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Semiclassical origin of the spectral gap for transfer operators of a partially expanding map

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IOP PUBLISHING NONLINEARITY

Nonlinearity 24 (2011) 1473–1498 doi:10.1088/0951-7715/24/5/005

Semiclassical origin of the spectral gap for transferoperators of a partially expanding map

Frédéric Faure

Institut Fourier, 100 rue des Maths, BP74 38402 St Martin d’Hères, France

E-mail: [email protected]

Received 25 February 2010, in final form 2 March 2011Published 31 March 2011Online at stacks.iop.org/Non/24/1473

Recommended by S Nonnenmacher

AbstractWe consider a specific family of skew product of partially expanding map on thetorus. We study the spectrum of the Ruelle transfer operator and show that in thelimit of high frequencies in the neutral direction (this is a semiclassical limit),the spectrum develops a spectral gap, for a generic map. This result has alreadybeen obtained by Tsujii (2008 Ergodic Theory Dyn. Syst. 28 291–317). Thenovelty here is that we use semiclassical analysis which provides a different andquite natural description. We show that the transfer operator is a semiclassicaloperator with a well-defined ‘classical dynamics’ on the cotangent space. Thisclassical dynamics has a ‘trapped set’ which is responsible for the Ruelleresonances spectrum. In particular, we show that the spectral gap is closelyrelated to a specific dynamical property of this trapped set.

Mathematics Subject Classification: 37D20, 37C30, 81Q20

S Online supplementary data available from stacks.iop.org/Non/24/1473/mmedia

1. Introduction

Chaotic behaviour of certain dynamical systems is due to hyperbolicity of the trajectories. Thismeans that the trajectories of two closed initial points will diverge from each other either inthe future or in the past (or both). As a result the behaviour of an individual trajectory seemscomplicated and unpredictable. However, evolution of a cloud of points seems more simple:it will spread and equidistributes according to an invariant measure, called an equilibriummeasure (or S.R.B. measure). Following this idea, Ruelle in the 1970 [Rue78, Rue86] hasshown that instead of considering individual trajectories, it is much more natural to considerevolution of densities under a linear operator called the Ruelle transfer operator or the PerronFrobenius operator.

0951-7715/11/051473+26$33.00 © 2011 IOP Publishing Ltd & London Mathematical Society Printed in the UK & the USA 1473

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1474 F Faure

For dynamical systems with strong chaotic properties, such as uniformly expandingmaps or uniformly hyperbolic maps, Ruelle, Bowen, Fried, Rugh and others, using symbolicdynamics techniques, have shown that the spectrum of the transfer operator has a discretespectrum of eigenvalues. This spectral description has an important meaning for the dynamicssince each eigenvector corresponds to an invariant distribution (up to a time factor). From thisspectral characterization of the transfer operator, one can derive other specific properties ofthe dynamics such as decay of time correlation functions, central limit theorem, mixing, etc.In particular, a spectral gap implies exponential decay of correlations.

This spectral approach has recently (2002–2005) been improved by Blank, Gouzel, Keller,Liverani [BKL02, GL05, Liv05] and Baladi and Tsujii [Bal05, BT07] (see [BT07] for somehistorical remarks), through the construction of functional spaces adapted to the dynamics,independent of any symbolic dynamics. The case of dynamical systems with continuous timeis more delicate (see [FMT07] for historical remarks). This is due to the direction of time flowwhich is neutral (i.e. two nearby points on the same trajectory will not diverge from each other).In 1998 Dolgopyat [Dol98, Dol02] showed the exponential decay of correlation functions forcertain Anosov flows, using techniques of oscillatory integrals and symbolic dynamics. In2004 Liverani [Liv04] adapted Dolgopyat’s ideas to his functional analytic approach, to treatthe case of contact Anosov flows. In 2005 Tsujii [Tsu08] obtained an explicit estimate for thespectral gap for the suspension of an expanding map. Then in 2008 Tsujii [Tsu10b, Tsu10a]obtained an explicit estimate for the spectral gap, in the case of contact Anosov flows.

This work is also closely related to the series of work [GLZ04, SZ04] where the authorsstudy the Zeta function and the density of Ruelle resonances associated with the hyperbolicflow on Schottky manifold and for the quadratic map. In [Chr] the author studies an hyperbolicclosed trajectory.

Semiclassical approach for transfer operators. It has appeared recently [FR06, FRS08] thatfor hyperbolic dynamics, the study of transfer operator is naturally a semiclassical problemin the sense that a transfer operator can be considered as a ‘Fourier integral operator’ andusing standard tools of semiclassical analysis1, some of its spectral properties can be obtainedfrom the study of ‘the associated classical symplectic dynamics’, namely the initial hyperbolicdynamics lifted on the cotangent space (the phase space).

The simple idea behind this, crudely speaking, is that a transfer operator transports a ‘wavepacket’ (i.e. localized both in space and in Fourier space) into another wave packet, and this isexactly the characterization of a Fourier integral operator. A wave packet is characterized bya point in phase space (its position and its momentum), hence one is naturally led to study thedynamics in phase space. Moreover, since any function or distribution can be decomposed asa linear superposition of wave packets, the dynamics of wave packets characterizes completelythe transfer operators.

Following this approach, in the papers [FR06, FRS08] we studied hyperbolicdiffeomorphisms. In [FS10] we studied uniformly hyperbolic flows on a compact manifold.The aim of this paper is to show that semiclassical analysis is well adapted for hyperbolicsystems with a neutral direction. We consider here the simplest model: a partially expandingmap f : (x, s) → f (x, s), i.e. a map on a torus (x, s) ∈ S1 × S1 with an expandingdirection (x ∈ S1x ) and a neutral direction (s ∈ S1s ) (the inverse map f −1 is k-valued, withk � 2). The results are presented in section 2. We summarize them in a few lines. Firstin order to reduce the problem and drop out the neutral direction, we use a Fourier analysis

1 We recommend reference [Tay96b, chapter 7] for the homogeneous semiclassical theory and [Mar02, EZ03] forthe semiclassical theory with a small h parameter.

Semiclassical origin of the spectral gap for transfer operators of a partially expanding map 1475

in s ∈ S1s and decompose the transfer operator F̂ on S1x × S1s (defined by F̂ ϕ := ϕ ◦ f ) asa collection of transfer operators F̂ν on the expanding space S1x only, with ν ∈ Z being theFourier parameter and playing the role of the semiclassical parameter. The semiclassical limitis |ν| → ∞.

Then we introduce a (multivalued) map Fν on the cotangent space (x, ξ) ∈ T ∗S1x whichis the canonical map associated with the transfer operator F̂ν . The fact that the initial map fis expanding along the space S1x implies that on the cylinder T

∗S1x trajectories starting froma large enough value of |ξ | escape towards infinity (|ξ | → ∞). We define the trapped set asthe compact set K = limn→∞ F−nν (K0) where K0 ⊂ T ∗S1 is an initial large compact set. Kcontains trajectories which do not escape towards infinity.

Using a standard semiclassical approach (with escape functions on phase space [HS86])we first show that the operator F̂ν has a discrete spectrum called Ruelle resonances (we haveto consider F̂ν in Sobolev space of distributions). This is theorem 2. This result is well known,but the semiclassical approach we use here is new.

Then we show that a specific hypothesis on the trapped set implies that the operator F̂νdevelops a ‘spectral gap’ in the semiclassical limit ν → ∞ (i.e. its spectral radius reduces).This is theorem 3 illustrated in figure 2. This theorem is very similar to theorem 1.1 in [Tsu08].With the semiclassical approach, this result is very intuitive: the basic idea (followed in theproof) is that an initial wave packet ϕ0 represented as a point on the trapped set K evolves inseveral wave packets (ϕj )j=1→k under the transfer operator F̂ν , but in general only one wavepacket remains on the trapped set K and the (k − 1) other ones escape towards infinity. Asa result the probability on the trapped set K decays by a factor 1/k. This is the origin of thespectral gap at 1/

√k in figure 2.

2. Model and results

2.1. A partially expanding map

Let g : S1 → S1 be a C∞ diffeomorphism (on S1 := R/Z). g can be written as g : R → Rwith g(x + 1) = g(x) + 1, ∀x ∈ R. Let k ∈ N, k � 2, and let the map E : S1 → S1 bedefined by

E : x ∈ S1 → E(x) = kg(x) mod 1. (1)Let

Emin := minx

(dE

dx

)(x) = k min

x

(dg

dx(x)

).

We will suppose that the function g is such that dgdx (x) >1k, i.e.

Emin > 1 (2)

so that E is a uniform expanding map on S1. The map E is then a k : 1 map (i.e. every pointy has k previous images x ∈ E−1(y)). Let τ : S1 → R be a C∞ function, and define a map fon T2 = S1 × S1 by2:

f :

(x

s

)−→

x ′ = E(x) = kg(x) mod 1

s ′ = s + 12π

τ(x) mod 1

. (3)

2 The factor 12π in front of τ(x) is just for convenience. It will simplify other expressions below.

1476 F Faure

Figure 1. Numerical evolution of an initial small cloud of points on the torus (x, s) ∈ T2 underthe map f , equation (3), at different time n = 0, 2, 10, 19. We have chosen here E(x) = 2x andτ(x) = cos(2πx). The initial cloud of points is centred around the point (0, 0). For small time n,the cloud of point is transported in the vertical direction s and spreads in the expanding horizontaldirection x. Due to instability in x and periodicity, the cloud fills the torus S1 × S1 for large timen. On the last image n = 19, one observes an invariant absolutely continuous probability measure(called SRB measure, equal to the Lebesgue measure in our example). It reveals the mixing propertyof the map f in this example (see supplementary data available at stacks.iop.org/Non/24/1/mmediaor www-fourier.ujf-grenoble.fr/∼faure/articles/09 html partially expanding maps).

The map f is also a k : 1 map. The map f is a very simple example of a compact groupextension of the expanding map E (see [Dol02, Pes04, p 17]). It is also a special example of apartially hyperbolic map3. See figure 1.

2.2. Transfer operator

Instead of studying individual trajectories which have chaotic behaviour, one prefer to studythe evolutions of densities induced by the map f . This is the role of the Perron–Frobeniustransfer operator F̂ ∗ on C∞(T2) given by:

(F̂ ∗ψ)(y) =∑

x∈f −1(y)

1

|Dxf |ψ(x), ψ ∈ C∞(T2). (4)

Indeed if the the function ψ has its support in the vicinity of x then the support of F̂ ∗ψis in the vicinity of y = f (x). To explain the Jacobian in the pre-factor, one checks4 that∫

T2(F̂ ∗ψ)(y) dy = ∫

T2ψ(x) dx, i.e. the total measure is preserved.

The operator F̂ ∗ extends to a bounded operator on L2(T2, dx). Its L2-adjoint written F̂ isdefined by (F̂ ∗ψ, ϕ)L2 = (ψ, F̂ϕ)L2 , with the scalar product (ψ, ϕ)L2 :=

∫T2

ψ(x)ϕ(x) dx.

One checks easily that F̂ has a simpler expression than F̂ ∗: it is the pull back operator, also

3 A even more general setting would be a C∞ map f : M → M on a compact Riemannian manifold M , which issupposed to be partially expanding, i.e. for any m ∈ M , the tangent space TmM decomposes continuously as

TmM = Eu(m) ⊕ E0(m),where Eu(m) is a (non-invariant) expanding direction (with respect to a Riemannian metric g):

|Dmf (vu)|g > |vu|g, ∀vu ∈ Eu(m)and E0(m) the neutral direction: there exist a non-zero global section v0 ∈ C∞(T M) such that v0(m) ∈ E0(m) andDf (v0) = v0. In our example (3), M = S1 × S1, the neutral section is v0 = (0, 1), and the expanding directionEu(m) is spanned by the vector (1, 0).4 Since y = f (x), then dy = |Dxf | dx, and∫

T2(F̂ ∗ψ)(y) dy =

∫T2

∑x∈f −1(y)

1

|Dxf |ψ(x)|Dxf | dx =∫

T2ψ(x) dx.

http://stacks.iop.org/Non/24/1/mmediahttp://www-fourier.ujf-grenoble.fr/~faure/articles/09_html_partially_expanding_maps


called the Koopman operator, or Ruelle transfer operator and given by

(F̂ψ)(x) = ψ(f (x)). (5)

2.3. The reduced transfer operator

The particular form of map (3) allows some simplifications. Observe that for a function ofthe form

ψ(x, s) = ϕ(x)ei2πνswith ν ∈ Z (i.e. a Fourier mode in s), then

(F̂ψ)(x, s) = ϕ(E(x))eiντ(x)ei2πνs .Therefore, the operator F̂ preserves the following decomposition in Fourier modes:

L2(T2) =⊕ν∈Z

Hν, Hν := {ϕ(x)ei2πνs, ϕ ∈ L2(S1)}. (6)

The space Hν and L2(S1) are unitary equivalent. For ν ∈ Z given, the operator F̂ restricted tothe space Hν ≡ L2(S1), written F̂ν is5

(F̂νϕ)(x) := ϕ(E(x))eiντ(x), ϕ ∈ L2(S1) ≡ Hν, (7)and with respect to the orthogonal decomposition (6), we can write

F̂ =⊕ν∈Z

F̂ν .

We will study the spectrum of this family of operators F̂ν , with parameter ν ∈ Z, andconsider more generally a real parameter ν ∈ R. We will see that the parameter ν is asemiclassical parameter, and ν → ∞ is the semiclassical limit. (if ν �= 0, ν = 1/h̄ in usualnotations [Mar02]).

Remarks.

• For ν = 0, F̂0 has an obvious eigenfunction ϕ(x) = 1, with eigenvalue 1. Except inspecial cases (e.g. τ = 0 or τ cohomologous to 0, see appendix A), there is no otherobvious eigenvalues for F̂ν in L2(S1) even in the semiclassical limit ν → ∞. By this,we mean that the author is not aware of the existence of analytical expressions for theeigenvalues.

2.4. Main results on the spectrum of the transfer operator F̂ν

We first observe that by duality6, the operator F̂ν defined in (7) extends to the distributionspace D′(S1):

F̂ν(α)(ϕ) = α(F̂ ∗ν (ϕ)), α ∈ D′(S1), ϕ ∈ C∞(S1), (8)where the L2-adjoint F̂ ∗ν is given by

(F̂ ∗ν ϕ)(y) =∑

x∈E−1(y)

e−iντ(x)

E′(x)ϕ(x), ϕ ∈ C∞(S1). (9)

5 Note that the operator F̂ν appears to be a transfer operator for the expanding map E with an additional weightfunction eiντ(x).6 The complex conjugation in (8) is because duality is related to scalar product on L2(S1) by α(ϕ) := ∫

S1 ϕα =(ϕ, α)L2 for any α, ϕ ∈ L2(S1).

1478 F Faure

Before giving the main results, recall that for m ∈ R, the Sobolev space Hm(S1) ⊂ D′(S1)consists in distributions (or continuous functions if m > 1/2) ψ such that their Fourier seriesψ̂(ξ) satisfy ‖ψ‖2Hm :=

∑ξ∈2πZ |〈ξ〉mψ̂(ξ)|2 < ∞, with 〈ξ〉 := (1+ξ 2)1/2. It can equivalently

be written ([Tay96a, p 271]).

Hm(S1) := 〈ξ̂〉−m(L2(S1))with the differential operator ξ̂ := −i ddx .

The following theorem is well known [Rue86]. We will, however, provide a new proofbased on semiclassical analysis.

Theorem 2 (Discrete spectrum of resonances). Let m < 0. For any ν ∈ Z, the operator F̂νleaves the Sobolev space Hm(S1) invariant, and

F̂ν : Hm(S1) → Hm(S1)

is a bounded operator and can be written

F̂ν = R̂ν + K̂ν, (10)where K̂ν is a compact operator, and R̂ν has a small norm:

‖R̂ν‖Hm � rm := 1E

|m|min

√k

Emin(11)

(the interesting situation is m � 0, since the norm ‖R̂ν‖Hm shrinks to zero for m → −∞).Therefore, F̂ν has an essential spectral radius less than rm, which means that F̂ν has

discrete (possibly empty) spectrum of generalized eigenvalues λi outside the circle of radiusrm (see [Tay96a, prop 6.9, p 499]). The eigenvalues λi are called Ruelle resonances. Togetherwith their associated eigenspace, they do not depend on m and are intrinsic to the transferoperator F̂ν .

The following theorem is analogous to theorem 1.1 in [Tsu08]. However, the approachand the proof we propose are different and rely on semiclassical analysis.

Theorem 3 (Spectral gap in the semiclassical limit). If the map f is partially captive(definition given in p 1489) (and m sufficiently negative), then the spectral radius of theoperator F̂ν : Hm(S1) → Hm(S1) does not depend on m and satisfies in the semiclassicallimit ν → ∞:

rs(F̂ν) �1√Emin

+ o(1), (12)

which is strictly smaller than 1 from (2). The following result gives a more precise informationabout the norm of the operator that is controlled uniformly in ν and will be used in theorem 5.For any ρ > 1√

Emin, there exists c > 0,n0 ∈ N, ν0 ∈ N, m0 < 0 such that for any |ν| � ν0,

m < m0 we have ‖F̂ n0ν ‖Hmν � ρn0 . Then for any n ∈ N,‖F̂ nν ‖Hmν � cρn, (13)

where ‖ · ‖Hmν is the Sobolev norm defined by ‖ψ‖2Hmν :=∑

ξ∈2πZ |〈 1ν ξ〉mψ̂(ξ)|2.

Remarks.

• This remark concerns the regularity of the eigenfunctions of F̂ν . Let λi be a generalizedeigenvalue of F̂ν . Let ϕi denotes a generalized eigenfunction of F̂ν associated with λi (i.e.F̂νϕi = λiϕi if λi is an eigenvalue). A corollary of theorem 2 is that for any m such thatm < m0 < 0 where m0 is given by rm0 = |λi | (as defined in (11)) then λi is an eigenvalueof F̂ν and therefore ϕi belongs to the Sobolev space Hm(S1).


Figure 2. Black dots are numerical computation of the eigenvalues λi ∈ C of F̂ν fordifferent values of ν ∈ N, and union of these in the last image. We have chosenhere E(x) = 2x i.e. k = 2, and τ(x) = cos(2πx). The external circle hasradius 1. The internal circle has radius 1/

√Emin = 1/

√2 and represents the upper

bound given in equation (12). As ν ∈ R moves continuously, the resonances move ina spectacular way (see supplementary data available at stacks.iop.org/Non/24/1/mmedia orwww-fourier.ujf-grenoble.fr/∼faure/articles/09 html partially expanding maps).

• By duality we have similar spectral results for the Perron Frobenius operator F̂ ∗ν in the dualspaces (Hm(S1))∗ = H−m(S1), with m < 0. The eigenvalues of F̂ ∗ν are λi . An associatedgeneralized eigenfunctions ϕ̃i of F̂ ∗ν belongs to H

−m(S1) if m < m0 < 0 where m0 isgiven by rm0 = |λi | as above, therefore ϕ̃i belongs to

⋂m

1480 F Faure

• From the definition of N (n) it is clear that N (n) � kn hence exp( 12 limn→∞( log N (n)n )) �√k. Also from the definition of Emin, it is clear that Emin � k and therefore the upper

bound in (14) is not sharp since it does not give the obvious bound rs(F̂ν) � 1 (see [FRS08,corollary 2]). In [Tsu08, theorem 1.4] Tsujii proves that exp( 12 limn→∞(

log N (n)n

)) <√

k

if and only if τ is not a co-boundary. This implies a spectral gap for the linear modelE(x) = kx.

• Note that the above results say nothing about the existence of Ruelle resonances λi . Inthe literature there are many results concerning the existence of resonances [Nau08].

• One observes numerically that for large ν ∈ R, the eigenvalues λi(ν) repulse eachother like eigenvalues of random complex matrices. This suggests that many importantquestions of quantum chaos (e.g. the conjecture of Random Matrices [Boh91]) alsoconcerns the Ruelle resonances of partially hyperbolic dynamics in the semiclassicallimit. The semiclassical limit is precisely the limit of small wavelength in the neutraldirection.

• Remarks on numerical computation of the Ruelle resonances: one diagonalizes the matrixwhich expresses the operator F̂ν in Fourier basis ϕn(x) := exp(i2πnx), n ∈ Z. For theexample of figure 2 one obtains 〈ϕn′ |F̂νϕn〉 = e−i2π 34 (2n−n′)J(2n−n′)(ν) where Jn(x) is theBessel function of first kind [AS54, 9.1.21 p 360]. Corollary 2 in [FR06] guarantiesthat the eigenvalues of the truncated matrix |n|, |n′| � N converges towards the Ruelleresonances as N → ∞.

• One can prove [Arn10] that in the semiclassical limit ν → ∞, the number of Ruelleresonances λi (counting multiplicities) outside a fixed radius λ is bounded by a ‘Weylupper bound’:

∀λ > 0, {i ∈ N, s.t. |λi | � λ} �(

ν

2π

)µ(K) + o(ν),

where µ(K) is the Lebesgue measure of the trapped set K defined later in equation (43).As usual in the semiclassical theory of non-self-adjoint operators, see [Sjö90, SZ07], theWeyl law gives an upper bound for the density of resonances but no lower bound. Seediscussions in [Non08, section 3.1].

2.5. Spectrum of F̂ν and dynamical correlation functions

In this section, in order to give some ‘physical meaning’ to the spectrum of F̂ν , we recallrelations between the spectral results of theorems 2 and 3 and the evolution of correlationfunctions [Bal00]. This will allow us to interpret the evolution and convergence of clouds ofpoints observed in figure 1.

We first give an immediate corollary of theorem 2. Let ν ∈ Z. For ε > 0, let m suchthat rm < ε, (rm is defined in (11)). We denote by π the spectral projector associated withF̂ν : Hm(S1) → Hm(S1) outside the disc of radius ε, and k̂ν := π̂ F̂ν , r̂ν := (1 − π̂)F̂ν . Thenwe have a spectral decomposition7

F̂ν = k̂ν + r̂ν , k̂ν r̂ν = r̂ν k̂ν = 0 (15)

7 Note that in (10) the operators R̂ν ,K̂ν are different from r̂ν , k̂ν . They are not a spectral decomposition of F̂ν .


and

1. The spectral radius of r̂ν is smaller than ε.2. k̂ν has finite rank. Its spectrum has generalized eigenvalues λi,ν (counting multiplicity),

the Ruelle resonances, with ε < |λi,ν |. The general Jordan decomposition of k̂ν can bewritten as

k̂ν =∑

i�0,|λi,ν |>ε

λi,ν di∑

j=1vi,j,ν ⊗ wi,j,ν +

di−1∑j=1

vi,j,ν ⊗ wi,j+1,ν (16)

with di the dimension of the Jordan block associated with the eigenvalue λi,ν , withvi,j,ν ∈ Hm, wi,j,ν ∈ H−m (wi,j,ν is viewed as a linear form on Hm by duality). Theysatisfy wi,j,ν(vk,l,ν) = δikδjl .

If ψ1, ψ2 ∈ C∞(S1), the correlation function at time n ∈ N is defined byCψ2,ψ1(n) := (F̂ ∗nν ψ2, ψ1)L2(S1) = (ψ2, F̂ nν ψ1)L2(S1),

which represents the function ψ2 evolved n times by the Perron–Frobenius operator F̂ ∗ν andtested against the test function ψ1.

Lemma 4. For any ψ1, ψ2 ∈ C∞(S1), ε > 0 such that ε �= |λi |, ∀i, one has for n → ∞,

Cψ2,ψ1(n) =∑

i�0,|λi,ν |>ε

min(n,di−1)∑k=0

Cknλn−ki,ν

di−k∑j=1

vi,j,ν(ψ2)wi,j+k,ν(ψ1) + ‖ψ1‖Hm‖ψ2‖H−mO(εn)

(17)

with any m such that rm < ε, and Ckn := n!(n−k)!k! .

Remarks.

• More generally equation (17) still holds for ψ1 ∈ Hm and ψ2 ∈ H−m, with m such thatrm < ε.

• The right-hand side of equation (17) is complicated by the possible presence of ‘Jordanblocks’. In the case where there is no Jordan block (di = 1, ∀i) it reads more simply: forn → ∞,

Cψ2,ψ1(n) =∑

i�0,|λi,ν |>ελni,νvi,ν(ψ2)wi,ν(ψ1) + O(ε

n).

Proof. For any ε > 0, let m such that rm < ε. For any n � 1 we have F̂ nν = k̂nν + r̂nν and‖r̂nν ‖Hm = O(εn). If ψ1 ∈ Hm, ψ2 ∈ H−m then we use the Cauchy–Schwartz inequality|(ψ, ϕ)H−m×Hm | � ‖ψ‖H−m‖ϕ‖Hm to write

Cψ2,ψ1(n) = (ψ2, F̂ nν ψ1)L2 (18)= (ψ2, k̂nνψ1)L2 + (ψ2, r̂nν ψ1)L2= (ψ2, k̂nνψ1)L2 + ‖ψ2‖H−m‖ψ1‖HmO(εn).

Using the Jordan Block decomposition of k̂ν , equation (16), we have

(ψ2, k̂nνψ1) =

∑i�0,|λi,ν |>ε

min(n,di−1)∑k=0

Cknλn−ki,ν

di−k∑j=1

(ψ2, vi,j,ν)wi,j+k,ν(ψ1), (19)

=∑

i�0,|λi,ν |>ε

min(n,di−1)∑k=0

Cknλn−ki,ν

di−k∑j=1

vi,j,ν(ψ2)wi,j+k,ν(ψ1). (20)

We have obtained equation (17). �

1482 F Faure

We can then draw some conclusions for the full map f and its transfer operator F̂ , definedin equation (5) instead of merely the restricted transfer operator F̂ν , ν ∈ Z. We recall firstthat λ0,0 = 1 is an obvious eigenvalue for F̂ν=0 with eigenfunction v0,0(x) = 1. If λ0,0 isnon-degenerate, we denote w0,0(x) ∈ C∞(S1) the dual eigenfunction, i.e. F̂ ∗ν=0w0,0 = w0,0.Theorem 5 (Mixing on T2). If the conclusion of theorem 3 for a spectral gap holds then forany ρ > 1√

Eminthere exists ν0 ∈ Z such that for any 1, 2 ∈ C∞(T2), for n → ∞,

C2,1(n) : = (F̂ ∗n2, 1)L2(T2) = (2, F̂ n1)L2(T2) (21)

=∑

ν,|ν|ρ

min(n,di−1)∑k=0

Cknλn−ki,ν

di−k∑j=1

vi,j,ν(2,ν)wi,j+k,ν(1,ν) (22)

+ O(ρn), (23)

where 1,ν ∈ C∞(S1) (respectively, 2,ν) are the components of 1 (respectively 2) withrespect to the decomposition (6). If moreover λ0,0 = 1 is the only eigenvalue of F̂ on the unitcircle (with multiplicity 1) with associated eigen-states v0,0 = 1, w0,0(x) ∈ C∞(S1), then themap f on T2 is ‘mixing’ i.e. for n → ∞:

C2,1(n) →( ∫

T2

2(x, s) dx ds)

( ∫T2

1dµSRB

)(24)

with the ‘equilibrium Sina–Ruelle–Bowen measure’ dµSRB = w0,0(x) dx ds.

Proof. For smooth functions 1, 2 ∈ C∞(T2) we have for any m, and any N � 0,‖1,ν‖Hmν = O( 1νN ), ‖2,ν‖H−mν = O( 1νN ). On the other hand, if the conclusion of theorem 3for a spectral gap holds then for any ρ > 1√

Eminthere exists ν0 ∈ Z such that for any ν, |ν| � ν0,

F̂ν has no spectrum outside the disc of radius ρ. Then (17) and (13) give: there exists c > 0,n0 ∈ N, ν0 ∈ N, m0 < 0 such that for any |ν| � ν0, m < m0, any n ∈ N,(2,ν , F̂

nν 1,ν)L2(S1) � ‖1,ν‖Hmν ‖2,ν‖H−mν ‖F̂ nν ‖Hmν , for |ν| � ν0

� CNνN

cρn.

Hence for N large enough,∑

|ν|�ν0 |(2,ν , F̂ nν 1,ν)L2(S1)| � O(ρn). We use lemma 4 withε = ρ and obtainC2,1(n) = (2, F̂ n1)L2(T2) =

∑ν∈Z

(2,ν , F̂nν 1,ν)L2(S1)

=∑

|ν|


Remarks

• In figure 1 one observes the mixing property of the dynamics and the equilibrium measureon the last image. In this case (E(x) = 2x is linear) we have w0,0(x) = 1 and theequilibrium measure is the Lebesgue measure dµSRB = dx ds.

• In formula (21), note that this is a finite sum other the eigenvalues outside the disc of radiusρ. This expression shows that the time behaviour of correlation functions is governed byan effective linear operator of finite rank up to O(ρn), n → ∞. This effective linearoperator is the spectral projection of F̂ on the outer eigenspaces such that |λi,ν | > ρ. Thesemiclassical theory used in this paper does not give direct information on this effectiveoperator, but on its existence and the fact that it is supported by the low Fourier modes.

3. Proof of theorem 2 on resonances spectrum

In this proof, we follow closely the proof of theorem 4 in [FRS08] although we deal here withexpanding map instead of hyperbolic map, and this simplifies a lot, since we can work withordinary Sobolev spaces and not anisotropic Sobolev spaces. Here ν ∈ Z is fixed.

3.1. Dynamics on the cotangent space T ∗S1

The first step is to realize that in order to study the spectral properties of the transfer operator,we have to study the dynamics lifted on the cotangent space. This basic idea has already beenexploited in [FRS08].

In equation (1), the map E : S1 → S1 is a k : 1 map, which means that every point y ∈ S1has k inverses denoted by xε ∈ E−1(y) and given explicitly by

xε = E−1ε (y) = g−1(

y

k+ ε

1

k

), with ε = 0, . . . , k − 1, y ∈ [0, 1[.

We will denote the derivative by E′(x) := dE/dx.Proposition 6. In equation (7) F̂ν is a Fourier integral operator (FIO) acting on C∞(S1).

The associated canonical map on the cotangent space (x, ξ) ∈ T ∗S1 ≡ S1 ×R is k-valuedand given by

F(x, ξ) = {F0(x, ξ), . . . , Fk−1(x, ξ)}, (x, ξ) ∈ S1 × R, (25)where for any ε = 0, . . . , k − 1,

Fε :

x → x

′ε = E−1ε (x) = g−1

(1

kx + ε

1

k

),

ξ → ξ ′ε = E′(x ′ε)ξ = kg′(x ′ε)ξ.(26)

Similarly the adjoint F̂ ∗ is a FIO whose canonical map is F−1. See figure 3.

The proof is just that the operator ϕ → ϕ ◦ E on C∞(S1) is one of the simplest exampleof Fourier integral operator, see [Mar02, example 2, p 150].

The term eiντ(x) in equation (7) does not contribute to the expression of F , since here νis considered as a fixed parameter, and therefore eiντ(x) acts as a pseudodifferential operator(equivalently as a FIO whose canonical map is the identity).

The map F is the map E−1 lifted on the cotangent space T ∗S1 in the canonical way.Indeed, if we denote a point (x, ξ) ∈ T ∗S1 ≡ S1 × R then using the usual formula fordifferentials y = E(x) = kg(x) ⇒ dy = E′(x) dx ⇔ ξ ′ = E′(x)ξ , we deduce the aboveexpression for F .

1484 F Faure

Figure 3. This figure is for k = 2. The map F = {F0, . . . , Fk−1} is 1 : k, and its inverse F−1is k : 1 on T ∗S1 ≡ S1 × R. The dynamics of F is also the dynamics of wave packets under thetransfer operator F̂ν .

Remarks

• The physical meaning for F̂ε being a Fourier Integral Operator (F.I.O.) and its relationwith the canonical map F can be well understood in term of dynamics of wave packets.We provide a detailed presentation of this in the next section.

• Observe that the dynamics of the map F on S1 × R has a quite simple property: the zerosection {(x, ξ) ∈ S1 × R, ξ = 0} is globally invariant and any other point with ξ �= 0escapes towards infinity (ξ → ±∞) in a controlled manner:

|ξ ′ε| � Emin|ξ |, ∀ε = 0, . . . , k − 1, (27)where Emin > 1 is given in (2).

3.2. Dynamics of wave packets

The reading of this section is not necessary for the rest of the paper. It is intended to thereader non-familiar with Fourier integral operators and associated canonical maps. We showthat if ϕ(x,ξ) is a wave packet ‘micro-localized’ at position (x, ξ) ∈ T ∗S1 of phase space,then ϕ′ := F̂νϕ(x,ξ) is a superposition of k wave packets at positions (x ′ε, ξ ′ε) = Fε(x, ξ),ε = 0, . . . , k − 1, given by the canonical map F . This is illustrated in figure 3.

Let h̄ > 0 be a parameter (called the semiclassical parameter), and for (x, ξ) ∈ R2 ≡ T ∗Rwe define ϕ(x,ξ) ∈ S(R) called a Gaussian wavepacket by

ϕ̃(x,ξ)(y) := Cei 1h̄ yξ e− 12h̄ (y−x)2 , C > 0.Note that in this last expression the factor eiy(ξ/h̄) is similar to a Fourier mode with Fourier

component ξ/h̄. In other words, the role of h̄ in this factor is just a scaling in Fourier space.This ‘artificial’ parameter h̄ is independent of ν in this section, whereas we will choose it suchthat h̄ = 1/ν in section 4.

We then construct the periodic function ϕ(x,ξ)(y) :=∑

k∈Z ϕ̃(x,ξ)(y − k) in C∞(S1), andchoose C such that ‖ϕ(x,ξ)‖L2(S1) = 1. We easily observe that for h̄ � 1, ϕ(x,ξ)(y) = O(h̄∞)is negligible if y �= x mod 1, where O(h̄∞) means O(h̄N) for any N > 0. Similarlythe h̄-Fourier transform of ϕ̃(x,ξ) (defined by (F ϕ̃(x,ξ))(η) := 1√2πh̄

∫e−i

ηy

h̄ ϕ̃(x,ξ)(y) dy) is

|(F ϕ̃(x,ξ))(η)| = C exp(− (η−ξ)22h̄ ) and is negligible if η �= ξ .


Alternatively this can be seen on the scalar product between two Gaussian wavepacketswhich is also easy to compute:

|(ϕ(y,η), ϕ(x,ξ))L2(S1)| = |(ϕ̃(y,η), ϕ̃(x,ξ))L2(R)| + O(e−c/h̄), c > 0= exp

(− 1

4h̄((x − y)2 + (ξ − η)2)

)+ O(e−c/h̄)

= O(h̄∞) if (y, η) �= (x, ξ) (28)Definition 7 ([Mar02], chapter 3). If ψh̄ ∈ L2(R), h̄ → 0, is a sequence of functions, we saythat ψ is micro-locally small near (x0, ξ0) ∈ T ∗R if

|(ϕ̃(y,η), ψ)L2(R)| = O(h̄∞)uniformly for (y, η) in a neighbourhood of (x0, ξ0). The complementary of such points (x0, ξ0)is called the micro-support of ψ .

For example (28) shows that the micro-support of a Gaussian wavepacket ϕ̃(x,ξ) is thepoint (x, ξ) ∈ T ∗R.Proposition 8. The micro-support of F̂νϕ(x,ξ) is the set of points

F(x, ξ) = {F0(x, ξ), . . . , Fk−1(x, ξ)} ∈ T ∗S1.We have indeed uniformly in a neighbourhood of (y, η) �= Fε(x, ξ), ∀ε = 0, . . . , k − 1:

|(ϕ(y,η), F̂νϕ(x,ξ))L2(S1)| = O(h̄∞).Proof. We have

(ϕ̃(y,η), F̂ν ϕ̃(x,ξ))L2(R) = C2∫

R

e−i1h̄ηze−

12h̄ (y−z)2 e+i

1h̄ξE(z)e−

12h̄ (x−E(z))2 eiντ(z) dz. (29)

In the semiclassical limit h̄ � 1, because of the quadratic terms in the exponential, this isclearly negligible if z �= y or x �= E(z) hence if y �= E−1ε (x) mod 1, ∀ε. But due to the highoscillatory terms with the phase function ϕ(z) := −ηz + ξE(z) this is also negligible fromthe non-stationary phase approximation8, if ϕ′(z) = −η + ξE′(z) �= 0 i.e. if η �= E′(y)ξ .Together this gives (y, η) �= Fε(x, ξ), ε = 0, . . . , k − 1. �

3.3. The escape function

The class of symbols9 Sm, with order m ∈ R, consists of functions on the cotangent spaceA ∈ C∞(S1 × R) such that

|∂αξ ∂βx A|∞ � Cα,β〈ξ〉m−|α|, 〈ξ〉 = (1 + ξ 2)1/2.Lemma 9. Let m < 0 and define the C∞ function on T ∗S1:

Am(x, ξ) := 〈ξ〉m ∈ Smwith 〈ξ〉 = (1+ξ 2)1/2. Am decreases with |ξ | and belongs to the symbol class Sm. Equation (27)implies that the function Am decreases strictly along the trajectories of F outside the zerosection:

∀R > 0, ∀|ξ | > R, ∀ε = 0, . . . , k − 1 Am(Fε(x, ξ))Am(x, ξ)

� C|m| < 1,

with C =√

R2 + 1

R2E2min + 1< 1. (30)

One has C → 1/Emin for R → ∞.8 Let ϕ ∈ C∞(R) real valued, such that ϕ′ �= 0 every where. It u ∈ C∞0 (R) then the integral

∫ei

1h̄ϕ(x)u(x)dx is

rapidly decreasing when h̄ → 0 [EZ03, chaptre 3].9 See [Tay96b, p 2].

1486 F Faure

Proof.

Am(Fε(x, ξ))

Am(x, ξ)= (1 + ξ

2)|m|/2

(1 + (ξ ′ε)2)|m|/2� (1 + ξ

2)|m|/2

(1 + E2minξ2)|m|/2

�(

1 + R2

1 + E2minR2

)|m|/2= C|m|. �

The symbol Am can be quantized into a pseudodifferential operator Âm (PDO for short)which is self-adjoint and invertible on C∞(S1) using the quantization rule ([Tay96b, p 2])

(Âϕ)(x) =∫

A(x, ξ)ei(x−y)ξϕ(y) dy dµ(ξ), (31)

with measure dµ(ξ) = ∑n∈Z δ(ξ − 2πn). In our simple case, this is very explicit: in Fourierspace, Âm is simply the multiplication by 〈ξ〉m.

Recall that the Sobolev space Hm(S1) is defined by ([Tay96a, p 271]):

Hm(S1) := Â−1m (L2(S1)).The following commutative diagram:

L2(S1)Q̂m→ L2(S1)

↓ Â−1m � ↓ Â−1mHm(S1)

F̂ν→ Hm(S1)shows that F̂ν : Hm(S1) → Hm(S1) is unitary equivalent to

Q̂m := ÂmF̂νÂ−1m : L2(S1) → L2(S1).We will therefore study the operator Q̂m. Note that Q̂m is defined a priori on a dense domain(C∞(S1)). Define

P̂ := Q̂∗mQ̂m = Â−1m (F̂ ∗ν Â2mF̂ν)Â−1m = Â−1m B̂Â−1m , (32)where appears the operator

B̂ := F̂ ∗ν Â2mF̂ν. (33)The Egorov theorem will help us to treat this operator (see [Tay96b, p 24]). This is a

simple but crucial step in the proof: as explained in [FRS08], the Egorov theorem is the maintheorem used in order to establish both the existence of a discrete spectrum of resonances andproperties of them. However, there is a difference with [FRS08]: for the expanding map weconsider here, the operator F̂ν is not invertible and the canonical map F is k-valued. Thereforewe have to state the Egorov theorem in an appropriate way (we restrict, however, the statementto our simple context).

Lemma 10 (Egorov theorem). B̂ := F̂ ∗ν Â2mF̂ν is a pseudodifferential operator with symbolin S2m given by

B(x, ξ) =( ∑

ε=0,...,k−1

1

E′(x ′ε)A2m(Fε(x, ξ))

)+ R (34)

with R ∈ S2m−1 has a subleading order.

Proof. As we explained in proposition 6, F̂ν and F̂ ∗ν are Fourier integral operators (FIO)whose canonical map are respectively F and F−1. The pseudodifferential operator (PDO)Âm can also be considered as a FIO whose canonical map is the identity. By composition wededuce that B̂ = F̂ ∗ν Â2mF̂ν is a FIO whose canonical map is the identity since F−1 ◦ F = I .


See figure 3. Therefore B̂ is a PDO. Using (7), (9) and (26) we obtain that the principal symbolof B̂ is ∑

ε=0,...,k−1

1

E′(x ′ε)A2m(Fε(x, ξ)). (35)

�

Remark. Contrary to (33), F̂νÂmF̂ ∗ν is not a PDO, but a FIO whose canonical map F ◦ F−1is k−valued (see figure 3).

Now by theorem of composition of PDO ( [Tay96b, p 11], (32) and (34) imply that P̂ is aPDO of order 0 with principal symbol:

P(x, ξ) = B(x, ξ)A2m(x, ξ)

=( ∑

ε=0,...,k−1

1

E′(x ′ε)A2m(Fε(x, ξ))

A2m(x, ξ)

).

Estimate (30) together with (2) give the following upper bound:

∀|ξ | > R, |P(x, ξ)| � C2|m|∑

ε=0,...,k−1

1

E′(x ′ε)� C2|m| k

Emin

(This upper bound goes to zero as m → −∞). From L2-continuity theorem for PDO wededuce that for any α > 0 (see [FRS08, lemma 38])

P̂ = k̂α + p̂αwith k̂α a smoothing operator (hence compact) and ‖p̂α‖ � C2|m| kEmin + α. If Q̂m = Û |Q̂| isthe polar decomposition of Q̂m, with Û unitary, then from (32) P̂ = |Q̂|2 ⇔ |Q̂| =

√P̂ and

the spectral theorem ([Tay96b, p 75]) gives that |Q̂| has a similar decomposition|Q̂| = k̂′α + q̂α

with k̂′α smoothing and ‖q̂α‖ � C|m|√

kEmin

+ α, with any α > 0. Since ‖Û‖ = 1 we deduce asimilar decomposition for Q̂m = Û |Q̂| : L2(S1) → L2(S1) and we deduce (10) and (11) forF̂ν : Hm → Hm. We also use the fact that C → 1/Emin for R → ∞ in (30).

The fact that the eigenvalues λi and their generalized eigenspaces do not depend on thechoice of space Hm is due to density of Sobolev spaces. We refer to the argument given in theproof of corollary 1 in [FRS08]. This completes the proof of theorem 2.

4. Proof of theorem 3 on spectral gap

We will follow step by step the same analysis as in the previous section. The main differencenow is that in theorem 3, ν � 1 is a semiclassical parameter. In other words, we just performa linear rescaling in cotangent space: ξh := h̄ξ with

h̄ := 1ν

� 1.Therefore, our quantization rule for a symbol A(x, ξh) ∈ Sm, equation (31) writes now(see [Mar02, p 22])

(Âϕ)(x) =∫

A(x, ξh)ei(x−y)ξh/h̄ϕ(y) dy dµ(ξh) (36)

with measure dµ(ξh) =∑

n∈Z δ(ξh − 2πnh̄). For simplicity we will write ξ for ξh below.

1488 F Faure

4.1. Dynamics on the cotangent space T ∗S1

If we consider again equation (29) then the multiplicative term eiντ(x) = eiτ(x)/h̄ contributesnow to the stationary phase approximation (the phase function is ϕ(z) = −ηz + ξE(z) + τ(z)so 0 = ϕ′(z) = −η + ξE′(z) + τ ′(z) implies η = E′(z)ξ + τ ′(z)) and modifies the expressionof the canonical map F . We obtain:

Proposition 11. In equation (7) F̂ν is a semiclassical Fourier integral operator acting onC∞(S1) (with semiclassical parameter h̄ := 1/ν � 1). The associated canonical map onthe cotangent space (x, ξ) ∈ T ∗S1 ≡ S1 × R is k-valued and given by

F(x, ξ) = {F0(x, ξ), . . . , Fk−1(x, ξ)}, (x, ξ) ∈ S1 × R, (37)

Fε :

x → x ′ε = E−1ε (x),ξ → ξ ′ε = E′(x ′ε)ξ +

dτ

dx(x ′ε),

ε = 0, . . . , k − 1. (38)

Similarly F̂ ∗ν is a FIO whose canonical map is F−1.

Note that for simplicity we have kept the same notation for the canonical map F althoughit differs from (26).

Since the map F is k-valued, a trajectory is a tree. Let us precise the notation:

Definition 12. For ε = (. . . ε3, ε2, ε1) ∈ {0, . . . , k − 1}N∗ , a point (x, ξ) ∈ S1 × R and timen ∈ N∗ let us denote

Fnε (x, ξ) := FεnFεn−1 . . . Fε1(x, ξ). (39)For a given sequence ε ∈ {0, . . . , k − 1}N∗ , a trajectory issued from the point (x, ξ) is

{Fnε (x, ξ), n ∈ N}.

Note that at time n ∈ N, there are kn points issued from a given point (x, ξ):Fn(x, ξ) := {Fnε (x, ξ), ε ∈ {0, . . . , k − 1}n}. (40)

The new term dτdx (x′ε) in the expression of ξ

′ε, equation (38), complicates significantly the

dynamics near the zero section ξ = 0. However, a trajectory from an initial point with |ξ |large enough still escape towards infinity:

Lemma 13. For any 1 < κ < Emin, there exists R � 0 such that for any |ξ | > R, anyε = 0, . . . k − 1,

|ξ ′ε| > κ|ξ |. (41)

Proof. From (38), one has ξ ′ε = E′(x ′ε)ξ + τ ′(x ′ε), so ξ ′ε − κξ = (E′(x ′ε) − κ)ξ + τ ′(x ′ε) �(Emin−κ)ξ+min τ ′ > 0 if ξ > − min τ ′(Emin−κ) � 0, and similarly ξ ′ε−κξ � (Emin−κ)ξ+max τ ′ < 0if ξ < − max τ ′

(Emin−κ) . �

We will denote the set

Z := S1 × [−R, R] (42)outside of which trajectories escape in a controlled manner (41). See figure 4.


Figure 4. The trapped set K in the cotangent space S1 × R. We have chosen here E(x) = 2x andτ(x) = cos(2πx).

4.2. The trapped set K

We will be interested now in the trajectories of F which do not escape towards infinity.

Definition 14. We define the trapped set

K :=⋂n∈N

(F−1)n(Z), (43)

which contains points for which a trajectory at least does not escape towards infinity. Seefigure 4. The definition of K does not depend on the compact set Z (if Z is chosen largeenough).

Since the map F is multivalued, some trajectories may escape from the trapped set. Wewill need a characterization of how many such trajectories succeed to escape: For n ∈ N, let

N (n) := max(x,ξ)

{Fnε (x, ξ) ∈ Z, ε ∈ {0, . . . , k − 1}n}. (44)

See figure 5 for an illustration of N (n). Of course N (n) � kn.

Definition 15. The map F (or f ) is partially captive if10

log N (n)n

−→n→∞ 0. (45)

This property is the hypothesis of theorem 3.

10 It can be shown that log N (n) is sub-additive and therefore limn→∞ inf( log N (n)n ) = limn→∞( log N (n)n )[RS72, p 217, ex 11].

1490 F Faure

Figure 5. This Figure illustrates the trajectories Fnε (x, ξ) issued from an initial point (x, ξ) aftertime n. Here k = 2 and n = 3. The property for the map F of being ‘partially captive’ accordingto definition 15 is related to the number of points N (n) which do not escape from the compact zoneZ after time n.

Remarks.

• N (n) defined in (44) is the number of trajectories which do not escape outside the vicinityZ of the trapped set before time n. Since there are kn trajectories which start from agiven point (x, ξ) ∈ S1 × R, the property for the map F to be ‘partially captive’, i.e.log N (n)

n−→n→∞ 0, means that ‘most’ of the trajectories escape from the trapped set K . See

figure 5. Another description of the trapped set K and of the partially captive propertywill be given in appendix B. Note that the function N (n), equation (44) depends on theset Z but property (45) does not.

• If the function τ is trivial in (38), i.e. τ = 0 , then obviously all the trajectories issuedfrom a point (x, ξ) on the line ξ = 0 remains on this line (the trapped set). Therefore

{Fnε (x, ξ) ∈ Z, ε ∈ {0, 1, . . . , k − 1}n} = kn

and the map F is not partially captive (but could be called ‘totally captive’). This is alsotrue if the function τ is a ‘co-boundary’, i.e. if τ(x) = η(E(x)) − η(x) with η ∈ C∞(S1)as discussed in appendix A.

• Tsujii has studied a dynamical system very similar to (38) in [Tsu01], but this model isnot volume preserving. He establishes there that the SRB measure on the trapped set isabsolutely continuous for almost every τ .

4.3. The escape function

Let m < 0 and consider the C∞ function on T ∗S1:

Am(x, ξ) := 〈ξ〉m for |ξ | > R + η,:= 1 for ξ � R,


where η > 0 is small and with 〈ξ〉 := (1 + ξ 2)1/2. Am decreases with |ξ | and belongs to thesymbol class Sm.

Equation (41) implies that the function Am decreases strictly along the trajectories of Foutside the trapped set (similarly to equation (30)):

∀|ξ | > R, ∀ε = 0, . . . , k − 1 Am(Fε(x, ξ))Am(x, ξ)

� C|m| < 1,

with C =√

R2 + 1

κ2R2 + 1< 1. (46)

And for any point we have the general bound:

∀ε = 0, . . . , k − 1, ∀(x, ξ) ∈ T ∗S1, Am(Fε(x, ξ))Am(x, ξ)

� 1. (47)

Using the quantization rule (36), the symbol Am can be quantized giving a pseudodifferentialoperator Âm which is self-adjoint and invertible on C∞(S1). In our case Âm is simply amultiplication operator by Am(ξ) in Fourier space.

Let us consider the (usual) Sobolev space

Hmν (S1) := Â−1m (L2(S1)).

Then F̂ν : Hmν (S1) → Hmν (S1) is unitary equivalent to

Q̂ := ÂmF̂νÂ−1m : L2(S1) → L2(S1).Let n ∈ N∗ (a fixed time which will be made large at the end of the proof) and define

P̂ (n) := Q̂∗nQ̂n = Â−1m F̂ ∗nν Â2mF̂ nν Â−1m . (48)Using Egorov theorem (lemma 10) and theorem of composition of PDO [EZ03, chaptre 4], weobtain that P̂ (n) is a PDO of order 0 with principal symbol

P (n)(x, ξ) = ∑

ε∈{0,...,k−1}n

1

E′n(x)A2m(F

nε (x, ξ))

A2m(x, ξ)

, (49)

where E′n(x) :=∏n

j=1 E′(E−jεj (x)) is the expanding rate of the trajectory at time n.

Equation (2) implies that E′n(x) � Enmin. Now we will bound this (positive) symbol fromabove, considering different cases for the trajectory Fnε (x, ξ), as illustrated in figure 5.

1. If (x, ξ) /∈ Z then (46) givesA2(F nε (x, ξ))

A2(x, ξ)= A

2(F nε (x, ξ))

A2(F n−1ε (x, ξ)). . .

A2(Fε(x, ξ))

A2(x, ξ)� (C2|m|)n (50)

therefore

P (n)(x, ξ) � kn

Enmin(C2|m|)n.

2. If (x, ξ) ∈ Z but Fn−1ε (x, ξ) /∈ Z then (A2◦Fnε )(x,ξ)

(A2◦Fn−1ε )(x,ξ) � C2|m| from (46). Using also (47)

we haveA2(F nε (x, ξ))

A2(x, ξ)= A

2(F nε (x, ξ))

A2(F n−1ε (x, ξ))· · · A

2(Fε(x, ξ))

A2(x, ξ)� C2|m|. (51)

3. In the other cases ((x, ξ) ∈ Z and Fn−1ε (x, ξ) ∈ Z) we can only use (47) to bound:A2(F nε (x, ξ))

A2(x, ξ)� 1 (52)

1492 F Faure

From definition (44) we have

{Fn−1ε (x, ξ) ∈ Z, ε ∈ {0, 1}n} � N (n − 1).For (x, ξ) ∈ Z , we split the sum equation (49) accordingly to cases 1,2 or 3 above. Note that(C2|m|)n � C2|m|. This gives

P (n)(x, ξ) � 1Enmin

((kn − N (n − 1))C2|m| + N (n − 1)) � B (53)with the bound

B :=(

k

Emin

)nC2|m| +

N (n − 1)Enmin

. (54)

Then

sup(x,ξ)

|P (n)(x, ξ)| � B

With L2 continuity theorem for pseudodifferential operators [Mar02] this implies that in thelimit h̄ → 0

‖P̂ (n)‖L2→L2 � B + On(h̄). (55)Polar decomposition of Q̂n gives

‖Q̂n‖L2→L2 � ‖|Q̂n|‖ =√

‖P̂ (n)‖ � (B + On(h̄))1/2. (56)We make now the assumption that F be partially captive, equation (45). For any c > 0,

and some n we have log N (n)n

< c hence N (n) < ecn. In other words for any ρ > 1√Emin

, there

exists n ∈ N such that N (n−1)Enmin

< ρ2n.

We let h̄ = 1/ν → 0 first, and after m → −∞ giving C |m| → 0, and also we let n belarge enough. Then from (56) and (54) we have obtained that for any ρ > 1√

Emin, there exists

n0 ∈ N, ν0 ∈ N, m0 < 0 such that for any |ν| � ν0, m < m0,‖F̂ n0ν ‖Hmν = ‖Q̂n0‖L2 � ρn0 . (57)

Also, there exists c > 0 independent of |ν| � ν0, such that for any r such that 0 � r < n0we have ‖Q̂r‖L2 < c. As a consequence for any n ∈ N we write n = kn0 + r with 0 � r < n0and

‖F̂ nν ‖Hmν = ‖Q̂n‖L2 � ‖Q̂n0‖kL2‖Q̂r‖L2 � ρn‖Q̂r‖L2

ρr� cρn

with c > 0 independent of |ν| � ν0. We have obtained (13).For any n the spectral radius of Q̂ satisfies [RS72, p 192]

rs(Q̂) � ‖Q̂n‖1/n � c1/nρ.So we get that for h̄ = 1/ν → 0,

rs(F̂ν) = rs(Q̂) � 1√Emin

+ o(1).

Without the assumption that F be partially captive, equation (45), we have more generally(N (n − 1)

Enmin

)1/2n= 1√

Eminexp

(1

2nlog N (n − 1)

)and

rs(Q̂) �√

1

Eminexp

(lim

n→∞ inf(

log N (n)n

))+ o(1). (58)

We have completed the proof of theorem 3.


Acknowledgments

The author thanks Masato Tsujii and Colin Guillarmou for interesting discussions. This workhas been supported by the ‘Agence Nationale de la Recherche’ under the grant JC05 52556.

Appendix A. Equivalence classes of dynamics

Let us make a simple and well-known observation about equivalent classes of dynamics. Themap f we consider in equation (3) depends on k ∈ N and on the functions E : S1 → S1,τ : S1 → R. To emphasize this dependence, we denote f(E,τ). The transfer operator (7) isalso denoted by F̂(E,τ).

In this appendix we characterize an equivalence class of functions (E, τ) such that in agiven equivalence class the maps f(E,τ) are C∞ conjugated together, the transfer operatorsF̂(E,τ) are also conjugated and the resonances spectrum are therefore the same.

Let η : S1 → R be a smooth function. Let us consider the map T : S1 × S1 → S1 × S1defined by

T (x, s) =(

x, s +1

2πη(x)

).

Then using (3) one obtains that

(T −1 ◦ f(E,τ) ◦ T )(x, s) =(

E(x), s +1

2π(τ(x) + η(x) − η(E(x)))

).

Therefore

(T −1 ◦ f(E,τ) ◦ T ) = f(E,ζ ),i.e. f(E,ζ ) ∼ f(E,τ), with

ζ = τ + (η − η ◦ E).The function τ has been modified by a ‘co-boundary term’ [KH95, p 100]. The function ζ issaid to be cohomologous to τ .

Lemma 16. With (7) we also obtain that the transfer operator F̂(E,ζ ) of f(E,ζ ) on C∞(S1) isgiven by

F̂(E,ζ ) = χ̂ F̂(E,τ)χ̂−1 (59)with the operator χ̂ : C∞(S1) → C∞(S1) defined by

(χ̂ϕ)(x) = eiνη(x)ϕ(x).Proof. (χ̂ F̂(E,τ)χ̂−1ϕ)(x) = (ϕ(E(x))e−iνη(E(x)))eiντ(x)eiνη(x) = (F̂(E,ζ )ϕ)(x). �

The conjugation (59) immediately implies that F̂(E,ζ ) and F̂(E,τ) have the same spectrumof Ruelle resonances.

Observe that χ̂ is a FIO whose associated canonical map on T ∗S1 ≡ S1 × R is given by(ν � 1 is considered as a semiclassical parameter):

χ : (x, ξ) ∈ (S1 × R) →(

x, ξ +dη

dx

)∈ (S1 × R).

Therefore at the level of canonical maps on T ∗S1:F(E,ζ ) = χ ◦ F(E,τ) ◦ χ−1. (60)

The conjugation (60) implies in particular that the corresponding trapped sets (43) arerelated by

K(E,ζ ) = χ(K(E,τ)).

1494 F Faure

Appendix B. Description of the trapped set

In this section we provide further description of the trapped set K defined in equation (43) aswell as the dynamics of the canonical map F restricted on it.

Appendix B.1. Dynamics on the cover R2

The dynamics of F on the cylinder T ∗S1 = S1 × R has been defined in equation (38). This isa multivalued map. It is convenient to consider the lifted dynamics on the cover R2 which is adiffeomorphism given by

F̃ :

x → x ′ = E−1(x),ξ → ξ ′ = E′(x ′)ξ + dτ

dx(x ′),

(61)

where E = kg : R → R is map (1) lifted on R. It is invertible from (2). Let us suppose forsimplicity that E(0) = 0.

One easily establishes the following properties of the map F̃ , illustrated in figure 6:

• The point

I := (0, ξI ) :=(

0, − τ′(0)

(E′(0) − 1))

is the unique fixed point of F̃ . It is hyperbolic with unstable manifold

Wu = {(0, ξ), ξ ∈ R}and stable manifold

Ws = {(x, S(x)), x ∈ R}, (62)where the C∞ function S(x) is defined by the following co-homological equation, deduceddirectly from (61)

S(E−1(x)) = E′(E−1(x))S(x) + τ ′(E−1(x)), S(0) = ξI .This gives

S(x) = 1E′(E−1(x))

(S(E−1(x)) − τ ′(E−1(x)))and recursively we deduce that

S(x) = −∞∑

p=1

1

E′(−p)(x)τ ′(E(−p)x), (63)

where

x(−p) := E(−p)(x) := (E−1 ◦ . . . ◦ E−1︸︷︷︸p

)(x) (64)

and

E′(−p)(x) := E′(x−p) . . . E′(x−2)E′(x−1) (65)is the product of derivatives. In the case of E(x) = 2x, one obtains simply

S(x) = −∞∑

p=1

1

2pτ ′

( x2p

). (66)


Figure 6. The fixed point I = (0, ξ0), the stable manifold Ws and unstable manifold Wu of thelifted map F̃ , equation (61), in the example E(x) = 2x, τ(x) = cos(2πx).

• If P : (x, ξ) ∈ R2 → (x mod 1, ξ) ∈ S1 ×R ≡ T ∗S1 denotes the projection, then trappedset K , defined in equation (43) is obtained by wrapping the stable manifold around thecylinder and taking the closure:

K = P(Ws) (67)

Compare figures 6 and 4.

• If X0 = (x0, ξ0) ∈ R2 is an initial point on the plane, and P(X0) denotes its image onthe cylinder, then at time n ∈ N, the kn evolutions of the point P(X0) under the map Fnare the images of the evolutions F̃ n(Xk) of the translated points Xp = X0 + (p, 0), withp = 0 → kn − 1:

Fn(P(X0)) = {P(F̃ n(Xp)), Xp = X0 + (p, 0), p ∈ {0, . . . , kn − 1}}

and more precisely, using notation of equation (39) for these points, one has the relation:

Fnε (P(X0)) = P(F̃ n(Xp)), (68)

where ε is the number p written in base k:

ε = εn−1 . . . ε1ε0 = pbase k ∈ {0, . . . , k − 1}n

Figure 7 illustrates this correspondence.

• For an initial point X0 = (x0, ξ0) ∈ R2, then Xn = (xn, ξn) = F̃ n(X0) satisfies

xn = E(−n)(x0), ξn − S(xn) = (E′(−n)(x0))(ξ0 − S(x0)) (69)

with E(−n)(x), E′(−n)(x) given by (64) and (65). Hence

|ξn − S(xn)| � Enmin|ξ0 − S(x0)|.

This last inequality describes how fast the trajectories above or below the separatrix Wsescape towards infinity in figure 6.

1496 F Faure

Figure 7. This picture shows how the dynamics of a point X = P(X0) ∈ S1 × R under the mapF is related by equation (68) to the dynamics of its lifted images Xk = X0 + (k, 0) under F̃ on thecover R2.

Appendix B.2. Partially captive property

Here we rephrase the property of partial captivity, definition 15, in terms of a property on theseparatrix function S(x) defined in equation (62) and given in equation (63).

Proposition 17. For n ∈ N, and R̃ > 0, let

ÑR̃(n) = max(x,ξ)∈R2

{p ∈ {0, . . . , kn − 1}, |ξ − S(x + p)| � R̃

E′(−n)(x + p)

}. (70)

Then the map F is partially captive (see definition p 1489) if and only if

limn→∞

log ÑR̃(n)n

= 0 (71)

for every R̃.

Proof. From (44) and using (68) and (69) one obtains

N (n) = max(x,ξ)

{Fnε ((x, ξ)) ∈ Z, ε ∈ {0, . . . , k − 1}n}= max

(x,ξ)

{|ξn,p| � R, p ∈ {0, . . . , kn − 1}}

with (xn,p, ξn,p) := F̃ n((x + p, ξ)) given byxn,p = E(−n)(x + p)

and

ξn,p = S(xn,p) + (E′(−n)(x + p))(ξ − S(x + p)).Therefore

|ξn,p| � R ⇔ | S(xn,p)E′(−n)(x + p)

+ ξ − S(x + p)| � RE′(−n)(x + p)

.


Figure 8. This picture represents the set Kcx ⊂ C defined by equation (72). Thetrapped set K at position x is obtained by the projection on the imaginary axis Kx =�(Kcx). In the supplementary data (available at stacks.iop.org/Non/24/1473/mmedia orwww-fourier.ujf-grenoble.fr/∼faure/articles/09 html partially expanding maps) one can observethe motion of the fractal Kcx as x ∈ R increases smoothly.

But S is a bound function on R, |S(x)| � Smax. Therefore the last equation is equivalentto |ξ − S(x + p)| � R̃

E′(−n)(x+p) with some R̃ > 0, and conversely. This implies that (71) isequivalent to (45). �

Appendix B.3. Fractal aspect of the trapped set

The characterization equation (70) concerns the discrete set of points S(x + m), m ∈ Z. Fromequation (67) these points are the slice of the trapped set K = ∪x∈S1Kx :

Kx = {S(x + m), m ∈ Z}.Since we present here merely an observation and not a proof, we consider, for simplicity,

the simple model with E(x) = 2x, and τ(x) = cos(2πx). From equation (66), these pointsare given by

S(x + m) =∞∑

p=1

2π

2psin

(2π

2p(x + m)

)= �

( ∞∑p=1

2π

2pexp

(i2π

2p(x + m)

)).

Therefore, the slice Kx is the projection on the imaginary axis of the following set:

Kcx = {Sc(x + m), m ∈ Z}, (72)

Sc(x + m) =∞∑

p=1

2π

2pei2π

12p (x+m).

In figure 8 we observe that Kcx is a fractal set. (a proof of this would require the theory of‘iterated function systems’ [Fal97]). Compare figures 8 with 4.

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1. Introduction2. Model and results2.1. A partially expanding map2.2. Transfer operator2.3. The reduced transfer operator2.4. Main results on the spectrum of the transfer operator "705E F2.5. Spectrum of "705E F and dynamical correlation functions

3. Proof of theorem 2 on resonances spectrum3.1. Dynamics on the cotangent space T*S13.2. Dynamics of wave packets3.3. The escape function

4. Proof of theorem 3 on spectral gap4.1. Dynamics on the cotangent space T*S14.2. The trapped set K4.3. The escape function

AcknowledgmentsAppendix A. Equivalence classes of dynamicsAppendix B. Description of the trapped setAppendix B.1. Dynamics on the cover R2Appendix B.2. Partially captive propertyAppendix B.3. Fractal aspect of the trapped set

References

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