Frédéric Faure
email: frederic.faure@ujf-grenoble.fr

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

Keywords: 2000 Mathematics Subject Classification:37D20 hyperbolic systems (expanding, Anosov, Axiom A, etc.) 37C30 Zeta functions, (Ruelle-Frobenius) transfer operators, and other functional analytic techniques in dynamical systems 81Q20 Transfer operator, Ruelle resonances, decay of correlations, Semi-classical analysis

. The novelty here is that we use semiclassical analysis which provides a different and quite natural description. We show that the transfer operator is a semiclassical operator with a well defined "classical dynamics" on the cotangent space. This classical dynamics has a "trapped set" which is responsible for the Ruelle resonances spectrum. In particular we show that the spectral gap is closely related to a specific dynamical property of this trapped set.

Introduction

Chaotic behavior of certain dynamical systems is due to hyperbolicity of the trajectories. This means that the trajectories of two closed initial points will diverge from each other either in the future or in the past (or both). As a result the behavior of an individual trajectory looks like complicated and unpredictable. However evolution of a cloud of points seems more simple: it will spread and equidistributes according to an invariant measure, called an equilibrium measure (or S.R.B. measure). Following this idea, D. Ruelle in the 70' [START_REF] Ruelle | Thermodynamic formalism. The mathematical structures of classical equilibrium[END_REF][START_REF] Ruelle | Locating resonances for axiom A dynamical systems[END_REF], has shown that instead of considering individual trajectories, it is much more natural to consider evolution of densities under a linear operator called the Ruelle Transfer operator or the Perron Frobenius operator.

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

This spectral approach has recently (2002)(2003)(2004)(2005) been improved by M. Blank, S. Gouëzel, G. Keller, C. Liverani [BKL02, GL05, Liv05] and V. Baladi and M. Tsujii [START_REF] Baladi | Anisotropic Sobolev spaces and dynamical transfer operators: C ∞ foliations[END_REF][START_REF] Baladi | Anisotropic Hölder and Sobolev spaces for hyperbolic diffeomorphisms[END_REF] (see [START_REF] Baladi | Anisotropic Hölder and Sobolev spaces for hyperbolic diffeomorphisms[END_REF] for some historical remarks), through the construction of functional spaces adapted to the dynamics, independent of any symbolic dynamics. The case of dynamical systems with continuous time is more delicate (see [START_REF] Field | Stability of mixing and rapid mixing for hyperbolic flows[END_REF] for historical remarks). This is due to the direction of time flow which is neutral (i.e. two nearby points on the same trajectory will not diverge from each other). In 1998 Dolgopyat [START_REF] Dolgopyat | On decay of correlations in Anosov flows[END_REF][START_REF] Dolgopyat | On mixing properties of compact group extensions of hyperbolic systems[END_REF] showed the exponential decay of correlation functions for certain Anosov flows, using techniques of oscillatory integrals and symbolic dynamics. In 2004 Liverani [START_REF] Liverani | On contact Anosov flows[END_REF] adapted Dolgopyat's ideas to his functional analytic approach, to treat the case of contact Anosov flows. In 2005 M. Tsujii [START_REF] Tsujii | Decay of correlations in suspension semi-flows of angle-multiplying maps[END_REF] obtained an explicit estimate for the spectral gap for the suspension of an expanding map. Then in 2008 M. Tsujii [START_REF] Tsujii | Quasi-compactness of transfer operators for contact Anosov flows[END_REF] obtained an explicit estimate for the spectral gap, in the case of contact Anosov flows.

Semiclassical approach for transfer operators: It also appeared recently [START_REF] Faure | Ruelle-pollicott resonances for real analytic hyperbolic map[END_REF][START_REF] Faure | A semiclassical approach for anosov diffeomorphisms and ruelle resonances[END_REF] that for hyperbolic dynamics, the study of transfer operator is naturally a semiclassical problem in the sense that a transfer operator can be considered as a "Fourier integral operator" and using standard tools of semiclassical analysis, some of its spectral properties can be obtained from the study of "the associated classical symplectic dynamics", namely the initial hyperbolic dynamics lifted on the cotangent space (the phase space).

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

Following this approach, in the papers [START_REF] Faure | Ruelle-pollicott resonances for real analytic hyperbolic map[END_REF][START_REF] Faure | A semiclassical approach for anosov diffeomorphisms and ruelle resonances[END_REF] we studied hyperbolic diffeomorphisms. The aim of the present paper is to show that semiclassical analysis is also well adapted (even better) for hyperbolic systems with neutral direction. We consider here the simplest model: a partially expanding map f : (x, s) → f (x, s), i.e. a map on a torus (x, s) ∈ S 1 × S 1 with an expanding direction x ∈ S 1

x and a neutral direction s ∈ S 1 s (the inverse map f -1 is k-valued, with k ≥ 2). The results are presented in section 2. We summarize them in few lines. First in order to reduce the problem and drop out the neutral direction, we use a Fourier analysis in s ∈ S 1 s and decompose the transfer operator F on S 1

x × S 1 s (defined by F ϕ := ϕ • f ) as a collection of transfer operators Fν on the expanding space S 1

x only, with ν ∈ Z being the Fourier parameter and playing the role of the semiclassical parameter. The semiclassical limit is |ν| → ∞.

Then we introduce a (multivalued) map F ν on the cotangent space (x, ξ) ∈ T * S 1 x which is the canonical map associated to the transfer operator Fν . The fact that the initial map f is expanding along the space S 1

x implies that on the cylinder T * S 1 x trajectories starting from a large enough value of |ξ| escape towards infinity (|ξ| → ∞). We define the trapped set as the compact set K = lim n→∞ F -n ν (K 0 ) where K 0 ⊂ T * S 1 is an initial large compact set. K contains trajectories which do not escape towards infinity.

Using a standard semiclassical approach (with escape functions on phase space [START_REF] Helffer | Résonances en limite semi-classique. (resonances in semiclassical limit)[END_REF]) we first show that the operator Fν as a discrete spectrum called Ruelle resonances (we have to consider Fν in Sobolev space of distributions). This is Theorem 1. 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 Fν develops a "spectral gap" in the semi-classical limit ν → ∞ (i.e. its spectral radius reduces). This is Theorem 2 illustrated on Figure 2. This Theorem is very similar to Theorem 1.1 in [START_REF] Tsujii | Decay of correlations in suspension semi-flows of angle-multiplying maps[END_REF]. With the semiclassical approach, this result is very intuitive: the basic idea (followed in the proof) is that an initial wave packet ϕ 0 represented as a point on the trapped set K evolves in several wave packets (ϕ j ) j=1→k under the transfer operator Fν , but in general only one wave packet remains on the trapped set K and the (k -1) other ones escape towards infinity. As a result the probability on the trapped set K decays by a factor 1/k. This is the origin of the spectral gap at 1/ √ k on Figure 2.

This work has been supported by "Agence Nationale de la Recherche" under the grant JC05 52556.

Model and results

A partially expanding map

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

E : x ∈ S 1 → E (x) = kg (x) mod 1 (1) Let E min := min x dE dx (x) = k min x dg dx (x)
We will suppose that the function g is such that

E min > 1 (2)
so that E is a uniform expanding map on S 1 . The map E is then a k : 1 map (i.e. every point y has k previous images x ∈ E -1 (y)). Let τ : S 1 → R be a C ∞ function, and define a map f on T 2 = S 1 × S 1 by:

f : x s -→ x ′ = E (x) = kg (x) mod 1 s ′ = s + 1 2π τ (x) mod 1 (3)
The map f is also a k : 1 map. The map f is a very simple example of a compact group extension of the expanding map E (see [START_REF] Dolgopyat | On mixing properties of compact group extensions of hyperbolic systems[END_REF], [Pes04, p.17]). It is also a special example of a partially hyperbolic map 2 . See figure 1. (3), at different time n = 0, 2, 10, 19. We have chosen here E (x) = 2x and τ (x) = cos (2πx). The initial cloud of points is centered 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 horizontal direction x. Due to instability in x and periodicity, the cloud fills the torus S 1 × S 1 for large time n. 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 property of the map f in this example.

Transfer operator

Instead of studying individual trajectories which have chaotic behavior, one prefer to study the evolutions of densities induced by the map f . This is the role of the Perron-Frobenius transfer operator F * on C ∞ T 2 given by:

F * ψ (y) = x∈f -1 (y) 1 |D x f | ψ (x) , ψ ∈ C ∞ T 2 . ( 4
)
Indeed if the the function ψ has its support in a vicinity of x then the support of F * ψ is in a vicinity of y = f (x). To explain the Jacobian in the prefactor, one checks 3 that T 2 F * ψ (y) dy = T 2 ψ (x) dx, i.e. the total measure is preserved.

The operator F * extends to a bounded operator on L 2 T 2 , dx . Its L 2 -adjoint written F is defined by F * ψ, ϕ

L 2 = ψ, F ϕ L 2 , with the scalar product (ψ, ϕ) L 2 := T 2 ψ (x) ϕ (x) dx. One
2 A even more general setting would be a C ∞ map f : M → M on a compact Riemannian manifold M , which is supposed to be partially expanding, i.e., for any m ∈ M , the tangent space TmM decomposes continuously as

TmM = Eu (m) ⊕ E 0 (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 E 0 (m) the neutral direction: there exist a non zero global section v 0 ∈ C ∞ (T M ) such that v 0 (m) ∈ E 0 (m) and Df (v 0 ) = v 0 . In our example (3), M = S 1 × S 1 , the neutral section is v 0 = (0, 1), and the expanding direction Eu (m) is spanned by the vector (1, 0).

3 Since y = f (x), then dy = |Dxf | dx, and

Z T 2 " F * ψ " (y) dy = X x∈f -1 (y) Z T 2 1 |Dxf | ψ (x) |Dxf | dx = Z T 2 ψ (x) dx
checks easily that F has a simpler expression than F * : it is the pull back operator, also called the Koopman operator, or Ruelle transfer operator and given by:

F ψ (x) = ψ (f (x)) (5)

The reduced transfer operator

The particular form of the map (3) allows some simplifications. Observe that for a function of the form ψ (x, s) = ϕ (x) e i2πνs with ν ∈ Z (i.e. a Fourier mode in s), then

F ψ (x, s) = ϕ (E (x)) e iντ (x) e i2πνs .
Therefore the operator F preserves the following decomposition in Fourier modes:

L 2 T 2 = ν∈Z H ν , H ν := ϕ (x) e i2πνs , ϕ ∈ L 2 S 1 (6)
The space H ν and L 2 S 1 are unitary equivalent. For ν ∈ Z given, the operator F restricted to the space

H ν ≡ L 2 S 1 , written Fν is 4 : Fν ϕ (x) := ϕ (E (x)) e iντ (x) ϕ ∈ L 2 S 1 ≡ H ν (7)
and with respect to the orthogonal decomposition (6), we can write:

F = ν∈Z Fν
We will study the spectrum of this family of operators Fν , with parameter ν ∈ Z, and consider more generally a real parameter ν ∈ R. We will see that the parameter ν is a semiclassical parameter, and ν → ∞ is the semiclassical limit. (if ν = 0, ν = 1/ in usual notations [START_REF] Martinez | An Introduction to Semiclassical and Microlocal Analysis[END_REF]).

Remarks:

• For ν = 0, F0 has an obvious eigenfunction ϕ (x) = 1, with eigenvalue 1. Except in special cases (e.g. τ = 0), there is no other obvious eigenvalues for Fν in L 2 S 1 .

Main results on the spectrum of the transfer operator Fν

We first observe that by duality, the operator Fν defined in (7) extends to the distribution space

D ′ S 1 : Fν (α) (ϕ) = α F * ν (ϕ) , α ∈ D ′ S 1 , ϕ ∈ C ∞ S 1 ,
where the L 2 -adjoint F * ν is given by

F * ν ϕ (y) = x∈E -1 (y) e -iντ (x) E ′ (x) ϕ (x) , ϕ ∈ C ∞ S 1 . (8)
Before giving the main results, remind that for m ∈ R, the Sobolev space H m S 1 ⊂ D ′ S 1 consists in distributions (or continuous functions if m > 1/2) such that their Fourier series ψ (ξ)

satisfy ξ∈Z ξ m ψ (ξ) 2 < ∞, with ξ := 1 + ξ 2 1/2
. It can equivalently be written [START_REF] Taylor | Partial differential equations[END_REF] p.271).

H m S 1 := ξ -m L 2 S 1
with the differential operator ξ := -i d dx . The following theorem is well known [START_REF] Ruelle | Locating resonances for axiom A dynamical systems[END_REF]. We will however provide a new proof based on semiclassical analysis.

Theorem 1. Discrete spectrum of resonances.

Let m < 0. The operator Fν leaves the Sobolev space H m S 1 invariant, and

Fν : H m S 1 → H m S 1
is a bounded operator and can be written

Fν = R + K (9)
where K is a compact operator, and R has a small norm:

R ≤ r m := 1 E |m| min k E min . ( 10
)
(the interesting situation is m ≪ 0, since the norm R shrinks to zero for m → -∞).

Therefore, Fν has an essential spectral radius less than r m , which means that Fν has discrete (eventually empty) spectrum of generalized eigenvalues λ i outside the circle of radius r m (see [START_REF] Taylor | Partial differential equations[END_REF] prop. 6.9 p.499]). The eigenvalues λ i are called Ruelle resonances. Together with their associated eigenspace, they do not depend on m and are intrinsic to the transfer operator Fν .

The following theorem is analogous to Theorem 1.1 in [START_REF] Tsujii | Decay of correlations in suspension semi-flows of angle-multiplying maps[END_REF]. However the approach and the proof we propose are different and rely on semiclassical analysis.

Theorem 2. Spectral gap in the semiclassical limit. if the map f is partially captive (definition given page 15) (and m small enough), then the spectral radius of the operator Fν : H m S 1 → H m S 1 does not depend on m and satisfies in the semi-classical limit ν → ∞:

r s Fν ≤ 1 √ E min + o (1) (11)
which is strictly smaller than 1 from (3).

Remarks:

• This remark concerns the regularity of the eigenfunctions of Fν . Let λ i be a generalized eigenvalue of Fν . Let ϕ i denotes a generalized eigenfunction of Fν associated to λ i (i.e. Fν ϕ i = λ i ϕ i if λ i is an eigenvalue). Then ϕ i belongs to H m for any m such that m < m 0 where m 0 is given by

r m0 = |λ i |. ν = 0 ν = 1 ν = 10 ν = 50 ν = 0 → 80
ν : H m S 1 → H m S 1 if m > 0. The eigenvalues of F * ν are λ i .
We have seen that the generalized eigenfunctions of Fν belong to different Sobolev spaces H m S 1 with m < 0. Eq.(10) says that m should satisfy

r m < |λ i |, so m → -∞ as |λ i | → 0. The situation is simpler for the generalized eigenfunctions of F * ν since they all belong to m>0 H m = H ∞ = C ∞ S 1 .
• In the proof of Theorem 2, we will obtain that a general bound for r s Fν (with no hypothesis on f ) is given by

r s Fν ≤ 1 √ E min exp 1 2 lim n∞ log N (n) n + o (1) (12)
where the function N (n) will be defined in Eq.(30). This bound is similar to the bound given in [Tsu08a, Theorem 1.1] by M. Tsujii.

• In [Tsu08a, Theorem 1.2] M. Tsujii shows that the partially captive property, i.e.

lim n∞ log N (n) n = 0, is true for almost all functions τ . • From the definition of N (n) it is clear that N (n) ≤ k n hence exp 1 2 lim n∞ log N (n) n ≤ √ k.
Also from the definition of E min , it is clear that E min ≤ k and therefore the upper bound in (12) is not sharp since it does not give the obvious bound r s Fν ≤ 1 (see [START_REF] Faure | A semiclassical approach for anosov diffeomorphisms and ruelle resonances[END_REF] corollary 2]). It is therefore tempting to conjecture that for almost all functions τ Eq.( 11) can be replaced by:

r s Fν ≤ 1 √ k + o (1)
• Notice that the above results say nothing about the existence of Ruelle resonances λ i . The work of F. Naud [START_REF] Naud | Entropy and decay of correlations for real analytic semi-flows[END_REF] are the first results concerning the existence of resonances λ i .

• One observes numerically that for large ν ∈ R, the eigenvalues λ i (ν) repulse each other like eigenvalues of random complex matrices. (See a movie on http://www-fourier.ujf-grenoble. fr/ ∼ faure/articles). This suggests that many important questions of quantum chaos (e.g. the conjecture of Random Matrices [START_REF] Bohigas | Random matrix theories and chaotic dynamics[END_REF]) also concerns the Ruelle resonances of partially hyperbolic dynamics in the semiclassical limit.

• Remarks on numerical computation of the Ruelle resonances: one diagonalizes the matrix which expresses the operator Fν in Fourier basis ϕ n (x) := exp (i2πnx), n ∈ Z. For the example of Figure 2 one gets • One can proves [AF09] that in the semi-classical limit ν → ∞, the number of Ruelle resonances λ i (counting multiplicities) outside a fixed radius λ is bounded by a "Weyl law":

ϕ n ′ | F ϕ n = e -i2π 3 4 (2n-n ′ ) J (2n-n ′ ) (ν) where J n (x)
∀λ > 0, ♯ {i ∈ N, s.t. |λ i | ≥ λ} ≤ ν 2π µ (K) + o (ν)
where µ (K) is the Lebesgue measure of the trapped set K defined later in Eq.(29). As usual in the semiclassical theory of non selfadjoint operators, see [START_REF] Sjöstrand | Geometric bounds on the density of resonances for semiclassical problems[END_REF][START_REF] Sjöstrand | Fractal upper bounds on the density of semiclassical resonances[END_REF], the Weyl law gives an upper bound for the density of resonances but no lower bound. See discussions in [Non08, section 3.1].

Spectrum of F and dynamical correlation functions

In this section, in order to give some "physical meaning" to the spectrum of Fν , we recall relations between the spectral results of Theorems 1,2 and the evolution of correlation functions [START_REF] Baladi | Positive transfer operators and decay of correlations[END_REF]. This will allow us to interpret the evolution and convergence of clouds of points observed in Figure 1.

Let ν ∈ Z. If ψ 1 , ψ 2 ∈ C ∞ S 1 , the correlation function at time n ∈ N is defined by: C ψ2,ψ1 (n) := F * n ν ψ 2 , ψ 1 L 2 = ψ 2 , F n ν ψ 1 L 2
which represents the function ψ 2 evolved n times by the Perron-Frobenius operator F * ν and tested against the test function ψ 1 .

The first spectral result of Theorem 1 implies that for any ε > 0, and large n (and assuming that the eigenvalues (λ i,ν ) i of Fν are simple for short; see [START_REF] Faure | A semiclassical approach for anosov diffeomorphisms and ruelle resonances[END_REF] for a more extended discussion)

C ψ2,ψ1 (n) = |λi,ν |>0 λ n i,ν v i,ν ψ 2 w i,ν (ψ 1 ) + O ε (ε n )
If the conclusion of Theorem 2 holds, this implies that for any ρ such that 1 √ Emin < ρ < 1, there exists ν 0 such that for any ν ≥ ν 0 , all the eigenvalues of Fν are bounded: |λ i,ν | < ρ < 1, ∀i. This gives an exponential decay of correlations for n → ∞ in these space Fν :

C ψ2,ψ1 (n) = O (ρ n )
It is known that if the function τ is not a co-boundary (i.e. if the map f is not equivalent to the trivial case τ = 0, as explained in Appendix A) then the map f is ergodic, which implies that all the eigenvalues λ i,ν are strictly less than one: |λ i,ν | < 1, ∀ν, ∀i, except for λ 0,0 = 1 associated to the eigenfunction ϕ (x) = 1. One deduces mixing property of the dynamics as observed in Figure 1.

Proof of theorem 1 on resonances spectrum

In this proof, we follow closely the proof of theorem 4 in [START_REF] Faure | A semiclassical approach for anosov diffeomorphisms and ruelle resonances[END_REF] although we deal here with expanding map instead of hyperbolic map, and this simplifies a lot, since we can work with ordinary Sobolev spaces and not anisotropic Sobolev spaces. Here ν ∈ Z is fixed.

Dynamics on the cotangent space T * S 1

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 been exploited in [START_REF] Faure | A semiclassical approach for anosov diffeomorphisms and ruelle resonances[END_REF].

In Eq.(1), the map E : S 1 → S 1 is a k : 1 map, which means that every point y ∈ S 1 has k inverses denoted by x ε ∈ E -1 (y) and given explicitly by

x ε = E -1 ε (y) = g -1 y k + ε 1 k , with ε = 0, . . . , k -1
We will denote the derivative by E ′ (x) := dE/dx.

Proposition 1. In Eq.( 7) Fν is a Fourier integral operator (FIO) acting on C ∞ S 1 . The associated canonical transform on the cotangent space (x, ξ) ∈ T * S 1 ≡ S 1 × R is k-valued and given by:

F (x, ξ) = {F 0 (x, ξ) , . . . , F k-1 (x, ξ)} , (x, ξ) ∈ S 1 × R ( 13
)
where for any ε = 0, . . . , k -1,

F ε : x → x ′ ε = E -1 ε (x) = g -1 1 k x + ε 1 k ξ → ξ ′ ε = E ′ (x ′ ε ) ξ = kg ′ (x ′ ε ) ξ (14)
Similarly the adjoint F * is a FIO whose canonical transformation is F -1 . See figure 3.

The proof is just that the operator ϕ → ϕ • E on C ∞ S 1 is one of the simplest example of Fourier integral operator, see [START_REF] Martinez | An Introduction to Semiclassical and Microlocal Analysis[END_REF] example 2 p.150.

The term e iντ (x) in Eq.( 7) does not contribute to the expression of F , since here ν is considered as a fixed parameter, and therefore e iντ (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 * S 1 in the canonical way. Indeed, if we denote a point (x, ξ) ∈ T * S 1 ≡ S 1 × R then using the usual formula for differentials y = E (x) = kg (x)⇒dy = E ′ (x) dx ⇔ ξ ′ = E ′ (x) ξ, we deduce the above expression for F .

Remarks

• The physical meaning for Fε being a Fourier Integral Operator is that if ϕ (x,ξ) is a wave packet "micro-localized" at position (x, ξ) ∈ T * S 1 of phase space (this makes sense for ξ ≫ 1, and means that the micro-support of ϕ is (x, ξ)), then ϕ ′ := Fν ϕ (x,ξ) will be a superposition of k wave packets at positions (x ′ ε , ξ ′ ε ) = F ε (x, ξ), ε = 0, . . . , k -1, i.e. with a very restricted micro-support, controlled by the canonical map F .

• Observe that the dynamics of the map F on S 1 × R has a quite simple property: the zero section (x, ξ) ∈ S 1 × R, ξ = 0 is globally invariant and any other point with ξ = 0 escapes towards infinity (ξ → ±∞) in a controlled manner:

|ξ ′ ε | ≥ E min |ξ| , ∀ε = 0, . . . , k -1 ( 15
)
where

E min > 1 is given in (2). 0 1 x ξ F -1 F F 0 (x, ξ) F 1 (x, ξ) (x, ξ) Figure 3: This figure is for k = 2. The map F = {F 0 , . . . , F k-1 } is 1:k, and its inverse F -1 is k:1 on T * S 1 ≡ S 1 × R.

The escape function

Let m < 0 and define the C ∞ function on T * S 1 :

A m (x, ξ) := ξ m ∈ S m
with ξ = 1 + ξ 2 1/2 . A m decreases with |ξ| and belongs to the symbol class 5 S m . Eq. ( 15) implies that the function A m decreases strictly along the trajectories of F outside the zero section:

∀R > 0, ∀ |ξ| > R, ∀ε = 0, . . . , k-1 A m (F ε (x, ξ)) A m (x, ξ) ≤ C |m| < 1, with C = R 2 + 1 R 2 E min + 1 < 1 (16) Proof. A m (F ε (x, ξ)) A m (x, ξ) = 1 + ξ 2 |m|/2 1 + (ξ ′ ε ) 2 |m|/2 ≤ 1 + ξ 2 |m|/2 (1 + E min ξ 2 ) |m|/2 ≤ 1 + R 2 1 + E min R 2 |m|/2 = C |m| 5 See [Tay96b] p.2.
The class of symbols S m , with order m ∈ R, consists of functions on the cotangent space

A ∈ C ∞ `S1 × R ´such that ˛∂α ξ ∂ β x A ˛∞ ≤ C α,β ξ m-|α| , ξ = `1 + ξ 2 ´1/2
The symbol A m can be quantized into a pseudodifferential operator Âm (PDO for short) which is self-adjoint and invertible on C ∞ S 1 using the quantization rule ([Tay96b] p.2)

Âϕ (x) = 1 2π A (x, ξ) e i(x-y)ξ ϕ (y) dydξ, (17)
but in our simple case, this is very explicit: in Fourier space, Âm is simply the multiplication by ξ m . Remind that the Sobolev space H m S 1 is defined by ([Tay96a] p.271):

H m S 1 := Â-1 m L 2 S 1
The following commutative diagram

L 2 S 1 Qm → L 2 S 1 ↓ Â-1 m ↓ Â-1 m H m S 1 Fν → H m S 1
shows that Fν :

H m S 1 → H m S 1 is unitary equivalent to Qm := Âm Fν Â-1 m : L 2 S 1 → L 2 S 1
We will therefore study the operator Qm . Notice that Qm is defined a priori on a dense domain (C ∞ S 1 ). Define

P := Q * m Qm = Â-1 m F * ν Â2 m Fν Â-1 m = Â-1 m B Â-1 m (18)
where appears the operator

B := F * ν Â2 m Fν (19)
The Egorov Theorem will help us to treat this operator (see [START_REF] Taylor | Partial differential equations[END_REF] p.24). This is a simple but crucial step in the proof: as explained in [START_REF] Faure | A semiclassical approach for anosov diffeomorphisms and ruelle resonances[END_REF], the Egorov theorem is the main Theorem used in order to establish both the existence of a discrete spectrum of resonances and properties of them. However there is a difference with [START_REF] Faure | A semiclassical approach for anosov diffeomorphisms and ruelle resonances[END_REF]: for the expanding map we consider here, the operator Fν is not invertible and the canonical map F is k-valued. Therefore we have to state the Egorov theorem in an appropriate way (we restrict however the statement to our simple context).

Lemma 1. (Egorov theorem). B := F

* ν Â2
m Fν is a pseudo-differential operator with symbol in S m given by:

B (x, ξ) =   ε=0,...,k-1 1 E ′ (x ′ ε ) A 2 m (F ε (x, ξ))   + R (20)
with R ∈ S m-1 has a subleading order.

Proof. As we explained in Proposition 1, Fν and F * 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 we deduce that B = F * ν Â2 m Fν is a FIO whose canonical map is the identity since F -1 • F = Id. See figure 3. Therefore B is a PDO. Using (7), ( 8) and ( 14) we obtain that the principal symbol of B is ε=0,...,k-1

1 E ′ (x ′ ε ) A 2 m (F ε (x, ξ)) (21)
Remark: contrary to (19), Fν Âm F * ν is not a PDO, but a FIO whose canonical map F • F -1 is k-valued (see figure 3). Now by theorem of composition of PDO ([Tay96b] p.11), ( 18) and (20) imply that P is a PDO of order 0 with principal symbol:

P (x, ξ) = B (x, ξ) A 2 m (x, ξ) =   ε=0,...,k-1 1 E ′ (x ′ ε ) A 2 m (F ε (x, ξ)) A 2 m (x, ξ)  
The estimate (16) together with (2) give the following upper bound

∀ |ξ| > R, |P (x, ξ)| ≤ C 2|m| ε=0,...,k-1 1 E ′ (x ′ ε ) ≤ C 2|m| k E min
(This upper bound goes to zero as m → -∞). From L 2 -continuity theorem for PDO we deduce that for any α > 0 (see [START_REF] Faure | A semiclassical approach for anosov diffeomorphisms and ruelle resonances[END_REF] Emin + α, with any α > 0. Since Û = 1 we deduce a similar decomposition for Qm = Û Q : L 2 S 1 → L 2 S 1 and we deduce (9) and (10) for Fν : H m → H m . We also use the fact that C → 1/E min for R → ∞ in (16).

The fact that the eigenvalues λ i and their generalized eigenspaces do not depend on the choice of space H m is due to density of Sobolev spaces. We refer to the argument given in the proof of corollary 1 in [START_REF] Faure | A semiclassical approach for anosov diffeomorphisms and ruelle resonances[END_REF]. This finishes the proof of Theorem 1.

Proof of theorem 2 on spectral gap

We will follow steps by steps the same analysis as in the previous section. The main difference now is that in Theorem 2, ν ≫ 1 is a semi-classical parameter. In other words, we just perform a linear rescaling in cotangeant space:

ξ h := ξ with := 1 ν ≪ 1.
Therefore, our quantization rule for a symbol A (x, ξ h ), Eq.( 17) writes now (see [START_REF] Martinez | An Introduction to Semiclassical and Microlocal Analysis[END_REF] p.22)

Âϕ (x) = 1 2π A (x, ξ) e i(x-y)ξ h / ϕ (y) dydξ h ( 22
)
For simplicity we will write ξ for ξ h below.

4.1 Dynamics on the cotangent space T * S 1

In Eq.( 7) the multiplicative term e iντ (x) = e iτ (x)/ acts now as a Fourier integral operator (FIO) and contributes to the transport (it was merely a P.D.O. for theorem 1 in Section 3 when ν was fixed). Its associated canonical transformation on T * S 1 = S 1 × R is (x, ξ) → x, ξ + dτ dx (x) (this is a direct consequence of stationary phase approximation in Fourier transform see [START_REF] Martinez | An Introduction to Semiclassical and Microlocal Analysis[END_REF] Examples 1,2 p.150]). We obtain: Proposition 2. In Eq.( 7) Fν is a semi-classical Fourier integral operator acting on C ∞ S 1 (with semi-classical parameter := 1/ν ≪ 1). The associated canonical transformation on the cotangent space (x, ξ) ∈ T * S 1 ≡ S 1 × R is k-valued and given by:

F (x, ξ) = {F 0 (x, ξ) , . . . , F k-1 (x, ξ)} , (x, ξ) ∈ S 1 × R (23) F ε : x → x ′ ε = E -1 (x) ξ → ξ ′ ε = E ′ (x ′ ε ) ξ + dτ dx (x ′ ε ) , ε = 0, . . . , k -1 (24)
Similarly F * is a FIO whose canonical transformation is F -1 .

Notice that for simplicity we have kept the same notation for the canonical transformation F although it differs from (14).

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

Definition 1. For ε = (. . . ε 3 , ε 2 , ε 1 ) ∈ {0, . . . , k -1} N *
, a point (x, ξ) ∈ S 1 × R and time n ∈ N * let us denote:

F n ε (x, ξ) := F εn F εn-1 . . . F ε1 (x, ξ) (25)
For a given sequence ε ∈ {0, . . . , k -1} N * , a trajectory issued from the point (x, ξ) is

{F n ε (x, ξ) , n ∈ N}.
Notice that at time n ∈ N, there are k n points issued from a given point (x, ξ):

F n (x, ξ) := {F n ε (x, ξ) , ε ∈ {0, . . . , k -1} n } (26)
The new term dτ dx (x ′ ε ) in the expression of ξ ′ ε , Eq.( 24), 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 2. For any 1 < κ < E min , there exists R ≥ 0 such that for any |ξ| > R, any ε = 0, . . . k-1,

|ξ ′ ε | > κ |ξ| (27) Proof. From (24), 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 ′ (E min -κ) ≥ 0, and similarly ξ ′ ε -κξ ≤ (Emin -κ) ξ + max τ ′ < 0 if ξ < -max τ ′ (E min -κ) .
We will denote the set:

Z := S 1 × [-R, R] (28)
outside of which trajectories escape in a controlled manner (27). See figure 4.

The trapped set K

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

Definition 2. We define the trapped set Since the map F is multivalued, some trajectories may escape from the trapped set. We will need a characterization of how many such trajectories succeed to escape:

K := n∈N F -1 n (Z) (29
For n ∈ N, let

N (n) := max (x,ξ) ♯ {F n ε (x, ξ) ∈ Z, ε ∈ {0, . . . , k -1} n } (30)
See Figure 5 for an illustration of

N (n). Of course N (n) ≤ k n . Definition 3. The map F (or f ) is partially captive if log N (n) n -→ n→∞ 0 ( 31
)
This property is the hypothesis of Theorem 2.

Remarks

• "F partially captive" 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 property will be given in Appendix B. Notice that the function N (n), Eq.(30) depends on the set Z but property (31) does not.

• If the function τ is trivial in (24), i.e. τ = 0 , then obviously all the trajectories issued from a point (x, ξ) on the line ξ = 0 remains on this line (the trapped set). Therefore

♯ {F n ε (x, ξ) ∈ Z, ε ∈ {0, 1, . . . , k -1} n } = k n
and the map F is not partially captive (but could be called "totally captive"). This is also true if the function τ is a "co-boundary", i.e. if τ

(x) = η (E (x)) -η (x) with η ∈ C ∞ S 1 as discussed in Appendix A.
• M. Tsujii has studied a dynamical system very similar to (24) in [START_REF] Tsujii | Fat solenoidal attractors[END_REF], but this model is not volume preserving. He establishes there that the SRB measure on the trapped set is absolutely continuous for almost every τ .

The escape function

Let m < 0 and consider the C ∞ function on T * S 1 :

A m (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 . A m decreases with |ξ| and belongs to the symbol class S m . Eq. ( 27) implies that the function A m decreases strictly along the trajectories of F outside the trapped set (similarly to Eq.( 16)):

∀ |ξ| > R, ∀ε = 0, . . . , k -1 A m (F ε (x, ξ)) A m (x, ξ) ≤ C |m| < 1, with C = R 2 + 1 κR 2 + 1 < 1 (32)
And for any point we have the general bound:

∀ (x, ξ) ∈ T * S 1 , A m (F ε (x, ξ)) A m (x, ξ) ≤ 1. ( 33
)
Using the quantization rule (22), the symbol A m can be quantized giving a pseudodifferential operator Âm which is self-adjoint and invertible on C ∞ S 1 . In our case Âm is simply a multiplication operator by A m (ξ) in Fourier space. Let us consider the (usual) Sobolev space Here k = 2 and n = 3. The property for the map F of being "partially captive" according to definition 3 is related to the number of points N (n) which do not escape from the compact zone Z after time n.

H m S 1 := Â-1 m L 2 S 1 (x, ξ) R -R Z k n = 8 N (n) = 2 ξ
Then Fν : H m S 1 → H m S 1 is unitary equivalent to Q := Âm Fν Â-1 m : L 2 S 1 → L 2 S 1
Let n ∈ N * (a fixed time which will be made large at the end of the proof) and define

P (n) := Q * n Qn = Â-1 m F * n ν Â2 m F n ν Â-1 m (34)
Using Egorov theorem (the semi-classical version of Lemma 1) and Theorem of composition of PDO, we obtain that P (n) is a PDO of order 0 with principal symbol

P (n) (x, ξ) =   ε∈{0,...,k-1} n 1 E ′ n (x) A 2 m (F n ε (x, ξ)) A 2 m (x, ξ)   (35)
where

E ′ n (x) := n j=1 E ′ E -j εj (x)
is the expanding rate of the trajectory at time n. Eq.(2) implies that E ′ n (x) ≥ E n min . Now we will bound this (positive) symbol from above, considering different cases for the trajectory F n ε (x, ξ), as illustrated on Figure 5.

1. If (x, ξ) / ∈ Z then (32) gives

A 2 (F n ε (x, ξ)) A 2 (x, ξ) = A 2 (F n ε (x, ξ)) A 2 F n-1 ε (x, ξ) . . . A 2 (F ε (x, ξ)) A 2 (x, ξ) ≤ C 2|m| n (36) therefore P (n) (x, ξ) ≤ k n E n min C 2|m| n 2. If (x, ξ) ∈ Z but F n-1 ε (x, ξ) / ∈ Z then (A 2 •F n ε )(x,ξ) (A 2 •F n-1 ε )(x,ξ) ≤ C 2|m| from (32). Using also (33) we have A 2 (F n ε (x, ξ)) A 2 (x, ξ) = A 2 (F n ε (x, ξ)) A 2 F n-1 ε (x, ξ) . . . A 2 (F ε (x, ξ)) A 2 (x, ξ) ≤ C 2|m| (37)
3. In the other cases ((x, ξ) ∈ Z and F n-1 ε (x, ξ) ∈ Z) we can only use (33) to bound:

A 2 (F n ε (x, ξ)) A 2 (x, ξ) ≤ 1 (38)
From definition (30) we have

♯ F n-1 ε (x, ξ) ∈ Z, ε ∈ {0, 1} n ≤ N (n -1) .
For (x, ξ) ∈ Z, we split the sum Eq.( 35) accordingly to cases 1,2 or 3 above. Notice that C 2|m| n ≤ C 2|m| . This gives

P (n) (x, ξ) ≤ 1 E n min (k n -N (n -1)) C 2|m| + N (n -1) ≤ B ( 39
)
with the bound

B := k E min n C 2|m| + N (n -1) E n min Then lim sup (x,ξ) P (n) (x, ξ) ≤ B
With L 2 continuity theorem for pseudodifferential operators this implies that in the limit → 0

P (n) ≤ B + O n ( ) (40)
Polar decomposition of Qn gives

Qn ≤ Qn = P (n) ≤ (B + O n ( )) 1/2
Then for any n the spectral radius of Q satisfies [START_REF] Reed | Mathematical methods in physics, vol I : Functional Analysis[END_REF]p.192]

r s Q ≤ Qn 1/n ≤ (B + O n ( )) 1/2n
Also notice that

N (n -1) E n min 1/2n = 1 √ E min exp 1 2n log N (n -1)
We let → 0 first, and after m → -∞ giving C |m| → 0, and also we let n → ∞. Then for = 1/ν → 0 we have 6 :

r s Q ≤ 1 E min exp lim n∞ inf log N (n) n + o (1) . ( 41
)
If we make the assumption that F be partially captive, Eq.(31), we get that for = 1/ν → 0,

r s Q ≤ 1 √ E min + o (1) .
We have finished the proof of Theorem 2.

6 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]

A Equivalence classes of dynamics

Let us make a simple and well known observation about equivalent classes of dynamics. The map f we consider in Eq.(3) depends on k ∈ N and on the functions E : S 1 → S 1 , τ : S 1 → R. To emphasize this dependence, we denote f (E,τ ) . The transfer operator (7) is also denoted by F(E,τ) .

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

Let η : S 1 → R be a smooth function. Let us consider the map T : S 1 × S 1 → S 1 × S 1 defined by

T (x, s) = x, s + 1 2π η (x)
Then using (3) one gets 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). With (7) we also obtain that the transfer operator F

(E,ζ) of f (E,ζ) on C ∞ S 1 is given by F(E,ζ) = χ F(E,τ) χ-1 (42)
with the operator χ : C ∞ S 1 → C ∞ S 1 defined by: ( χϕ) (x) = e iνη(x) ϕ (x) .

Proof. χ F(E,τ) χ-1 ϕ (x) = ϕ (E (x)) e -iνη(E(x)) e iντ (x) e iνη(x) = F(E,ζ) ϕ (x).

The conjugation (42) immediately implies that F(E,ζ) and F(E,τ) have the same spectrum of Ruelle resonances.

Observe that χ is a O.I.F whose associated canonical transformation on T * S 1 ≡ S 1 × R is given by (ν ≫ 1 is considered as a semi-classical parameter):

χ : (x, ξ) ∈ S 1 × R → x, ξ + dη dx ∈ S 1 × R .
Therefore at the level of canonical transforms on T * S 1 :

F (E,ζ) = χ • F (E,τ ) • χ -1 (43)
The conjugation (43) implies in particular that the corresponding trapped sets (29) are related by

K (E,ζ) = χ K (E,τ )

B Description of the trapped set

In this section we provide further description of the trapped set K defined in Eq.(29) as well as the dynamics of the canonical map F restricted on it.

B.1 Dynamics on the cover R 2

The dynamics of F on the cylinder T * S 1 = S 1 × R has been defined in Eq.( 24). This is a multivalued map. It is convenient to consider the lifted dynamics on the cover R 2 which is a diffeomorphism given by F :

x

→ x ′ = E -1 (x) ξ → ξ ′ = E ′ (x ′ ) ξ + dτ dx (x ′ ) (44)
where E = kg : R → R is the map (1) lifted on R. It is invertible from (2). Let us suppose for simplicity that E (0) = 0.

One easily establish the following properties of the map F , illustrated on figure 6:

• The point

I := (0, ξ I ) := 0, - τ ′ (0) (E ′ (0) -1)
is the unique fixed point of F . It is hyperbolic with unstable manifold

W u = {(0, ξ) , ξ ∈ R} and stable manifold W s = {(x, S (x)) , x ∈ R} (45)
where the C ∞ function S (x) is defined by the following co-homological equation, deduced directly from ( 44)

S E -1 (x) = E ′ E -1 (x) S (x) + τ ′ E -1 (x) , S (0)
= ξ I .
The first equation can be written as

S (x) = 1 E ′ (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 (46)
where

x (-p) := E (-p) (x) :=   E -1 • . . . • E -1 p   (x) (47)
and

E ′(-p) (x) := E ′ (x -p ) . . . E ′ (x -2 ) E ′ (x -1 ) (48)
is the product of derivatives. In the case of E (x) = 2x, one gets simply

S (x) = - ∞ p=1 1 2 p τ ′ x 2 p (49) • If P : (x, ξ) ∈ R 2 → (x mod 1, ξ) ∈ S 1 × R ≡ T * S 1
denotes the projection, then trapped set K, defined in Eq.( 29) is obtained by wrapping the stable manifold around the cylinder and taking the closure:

K = P (W s ) (50)
Compare Figures 6 and Figure 4.

• If X 0 = (x 0 , ξ 0 ) ∈ R 2 is an initial point on the plane, and P (X 0 ) denotes its image on the cylinder, then at time n ∈ N, the k n evolutions of the point P (X 0 ) under the map F n are the images of the evolutions F n (X k ) of the translated points X p = X 0 + (p, 0), with p = 0 → k n -1:

F n (P (X 0 )) = P F n (X p ) , X p = X 0 + (p, 0) , p ∈ [0, k n [
and more precisely, using notation of Eq.( 25) for these points, one has the relation:

F n ε (P (X 0 )) = P F n (X p ) ( 51
)
where ε is the number p written in base k: • For an initial point X 0 = (x 0 , ξ 0 ) ∈ R 2 , then X n = (x n , ξ n ) = F n (X 0 ) satisfies

ε = ε n-1 . . .
x n = E (-n) (x 0 ) , ξ n -S (x n ) = E ′(-n) (x 0 ) (ξ 0 -S (x 0 )) ( 52
)
with E (-n) (x), E ′(-n) (x) given by ( 47), (48). Hence

B.2 Partially captive property

Here we rephrase the property of partial captivity, Definition 3, in terms of a property on the separatrix function S (x) defined in Eq.( 45) and given in Eq. ( 46).

For simplicity, we consider from now on the simple model with a linear expanding map E (x) = kx.

X X 0

X 1 X 3 F 2 00 (X)

x Projection X 2 P F 0 (X) F 2 10 (X)

F 1 (X)
F 2 11 (X) x ⊂ C defined by Eq.(55). The trapped set K at position x is obtained by the projection on the imaginary axis K x = ℑ (K c

F 2 01 (X) ξ ξ F 2 (X 3 ) F 2 (X 1 ) F (X 3 ) F (X 1 ) F (X 0 ) F (X 2 ) F 2 (X 2 ) F 2 (X 0 ) x π 0 -π 0 1
x ). On the web page http://www-fourier.ujf-grenoble.fr/ ∼ faure/articles one can observe the motion of the fractal K c

x as x ∈ R increases smoothly.

Figure 1 :

 1 Figure1: Numerical evolution of an initial small cloud of points on the torus (x, s) ∈ T 2 under the map f , Eq.(3), at different time n = 0, 2, 10, 19. We have chosen here E (x) = 2x and τ (x) = cos (2πx). The initial cloud of points is centered 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 horizontal direction x. Due to instability in x and periodicity, the cloud fills the torus S 1 × S 1 for large time n. 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 property of the map f in this example.

Figure 2 :

 2 Figure 2: Black dots are numerical computation of the eigenvalues λ i of Fν for different values of ν ∈ N, and union of these in the last image. We have chosen here E (x) = 2x i.e. k = 2, and τ (x) = cos (2πx). The external red circle has radius 1. The internal green circle has radius 1/ √ E min = 1/ √ 2 and represents the upper bound given in Eq.(11). As ν ∈ R moves continuously, the resonances move in a spectacular way. This can be seen on a movie on http://www-fourier. ujf-grenoble.fr/ ∼ faure/articles

  is the Bessel function of first kind [AS54, 9.1.21 p 360]. Corollary 2 in [FR06] guaranties that the eigenvalues of the truncated matrix |n| , |n ′ | ≤ N converges towards the Ruelle resonances as N → ∞.

  Lemma 38) P = kα + pα with kα a smoothing operator (hence compact) and pα ≤ C 2|m| k Emin + α. If Qm = Û Q is the polar decomposition of Qm , with Û unitary, then from (18) P = Q 2 ⇔ Q = P and the spectral theorem ([Tay96b] p.75) gives that Q has a similar decomposition Q = k′ α + qα with k′ α smoothing and qα ≤ C |m| k

)Figure 4 :

 4 Figure 4: The trapped set K in the cotangent space S 1 × R. We have chosen here E (x) = 2x and τ (x) = cos (2πx).

Figure 5 :

 5 Figure 5: This Figure illustrates the trajectories F n ε (x, ξ) issued from an initial point (x, ξ).Here k = 2 and n = 3. The property for the map F of being "partially captive" according to definition 3 is related to the number of points N (n) which do not escape from the compact zone Z after time n.

Figure 7

 7 Figure 7 illustrates this correspondence.

Figure 6 :

 6 Figure 6: The fixed point I = (0, ξ 0 ), the stable manifold W s and unstable manifold W u of the lifted map F , Eq.(44), in the example E (x) = 2x, τ (x) = cos (2πx).

Figure 7 :Figure 8 :

 78 Figure 7: This picture shows how the dynamics of a point X = P (X 0 ) ∈ S 1 × R under the map F is related by Eq.(51) to the dynamics of its lifted images X k = X 0 + (k, 0) under F on the cover R 2 .

Notice that the operator Fν appears to be a transfer operator for the expanding map E with an additional weight function e iντ (x) .