PREQUANTUMTRANSFEROPERATORFORSYMPLECTICANOSOVDIFFEOMORPHISM 3752015

[smfthm]Proposition

SOCIÉTÉ MATHÉMATIQUE DE FRANCE

ASTÉRISQUE

375

2015

PREQUANTUM TRANSFER OPERATOR

FOR SYMPLECTIC ANOSOV DIFFEOMORPHISM

Frédéric FAURE and Masato TSUJII

Numéro 375, 2015

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ISSN 0303-1179 ISBN 978-2-85629-823-7

SOCIÉTÉ MATHÉMATIQUE DE FRANCE

ASTÉRISQUE

375

2015

PREQUANTUM TRANSFER OPERATOR

FOR SYMPLECTIC ANOSOV DIFFEOMORPHISM

Frédéric FAURE and Masato TSUJII

100 rue des Maths, BP74

FR-38402 St Martin d’Hères, France [email protected]

http://www-fourier.ujf-grenoble.fr/~faure

Masato Tsujii

Department of Mathematics, Kyushu University Moto-oka 744, Nishi-ku

Fukuoka, 819-0395, Japan [email protected]

Classification mathématique par sujet(2000). —37D20, 37D35, 37C30, 81Q20, 81Q50.

Mots-clefs. — Opérateurs de transfert, résonances de Ruelle, décroissance des corrélations, analyse semi-classique.

SYMPLECTIC ANOSOV DIFFEOMORPHISM

by Frédéric FAURE and Masato TSUJII

Abstract. — We define the prequantization of a symplectic Anosov diffeomorphism f : M → M as a U(1) extension of the diffeomorphism f preserving a connection related to the symplectic structure on M. We study the spectral properties of the associated transfer operator with a given potential V ∈C^∞(M), calledprequantum transfer operator. This is a model of transfer operators for geodesic flows on negatively curved manifolds (or contact Anosov flows).

We restrict the prequantum transfer operator to theN-th Fourier mode with re- spect to theU(1) action and investigate the spectral property in the limitN → ∞, regarding the transfer operator as a Fourier integral operator and using semi-classical analysis. In the main result, under some pinching conditions, we show a “band struc- ture” of the spectrum, that is, the spectrum is contained in a few separated annuli and a disk concentric at the origin.

We show that, with the special (Hölder continuous) potentialV0= ¹₂log|detDf|E_u|, whereEu is the unstable subspace, the outermost annulus is the unit circle and sep- arated from the other parts. For this, we use an extension of the transfer operator to the Grassmanian bundle. Using Atiyah-Bott trace formula, we establish the Gutzwiller trace formula with exponentially small reminder for large time. We show also that, for a potential V such that the outermost annulus is separated from the other parts, most of the eigenvalues in the outermost annulus concentrate on a circle of radiusexp (hV −V0i)whereh.idenotes the spatial average on M. The number of the eigenvalues in the outermost annulus satisfies a Weyl law, that is, N^dVol (M)in the leading order withd= ¹₂dimM.

We develop a semiclassical calculus associated to the prequantum operator by defin- ing quantization of observablesOp_N(ψ) as the spectral projection of multiplication operator byψto this outer annulus. We obtain that the semiclassical Egorov formula of quantum transport is exact. The correlation functions defined by the classical trans- fer operator are governed for large time by the restriction to the outer annulus that we call the quantum operator. We interpret these results from a physical point of view as the emergence of quantum dynamics in the classical correlation functions for large

time. We compare these results with standard quantization (geometric quantization) in quantum chaos.

Résumé(Opérateur de transfert préquantique pour un difféomorphisme symplectique Ano- sov.)— On définit la prequantification d’un difféomorphisme symplectique et Anosov f : M → M comme étant une extension U(1) de f qui préserve une connexion dont la courbure est la forme symplectique sur M. On étudie les propriétés spec- trales de l’opérateur de transfert associé avec un potentielV ∈C^∞(M). On l’appelle l’opérateur de transfert préquantique. C’est un modèle pour les opérateurs associés au flot géodésique sur les variétés de courbure négative (ou les flots Anosov de contact).

On restreint l’opérateur de transfert au mode de FourierN par rapport à l’action de U(1) et on étudie ses propriétés spectrales dans la limite N → ∞, en considé- rant l’opérateur de transfert comme un opérateur intégral de Fourier et en utilisant l’analyse semi-classique. Le résultat principal, avec des conditions de pincements, montre que le spectre a une “structure en bandes”, c’est à dire qu’il est contenu dans des anneaux séparés et concentriques à l’origine.

On montre qu’avec le potentiel spécial (et seulement Hölder continu) V0 =

1

2log|detDf|_Eu|, où E_u est l’espace instable, la bande la plus externe est le cercle unité et est séparé des autres bandes par un gap uniforme enN. Pour cela on utilise une extension de l’opérateur de transfert au fibré de Grassmann. En utilisant la formule des traces de Atiyah-Bott, on établit une formule des traces de Gutzwiller avec un reste décroissant exponentiellement vite en temps longs. Pour un potentiel V général, et pour N → ∞, la plupart des valeurs propres de la bande externe se concentrent et s’équidistribuent sur le cercle de rayon exp (hV −V₀i_M) où h.i_M signifie la moyenne sur M. Le nombre de valeurs propres sur la bandes externe satisfait la loi de Weyl c’est à direN^dVol (M)à l’ordre dominant, avecd=¹₂dimM. On développe un calcul semi-classique associé à l’opérateur préquantique en dé- finissant une quantification des observables Op_N(ψ) comme étant la projection de l’opérateur multiplication parψ sur l’espace spectral de la bande extérieure. On ob- tient une formule de transport de “type Egorov” qui est exacte. Les fonctions de corrélations définies par l’opérateur de transfert sont gouvernées en temps long par l’opérateur restreint à la bande externe que l’on appelle opérateur quantique. On in- terprète ces résultats d’un point de vue physique comme l’émergence de la dynamique quantique dans les fonctions de corrélations classiques en temps longs. On compare ces résultats avec la quantification géométrique (standard) en chaos quantique.

1. Introduction and results . . . 1

1.1. Introduction . . . 1

1.2. Definitions . . . 5

1.3. Results on the spectrum of the prequantum operatorFˆN . . . 11

1.4. Spectral results for extended models on the Grassmanian bundle . . . 17

1.5. Gutzwiller trace formula . . . 22

1.6. Dynamical correlation functions and emergence of quantum dynamics 25 1.7. Semiclassical calculus on the quantum space . . . 29

2. Semiclassical description of the prequantum operator FˆN . . . 35

2.1. The associated canonical mapF :T^∗M →T^∗M . . . 35

2.2. The trapped setK . . . 41

2.3. Microlocal description near the trapped set. Sketch of proof of the main theorem . . . 47

3. Resonances of linear expanding maps . . . 51

3.1. Bargmann transform . . . 51

3.2. Action of linear transforms . . . 55

3.3. The weighted spacesL² R^2D,(W_~^r)² . . . 58

3.4. Spectrum of transfer operator for linear expanding map . . . 60

3.5. Proof of Claim (2) in Proposition 3.4.6 . . . 65

4. Resonances of hyperbolic linear prequantum maps . . . 73

4.1. Prequantum transfer operator onR^2d . . . 73

4.2. Prequantum transfer operator for a symplectic affine map onR^2d . . . 74

4.3. Prequantum transfer operator for a linear hyperbolic map . . . 78

4.4. Anisotropic Sobolev space . . . 79

4.5. The structure of prequantum transfer operator for hyperbolic symplectic linear map . . . 81

4.6. An affine transformation group

A

. . . 84

5. Nonlinear prequantum maps on R^2d . . . 85

5.1. Truncation operations in the real space . . . 86

5.2. Decomposition of the projection operatort^(k)

~ into localized rank one

projectors and estimates on trace norm . . . 90

5.3. Truncation operations in the phase space . . . 93

5.4. Prequantum transfer operators for non-linear transformations close to the identity map . . . 95

6. Band structure of the spectrum ofFˆN. (Proof of Theorems 1.3.4 and 1.7.5) . . . 105

6.1. Structure of the prequantum transfer operatorFˆN . . . 105

6.2. Local charts onM and local trivialization of the bundleP . . . 109

6.3. The prequantum transfer operator decomposed on local charts . . . 112

6.4. The anisotropic Sobolev spaces . . . 115

6.5. The main propositions . . . 118

6.6. Proof of Theorem 6.1.1 . . . 124

6.7. Proof of Theorem 1.7.5 . . . 128

7. The Grassmann Extension. (Proof of Theorems 1.4.9–1.4.12) . 135 7.1. Discussion about the linear model . . . 136

7.2. Non-linearity . . . 139

7.3. Proof of the main theorems in the setting of Grassmanian extension 144 7.4. Relation between the operatorsFˆN andFeN (and proof of Theorem 1.4.11) . . . 153

8. Consequences of ergodicity. (Proof of Theorem 1.3.11) . . . 159

8.1. Time average and Birkhoff’s ergodic theorem . . . 159

8.2. Proof of concentration of the resonance to the circle|z|=e^hV−V0i . . 162

8.3. Proof of equidistribution of the arguments of the resonances . . . 164

9. Gutzwiller trace formula. (Proof of Th. 1.5.1) . . . 167

9.1. The Atiyah-Bott trace formula . . . 167

9.2. The Gutzwiller Trace formula from the Atiyah-Bott trace formula . . 169

9.3. Restriction to the external band . . . 173

Appendix . . . 181

10. The rough Laplacian and geometric quantization . . . 183

10.1. The covariant derivativeD . . . 183

10.2. The rough Laplacian∆ . . . 186

10.3. Geometric quantization of a symplectic map . . . 190

11. Spectrum of the rough Laplacian in clusters (proof of Theorems 10.2.2 and 10.3.2) . . . 193

11.1. The harmonic oscillator onR^D . . . 193

11.2. The rough Laplacian onR^2d . . . 195

11.3. The cluster structure of the spectrum of the rough Laplacian . . . 196

11.4. Proof of the second part of Theorem 10.2.2 . . . 198

12. Quantum operator and geometric quantization (proof of Theorem 10.3.2) . . . 203

12.1. Expression of the metaplectic correction (proof of Th. 10.3.4) . . . 206

13. Proofs of Theorem 1.2.4 and Lemma 6.6.2 . . . 211

13.1. Proof of Theorem 1.2.4 . . . 211

13.2. Proof of Lemma 6.6.2 . . . 214

Bibliography . . . 217

INTRODUCTION AND RESULTS

1.1. Introduction

We consider a smooth symplectic Anosov diffeomorphismf :M →M on a2d-di- mensional closed symplectic manifold(M, ω)as a standard model of “chaotic” dynam- ical system. Following the geometric quantization procedure introduced by Kostant, Souriau and Kirillov in 1970s’, we consider the prequantum bundleπ:P→M. This is the U(1)-principal bundle overM equipped with a connection whose curvature is (−2πi)·ω. Then we introduce the prequantum mapf˜:P → P as the U(1)-equiv- ariant extension of the map f preserving the connection. The prequantum map f˜ thus defined is known to be exponentially mixing⁽¹⁾, that is, any smooth probability density which evolves under the iteration off˜converges weakly towards the uniform equilibrium distribution onP and the speed of convergence is exponentially fast if it is measured against smooth observables. We study the fluctuations in this convergence to the equilibrium by investigating spectral properties of the transfer operatorFˆasso- ciated to the prequantum mapf˜simply defined asF uˆ :=e^V◦π·u◦f˜⁻¹foru∈C^∞(P) and withV ∈C^∞(M)a given function called potential. Following the approach taken by David Ruelle in his study of expanding dynamical systems[64], we first show that the transfer operator displays discrete spectrum, which is sometimes called Ruelle- Pollicott resonances. Precisely we consider the restrictionFˆN of the transfer operator Fˆ to the N-th Fourier mode with respect to the U(1) action on P and show that its natural extension to appropriate generalized Sobolev spaces of distributions has discrete spectrum. This result concerning discrete spectrum is already known in the preceding works [65],[13, 37, 38],[7, Theorem 1.1],[29, Theorem 1] and will be re- called in Theorem 1.3.1. In this paper we are mainly concerned with the limitN → ∞ of high Fourier modes. We will use the standard notation of semiclassical analysis and put throughout this paper:

~:= 1 2πN.

(1)Exponential mixing of the mapf˜is already known [21] but is also a direct consequence of results presented in this paper.

The new result of this paper is in Theorem 1.3.4, where we show that the spectrum ofFˆN has a particular band structure: for everyN large enough, there is an annulus that contains finitely many (but increasing to infinity as N → ∞) eigenvalues; they are separated from the rest of the internal spectrum by a gap under some pinching conditions. The pinching conditions involve the joint fluctuations of the hyperbolic exponents together with the potential function V. These results are illustrated in Figure 1.3.1. We denote by

F

ˆ_~ ^:

H

_~ ^→

H

_~ the spectral restriction of the prequan- tum transfer operator FˆN on its external annulus. The band structure means that the convergence to the equilibrium mentioned above, restricted to the N-th Fourier mode, is described by this finite rank operator

F

ˆ~up to relatively small exponentially decaying errors. We show in Theorem 1.3.8 that the dimension of

H

~is proportional toN^dasymptotically as⁽²⁾N → ∞. These results are generalizations of the results in [25] for the linear Arnold cat map to the case of general non-linear symplectic Anosov diffeomorphisms.

1.1.1. Motivations of the study. — From the construction above, the prequan- tum map f˜: P → P is partially hyperbolic, that is, hyperbolic in the directions transverse to the fibers but is neutral (because of equivariance) in the direction of the fibers. This is illustrated on Figure 1.2.2. Also note that f˜preserves the con- nection one form on the prequantum bundle P which is a contact form on P. (See Remark 1.2.7.) These properties of the prequantum map are very similar to those of the time-t-map of the geodesic flow φt:T₁^∗

M

^→T1^∗

M

on a closed Riemannian man- ifold

M

with negative curvature, acting on the unit cotangent bundle T₁^∗

M

^{. In the}

latter case the time-t-map of the geodesic flow is partially hyperbolic and preserves the canonical Liouville contact one form ξdxonT₁^∗

M

. (See [52, 71, 72, 30]). With this point of view, the prequantum transfer operator can be considered as a model of the transfer operators for the geodesic flows on negatively curved manifolds. One of our objective behind the present work is to show some band structure of the spec- trum for the case of geodesic flow and extend other results presented in this paper to that case [34, 31, 32]. In the special case of manifolds with constant curvature, such a band structure is readily observed from the classical theorem of Selberg on zeta functions [66] (see [24] for a detailed study in higher dimensions also).

Another motivation already discussed in [25] in a special case is the following obser- vation: The finite rank operator ˆ

F

~which describes the long time classical correlation functions of the map f˜has the properties of a “quantum map” i.e., a “quantization of f” but with additional interesting properties. It satisfies the Gutzwiller trace for- mula with an error term which decreases exponentially fast in large time, an exact Egorov theorem, etc. For these reasons we call

F

ˆ~:

H

~→

H

~the “natural quantum operator” because we show in Section 1.5.1 that its spectrum is determined from the set of periodic orbits off. Surprisingly this “quantization” or quantum behavior, ap- pears here dynamically (after long time) in the classical correlation functions of the

(2)The precise value ofdimH_~is given by an index formula of Atiyah-Singer in Th. 1.3.8.

“classical” map f˜: the finite dimensional “quantum space”

H

~ in which ˆ

F

~ acts is defined from the dynamics. There are many open questions in “quantum chaos” for example related to “unique quantum ergodicity” or “random matrix theory” [60, 27].

These questions can be posed for the family of quantum operators ˆ

F

~

~

considered here and maybe their special properties with respect to the dynamics may help. In Section 10.3 we compare the operator

F

ˆ~ with more usual quantum operators that are obtained from geometric (Toeplitz) quantization of the mapf after the choice of an almost complex polarization [54].

1.1.2. Semiclassical approach. — The general method that we use to obtain the main results is semiclassical analysis⁽³⁾. We regard the prequantum transfer opera- tor as a Fourier Integral Operator (FIO), which means that we consider its action on wave packets in the high frequency limit N → ∞. From the general idea in semiclassical analysis, this action is effectively described by the associated canoni- cal map (Df^∗)⁻¹ on the cotangent space T^∗M equipped with the symplectic struc- tureΩ =dx∧dζ+π^∗ω(wheredx∧dζ stands for the canonical symplectic structure on T^∗M and π^∗ω is the pull-back of ω on T^∗M). For the action of the canonical map(Df^∗)⁻¹, the non-wandering set is the zero sectionK⊂T^∗M and is called the trapped set. The additional term π^∗ω in Ωmakes K a symplectic submanifold. The trapped set is therefore symplectic and normally hyperbolic. We will see that these facts are the core of our argument and give the band structure of the spectrum in the main theorem. The explicit use of semiclassical calculus (Egorov theorem etc) for Ruelle resonances of hyperbolic dynamics has been introduced in [29, 30] (after some suggestions in [28] with the stationary phase formula). The semiclassical study of symplectic and normally hyperbolic trapped set is also done recently in [61] where S. Nonnenmacher and M. Zworski show a spectral gap (for more general models) and in [23] where S. Dyatlov shows a band structure for the resonances of waves around black holes.

1.1.3. Organization of the paper. — In Section 1.2 we define precisely the pre- quantum mapf˜and the prequantum transfer operatorFˆ that are associated to the symplectic Anosov mapf. In Section 1.3 we present the main results concerning the discrete spectrum of FˆN (after Fourier decomposition of Fˆ on Fourier component N = 1/(2π~)∈Z) and acting on a Hilbert space

H

^r_~ called the anisotropic Sobolev space. In Definition 1.3.6 we define the quantum operator ˆ

F

~ as the spectral re- striction of the operatorFˆN on its external band. The associated spectral projector is denoted by Π_~. In Section 1.4 we show that with the special choice of potential V₀= ¹₂log|detDf|Eu|, whereE_u is the unstable subspace, the external band of res- onances concentrates on the unit circle in the limit~→0, see Figure 1.4.2. However the difficulty is that this potential V0 is only Hölder continuous on M and for that reason we need to consider the extension of the transfer operator to the Grassmanian

(3)The lecture notes [33] give a partial overview of this approach for hyperbolic dynamics.

bundle, giving an equivalent but smooth potential V˜₀. In Section 1.5 we show that the quantum operator ˆ

F

ⁿ_~ satisfies the Gutzwiller trace formula with an error that decays exponentially fast as n → ∞. We discuss the fact that this property deter- mines the spectrum of ˆ

F

~and deduce that the family of operators

ˆ

F

~

~

is a kind of

“natural (or intrinsic) quantization” of the Anosov symplectic mapf. In Section 1.6, we explore the properties of this quantum operator

F

ˆ~: in Theorem 1.6.3 it is shown that ˆ

F

~ describes the exponential decay of correlations of the prequantum Anosov map f˜. In Section 1.7 we show in which sense the quantum operator

F

ˆ_~ ^{is a kind} of “quantum map”: it satisfies an exact Egorov formula with respect to an algebra of quantum observables Op_~(ψ). For this, we define a new kind of quantization proce- dure Op_~ : ψ ∈ C^∞(M) → Op_~(ψ) ∈ End (

H

_~⁾ which satisfies most of the usual

“axioms of quantization”. In particular the spectral projector on the external band is Π_~ = Op_~(1). In Theorem 1.7.5, Op_~(ψ)is expressed as an integral over x∈ M of ψ(x)·π_x where π_x is a rank one projector over a “localized wave packet”⁽⁴⁾ at position x ∈ M. Subsequent sections contain the proofs. In Chapter 2, we present the main ideas of semiclassical analysis used in the proofs. The global strategy of the proof is explained in Section 2.3. In Chapter 3, we study the resonance spectrum of linear expanding maps, which is interesting by itself. In Chapter 4, we study the spectrum of hyperbolic linear prequantum maps onR^2d which can be considered as a local and linearized version of the global model onM. In Chapter 5 we study non linearities in order to show that they can be neglected in the limit~→0and that the global model can be understood as a patch of local linearized models. Chapters 6 to 9 are devoted to proofs of the main theorems. In Chapter 10 and 11 in the appendix, we consider the usual geometric (Toeplitz) quantization of the symplectic mapf and compare it with the quantum operator ˆ

F

~ or “natural quantization” that we have introduced. We show that both quantizations coincide up to a small error in the limit

~→0, provided that we put appropriate correction terms.

1.1.4. Acknowledgments. — F. Faure would like to thank Yves Colin de Verdière, Louis Funar, Sébastien Gouëzel, Colin Guillarmou, Malik Mezzadri, Johannes Sjös- trand for interesting discussions related to this work. M. Tsujii would like to thank Carlangelo Liverani, Michael Benedicks, Setsuro Fujiie and Shu Nakamura for encour- agements to this work. Both of the authors would like to express their gratitude to Viviane Baladi who took very careful and thoughtful care of this paper as an editor and also to the anonymous referees who gave detailed, critical and constructive com- ments to the first version of this paper, which is indispensable to improvement of the manuscript. During the period of this research project, M. Tsujii has been supported by Grant-in-Aid for Scientific Research (B) (No.22340035) from Japan Society for the Promotion of Science. F. Faure has been supported by Agence Nationale de la Recherche under the grants JC05_52556 and ANR-08-BLAN-0228-01.

(4)This localization property is with respect to the anisotropic Sobolev space H^r_~.

1.2. Definitions

1.2.1. Symplectic Anosov map. — LetM be aC^∞closed connected symplectic manifold of dimension2dwith symplectic two formω. Letf :M →M be aC^∞sym- plectic Anosov diffeomorphism, i.e., aC^∞Anosov diffeomorphism such thatf^∗ω=ω.

Throughout the paper we will denote dVolω:= 1

d!ω^∧d= 1

d!ω∧ · · · ∧ω

| {z }

d

the symplectic volume form onM. We recall the definition of an Anosov diffeomor- phism:

Definition 1.2.1. — [48, p.263] A diffeomorphism f : M →M is said to be Anosov if there exists a continuous Riemannian metric g on M, an f-invariant continuous decomposition of T M,

(1.2.1) TxM =Eu(x)⊕Es(x), ∀x∈M and a constantλ >1, such that, for anyx∈M, hold

|D_xf(v_s)|_g < 1

λ|v_s|_g ∀v_s∈E_s(x), and (1.2.2)

D_xf⁻¹(v_u)

_g < 1

λ|v_u|_g ∀v_u∈E_u(x).

The subbundleEs(resp.Eu) in which f is uniformly contracting (resp. expanding) is called the stable (resp. unstable) sub-bundle. See Figure 1.2.1.

f

x

Es(x)

M f(x)

Eu(x)

Figure 1.2.1. Anosov mapf from Definition 1.2.1.

Remark 1.2.2. — (1) The subspacesEu(x)andEs(x)do not depend smoothly on the pointxin general. However it is known that they are Hölder continuous inxwith some Hölder exponent [48, Section 19]. In what follows, we assume that the Hölder exponent is

(1.2.3) 0< β≤1.

The subspacesE_u(x)andE_s(x)are Lagrangian⁽⁵⁾linear subspace ofT_xM and both have dimensiond.

(2) TheArnold cat map[3] is a simple example of a symplectic Anosov diffeomor- phism on the torusx= (q, p)∈T²=R²/Z²,

(1.2.4) f0

q p

!

= 2 1

1 1

! q p

!

mod Z².

It preserves the symplectic formω=dq∧dp. Ifh:M →M is a diffeomorphism close enough to identity in theC¹ norm and preserves the symplectic formω, theperturbed cat map

(1.2.5) f(x) :=h(f₀(x))

is also a (probably non-linear) symplectic Anosov diffeomorphism[48, p.266].

Similarly, we get examples of symplectic Anosov diffeomorphisms onT^2dfrom any symplectic linear mapf₀∈Sp_2d(Z)with no eigenvalues on the unit circle.

1.2.2. The prequantum bundle and the lift mapf˜. — A prequantum bundle is aU(1)-principal bundleP equipped with a specific connection. In a few paragraphs below, we recall the definition of aU(1)-principal bundle and that of a connection on it. For the detailed account, we refer [75]. The one-dimensional unitary groupU(1)is the multiplicative group of complex numbers of the forme^iθ, θ∈R. AU(1)-principal bundleP overM is a manifold with a free action ofU(1), written

(1.2.6) p∈P → e^iθp

∈P,

such that the quotient space isM =P/U(1). We writeπ:P →M for the projection map. From the definition, theU(1)-principal bundleP has a local product structure over M: there exist a finite cover of M by simply connected open subsets Uα ⊂M, α∈I, and smooth sectionsτ_α:U_α→P on each ofU_α, called alocal smooth section; Alocal trivialization ofP overUαis defined as the diffeomorphism

(1.2.7) Tα:

(U_α×U(1) →π⁻¹(U_α) x, e^iθ

→e^iθτα(x).

AconnectiononP is a differential one formA∈C^∞ P,Λ¹⊗(iR)

onP with values in the Lie algebrau(1) =iRwhich is invariant by the action ofU(1)and normalized so that

(1.2.8) A

∂

∂θ

=i

(5) To prove that Es(x) is Lagrangian, let u, v ∈ Es(x); we have ω(u, v) = ω(Dxfⁿ(u), Dxfⁿ(v)) →

n→+∞0. Similarly forEu. SoEsandEuare isotropic subspaces,Eu⊕Es= T M, hence they are Lagrangian.

where _∂θ^∂ denotes the vector field on P generating the action ofU(1). Consequently the pull-back of the connectionAonP by the trivialization map (1.2.7) is written as

(1.2.9) T_α^∗A=idθ−i2πηα

where ηα ∈ C^∞ Uα,Λ¹

is a one-form on Uα which depends on the choice of the local sectionτα. A different local sectionτβ:Uβ →P withUαT

Uβ 6=∅is written as τ_β=e^iχτ_αwith a functionχ:U_αTU_β→Rand hence the connectionApulled-back by the corresponding trivializationTβ is written as (1.2.9) but with⁽⁶⁾

(1.2.10) ηβ=ηα− 1

2πdχ on Uα∩Uβ.

The curvature of the connection A is the two form Θ = dA on P. In the local trivialization (1.2.7), we haveT_α^∗Θ =−i2π(dηα)and (1.2.10) implies thatdηα=dηβ. Therefore the curvature two form is written as

Θ =−i(2π) (π^∗ω)˜

whereω˜ =dηα is a closed two form onM independent of the trivialization.

Since there is a given symplectic two form ω on M in our setting, we naturally require below in (1.2.13) that the two form ω˜ coincides with the symplectic formω and then

(1.2.11) ω=dηα.

For the construction of the prequantum bundle and prequantum transfer operator, we will need the following two assumptions:

— Assumption 1:The cohomology class[ω]∈H²(M,R)represented by the sym- plectic formω is integral, that is,

(1.2.12) [ω]∈H²(M,Z).

— Assumption 2:The integral homology groupH₁(M,Z)has no torsion part and 1is not an eigenvalue of the linear map f_∗ :H1(M,R)→H1(M,R)induced byf :M →M.

Remark 1.2.3. — The second assumption above is not restrictive and may not be necessary. In fact Assumption 2 is conjectured to be true in general. For the case M =T^2d, this is always satisfied [36, 55, 9].

Theorem 1.2.4. — Under Assumption 1 above, there exists a U(1)-principal bundle π :P → M and a connectionA ∈C^∞ P,Λ¹⊗(iR)

on P such that the curvature two formΘ =dAsatisfies

(1.2.13) Θ =−i(2π) (π^∗ω).

(6)Proof:p∈π⁻¹ Uα∩Uβ

is writtenp=e^iθτα(x) =e^iθ0τβ(x)henceθ=θ⁰+χ. ThenT_α^∗A= idθ−i2πηα=idθ⁰−i2πηβ impliesdχ−2πηα=−2πη_β henceηβ=ηα− ¹

2πdχ.

Under Assumption 2 in addition, there exists a connection A so that there exists an equivariant lift f˜:P →P of the mapf :M →M preserving the connectionA, that is:

(1.2.14)

π◦f˜

(p) = (f◦π) (p), ∀p∈P : ˜f is a lift off. (1.2.15)

f e˜ ^iθp

=e^iθf˜(p), ∀p∈P,∀θ∈R : ˜f is equivariant w.r.t. the U(1)action.

(1.2.16) f˜^∗A=A : ˜f preserves the connection A.

(See Figure 1.2.2.)

The proof of Theorem 1.2.4 is given in Section 13.1.

Definition 1.2.5. — The U(1)-principal bundle π : P → M equipped with the con- nection A ∈ C^∞ P,Λ¹⊗(iR)

satisfying (1.2.13) is called the prequantum bundle over the symplectic manifold (M, ω). The map f˜:P →P satisfying the conditions (1.2.14),(1.2.15) and (1.2.16) is called theprequantum map.

P

0

x

π

Ker(A)

0

f

M

dA≡ −i(2π)dq∧dp θ

p

A≡ −i2πqdp+idθ

=−i(2π)ω p

f˜ 2π

q

Figure 1.2.2. A picture of the prequantum bundleP →M in the case of M = T², with connection one form A and the prequantum map f˜: P →P which is a lift off :M →M. A fiberPx≡U(1)overx∈M is represented here as a segment[0,2π). The plane at a pointprepresents the horizontal spaceHpP = Ker (Ap) which is preserved by f. These planes˜ form a non integrable distribution with curvature given by the symplectic formω.

Remark 1.2.6. — ”Uniqueness of the prequantum bundle and the prequantum map”.

Under Assumption 1 and condition (1.2.13), the prequantum bundle P exists and is unique (as a smooth manifold) because it is determined by its first Chern class c1(P) = [ω]∈ H² M,Z²

, see [74]. However the connection A on the prequantum bundleP is not unique. But under Assumption 2 and conditions (1.2.13) and (1.2.16), we explicitly show in the proof of the theorem above, in Section 13.1, that there are

finitely many connections A up to bundle isomorphisms onP, and they differ from each other by a flat connection. Once the prequantum bundleP and the connection A on it are given, the lifted map f˜is unique up to a global phase e^iθ0 ∈U(1); i.e., another map ˜g satisfying the conditions (2) of Theorem 1.2.4 is given by g˜=e^iθ0f˜ for somee^iθ0 ∈U(1)(and conversely).

Remark 1.2.7. — Letα:= _2πⁱ A. Then the differential(2d+ 1)-form

(1.2.17) µP := 1

d!α∧(dα)^∧d

is a non-degenerate volume form onP. This means thatαis acontact one formonP preserved byf˜.

Suppose thatx∈M is a periodic point of the mapf with period n∈N,n≥1, i.ex=fⁿ(x). Then ifp∈π⁻¹(x)is in the fiber, the condition (1.2.14) implies that f˜ⁿ(p)∈π⁻¹(x)lies in the same fiber and therefore differs frompby a phase:

(1.2.18) f˜ⁿ(p) =e^i2πSn,xp

with Sn,x ∈ R/Z called the action of the periodic point x, see Figure 1.2.3. These actions are important quantities in semiclassical analysis and will appear in the Gutzwiller trace formula in (1.5.2).

f˜

p f˜

P

M f˜ⁿ(p)

e^i2πSn,x

f

f x=fⁿ(x)

Figure 1.2.3. Action of a periodic pointx=fⁿ(x)defined in (1.2.18).

1.2.3. The prequantum transfer operator Fˆ and the reduced operatorFˆ_N.

— As usual in dynamical system theory, we consider the transfer operator associated to the prequantum mapf˜:

Definition 1.2.8. — LetV ∈C^∞(M)be a real-valued smooth function, calledpoten- tial. Theprequantum transfer operator is defined as

(1.2.19) Fˆ :

(C^∞(P) →C^∞(P) u →Fˆ(u) =e^V◦π

u◦f˜⁻¹

whereV ◦π∈C^∞(P)is the functionV lifted on P.

Remark 1.2.9. — The fact that f˜⁻¹ appears instead of f˜in (1.2.19) is a matter of choice. In our choice,f˜maps the support of uto that ofF u.ˆ

From the equivariance property (1.2.15), the prequantum transfer operator com- mutes with the action of U(1)on functions on P and therefore is naturally decom- posed into each Fourier mode with respect to theU(1)action:

Definition 1.2.10. — For a given N ∈ Z, we consider the space of functions in the N-th Fourier mode

(1.2.20) C_N^∞(P) :=

u∈C^∞(P) | ∀p∈P,∀θ∈R, u e^iθp

=e^{iN θ}u(p) . The prequantum transfer operatorFˆ restricted toC_N^∞(P)is denoted by:

(1.2.21) FˆN := ˆF/C_N^∞(P): C_N^∞(P)→C_N^∞(P).

Remark 1.2.11. — The complex conjugation mapsC_N^∞(P)toC_−N^∞ (P)and commutes withFˆ. It is therefore enough to studyFˆN withN ≥0.

Remark 1.2.12. — The space of equivariant functionsC_N^∞(P)defined in (1.2.20) can be identified with the space of smooth sections of an associated Hermitian complex line bundleL^⊗N overM (i.e., theN tensor power of a line bundleL→M) equipped with a covariant derivative D. This line bundle is called theprequantum line bundle.

We have

C_N^∞(P)∼=C^∞ M, L^⊗N .

See [69, p.502, eq.(6.1)]. In order to simplify the presentation we will not use this identification in this paper although it will be present implicitly. Notice however that most of references about geometric quantization use the “line bundle terminology”

[75].

In this paper the main object of study is the spectrum of the operatorFˆ_N, (1.2.21), in the limitN → ∞. ForN >0, we set

(1.2.22) ~= 1

2πN.

This new variable~ is in one-to-one correspondence toN, and~→+0as N → ∞.

We introduce it for convenience in referring some argument in semi-classical analysis where~is regarded as the Plank’s constant and the limit ~→+0is considered.

Remark 1.2.13. — In the following, we will confuse the parameters N and ~ in the notation. For instance, the operatorFˆN will be writtenFˆ_~sometimes.

1.3. Results on the spectrum of the prequantum operator Fˆ_N

The following theorem has been obtained essentially in the works of Rugh [65], Liverani et al.[13, 37], Baladi et al.[7, Theorem 1.1], Faure et al. [29, Theorem 1].

The method employed in the present paper is close to the semiclassical approach given in [29, Theorem 1]. Before giving the theorem, let us mention that the transfer operator FˆN has been defined on the space of smooth functions C_N^∞(P)and can be extended by duality to the distributions space

D

⁰N(P)(i.e., the dual space ofC_N^∞(P)).

Theorem 1.3.1(Discrete spectrum of prequantum transfer operators). — For any N ∈ Z, there exists a family of Hilbert spaces

H

^rN(P) for arbitrarily large r > 0, called anisotropic Sobolev spaces, satisfying

C_N^∞(P)⊂

H

^rN(P)⊂

D

⁰N(P), such that the operatorFˆ_N extends to a bounded operator

FˆN :

H

^rN(P)→

H

^rN(P), and its essential spectral radiusress

FˆN

is bounded above byεr:=λ^−rmaxx e^V(x) where λ is the constant in (1.2.2). (Notice that εr → 0 as r → +∞.) Hence the spectrum of Fˆ_N in |z| > ε_r consists of discrete eigenvalues with finite multiplicity.

Those discrete eigenvalues ofFˆN (and their associated eigenspaces) are independent on the choice ofrin

H

^rN(P)and are therefore intrinsic to the Anosov diffeomorphism f and the potential function V. (See Remark 1.3.2 below.) The discrete eigenvalues Res

FˆN

:={λi}_i⊂C^∗ are called Ruelle-Pollicott resonances. The definition of the space

H

^rN(P) depends on the Anosov diffeomorphismf but does not depend on the potential functionV.

Remark 1.3.2. — Independence of the discrete eigenvalues from the choice of the func- tion space

H

^rN(P) (or r) is stated more precisely as follows. Let r⁰ > r. From the theorem above, we observe the discrete eigenvalues of FˆN acting on

H

^rN⁰(P) and

H

^rN(P) respectively in |z| > εr⁰ and in |z| > εr. Such sets of discrete eigenvalues coincides (up to multiplicity) in the intersection|z| ≥max{εr⁰, ε_r}=ε_r. By letting r large, we may find more and more eigenvalues in smaller neighborhood of the origin.

The Ruelle-Pollicott resonancesRes FˆN

:={λi}_i⊂C^∗are the discrete eigenvalues thus found. See [8, Appendix],[29, Cor. 1.3] for more general argument about this independence ofRes

FˆN

from the choice of the function spaces.

The main new result of this paper is the next theorem. It is illustrated in Fig- ure 1.3.1. Let us prepare some notations. For a linear invertible map L between normed linear spaces, we will use the notation

(1.3.1) kLk_max:=kLk, kLk_min:=

L⁻¹

−1

e^hDi

A1 A0

r⁺₀ r⁻₀

r⁻₁ r⁺₁ εr

Figure 1.3.1. Theorem 1.3.1 shows that the spectrum ofFˆN consists of discrete eigenvalues called Ruelle-Pollicott resonances (the dots on the fig- ure). Theorem 1.3.4 shows that forNlarge enough it is structured in bands (i.e., eigenvalues are in the grey annuli) and that the resolvent is bounded between the bands, uniformly with respect to ~ = 1/(2πN). In the ex- ternal band A0, the number of resonances is given by the Weyl formula

1

(2π~)^dVolω(M)at leading order, Eq.(1.3.12). The precise number is given by the Atiyah-Singer formula in Theorem 1.3.8. Theorem 1.3.11 shows that in this external bandA0almost all the resonances are distributed uniformly on the circle of radiuse^hDiin the limitN→ ∞. The spectral restriction of the operatorFˆN on this external band will be called thequantum operator and denoted ˆF~in Definition 1.3.6. The spectral projector isΠ_~.

wherekLkdenotes the usual operator norm. We define the special potential of refer- ence

(1.3.2) V0(x) := 1

2log

detDf|_Eu(f−1(x))

.

Remark 1.3.3. — The unstable foliation E_u(x) is not smooth in x in general (see Remark 1.2.2(1)) which implies that this function V0(x) is not smooth but Hölder continuous inx.

We then consider the difference

(1.3.3) D:=V −V₀ ∈C^β(M)

which is also a Hölder continuous function onM. This function D will be called the effective damping function. It will appear in many results below. Finally we denote

by

(1.3.4) D_n(x) :=

n

X

j=1

D f^j(x) the Birkhoff sum of the damping function.

Theorem 1.3.4(Band structure of the spectrum ofFˆN). — For any ε > 0, there exist Cε>0 andNε≥1 such that for anyN ≥Nε

(1) the Ruelle-Pollicott resonances ofFˆN is contained in a small neighborhood of the annuli

A

k:={r_k⁻≤ |z| ≤r_k⁺}for integersk≥0:

(1.3.5) Res

Fˆ_N

⊂ [

k≥0

r_k⁻−ε <|z|< r⁺_k +ε

| {z }

ε-neighborhood ofAk

with

r⁻_k := lim

n→∞ inf

x∈M

e^n1Dn(x)kDf_xⁿ|Euk^−k/n_max , (1.3.6)

r⁺_k := lim

n→∞sup

x∈M

e^n1Dn(x)kDf_xⁿ|E_uk^−k/n_min .

(2) Suppose that r_k⁺+ε < r_k−1⁻ −ε for some k ≥ 1. For any z ∈ C such that r_k⁺+ε <|z|< r⁻_k−1−ε(that is, in a gap between the annuli

A

k), the resolvent ofFˆ_N on

H

^rN(P)is controlled uniformly with respect toN:

(1.3.7)

z−FˆN

−1

≤Cε. This is true also forz∈Cwith|z|> r₀⁺+ε.

The proof of Theorem 1.3.4 is given in Chapter 6, where we will provide a more detailed result, Theorem 6.1.1, on the structure of the operatorsFˆN and derive The- orem 1.3.4 as a consequence.

Remark 1.3.5. — (1) The limits on the right hand sides of (1.3.6) exist. In fact, because the sequences

an := inf

x∈M(Dn(x)−klogkDf_xⁿ|E_uk_max), bn:= sup

x∈M

(Dn(x)−klogkDf_xⁿ|E_uk_min) are superadditive (i.e., an+am ≤ an+m) and subadditive (i.e., bn +bm ≥ bn+m) respectively, Fekete’s Lemma guaranties existence of the limitslogr⁻_k = limn→∞an/nandlogr⁺_k = limn→∞bn/n.

(2) SincekDf_xⁿ|_Euk^1/n_max ≥ kDf_xⁿ|_Euk^1/n_min > λ >1, from (1.2.2), we haver_k⁻ ≤r_k⁺ and alsor⁻_k+1< r_k⁻ andr⁺_k+1< r_k⁺ for everyk≥0. However we don’t always have r⁺_k+1 < r⁻_k and therefore the annuli

A

k may intersect each other. In general, only a finite number of annuli

A

k will be distinguished.

(3) From Theorem 1.3.4 it is tempting to take the potential V = V₀ defined in (1.3.2) which would indeed giveD = 0hencer⁺₀ =r⁻₀ = 1 in (1.3.6). In that case the external band

A

0would be the unit circle, separated from the internal band

A

1 by a “spectral gap” r₁⁺ given by

r⁺₁ = lim

n→∞sup

x∈M

kDf_xⁿ|E_uk^−1/n_min

< 1 λ <1.

However Theorem 1.3.4 does not apply in this case because the functionV0

is not smooth inxas required. This is the purpose of the next Section 1.4 to show how to handle this non smooth potentialV₀ and still get spectral results similar to Theorem 1.3.4. For the moment let us remark that ifEe_u⊂T M is a smooth approximation of the unstable sub-bundleEu⊂T M in C⁰ norm and if one chooses the potential:

(1.3.8) V (x) = 1

2log

detDf_x|E˜u(f⁻¹(x))

then one can haver₀⁻, r₀⁺ (arbitrarily) close to one and the annulus

A

0 of the external band gets isolated from the other ones, that is,

A

0T

A

k =∅fork6= 0.

(4) In the simple case of the linear hyperbolic map on the torus T² in (1.2.4) withV (x) = 0, we have r_k⁺ =r⁻_k =λ^−k−12 with λ=Df_0/Eu = ³⁺

√ 5 2 '2.6 (constant), and each annulus

A

k is a circle. In this case Theorem 1.3.4 has been obtained in [25] and is depicted in [25, Figure (1-b)]. In this linear case the eigenvalues exactly lie on those circles

A

k. If one choosesV (x) =V₀(x) =

1 2log

detDf|_Eu(f−1(x))

= ¹₂logλthe external band

A

0 is the unit circle and it is shown in [25] that the Ruelle-Pollicott resonances on the external band coincide with the spectrum of the quantized map usually called the “quantum cat map”.

(5) The estimate (1.3.7) on the resolvent will be useful in Section 1.6 (Theo- rem 1.6.3 and its proof) to express dynamical correlation functions.

(6) By taking the limitn→ ∞in (1.3.6), the values of r^±_k do not depend on the choice of the metric g and volume form on M, although the function V0 in (1.3.2) depends on the choice of volume forms onEu.

From now on, we suppose thatr₁⁺< r₀⁻, i.e., that the external annulus

A

0defined in Theorem 1.3.4 is isolated from other annuliS

k≥1

A

k. We have seen in Remark 1.3.5(2) above that we can achieve this situation by a suitable choice of the potentialV(x).

Definition 1.3.6. — Assumer⁺₁ < r⁻₀ and thatε >0is small enough such thatr⁺₁+ε <

r⁻₀ −ε. LetN_ε be given as in Theorem 1.3.4. ForN ≥N_ε let (1.3.9) Π_~:

H

^rN(P)→

H

^rN(P)

be the spectral projector of the operator Fˆ_N on its external band, i.e., the ε−neighborhood of

A

0. The finite dimensional subspace

H

~:= Im (Π_~)

is called thequantum space. Let

(1.3.10)

F

ˆ~= Π_~◦FˆN = ˆFN ◦Π_~ :

H

~→

H

~

be the (finite dimensional) spectral restriction ofFˆN on the external annulus

A

0. We call this operator

F

ˆ~thequantum operator.

Remark 1.3.7. — We will justify in Section 1.6 this name of “quantum operator”.

The next theorem gives the exact value of the number of eigenvalues in (neighbor- hoods of) the external annulus

A

0 in terms of topological invariants.

Theorem 1.3.8(Index formula for the number of resonances andWeyl law). — If the ex- ternal annulus

A

0 is isolated, i.e.,r⁺₁ +ε < r₀⁻−ε for someε >0, then the number of resonancesRes

FˆN

in theε-neighborhood of the external annulus

A

0 is given by the Atiyah-Singer index formula:forN ≥N_ε,

(1.3.11) dim

H

_~⁼

Z

M

e^{N ω}Todd (T M)

2d

where

e^{N ω} = 1 +N ω+· · ·+N^dω^∧d d!

is the Chern character and Todd (T M) = det

Ω (T M)

1−exp (−Ω (T M))

= 1 +Ω (T M)

2 +· · · ∈H_DR^• (M) is the Todd class of the tangent bundle defined from the Riemannian curvatureΩ (T M) and[.]_2d denotes the restriction to 2d-forms. The leading term gives the “Weyl law”:

(1.3.12) dim

H

~=N^dVol_ω(M) +

O

^Nd−1

with Volω(M) :=R

M 1

d!ω^∧d=R

MdVolω being the symplectic volume ofM.

Theorem 1.3.8 above follows from Theorem 10.2.2 where we will introduce a differ- ential operator ∆ =D^∗D acting inC_N^∞(P)called the rough Laplacian. Beware that this Laplacian operator∆does not depend at all on the dynamics off :M →M. It depends only on the geometry of the bundleP →M. In Theorem 10.2.2, we will show that the low energy spectrum of this positive and self adjoint operator∆ in L²_N(P) has clusters⁽⁷⁾(or band spectrum on the real line) forN 1and that the cardinality of the eigenvalues in the cluster at the lowest level equals the quantity on the right hand side of the formula (1.3.11). The latter is actually a consequence of a theorem in geometry. We will also show that the rank of the projectorΠ_~coincides with the rank of the spectral projector of ∆ for the lowest cluster of eigenvalues. We thus obtain

(7)In physics,∆is a magnetic Laplacian and these clusters are called Landau levels.

the formula (1.3.11). Then the Weyl formula (1.3.12) is a direct consequence. Indeed we have

e^{N ω}Todd (T M)

2d=N^dω^d/d! +O N^d−1

and hence Z

M

e^{N ω}Todd (T M)

2d =N^dVolω(M) +

O

^Nd−1.

Remark 1.3.9. — We will obtain the Weyl law (1.3.12) in Corollary 1.7.16 with a worse remainder but with a simpler proof that does not invoke the Atiyah-Singer formula.

Remark 1.3.10. — In the case of M =T² which correspond to example (1.2.4) and treated in [25], the projectorΠ_~has exactlyrank (Π_~) =N. Indeed, for surfacesM of genusg, we haveTodd (T M) = 1+^{c1(T M)}₂ with first Chern numberR

Mc1(T M) = 2−

2g(the Gauss-Bonnet integral formula). Hencerank (Π_~) =R

M(N ω) +R

Mc1(T M) = N forM =T²with genusg= 1.

For the next theorem, recall the definition of the effective damping functionD(x) = V (x)−V₀(x)in (1.3.3).

Theorem 1.3.11(Asymptotic distribution of the resonances). — Assumer₁⁺< r₀⁻. In the limit~→0, most of eigenvalues of the quantum operator

F

ˆ~ asymptotically concen- trate and equidistributeon the circle of radius e^hDi with

(1.3.13) hDi:= 1

Volω(M) Z

M

D(x) dVol_ω(x).

(See Figure 1.3.1). More precisely, writingRes ˆ

F

~

= Res Fˆ_N

∩ {r₀⁻−ε <|z|<

r⁺₀ +ε} for the eigenvalues of ˆ

F

~ counted with multiplicity, we have, for anyε >0, that

lim

~→0

]n Res

F

ˆ~

∩

|z| −e^hDi < ε o ]n

Res ˆ

F

~

o = 1 and, for any0≤θ1< θ2≤2π, that

lim

~→0

]n Res

ˆ

F

~

∩ {θ1<arg (z)< θ2}o ]n

Res ˆ

F

~

o = θ₂−θ₁ 2π . The proof of Theorem 1.3.11 will be given in Chapter 8 .

Remark 1.3.12. — hDidoes not depend on the choice of volume form onEuused to define the functionD.

Remark 1.3.13. — The proof of Theorem 1.3.11 uses ergodicity of the map f :M → M and follows a technique presented by J. Sjöstrand in [68] for the damped wave equation. Using mixing and large deviations properties of the mapfit may be possible to improve the results and obtain that the number of resonances outside the annulus {||z| −R|< ε} is O N^d−δ

with some δ > 0, as obtained by N. Anantharaman in

[2] for the damped wave equation. Also we expect that the resonances in the internal bands

A

k concentrate to a few circles, at least under some pinching conditions that ensure disjointedness of

A

k from other bands.

1.4. Spectral results for extended models on the Grassmanian bundle In this section we extend the previous results to a family of prequantum transfer operators more general than that considered in Theorem 1.3.4 in the sense that we will admit some functions V for the potential, defined in (1.4.3) below, that may be only Hölder continuous. This is the case of V0 given in (1.3.2). The trick is to consider the dynamics off :M →M lifted on thed-dimensional Grassmanian bundle p:G_d(T M)→M. We consider a smooth potential functionV˜ onG_d(T M)and work with the associated transfer operators. The functionV˜ defines a (Hölder continuous) potential functionV := ˜V ◦E_u onM whereE_u is the unstable distribution regarded as a (Hölder continuous) section ofGd(T M). We explain now this construction.

1.4.1. The Grassmanian bundle G→M and the lifted map f_G. — Recall from Remark 1.2.2(1) that the unstable subspace Eu(x)⊂ TxM at each x ∈ M is of dimensiond= ¹₂dimM. For this reason, we consider the Grassmann bundle that consists of all thed-dimensional linear subspaces ofT_xM for allx∈M.

Definition 1.4.1. — At a given pointx∈M, theGrassmanian⁽⁸⁾G_d(T_xM)is the com- pact manifold of dimensiond²formed by alld-dimensional linear subspaces ofTxM. TheGrassmanian bundleis the fiber bundle

(1.4.1) p:Gd(T M)→M

whose base space isMand the fiber over a pointx∈M is the GrassmanianGd(TxM).

For simplicity we will writep:G→M for this bundle andG_x:=G_d(T_xM)for the fiber.

Definition 1.4.2. — The differential Dfx of the diffeomorphism f :M →M maps a linear subspace ofT_xM to a linear subspace ofT_f(x)M, hence a point ofG_xto a point ofG_f(x). In other wordsf induces a natural lifted map

(1.4.2) f_G=Df : G→G

still (abusively) denoted byDf. See Figure 1.4.1. By definition we have a commutative diagram:

G −−−−→^fG G

p



 y

p



 y M −−−−→^f M.

(8)Gd(TxM)is naturally identified with a homogeneous spaceO(2d)/(O(d)×O(d)).