = ( J comp . ) Ω ( v , v ) = ( V Lag . ) . Hence E ⊥ g = J ( E ) and E = J (cid:0) E ⊥ g (cid:1) , since J = − Id . From compatibility J is a symplecticmap hence E ⊥ g is Lagrangian as well. 64 ζN K ( x, ξ ) π K π N ˇΩ ∗ ( ζ )ˇΩ( ν ) νxξ F ∗ Figure C.1: Picture for the orthogonal decomposition F ⊕ F ∗ = K ⊥ ⊕ N in (C.24). C.5 Bergman projector on a symplectic vector space ( F, Ω , g ) Let ( F, Ω , g ) be a vector space with a symplectic structure Ω and compatible metric g . Definition C.16.
Let P F : S ( F ) → S ( F ) be the operator defined by its Schwartzkernel: for any ρ, ρ (cid:48) ∈ F , (cid:104) δ ρ (cid:48) |P F δ ρ (cid:105) = exp (cid:18) − i ρ (cid:48) , ρ ) − (cid:107) ρ (cid:48) − ρ (cid:107) g (cid:19) . (C.22) Lemma C.17. P F is an orthogonal projector in L (cid:16) F, dρ (2 π ) n (cid:17) , called the Bergman pro-jector .Proof.
We choose any E ⊂ F Lagrangian linear subspace. Then P F = P F, (R) is the Bergmanprojector in radial gauge defined in (C.13). C.6 The metaplectic decomposition of F ⊕ F ∗ Let ( F, Ω) be a linear symplectic space. Let us denote Ω the canonical symplectic form on F ⊕ F ∗ as in (C.4). Lemma C.18.
Let ( F, Ω) be a linear symplectic space and K := graph (cid:0) ˇΩ (cid:1) , N := graph (cid:0) ˇΩ ∗ (cid:1) . (C.23)We have F ⊕ F ∗ = K ⊥ ⊕ N. (C.24)where the right hand side is an Ω -orthogonal decomposition into linear symplectic sub-spaces. See Figure C.1. Explicitly for any ( x, ξ ) ∈ F ⊕ F ∗ , ( x, ξ ) = (cid:0) ν, ˇΩ ( ν ) (cid:1) + (cid:0) ζ, ˇΩ ∗ ( ζ ) (cid:1) ∈ K ⊕ N is given by ν = 12 (cid:0) x + ˇΩ − ( ξ ) (cid:1) (C.25) ζ = 12 (cid:0) x − ˇΩ − ( ξ ) (cid:1) (C.26) Remark
C.19 . The metaplectic decomposition (C.24) is central in this paper. It has also beenused in [9, Prop.6] and in [15, Prop.2.2.9]. 65 roof.
Let u, v ∈ F . (cid:104) v | ˇΩ ∗ u (cid:105) = (cid:104) ˇΩ v | u (cid:105) = Ω ( v, u ) = − Ω ( u, v ) = −(cid:104) ˇΩ u | v (cid:105) hence ˇΩ ∗ u = − ˇΩ u , hence ˇΩ ∗ = − ˇΩ . In coordinates Ω = dp ⊗ dq − dq ⊗ dp , hence ˇΩ ( q,p ) = pdq − qdp and d ˇΩ ( q,p ) = 2 dp ∧ dq = 2Ω . If ( x, ξ ) = (cid:0) x, ˇΩ ( x ) (cid:1) ∈ graph (cid:0) ˇΩ (cid:1) and ( x (cid:48) , ξ (cid:48) ) = (cid:0) x (cid:48) , ˇΩ ∗ ( x (cid:48) ) (cid:1) ∈ graph (cid:0) ˇΩ ∗ (cid:1) then Ω (( x, ξ ) , ( x (cid:48) , ξ (cid:48) )) = (cid:104) ξ | x (cid:48) (cid:105) − (cid:104) x | ξ (cid:48) (cid:105) = (cid:104) ˇΩ ( x ) | x (cid:48) (cid:105) − (cid:104) x | ˇΩ ∗ ( x (cid:48) ) (cid:105) = (cid:104) ˇΩ ( x ) | x (cid:48) (cid:105) + (cid:104) x | ˇΩ ( x (cid:48) ) (cid:105) = Ω ( x, x (cid:48) ) + Ω ( x (cid:48) , x ) = 0 Moreover if ( x, ξ ) = (cid:0) x, ˇΩ ( x ) (cid:1) = (cid:0) x, ˇΩ ∗ ( x ) (cid:1) ∈ graph (cid:0) ˇΩ (cid:1) ∩ graph (cid:0) ˇΩ ∗ (cid:1) then ˇΩ ( x ) = ˇΩ ∗ ( x ) = − ˇΩ ( x ) hence ˇΩ ( x ) = 0 hence x = 0 . This implies that graph (cid:0) ˇΩ (cid:1) , graph (cid:0) ˇΩ ∗ (cid:1) are symplectic.Let us prove (C.25). We write (cid:40) x = ν + ζξ = ˇΩ ( ν ) + ˇΩ ∗ ( ζ ) = ˇΩ ( ν − ζ ) ⇔ (cid:40) ν = (cid:0) x + ˇΩ − ( ξ ) (cid:1) ζ = (cid:0) x − ˇΩ − ( ξ ) (cid:1) Lemma C.20.
Let ( F, Ω , g ) be a vector space with a symplectic structure Ω and com-patible metric g . Then (C.24) is orthogonal for the metric g = g ⊕ g − as well. Themaps π K : (cid:0) x, ˇΩ ( x ) (cid:1) ∈ ( K, Ω , g ) → x ∈ ( F, , g ) (C.27) π N : (cid:0) x, ˇΩ ∗ ( x ) (cid:1) ∈ ( N, Ω , g ) → x ∈ ( F, − , g ) are isomorphism for the respective bilinear forms.Proof. K ⊥ ⊕ N is also orthogonal for the metric g because g (( x, ξ ) , ( x (cid:48) , ξ (cid:48) )) = g ( x, x (cid:48) ) + g − ( ξ, ξ (cid:48) ) = g ( x, x (cid:48) ) + g − (cid:0) ˇΩ ( x ) , ˇΩ ∗ ( x (cid:48) ) (cid:1) = g ( x, x (cid:48) ) + g − (cid:0) ˇΩ ( x ) , − ˇΩ ( x (cid:48) ) (cid:1) = g ( x, x (cid:48) ) − g − (cid:0) ˇΩ ( x ) , ˇΩ ( x (cid:48) ) (cid:1) = 0 because g − (cid:0) ˇΩ ( x ) , ˇΩ ( x (cid:48) ) (cid:1) = g ( x, x (cid:48) ) from compatibility between Ω and g .If ( x, ξ ) = (cid:0) x, ˇΩ ( x ) (cid:1) ∈ K and ( x (cid:48) , ξ (cid:48) ) = (cid:0) x (cid:48) , ˇΩ ( x (cid:48) ) (cid:1) ∈ K then Ω (( x, ξ ) , ( x (cid:48) , ξ (cid:48) )) = (cid:104) ξ | x (cid:48) (cid:105) − (cid:104) x | ξ (cid:48) (cid:105) = (cid:104) ˇΩ ( x ) | x (cid:48) (cid:105) − (cid:104) x | ˇΩ ( x (cid:48) ) (cid:105) = (cid:104) ˇΩ ( x ) | x (cid:48) (cid:105) − (cid:104) ˇΩ ∗ ( x ) | x (cid:48) (cid:105) = (cid:104) ˇΩ ( x ) | x (cid:48) (cid:105) + (cid:104) ˇΩ ( x ) | x (cid:48) (cid:105) = 2Ω ( x, x (cid:48) ) g (( x, ξ ) , ( x (cid:48) , ξ (cid:48) )) = g ( x, x (cid:48) ) + g − ( ξ, ξ (cid:48) ) = g ( x, x (cid:48) ) + g − (cid:0) ˇΩ ( x ) , ˇΩ ( x (cid:48) ) (cid:1) = 2 g ( x, x (cid:48) ) and a similar computation on N , if ˇΩ is replaced by ˇΩ ∗ = − ˇΩ .Let B F : S ( F ) → S ( F ⊕ F ∗ ) the Bargman transform. Let P F ⊕ F ∗ := B F B † F . Let P K , P N defined as in (C.22) for the respective spaces ( K, Ω K , g K ) , ( N, Ω N , g N ) defined in (C.23) withmetrics induced from Ω , g . From (C.24) we have the natural identification S ( F ⊕ F ∗ ) = S ( K ⊕ N ) = S ( K ) ⊗ S ( N ) . Lemma C.21.
We have P F ⊕ F ∗ = P K ⊗ P N (C.28) Proof.
Since Ω F ⊕ F ∗ = Ω K ⊕ Ω N and g = g K ⊕ g N , (C.28) follows from the expression of thekernel (C.22) that coincide for both sides. 66 .7 Metaplectic decomposition of a linear symplectic map Φ : ( F , Ω ) → ( F , Ω ) Let ( F , Ω ) and ( F , Ω ) be linear symplectic spaces. Let Φ : ( F , Ω ) → ( F , Ω ) a linearsymplectic map, Φ ◦ = S ( F ) → S ( F ) the pull back operator and ˜Φ := Φ − ⊕ Φ ∗ : F ⊕ F ∗ → F ⊕ F ∗ the induced map on cotangent spaces. Since Φ is symplectic we have that ˜Φ preservesthe decompositions (C.24), F j ⊕ F ∗ j = K j ⊕ N j , with j = 1 , and we denote its components by ˜Φ = Φ K ⊕ Φ N . (C.29)The Metaplectic correction Υ (Φ) > has been defined in (C.16). Theorem C.22. [9][15]Let
Φ : ( F , Ω ) → ( F , Ω ) a linear symplectic map and compat-ible metrics g j on F j , j = 1 , . We have B F Φ ◦ B † F = ˜Op (Φ K ) ⊗ ˜Op (Φ N ) (C.30) with unitary operators ˜Op (Φ K ) := (Υ (Φ)) / (cid:16) P † K Φ −◦ K P K (cid:17) : Im ( P K ) → Im ( P K )˜Op (Φ N ) := (Υ (Φ)) / (cid:16) P † N Φ −◦ N P N (cid:17) : Im ( P N ) → Im ( P N ) B F Φ ◦ B † F : Im (cid:0) P F ⊕ F ∗ (cid:1) → Im (cid:0) P F ⊕ F ∗ (cid:1) Proof.
We have Φ ◦ = ( C. Υ (Φ) B † F ˜Φ −◦ B F = Υ (Φ) B † F (cid:0) Φ −◦ K ⊗ Φ −◦ N (cid:1) B F B F Φ ◦ B † F = Υ (Φ) P F ⊕ F ∗ (cid:0) Φ −◦ K ⊗ Φ −◦ N (cid:1) P F ⊕ F ∗ = Υ (Φ) (cid:16) P † K Φ −◦ K P K (cid:17) ⊗ (cid:16) P † N Φ −◦ N P N (cid:17) = ˜Op (Φ K ) ⊗ ˜Op (Φ N ) Remark
C.23 . Lemma C.22 is central to the analysis in this paper. It is a factorization formulafor the map Φ and somehow gives a square root of Φ ◦ that is emphasized by the next definition.67 efinition C.24. Let
Φ : ( F , Ω ) → ( F , Ω ) a linear symplectic map and g j a compat-ible metric on F j , j = 1 , .• We define the metaplectic operator, or quantization of Φ by ˜Op (Φ) := (Υ (Φ)) / P F Φ −◦ P F : Im ( P F ) → Im ( P F ) (C.31)with the Metaplectic correction Υ (Φ) > defined in (C.16). This is a unitaryoperator.• If E ⊂ F , E ⊂ F are Lagrangian linear subspace, and using the identification(C.21), we define Op (Φ) : = B † ˜Op (Φ) B : S ( E ) → S ( E ) (C.32) = (Υ (Φ)) / B † Φ −◦ B (C.33)that is unitary conjugated to ˜Op (Φ) / Im B → Im B . Equivalently ˜Op (Φ) = B Op (Φ) B † . Proposition C.25.
For any linear symplectic maps Φ , : ( F , Ω ) → ( F , Ω ) , Φ , :( F , Ω ) → ( F , Ω ) with compatible metrics g j on F j , j = 1 , , we have ˜Op (Φ , Φ , ) = ˜Op (Φ , ) ˜Op (Φ , ) . (C.34) i.e. ˜Op is a contravariant functor.Proof. We have (cid:16) B F Φ ◦ , B † F (cid:17) (cid:16) B F Φ ◦ , B † F (cid:17) = ( C. B F Φ ◦ , Φ ◦ , B † F = B F (Φ , Φ , ) ◦ B † F and (C.30) gives (C.34).The next lemma considers a special case for the symplectic map Φ . Lemma C.26. [15, proof of Prop 4.3.1 p.79]Let ( E , g ) , ( E , g ) be Euclidean vectorspaces, and φ : E → E an invertible linear map. We denote φ ◦ : S ( E ) → S ( E ) thepull-back operator and Φ := φ − ⊕ φ ∗ : E ⊕ E ∗ → E ⊕ E ∗ the induced symplectic mapon cotangent spaces. Op (Φ) has been defined in (C.32) and ˜Op (Φ) in (C.31). We have
Op (Φ) = | det φ | / φ ◦ . (C.35) ˜Op (Φ) = | det φ | / B φ ◦ B † Proof.
We have Φ − := φ ⊕ φ ∗− . Hence Υ (Φ) = (cid:18) det (cid:18) (cid:16) Id + (cid:0) Φ − (cid:1) † Φ − (cid:17)(cid:19)(cid:19) / = (cid:18) det (cid:18) (cid:0) Id + φ † φ (cid:1)(cid:19)(cid:19) / (cid:18) det (cid:18) (cid:0) Id + φ ∗− † φ ∗− (cid:1)(cid:19)(cid:19) / (C.36) = Υ ( φ ) Υ (cid:0) φ − (cid:1) = ( C. | det φ | (Υ ( φ )) (C.37)68hen Op (Φ) = ( C. (Υ (Φ)) / B † Φ −◦ B = ( C. (Υ (Φ)) / (Υ ( φ )) − φ ◦ = ( C. | det φ | / φ ◦ C.8 Some useful decomposition
Here we give a relation that is trivial here in the linear setting, but is used in the main textas an approximation, in the proof of Theorem 6.8. Let us denote ˜ F = F ⊕ F ∗ = ( C. K ⊕ N and exp : T ˜ F → ˜ F the exponential map. Now we restrict the base to K ⊂ ˜ F and consider N ⊂ T K ˜ F as a subbundle N → K of T K ˜ F → K . We denote exp N : N → ˜ F the exponentialmap that is an isomorphism. It gives the (twisted) operators (cid:103) exp ◦ : S (cid:16) ˜ F (cid:17) → S (cid:16) T ˜ F (cid:17) , (cid:94) (cid:16) exp − N/ (cid:17) ◦ : S ( N ) → S (cid:16) ˜ F (cid:17) . We also use the restriction operators r : S (cid:16) T ˜ F (cid:17) → S (cid:16) ˜ F (cid:17) , r /K : S (cid:16) T ˜ F (cid:17) → S (cid:16) T K ˜ F (cid:17) , r N : S (cid:16) T K ˜ F (cid:17) → S ( N ) . We have Id S ( ˜ F ) = r (cid:103) exp ◦ = (cid:94) (cid:0) exp − N (cid:1) ◦ r N r /K (cid:103) exp ◦ , where the second can be seen as a generalization of the first for the respective decompositions ˜ F = { } ⊕ ˜ F = K ⊕ N . More generally Lemma C.27.
For the symplectic map ˜Φ = ( C. Φ K ⊕ Φ N : ˜ F → ˜ F , using ˜Op (Φ) : S (cid:16) T ˜ F (cid:17) → S (cid:16) T ˜ F (cid:17) and ˜Op /K (Φ) : S (cid:16) T K ˜ F (cid:17) → S (cid:16) T K ˜ F (cid:17) as bundle map operators, wehave the decomposition r ˜Op (Φ) (cid:103) exp ◦ = (cid:94) (cid:0) exp − N (cid:1) ◦ r N ˜Op /K (Φ) r /K (cid:103) exp ◦ . (C.38) C.9 Taylor operators T k on S ( E ) Let E an Euclidean vector space, n = dim E . If ( e , . . . e n ) is a basis of E , we write x = (cid:80) ni =1 x i e i ∈ E . A polynomial on E is P ( x ) := p ( x , . . . x n ) with p ∈ C [ x , . . . x n ] a polynomialon R n . For α = ( α , . . . , α n ) ∈ N n , we write x α := x α . . . x α n n a monomial of degree | α | := α + . . . α n . For k ∈ N , we denote Pol k ( E ) the space of homogeneous polynomials of degree k ,that is independent on the basis. P ∈ Pol k ( E ) can be written P ( x ) = (cid:80) α ∈ N k , | α | = k P α x α withcomponents P α ∈ C . We denote α ! := α ! . . . α n ! and δ ( α ) := δ ( α )0 ( x ) . . . δ ( α n )0 ( x n ) the Diracdistribution on E (with α -derivatives). We have that for any α, α (cid:48) ∈ N n , (cid:104) α (cid:48) ! δ ( α (cid:48) )0 | x α (cid:105) = δ α (cid:48) = α .Hence the set ( x α ) α forms a basis of Pol ( E ) and (cid:0) (cid:104) δ ( α ) | . (cid:105) (cid:1) α is the dual basis. See Section 10for comments about the equivalence Pol k ( E ) ≡ Sym (cid:16) ( E ∗ ) ⊗ k (cid:17) . Definition C.28.
For k ∈ N , let T k : S ( E ) → Pol k ( E ) ⊂ S (cid:48) ( E ) be the projector onto homogeneous polynomials of degree k i.e. with respect to a basisof E , T k = (cid:88) α ∈ N k , | α | = k x α (cid:104) α ! δ ( α )0 | . (cid:105) (C.39)69e have T k (cid:48) T k = T k δ k (cid:48) = k . The next lemma shows how this relation is changed by a small norm operator if we insert atruncation at large distance σ from the origin. We put χ σ ( ρ ) = 1 for | ρ | g ≤ σ , χ σ ( ρ ) = 0 for | ρ | g > σ and Op ( χ σ ) := B † M χ σ B . Lemma C.29.
For k, k (cid:48) ∈ N , we have ∀ N, ∃ C N > , ∀ σ > , R σ := Op ( χ σ ) ( T k (cid:48) Op ( χ σ ) T k − T k δ k (cid:48) = k ) Op ( χ σ ) ∈ Ψ Id , g /σ (cid:16) C N σ − N (cid:10) dist g /σ ( ρ, (cid:11) − N (cid:17) . Consequently (cid:107) R σ (cid:107) ≤ C N σ − N for any N > , with C N > .Proof. We have (cid:104) δ ρ (cid:48) |B R B † δ ρ (cid:105) = ( C. χ σ ( ρ (cid:48) ) χ σ ( ρ ) (C.40) (cid:88) α (cid:48) ∈ N k , | α (cid:48) | = k (cid:48) ,α ∈ N k , | α | = k, (cid:104) ϕ ρ (cid:48) | x α (cid:48) (cid:105) (cid:18) (cid:104) α (cid:48) ! δ ( α (cid:48) )0 | Op ( χ σ ) x α (cid:105) − δ α (cid:48) = α (cid:19) (cid:104) α ! δ ( α )0 | ϕ ρ (cid:105) . (C.41)We have ∀ N, ∃ C N , ∀ σ > , (cid:12)(cid:12)(cid:12)(cid:12) (cid:104) α (cid:48) ! δ ( α (cid:48) )0 |B † M χ σ B x α (cid:105) − δ α (cid:48) = α (cid:12)(cid:12)(cid:12)(cid:12) ≤ C N σ − N , and due to truncations χ σ ( ρ (cid:48) ) χ σ ( ρ ) in (C.40) we deduce that (cid:12)(cid:12) (cid:104) δ ρ (cid:48) |B † R B δ ρ (cid:105) (cid:12)(cid:12) ≤ (cid:10) dist g /σ ( ρ (cid:48) , ρ ) (cid:11) − N C N σ − N (cid:10) dist g /σ ( ρ, (cid:11) − N , i.e. R σ ∈ Ψ Id , g /σ (cid:16) C N σ − N (cid:10) dist g /σ ( ρ, (cid:11) − N (cid:17) from definition 3.18 and (cid:107) R σ (cid:107) ≤ C N σ − N fromexample (3.19). Lemma C.30. If ( F, Ω , g ) is a linear symplectic space with compatible metric g , E , E ⊂ F are Lagrangian such that E ⊕ E = F , there exists S E ,E : F → F linear symplecticsuch that S E ,E ( E ) = E , S E ,E (cid:16) E ⊥ g (cid:17) = E . (C.42) Proof. If ( e , . . . e n ) is a basis of E , let ( f , . . . f n ) be a basis of E given by the dual basis of (cid:0) ˇΩ f , . . . ˇΩ f n (cid:1) of E ∗ i.e. such that Ω ( e j , f k ) = δ j = k . Define S by S ( e j ) = e j and S ( J ( e j )) = f j . Definition C.31.
Let ˜ T ( k ) E ,E := ˜Op ( S E ,E ) B E T k B † E ˜Op ( S E ,E ) † (C.43)that is the projector onto “degree k Polynomials on E , parallel to E ” C.10 Analysis on T ( E ⊕ E ∗ ) In this section we “lift” the analysis from E ⊕ E ∗ to T ( E ⊕ E ∗ ) . This has no much meaning andinterest for vector spaces, but it will be useful in this paper as a linearized model for manifolds.70 .10.1 Bargman transform Let ( E, g ) be a linear Euclidean space. Let F := E ⊕ E ∗ with induced metric g = g ⊕ g − . Onehas T F = T ( E ⊕ E ∗ ) = ( E ⊕ E ∗ ) ⊕ ( E ⊕ E ∗ ) . The exponential map is exp : (cid:40) T F → F ( ρ, ρ (cid:48) ) → ρ + ρ (cid:48) Let us define the “twisted pull back operator” (cid:103) exp ◦ : S ( F ) → S ( T F ) by its Schwartz kernel asfollows. For ( ρ , ρ (cid:48) ) ∈ T F , ρ ∈ F , ρ = ( x , ξ ) , ρ (cid:48) = ( x (cid:48) , ξ (cid:48) ) , (cid:104) δ ρ ,ρ (cid:48) | (cid:103) exp ◦ δ ρ (cid:105) = δ ρ − ( ρ + ρ (cid:48) ) e − iξ x (cid:48) We define the “restriction” operator r : S ( T F ) → S ( F ) by its Schwartz kernel (cid:104) δ ρ | r δ ρ ,ρ (cid:48) (cid:105) = δ ρ − ρ δ ρ (cid:48) Let the “horizontal space” be H := F ⊕ E. (C.44)If B : S ( E ) → S ( F ) is the Bargman transform defined in (C.8), we define the partial Bargmantransform B : S ( H ) → S ( T F ) by B ( u ( ρ ) ⊗ v ( x (cid:48) )) = u ( ρ ) ⊗ ( B v ) ( x (cid:48) ) , i.e. action on the second part only.Let σ > and let us introduce the cut-off function that truncates at distance σ in the fiber F : χ σ : S ( T F ) → S (cid:48) ( T F ) u ( ρ, ρ (cid:48) ) → (cid:40) u ( ρ, ρ (cid:48) ) if (cid:107) ρ (cid:48) (cid:107) g ≤ σ otherwiseLet B ∆ χ := B † χ σ (cid:103) exp ◦ : S ( F ) → S ( H ) B := r B : S ( H ) → S ( F ) C.10.2 Linear map
Let A : E → E a linear invertible map and ˜ A := A − ⊕ A ∗ the induced map on F := E ⊕ E ∗ .Let A ◦ : S ( E ) → S ( E ) the pull back operator and A H := ˜ A − ⊕ A : H → H the inducedmap on H in (C.44). Lemma C.32.
Let us consider the difference operator R := B A ◦ H B ∆ χ − B A ◦ B † : S ( F ) → S ( F ) . Then ∀ N > , ∃ C N > such that R ∈ Ψ ˜ A (cid:0) C N σ − N (cid:1) , i.e. its Schwartz kernel (cid:104) δ ρ (cid:48) | Rδ ρ (cid:105) decays very fast outside the graph of ˜ A and on the graph it decays as C N σ − N for any N > .Remark C.33 . In particular, for σ = ∞ , i.e. no cut-off, then R = 0 . Another interestingparticular case is A = Id . 71 roof. We compute the Schwartz kernel. (cid:104) δ ρ (cid:48) | B A ◦ H B ∆ χ δ ρ (cid:105) = (cid:104) δ ρ (cid:48) | r BA ◦ H B † χ σ (cid:103) exp ◦ δ ρ (cid:105) = (cid:90) (cid:104) δ ρ | r δ ρ ,ρ (cid:48) (cid:105)(cid:104) δ ρ ,ρ (cid:48) | BA ◦ H B † δ ρ ,ρ (cid:48) (cid:105) χ σ ( ρ (cid:48) ) (cid:104) δ ρ ,ρ (cid:48) | (cid:103) exp ◦ δ ρ (cid:105) = (cid:90) (cid:0) δ ρ (cid:48) − ρ δ ρ (cid:48) (cid:1) (cid:0) δ ˜ Aρ − ρ (cid:104) ϕ ρ (cid:48) | A ◦ ϕ ρ (cid:48) (cid:105) (cid:1) χ σ ( ρ (cid:48) ) (cid:16) δ ρ − ( ρ + ρ (cid:48) ) e − iξ x (cid:48) (cid:17) = (cid:104) ϕ | A ◦ ϕ ρ − ˜ A − ρ (cid:48) (cid:105) e − i ( ( A ∗ ) − ξ (cid:48) ) ( x − Ax (cid:48) ) χ σ (cid:16) ρ − ˜ A − ρ (cid:48) (cid:17) because Dirac measures gave ρ (cid:48) = ρ − ρ = ρ − ˜ A − ρ (cid:48) , ξ = ( A ∗ ) − ξ (cid:48) and x (cid:48) = x − Ax (cid:48) . Then (cid:104) δ ρ (cid:48) | B A ◦ H B ∆ χ δ ρ (cid:105) = (cid:104) ϕ | A ◦ ϕ ρ − ˜ A − ρ (cid:48) (cid:105) e iξ (cid:48) ( x (cid:48) − A − x (cid:48) ) χ σ (cid:16) ρ − ˜ A − ρ (cid:48) (cid:17) = ( C. (cid:104) ϕ | A ◦ T −◦ x − Ax (cid:48) (cid:16) F T −◦ ξ − A ∗− ξ (cid:48) F (cid:17) ϕ (cid:105) e iξ (cid:48) ( x (cid:48) − A − x (cid:48) ) χ σ (cid:16) ρ − ˜ A − ρ (cid:48) (cid:17) = ( C. (cid:104) ϕ | A ◦ (cid:16) F T −◦− A ∗− ξ (cid:48) F (cid:17) T −◦− Ax (cid:48) T −◦ x (cid:0) F T −◦ ξ F (cid:1) ϕ (cid:105) e − i ( − A ∗− ξ (cid:48) ) x e iξ (cid:48) ( − A − x (cid:48) ) χ σ (cid:16) ρ − ˜ A − ρ (cid:48) (cid:17) = = ( C. (cid:104) ϕ | (cid:0) F T −◦− ξ (cid:48) F (cid:1) T −◦− x (cid:48) A ◦ T −◦ x (cid:0) F T −◦ ξ F (cid:1) ϕ (cid:105) χ σ (cid:16) ρ − ˜ A − ρ (cid:48) (cid:17) = (cid:104) ϕ ρ (cid:48) | A ◦ ϕ ρ (cid:105) χ σ (cid:16) ρ − ˜ A − ρ (cid:48) (cid:17) = (cid:104) δ ρ (cid:48) |B A ◦ B † δ ρ (cid:105) χ σ (cid:16) ρ − ˜ A − ρ (cid:48) (cid:17) If (cid:13)(cid:13)(cid:13) ρ − ˜ A − ρ (cid:48) (cid:13)(cid:13)(cid:13) ≤ σ then (cid:104) δ ρ (cid:48) | Rδ ρ (cid:105) = 0 otherwise ∀ N > , ∃ C N > , (cid:12)(cid:12) (cid:104) ϕ | A ◦ ϕ ρ − ˜ A − ρ (cid:48) (cid:105) (cid:12)(cid:12) ≤ C A C N σ − N . C.10.3 Taylor projectors
For ρ ∈ F , ˆ T ρ has been defined in (C.1). Let us define T ( k ) H := ˆ T − ρ T k ˆ T ρ : S ( H ) → S ( H ) where ρ ∈ F denotes the first variable and the operators acts on the second variable x (cid:48) ∈ E only. Lemma C.34.
Let us consider the difference operator R := B T ( k ) H B ∆ χ − B T k B ∗ : S ( F ) → S ( F ) . Then for (cid:107) ρ − ρ (cid:48) (cid:107) ≤ σ we have (cid:104) δ ρ (cid:48) | Rδ ρ (cid:105) = 0 .Remark C.35 . In particular for σ = ∞ , i.e. no cut-off, then R = 0 . Proof.
We repeat the lines of proof of Lemma C.32. (cid:104) δ ρ (cid:48) | B T ( k ) H B ∆ χ δ ρ (cid:105) = (cid:104) δ ρ (cid:48) | r BT ( k ) H B ∗ χ σ (cid:103) exp ◦ δ ρ (cid:105) = (cid:104) ϕ | ˆ T − ρ (cid:48) T k ˆ T ρ (cid:48) ϕ ρ − ρ (cid:48) (cid:105) e iξ (cid:48) ( x (cid:48) − x ) χ σ ( ρ − ρ (cid:48) )= ( C. (cid:104) ϕ | ˆ T † ρ (cid:48) e − iξ (cid:48) x (cid:48) T k ˆ T ρ (cid:48) ϕ ρ − ρ (cid:48) (cid:105) e iξ (cid:48) ( x (cid:48) − x ) χ σ ( ρ − ρ (cid:48) )= (cid:104) ϕ ρ (cid:48) | T k ϕ ρ (cid:105) χ σ ( ρ − ρ (cid:48) ) = (cid:104) δ ρ (cid:48) |B T k B † δ ρ (cid:105) χ σ ( ρ − ρ (cid:48) ) Linear expanding maps
Let ( E, g ) a finite dimensional vector space with Euclidean metric g . In this section we considera linear invertible and expanding map φ : ( E, g ) → ( E, g ) i.e. ∃ < λ < , (cid:13)(cid:13) φ − (cid:13)(cid:13) < λ. The purpose of this section is to study the spectrum of the following operator
Op (Φ) = ( C. | det φ | / φ ◦ : S ( E ) → S ( E ) on an adequate Hilbert space that contains S ( E ) ( Op (Φ) is unitary in L ( E ) ). Recall that Op (Φ) : = ( C. (Υ (Φ)) / B † Φ −◦ B with Φ := φ − ⊕ φ ∗ : E ⊕ E ∗ → E ⊕ E ∗ being the induced mapon cotangent space and B : S ( E ) → S ( E ⊕ E ∗ ) defined in (C.8).Consider T k the finite rank projector defined in (C.39). The vector space Im ( T k ) ⊂ S (cid:48) ( E ) is finite dimensional. Since φ is a linear map, we have for any k ∈ N , [ φ ◦ , T k ] = 0 , (D.1)hence Op (Φ) : Im ( T k ) → Im ( T k ) is invariant. Let γ ± k := lim t →±∞ log (cid:13)(cid:13)(cid:13) Op (cid:0) Φ t (cid:1) / Im( T k ) (cid:13)(cid:13)(cid:13) /t . (D.2)That is computable form the eigenvalues of φ see Remark [17, rem. 3.4.7 page 64.]. For every k ∈ N , we have γ − k ≤ γ + k , γ ± k +1 ≤ γ ± k . The spectrum of Op (Φ t ) : Im ( T k ) → Im ( T k ) is discreteand contained in the annulus (cid:110) z ∈ C , e tγ − k ≤ | z | ≤ e tγ + k (cid:111) . However we want to understand theaction of Op (Φ) on every function in S ( E ) . For K ∈ N , let T ≥ ( K +1) := Id S ( E ) − (cid:32) K (cid:88) k =0 T k (cid:33) . We will show below in Proposition D.3 that (cid:13)(cid:13)
Op (Φ t ) T ≥ ( K +1) (cid:13)(cid:13) H W ( E ) ≤ Ce tγ + K +1 for some ade-quate norm that we first define. D.1 Anisotropic Sobolev space H W ( E ) We use the “Japanese bracket” (cid:104) x (cid:105) defined in (5.1). Let h > , < γ < , R > . We define aweight function W : E ⊕ E ∗ → R + as follows. For ( x, ξ ) ∈ E ⊕ E ∗ , let h γ ( x, ξ ) = h (cid:68) (cid:107) ( x, ξ ) (cid:107) g ⊕ g − (cid:69) − γ and W ( x, ξ ) := (cid:68) h γ ( x, ξ ) (cid:107) x (cid:107) g (cid:69) R (cid:68) h γ ( x, ξ ) (cid:107) ξ (cid:107) g − (cid:69) R (D.3) Definition D.1.
For u ∈ S ( E ) , we define the norm (cid:107) u (cid:107) H W ( E ) := (cid:107)WB u (cid:107) L ( E ⊕ E ∗ ) , and the Sobolev space H W ( E ) := { u ∈ S ( E ) } (D.4)where the completion is with the norm (cid:107) u (cid:107) H W ( E ) .73 emark D.2 . • We have for R = 0 , H W ( E ) = L ( E ) .• The order is r = R (1 − γ ) as in (5.8).• In the lecture notes [10] we consider W as above with γ = 0 . D.2 Result
The result that we will show is
Proposition D.3.
For any (cid:15) > and K ∈ N , for R (cid:29) large enough in (D.3), ∃ C (cid:15) > , ∀ t ≥ , (cid:13)(cid:13) Op (cid:0) Φ t (cid:1) T ≥ ( K +1) (cid:13)(cid:13) H W ( E ) ≤ C (cid:15) e ( γ + K +1 + (cid:15) ) t . (D.5) with γ + K +1 defined in (D.2). D.2.1 Proof of Proposition D.3
Let ˜Op (Φ) := (Υ (Φ)) / P † Φ −◦ P and ˜Op W (Φ) := W ˜Op (Φ) W − . From (C.35) we have the commutative diagram: H W ( E ) B −→ L ( E ⊕ E ∗ ; W ) W −→ L ( E ⊕ E ∗ ) ↓ Op (Φ) ↓ ˜Op (Φ) ↓ W ˜Op (Φ) W − H W ( E ) B −→ L ( E ⊕ E ∗ ; W ) W −→ L ( E ⊕ E ∗ ) (D.6)where horizontal arrows are isometries by definition. Hence, the study of ˜Op (Φ t ) : H W ( E ) →H W ( E ) is equivalent to study ˜Op W (Φ t ) : L ( E ⊕ E ∗ ) → L ( E ⊕ E ∗ ) which is more conve-nient. Lemma D.4. “ Decay property of W with respect to Φ ”. There exists Λ > , C > ,such that for any t ≥ , there exists C t such that for any ρ ∈ E ⊕ E ∗ W (Φ t ( ρ )) W ( ρ ) ≤ C ≤ Ce − Λ t if (cid:107) ( x, ξ ) (cid:107) g ≥ C t (D.7) ” Temperate property of W ”: ∃ C > , N ≥ , ∀ ρ, ρ (cid:48) , W ( ρ (cid:48) ) W ( ρ ) ≤ C (cid:104) h γ ( ρ ) dist g ( ρ (cid:48) , ρ ) (cid:105) N . (D.8) Proof.
See [17, Thm 5.9]. It gives
Λ = λ (1 − γ ) R . Truncation in phase space near the trapped set
Let χ ∈ C ∞ c ( R + ; [0 , such that χ ( x ) = 1 if x ≤ ,χ ( x ) = 0 if x ≥ , Let σ > . For ρ ∈ E ⊕ E ∗ let χ σ ( ρ ) := χ (cid:18) (cid:107) ρ (cid:107) g σ (cid:19) . emma D.5. We have ∀ R > , ∀ (cid:15) > , ∃ C > , ∀ t ≥ , ∃ σ t > , ∀ σ > σ t , (cid:13)(cid:13)(cid:13) ˜Op W (cid:0) Φ t (cid:1) (1 − χ σ ) (cid:13)(cid:13)(cid:13) L ≤ Ce ( − Λ+ (cid:15) ) t . (D.9) where Λ is given in (D.7) that depends on R . The operator ˜Op W (Φ t ) χ σ is Trace classin L . Consequently for any t > , r ess (cid:16) ˜Op W (cid:0) Φ t (cid:1)(cid:17) ≤ e − Λ t . (D.10) Proof.
For (D.9), see [17, Lemma 5.14].
Lemma D.6.
Let k ∈ N . If k + 1 < r in (5.8), then T k : H W ( E ) → H W ( E ) is a boundedoperator.Proof. We check that in (C.39), (cid:107) x α (cid:107) H W ( E ) < ∞ and (cid:13)(cid:13)(cid:13) δ ( α )0 (cid:13)(cid:13)(cid:13) H W ( E ) ∗ < ∞ . Lemma D.7.
Let K ∈ N . For t > , we have r spec (cid:0) Op (cid:0) Φ t (cid:1) T ≥ ( K +1) (cid:1) = e tγ + K +1 Proof.
First, from (D.10) and because
Op (Φ t ) T k is finite rank, we have that r ess (cid:0) Op (cid:0) Φ t (cid:1) T ≥ ( K +1) (cid:1) = e − Λ t , i.e. Op (Φ t ) T ≥ ( K +1) has discrete spectrum outside the disk of radius e − Λ t . From Taylor-Lagrange remainder formula, for any u, v ∈ S ( E ) , we have (cid:12)(cid:12) (cid:104) v | Op (cid:0) Φ t (cid:1) T ≥ ( K +1) u (cid:105) L (cid:12)(cid:12) ≤ Ce tγ + K +1 (cid:107) u (cid:107) C K +1 (cid:13)(cid:13) x K +1 v (cid:13)(cid:13) L . (D.11)i.e. correlation functions decay faster than e tγ + K +1 . We deduce that Op (Φ t ) T ≥ ( K +1) has nospectrum on | z | > e tγ + K +1 .Finally Lemma D.7 is equivalent to Proposition D.3. D.3 Discrete Ruelle spectrum in a simple toy model
We give here a very simple example that illustrates Section D and some mechanisms that playa role in this paper. On R x let us consider the vector field X = − x∂ x (D.12)that gives a flow φ t ( x ) = e − t x that is contracting for t > , hence we think E s = R x as astable direction. In order to study the pull back operator (cid:0) e tX u (cid:1) ( x ) = u ( φ t ( x )) = u ( e − t x ) onfunctions u ∈ S ( R ) , we consider the induced (pull-back) flow on the cotangent space T ∗ E s = T ∗ R given by ˜ φ t ( x, ξ ) = (cid:0) e t x, e − t ξ (cid:1) . We observe that for any k ∈ N , the monomial x k is an eigenfunction of X (hence of e tX ) witheigenvalue ( − k ) : Xx k = ( − k ) x k x ∈ R ≡ E s X = − x∂ x Re( z )Im( z ) − R + 1 / o (1) − R + 1 / o (1) − R → + ∞ − k, ... Figure D.1: In blue, discrete spectrum of the operator (D.12), X = − x∂ x in H W ( R ) . In brown,the essential spectrum moves far away if R → + ∞ .and the spectral projector is Π k = x k (cid:104) k ! δ ( k ) | . (cid:105) , where δ ( k ) is the k -th derivative of the Dirac distribution (notice indeed from (cid:104) k ! δ ( k ) | x k (cid:48) (cid:105) = δ k = k (cid:48) that (cid:0) δ ( k ) (cid:1) k forms a dual basis to (cid:0) x k (cid:48) (cid:1) k (cid:48) ).However x k , δ ( k ) do not belong to L ( R ) . In the Hilbert space L ( R ) one has X † = − X +1 ⇔ (cid:0) X − (cid:1) † = − (cid:0) X − (cid:1) that implies that the spectrum of X is on the vertical axis + i R withsome essential spectrum. Some better Hilbert space H W ( R ) is constructed as in (D.4). For < h (cid:28) , R ≥ we define the “escape function” or “Lyapounov function” for ˜ φ t that is W ( x, ξ ) = (cid:68) √ hξ (cid:69) R (cid:68) √ hx (cid:69) R . Indeed, it satisfies W◦ ˜ φ t W ≤ C everywhere and W◦ ˜ φ t W ≤ e − Rt far from the “trapped set” or “nonwandering set” (0 , . In the Hilbert space H W ( R ) := Op (cid:0) W − (cid:1) L ( R ) , for − R + < − k , we have that x k ∈ H W ( R ) and (cid:107) Π k (cid:107) H W ≤ C is a bounded operator. Seefigure D.1.For large time t (cid:29) , the emerging behavior of e tX u is given by e tX u = Π u + O (cid:0) e − t (cid:1) = u (0) + O (cid:0) e − t (cid:1) i.e. projection onto the constant function, the remainder is in operator norm. References [1] V.I. Arnold.
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