Cyril Karamaoun

Amaryllis Haccuria

Alain Michils

Benoît Haut

Alain Van Muylem

Experimental and theoretical impact of hypertonic saline induction on exhaled nitric oxide

à la diffusion de documents scientifiques de niveau recherche, publiés ou non, émanant des établissements d'enseignement et de recherche français ou étrangers, des laboratoires publics ou privés.

Experimental and theoretical impact of hypertonic saline induction on exhaled nitric oxide Background Experimental [Haccuria, 2014] and simulation [Karamaoun, 2016] results showed that exhaled nitric oxide (FE NO ) is affected by airway caliber.

Hypertonic saline induction (HSI) likely affects airway caliber by increasing the level of liquid secretion.

Consequently, HSI may also affect FE NO value even in subjects without airway hyperresponsiveness.

Aim

Cyril Karamaoun 1 , Amaryllis Haccuria 2 , Alain Michils 2 , Benoît Haut 1 and Alain Van Muylem 0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21

Central airways

Small airways Acinus

To experimentally and theoretically assess the influence of HSI on FE NO in healthy subjects without airways hyperreactivity.

Simulations Experiments

A detailed model of airway epithelium and NO transport is used [Karamaoun, 2016], featuring a liquid layer lining airway epithelium.

In 14 healthy subjects (10F/4M, 28±8 y), we measured before and after 20 minutes of HSI:

 FE NO
 FEV 1 as a marker of central airways caliber  The phase III slope (S) of the single breath washout (SBWO) using gases with different diffusivities, helium (He) and sulfur hexafluoride (SF 6 ), as a marker of peripheral airway impairment

Methods

Results

Simulations Experiments

Discussion

 HSI did not affect central airways whereas peripheral heterogeneities were increased along with a huge FE NO decrease.

 Model simulations predicted that the occurrence of a liquid layer specifically in peripheral airways may decrease FE NO to the same extent than experimental observations.  Experiments confirmed model predictions that FE NO , after HSI, reacted to specific peripheral alterations in healthy subjects.

Conclusion

 This highlights the potential role of FE NO as a marker of peripheral airway calibre change.

Central airways

Generation number Acinus Small airways % of NO production Corresponding author: ckaramao@ulb.ac.be 1 TIPs (Transfers, Interfaces and Processes), Université libre de Bruxelles, Brussels, Belgium 2 Chest Department, Erasme University Hospital, Université libre de Bruxelles, Brussels, Belgium


Lucas Vacossin

RESOLVENT ESTIMATES IN STRIPS FOR OBSTACLE SCATTERING IN 2D AND LOCAL ENERGY DECAY FOR THE WAVE EQUATION

In this note, we are interested in the problem of scattering by J strictly convex obstacles satisfying a no-eclipse condition in dimension 2. We use the result of [Vac22] to obtain polynomial resolvent estimates in strips below the real axis. We deduce estimates in O(|λ| log |λ|) for the truncated resolvent on the real line and give an application to the decay of the local energy for the wave equation.

It is known that the resolvent R(λ) of the Dirichlet Laplacian -∆ Ω in Ω continues meromorphically to the logarithmic cover of C (see for instance [START_REF] Dyatlov | Mathematical Study of Scattering Resonances[END_REF], Chapter 4, Theorem 4.4). More precisely, if χ ∈ C ∞ c (R 2 ) is equal to one in a neighborhood of O, (1.1)

χR(λ)χ = χ(-∆ Ω -λ 2 ) -1 χ : L 2 (Ω) → L 2 (Ω)
is holomorphic in the region {Im λ > 0} and it continues meromorphically to the logarithmic cover Λ of C. Its poles are the scattering resonances and do not depend on χ. In [START_REF] Vacossin | Spectral gap for obstacle scattering in 2d[END_REF], the following result has been proved :

Theorem 1. There exist γ > 0 and λ 0 > 0 such that there is no resonance in the region

[λ 0 , +∞[+i[-γ, 0]
seen as a region in the first sheet of Λ.

In this note, we reuse the arguments of [START_REF] Nonnenmacher | Fractal weyl law for open quantum chaotic maps[END_REF] and the main estimate in [START_REF] Vacossin | Spectral gap for obstacle scattering in 2d[END_REF] (Proposition 4.1) to obtain estimates for the cut-off resolvent (1.1) in this region. We will rather state these resolvent estimates in a semiclassical form, so that it can also be applied to more general semiclassical problems such as the scattering by a smooth compactly potential (see [START_REF] Nonnenmacher | From open quantum systems to open quantum maps[END_REF], Section 2, for precise assumptions and [START_REF] Vacossin | Spectral gap for obstacle scattering in 2d[END_REF], Section 2.2 for applications of Theorem 1 in scattering by a potential under these assumptions). In obstacle scattering, the semiclassical problem is simply a rescaling : we are interested in the semiclassical operator

P (h) = -h 2 ∆ Ω -1 , 0 < h ≤ h 0
and spectral parameter z ∈ [-δ, δ] + i[-Kh, Kh] for some fixed K > 0 and some δ > 0. We note

(1.2) R h (z) = (P (h) -z) -1
continued meromorphically from Im z > 0 to z ∈ [-δ, δ] + i[-Kh, Kh]. We prove :

Theorem 2. Suppose that P (h) = -h 2 ∆ Ω -1 where -∆ Ω is the Dirichlet Laplacian in Ω, or

P (h) = -h 2 ∆ + V -E 0 where V ∈ C ∞ c (R 2
) and E 0 ∈ R * + satisfying the assumptions of [START_REF] Nonnenmacher | From open quantum systems to open quantum maps[END_REF], recalled in 2.3.1. Let χ ∈ C ∞ c (R 2 ) be equal to one in a neighborhood of O (in the case of obstacle scattering) or supp V (in the case of scattering by a potential). Fix K > 0. There exists δ 0 > 0, γ > 0, C > 0, h 0 > 0 and β ≥ 0 such that for all 0 < h ≤ h 0 , P (h) has no resonance in

(1.3) D h := {z ∈ C, Re z ∈ [-δ 0 , δ 0 ], -γh ≤ Im z ≤ Kh}
and for all z ∈ D h ,

(1.4) ||χR h (z)χ|| L 2 →L 2 ≤ Ch -β
Remark. In the case of the obstacles, with these notations, for δ small enough and h small enough, z is related to the spectral parameter λ h (z) by the relation λ h (z) 2 = h -2 (1 + z). As a consequence, λ h (z) lies in a neighborhood of 1/h in Λ. In particular, it lives in the first sheet of Λ, that is arg λ h (z) ∈] -π/2, π/2[.

Applications.

Decay of the local energy for the wave equation. As a first application, we obtain a decay rate O(t -2 ) for the local energy of the wave equation outside the obstacles. The link between resolvent estimates and energy decay is quite standard now (see for instance [START_REF] Zworski | Semiclassical Analysis[END_REF], Chapter 5, [START_REF] Lebeau | Equation des ondes amorties[END_REF]). In the particular case of obstacle scattering, Ikawa showed exponential decay in dimension 3, for the case of two obstacles ( [START_REF] Ikawa | Decay of solutions of the wave equation in the exterior of two convex obstacles[END_REF]) and for more obstacles under a dynamical assumption ( [START_REF] Ikawa | Decay of solutions of the wave equation in the exterior of several convex bodies[END_REF]) involving the topological pressure P (s) of the billiard flow. This assumption requires the pressure to be strictly negative at s = 1/2 (see also [START_REF] Nonnenmacher | Quantum decay rates in chaotic scattering[END_REF]). In the case of dimension 2 (and more generally, of even dimensions), one cannot expect such an exponential decay, due to the logarithmic singularity at 0 for the free resolvent and the fact that the strong Huygens principle does not hold. Even for the free case, the bound for the local energy is O(t -2 ). This is the bound we obtain here, assuming that the initial data are sufficiently regular :

Theorem 3. There exists k ∈ N such that for all R > 0, there exists C R > 0 such that the following holds: let (u 0 , u 1 ) ∈ H k+1 (Ω) ∩ H 1 0 (Ω) × H k (Ω) be initial data supported in B(0, R) ∩ Ω and consider the unique solution of the Cauchy problem

       ∂ 2 t u(t, x) -∆u(t, x) = 0 in Ω u(t, x) = 0 on ∂Ω u t=0 = u 0 ∂ t u t=0 = u 1
Then, for t ≥ 1, the local energy in the ball B(0, R), E R (t), satisfies the bound

E R (t) := B(0,R)∩Ω |∇u(t)| 2 + |∂ t u(t)| 2 ≤ C R t 2 ||u 0 || 2 H k+1 + ||u 1 || 2 H k
Theorem 3 is a consequence of Theorem 2. This fact is proved in Section 3 and the proof uses the strategy of [START_REF] Burq | Décroissance de l'énergie locale de l'équation des ondes pour le problème extérieur et absence de résonance au voisinage du réel[END_REF].

Resolvent estimates on the real line. Polynomial resolvent bounds in strips are known to imply better bounds on the real line, by using a semiclassical maximum principle (see for instance [START_REF] Burq | Smoothing effect for Schrödinger equations[END_REF], Lemma 4.7, or [START_REF] Ingremeau | Sharp resolvent bounds and resonance-free regions[END_REF]). As a consequence, we deduce the following estimates on the real line :

Corollary 1.1. Let P (h) be one of the operators described in Theorem 2 and let χ ∈ C ∞ c (R 2 ) as in this Theorem. There exits C 0 > 0, δ 0 > 0 and h 0 > 0 such that for all 0 < h ≤ h 0 and for all z ∈ [-δ 0 , δ 0 ],

||χ(P (h) -z) -1 χ|| L 2 →L 2 ≤ C 0 | log h| h
Remark. As a direct corollary of the proof of Lemma 4.7 in [START_REF] Burq | Smoothing effect for Schrödinger equations[END_REF], we can obtain a more general bound : for h > 0 small enough,

(1.5) ||χ(P (h) -z) -1 χ|| L 2 →L 2 ≤ C 0 | log h|h -1+σ| Im z|/h , z ∈ [-δ 0 , δ 0 ] + i[-γh, 0]
where σ > 0. With this method, based on the maximum principle for analytic functions, the value of σ in not explicit. In fact, our proof gives a bound of the form

||χ(P (h) -z) -1 χ|| L 2 →L 2 ≤ C| log h|h -1-M1-M2| Im z|/h , z ∈ [-δ 0 , δ 0 ] + i[-γh, 0]
where M 2 only depends on constants related to the billiard map (see (2.23)). The extra M 1 is a consequence of the method we use, based on the use of an escape function. It is possible that a more careful analysis could allow to get rid of this extra M 1 and we could straighlty obtain a bound of the form (1.5).

This kind of estimates is known to be useful to prove smoothing effects for the Schrödinger equation and to obtain Strichartz estimates, which turns out to be crucial for the local-well posedness of the non-linear Schrödinger equation (see for instance [START_REF] Burq | Smoothing effect for Schrödinger equations[END_REF], [START_REF] Burq | On nonlinear Schrödinger equations in exterior domains[END_REF]). Let's for instance mention the following smoothing estimates (see the references above for the proof and for pointers to the literature concerning these estimates) :

Corollary 1.2. Let Ω be as in Theorem 2 and let e -it∆Ω be the Schrödinger propagator of the Dirichlet Laplacian -∆ Ω in Ω. Then, for any ε > 0 and for any χ ∈ C ∞ c (R 2 ) equal to one in a neighborhood of O, there exists C > 0 such that for any u 0 ∈ L 2 (Ω),

||χe -it∆Ω χu 0 || L 2 (Rt,H 1/2-ε (Ω)) ≤ C||χu 0 || L 2
Organization of the paper. In Section 2, we prove Theorem 2 using the crucial estimate proved in [START_REF] Vacossin | Spectral gap for obstacle scattering in 2d[END_REF] and recalling the reduction to open quantum maps performed in [START_REF] Nonnenmacher | From open quantum systems to open quantum maps[END_REF] and [START_REF] Nonnenmacher | Fractal weyl law for open quantum chaotic maps[END_REF]. The main semiclassical ingredients of the above paper are recorded in Appendix A. Section 3 is devoted to the proof of the local energy decay for the wave equation.

Acknowledgment. The author would like to thank Stéphane Nonnenmacher for his careful reading of a preliminary version of this work and Maxime Ingremeau for suggesting to write this note.

Proof of Theorem 2

In this section, we prove the main resolvent estimate of this note. The central point, concerning a resolvent bound for open hyperbolic quantum maps, is common to the case of obstacle scattering and scattering by a potential. However, the reduction to open quantum maps differs in the two above situations, this is why we distinguish the two cases. We begin by recalling the definitions of open quantum maps from [START_REF] Nonnenmacher | From open quantum systems to open quantum maps[END_REF] and [START_REF] Nonnenmacher | Fractal weyl law for open quantum chaotic maps[END_REF] and the result of [START_REF] Vacossin | Spectral gap for obstacle scattering in 2d[END_REF] leading to a crucial resolvent bound.

2.1.

Resolvent bound for open quantum maps.

Y = J j=1 Y j ⊂ J j=1 R and consider U = J j=1 U j ⊂ J j=1 T * R d ; where U j T * Y j are open sets The Hilbert space L 2 (Y ) is the orthogonal sum J i=1 L 2 (Y i ). For j = 1, . . . , J, consider open disjoint subsets ‹ D ij U j , 1 ≤ i ≤ J
, the departure sets, and similarly, for i = 1, . . . , J consider open disjoint subsets A ij U i , 1 ≤ j ≤ J, the arrival sets (see Figure 2). We assume that there exist smooth symplectomorphisms (2.1)

F ij : ‹ D ij → F ij Ä ‹ D ij ä = A ij
We note F for the global smooth map F : ‹ D → A where A and ‹ D are the full arrival and departure sets, defined as

A = J i=1 J j=1 A ij ⊂ J i=1 U i ‹ D = J j=1 J i=1 ‹ D ij ⊂ J j=1 U j
We define the outgoing (resp. incoming) tail by

T + := {ρ ∈ U ; F -n (ρ) ∈ U, ∀n ∈ N} (resp. T -:= {ρ ∈ U ; F n (ρ) ∈ U, ∀n ∈ N}).
We assume that they are closed subsets of U and that the trapped set

(2.2) T = T + ∩ T - is compact.
We also assume that T is totally disconnected. For i, j ∈ {1, . . . , J}, we note

T i = T ∩ U i , D ij = {ρ ∈ T j ; F (ρ) ∈ T i } ⊂ ‹ D ij and A ij = {ρ ∈ T i ; F -1 (ρ) ∈ T j } ⊂ A ij
Remark. It is possible that for some values of i and j, ‹ D ij = ∅. For instance, when dealing with the billiard map (see (2.19)), the sets ‹ D ii are all empty.

-1.5 -1.5 -1 -1 -0.5 -0.5 0.5 0.5 1 1 1.5 1.5 2 2 -1 -1 -0.5 -0.5 0.5 0.5 1 1 0 0 (a) Single hyperbolic fixed point.

-1 / 3 -1 / 3 1 / 3 1 / 3 2 / 3 2 / 3 1 1 4 / 3 4 / 3 5 / 3 5 / 3 2 2 -1 / 3 -1 / 3 1 / 3 1 / 3 2 / 3 2 / 3 1 1 0 0 (b)
An open baker's map.

Figure 3. Examples when J = 1. The departure sets are in blue, the arrival sets in red. In the first example,

U 1 =] -1, 1[ 2 ⊂ T * R , D 11 =] -1, 1[×] -1/2, 1/2[, A 11 =]-1/2, 1/2[×]-1, 1[ with F (x, ξ) = (x/2, 2ξ).
The trapped set is reduced to a single hyperbolic fixed point. The second example is built on the model of an open baker's map. We have

U 1 =]0, 1[ 2 ⊂ T * R , D 11 =]0, 1[×]0, 1/3[∪]0, 1[×]2/3, 1[, A 11 =]0, 1/3[×]0, 1[∪]2/3, 1[×]0, 1[.
In such a model, the map F is piecewise affine and given by F (x + a, ξ) = (3x, a + ξ/3) for a ∈ {0, 2}, (x, ξ) ∈]0, 1[ 2 .

We then make the following hyperbolic assumption.

(2.3) T is a hyperbolic set for F Namely, for every ρ ∈ T , we assume that there exist stable/unstable tangent spaces E s (ρ) and E u (ρ) such that :

• dim E s (ρ) = dim E u (ρ) = 1 • T ρ U = E s (ρ) ⊕ E u (ρ)
• there exists λ > 0, C > 0 such that for every v ∈ T ρ U and any n ∈ N,

v ∈ E s (ρ) =⇒ ||d ρ F n (v)|| ≤ Ce -nλ ||v|| (2.4) v ∈ E u (ρ) =⇒ ||d ρ F -n (v)|| ≤ Ce -nλ ||v|| (2.5)
where || • || is a fixed Riemannian metric on U . The decomposition of T ρ U into stable and unstable spaces is assumed to be continuous.

Here ends the description of the classical map. See Figure 3 for simple examples of such open hyperbolic maps. We then associate to F open quantum hyperbolic maps, which are its quantum counterparts.

Definition 2.1. Fix δ ∈ [0, 1/2[. We say that T = T (h) is an open quantum hyperbolic map associated with F , and we note T = T (h) ∈ I δ (Y × Y, Gr(F ) ) if : for each couple (i, j) ∈ {1, . . . , J} 2 , there exists a semiclassical Fourier integral operator

T ij = T ij (h) ∈ I δ (Y j ×Y i , Gr(F ij ) ) associated with F ij in the sense of definition A.3, such that T = (T ij ) 1≤i,j≤J : J i=1 L 2 (Y i ) → J i=1 L 2 (Y i ) In particular WF h (T ) ⊂ A × ‹ D. We note I 0 + (Y × Y, Gr(F ) ) = δ>0 I δ (Y × Y, Gr(F ) ).
We will say that T ∈ I 0 + (Y × Y, Gr(F ) ) is microlocally invertible near T if there exists a neighborhood U ⊂ U of T and an operator

T ∈ I 0 + (Y × Y, Gr(F -1 ) ) such that, for every u = (u 1 , . . . , u J ) ∈ L 2 (Y ) ∀j ∈ {1, . . . , J}, WF h (u j ) ⊂ U ∩ U j =⇒ T T u = u + O(h ∞ )||u|| L 2 , T T u = u + O(h ∞ )||u|| L 2
Suppose that T is microlocally invertible near T and recall that T * T ∈ Ψ 0 + (Y ) ⊂ Ψ 1/4 (Y ) (this choice δ = 1/4 is arbitrary). Then, we can write

T * T = Op h (a h ) + O(h 1/4 ) L 2 →L 2
where a h is a smooth principal symbol in the class S 0 + (U ) (the definition of this symbol class is recalled in the appendix). We note α h = |a h | and call it the amplitude of T . Since T is microlocally invertible near T , |a h | > c 2 near T , for some h-independent constant c > 0, showing that α h is smooth and larger than c in a neighborhood of T .

Remark. If T has amplitude α, at first approximation, T transforms a wave packet u ρ0 of norm 1 centered at a point ρ 0 lying in a small neighborhood of T into a wave packet of norm α(ρ 0 ) centered at the point F (ρ 0 ).

2.1.2. Crucial resolvent bound. We now consider M (h) an open quantum hyperbolic map, associated with F . We suppose that M (h) is microlocally invertible near T . Additionally, we make the following assumption : there exists L > 0 and

φ 0 ∈ C ∞ c (T * Y, [0, 1]) such that supp(φ 0 ) is contained in a compact neighborhood W of T , W ⊂ ‹ D, φ 0 = 1 in a neighborhood of T and (2.6) M (h)(1 -Op h (φ 0 )) = O(h L )
Let us note α h the amplitude of M (h) and ||α h || ∞ its sup norm in W. It is a priori h-dependent, but it is uniformly bounded in h. Proposition 4.1 in [START_REF] Vacossin | Spectral gap for obstacle scattering in 2d[END_REF] then states that :

Proposition 2.1. Suppose that M (h) satisfies the above assumptions. There exists δ > 0, γ > 0, h 0 > 0 and a family of integer N (h) ∼ δ| log h|, defined for 0 < h ≤ h 0 , such that for all 0 < h ≤ h 0 ,

(2.7)

||M (h) N (h) || L 2 →L 2 ≤ h γ ||α h || N (h)

∞

Remark.

• Strictly speaking, the result of [START_REF] Vacossin | Spectral gap for obstacle scattering in 2d[END_REF] applies to operators of the form T (h) Op h (α) where T is microlocally unitary near T . We can reduce (2.7) to this case. Indeed, locally near every point ρ 0 ∈ T , M (h) takes this form, and T is totally disconnected, so that M (h) takes this form in a small neighborhood of T . Finally, as showed in [START_REF] Vacossin | Spectral gap for obstacle scattering in 2d[END_REF] (Subsection 4.2), the behavior of M (h) outside any neighborhood of T contributes as a O(h ∞ ) in (2.7), as soon as N (h) is bigger than a fixed N 0 depending on this neighborhood. • Note that the constant δ and γ are purely dynamical, that is, depend only on the dynamics of F near T . Indeed, δ is defined in Section 4.1 in [START_REF] Vacossin | Spectral gap for obstacle scattering in 2d[END_REF] using only dynamical parameters, such as the Jacobian of F . Concerning γ, it is implicitly defined using the porosity of the trapped set (see Section 6 in [START_REF] Vacossin | Spectral gap for obstacle scattering in 2d[END_REF]). h 0 depends on α (through a finite number of seminorms). This remark will turn out to be important when dealing with scattering by a potential.

This estimate, which is the crucial point in [START_REF] Vacossin | Spectral gap for obstacle scattering in 2d[END_REF] to prove the spectral gap naturally leads to a resolvent bound for (Id -M (h)) -1 : Proposition 2.2. Suppose that M (h) satisfies the above assumptions. Let γ and δ be given in Proposition 2.1 and assume that for some

h 1 > 0, for all 0 < h ≤ h 1 , (2.8) ||α h || ∞ < exp γ δ Let us consider A ≥ 1 such that for all 0 < h ≤ h 1 , ||α h || ∞ ≤ A. Then, there exists h 0 ∈]0, h 1 ] such that for all 0 < h ≤ h 0 , (2.9) (Id -M (h)) -1 L 2 →L 2 ≤ 2δ| log h|h -δ log A Proof. First recall that M * M ∈ Ψ 0 + with σ 0 (M * M ) = α 2 h and M = O(h L ) microlocally outside W.
Hence, we can estimate the operator norm of M (h) (see [START_REF] Zworski | Semiclassical Analysis[END_REF], Theorem 13.13),

||M (h)|| L 2 →L 2 ≤ ||α h || ∞ + O(h η )
where η is any fixed number in ]0, 1[. Let N (h) ∼ δ| log h| be the family of integers given by Proposition 2.1. Without loss of generality, we may assume that N (h) ≤ δ| log h|. We use the fact that

h γ ||α h || N (h) = o(1) when h → 0 if ||α h || ∞ < e γ δ . As a consequence, Id -M N (h) is invertible for h small enough with (2.10) ||(Id -M N (h) ) -1 || ≤ 3 2 , 0 ≤ h 1 This implies that I -M is invertible with inverse (2.11) (Id -M ) -1 = (Id +M + • • • + M N (h)-1 )(Id -M N (h) ) -1
We hence estimate

|| Id +M + • • • + M N (h)-1 || ≤ N (h)(||α h || ∞ + O(h η )) N (h) ≤ N (h)(A + O(h η ))) N (h) ≤ δ| log h|h -δ log A (1 + o(1)) ≤ 4 3 δ| log h|h -δ log A
if h is small enough. Using (2.11), we multiply with (2.10) and find the required inequality.

Remark. The constant 2 can be changed into any 1 + ε by changing h 0 into h 0 (ε).

If lim inf h→0 ||α h || ∞ > 1 we can get rid of the log h term by changing it into a constant depending on ||α h || ∞ . More precisely, a better estimate of the sum can show that

(Id -M (h)) -1 L 2 →L 2 ≤ 2 ||α h || ∞ -1 h -δ log ||α h ||∞
The main interest of the estimate in Proposition 2.2 is that it gives a uniform estimate in the limit

||α h || ∞ → 1.
2.2. Proof in the case of obstacle scattering. In this subsection, we recall the main ingredients of [START_REF] Nonnenmacher | Fractal weyl law for open quantum chaotic maps[END_REF] and prove the resolvent estimate of Theorem 2 in obstacle scattering.

i = j = k, O i does not intersect the convex hull of O j ∪ O k . Let P (h) = -h 2 ∆ Ω -1 and fix a cut-off function χ ∈ C ∞ c (R 2 ) equal to one in a neighborhood of O. First note that by a simple scaling argument, it is enough to prove (1.4) for z ∈ {z ∈ D(0, Kh), Im z ≥ -γh} for any K > 0 fixed. Complex scaling. We fix R χ > 0 such that supp χ ⊂ B(0, R χ ). For a parameter θ ∈]0, π/2[, we consider a complex deformation Γ θ ⊂ C 2 of R 2 such that for some R > R χ , Γ θ ∩ B C 2 (0, R χ ) = R 2 ∩ B R 2 (0, R χ ) Γ θ ∩ C 2 \ B C 2 (0, R ) = e iθ R 2 ∩ C 2 \ B C 2 (0, R ) Γ θ = f θ (R 2 ) ; f θ : R 2 → C 2 injective
By identifying R 2 and Γ θ through f θ , we note ∆ θ the corresponding complex-scaled free Laplacian, and ∆ Ω,θ the complex scaled Laplacian on H 2 (Ω) ∩ H1 0 (Ω). We fix K > 0 (which can be chosen arbitrarily large) and for z ∈ D(0, Kh), we note (2.12)

P • (z) = -h 2 ∆ • -1 -z with either • = θ or • = Ω, θ.
We note the associated resolvent, when they are defined,

R Ω,θ (z) : L 2 (Ω) → H 2 (Ω) ; R θ (z) : L 2 (R 2 ) → H 2 (R 2 )
Remark. With these notations, the parameter λ of the usual resolvent R(λ) takes the form

λ = λ h (z) = h -1 (1 + z) 1/2 with z ∈ D(0, Kh) ⊂ D(0, 1) if h small
enough, so that the square root is well defined and gives a holomorphic function of z.

Thanks to the usual properties of the complex scaling method (see for instance [START_REF] Dyatlov | Mathematical Study of Scattering Resonances[END_REF], Section 4.5 in Chapter 4 and the references given there), we have :

• The operators P θ (z) and P Ω,θ (z) are Fredholm operators of index 0;

• z is a pole of R Ω,θ (z) if and only if λ h (z) is a scattering resonance ; • For z not a pole of R Ω,θ (z)
, in virtue of the properties of χ and Γ θ , we have (recall the definitions of R(λ) and R h (z) in (1.1) and (1.2) respectively),

χR Ω,θ (z)χ = χR h (z)χ = h -2 χR(λ h (z))χ
• Finally, we recall that we have the following standard estimate for R θ (z) (see for instance [START_REF] Dyatlov | Mathematical Study of Scattering Resonances[END_REF], Theorem 6.10)

(2.13) ||R θ (z)|| L 2 (R 2 )→H 2 h (R 2 ) ≤ Ch -1 ; z ∈ D(0, Kh) In particular, it tells that R θ is holomorphic in D(0, Kh). Here, H 2 h (R 2 ) is a semiclassical Sobolev space i.e. H 2 (R 2 ) with the norm ||u|| H 2 h (R 2 ) = ||(1 -h 2 ∆)u|| L 2 .
To prove Theorem 2, it is then enough to give a bound for χR Ω,θ (z)χ : L 2 (Ω) → L 2 (Ω) in the corresponding region.

Reduction to the boundary of the obstacles. Following [NSZ14] (Section 6), we introduce the following operators to obtain a reduction to the boundary. For j = 1, . . . , J, let (2.14)

γ j : u ∈ H 2 (Ω) → u| ∂Oj ∈ H 3/2 (∂O j )
be the (bounded) trace operator and γu = (γ j u)

j ∈ H 3/2 (∂O) := H 3/2 (∂O 1 ) × • • • × H 3/2 (∂O J ), and let
(2.15)

H j (z) : H 3/2 (∂O j ) → H 2 (R 2 \ O j ) extension by 0 -→ L 2 (R 2 )
be the Poisson operator, defined, for v ∈ H 3/2 (∂O j ), as the solution to the problem

ß P θ (z)H j (z)v = 0 in R 2 \ O j γ j H j (z)v = v. u = H j (z)
v is a solution of the problem P θ (z)u = 0 with outgoing properties. So as P θ (z), H j (z) implicitly depends on h.

For - → v = (v 1 , . . . , v J ) ∈ H 3/2 (∂O), we set H(z) - → v = J j=1 H j (z)v j
Let us define the following operator-valued matrix M(z) :

H 3/2 (∂O) → H 3/2 (∂O) by the relation (2.16) Id -M(z) = γH(z)
We state a few facts concerning these operators. In the following lemma, we give estimates involving the semiclassical version of the Sobolev spaces

H 2 (R 2 \ O j ) and H 3/2 (∂O j ), denoted H 2 h (R 2 \ O j ) and H 3/2
h (∂O j ) respectively. Lemma 2.1. For j = 1, . . . , J, there exists C > 0 such that for all 0 < h ≤ 1, the norm of the bounded operator

γ j from H 2 h (R 2 \ O j ) to H 3/2 h (∂O j ) satisfies ||γ j || H 2 h (R 2 \Oj )→H 3/2 h (∂Oj ) ≤ Ch -1/2
Proof. Using a partition of unity argument and local charts, it is sufficient to prove that the above result holds with R 2 \ O j replaced by R × R * + and ∂O j replaced by R. In this setting, we note γ the associated trace operator. First, we extend an element u ∈ H

2 h (R × R * + ) to an element ũ ∈ H 2 h (R × R) such that ||ũ|| H 2 h ≤ C||u|| H 2 h (see for instance [Eva10],
Chapter 5, Section 4 : in the proof of Theorem 1, one can extend u ∈ H 2 h (R × R * + ) with the formula : for y > 0, u(x, -y) = -3u(x, y) + 4u(x, -y/2)). Then we observe that, with F 1 h (resp. F h ) the semiclassical unitary Fourier transform in 1D (resp. 2D),

F 1 h (γu)(ξ) = 1 (2πh) 1/2 R F h (ũ)(ξ, η)dη
From, this we get (2.17)

||γu|| H 3/2 h ≤ Ch -1/2 ||ũ|| H 2 h Indeed, by Cauchy-Schwarz, we have R F h (ũ)(ξ, η)dη 2 ≤ Å R |F h (ũ)(ξ, η)| 2 (1 + ξ 2 + η 2 ) 2 dη ã Å R (1 + ξ 2 + η 2 ) -2 dη ã ≤ Å R |F h (ũ)(ξ, η)| 2 (1 + ξ 2 + η 2 ) 2 dη ã (1 + ξ 2 ) -3/2 R (1 + η 2 ) -2 dη
We find (2.17) by multiplying by (1 + ξ 2 ) 3/2 and integrating over ξ. This concludes the proof.

Lemma 2.2. For j = 1, . . . , J, for any K > 0, there exists h 0 > 0 such that for all 0 < h ≤ h 0 , χH j (z) is holomorphic in D(0, Kh) and satisfies for some C > 0 independent of h, and for z ∈ D(0, Kh),

||χH j (z)|| H 3/2 h (∂Oj )→L 2 (R 2 \Oj ) ≤ Ch -1/2
Proof. We follow the main lines of the proof of Lemma 6.1 in [START_REF] Nonnenmacher | Fractal weyl law for open quantum chaotic maps[END_REF]. First, let us introduce an extension operator T h j :

H 3/2 h (∂O j ) → H 2 h (R 2 ) such that for v ∈ H 3/2 h (∂O j ), T h j v is supported in a small neighborhood of ∂O j and T h j = O(h 1/2 ) : H 3/2 h (∂O j ) → H 2 h (R 2
) This is possible, for instance by taking the extension operator given in the proof of Lemma 6.1 in [START_REF] Nonnenmacher | Fractal weyl law for open quantum chaotic maps[END_REF]. Another approach consists in using a partition of unity and local charts to replace ∂O j by R, as in the proof of Lemma 2.1. Then, one can consider the following operator

T h : v ∈ H 3/2 h (R) → T h v ∈ H 2 h (R 2 ) ; T h v(x, y) = χ x h hD y v (y) where χ ∈ C ∞ c (R), χ(0) = 1. Then, T h v(0, y) = v and one has ||T h v|| H 2 h (R 2 ) ≤ Ch 1/2 ||v|| H 3/2 h (R)
Indeed, one has

F h (T h v)(ξ, η) = h 1/2 η -1 F 1 1 (χ) Å ξ η ã × F 1 h (v)(η)
and hence

||T h v|| 2 H 2 h (R 2 ) = R 2 |F h (T h v)(ξ, η)| 2 (1 + ξ 2 + η 2 ) 2 dξdη ≤ h η -2 F 1 1 (χ) Å ξ η ã 2 F 1 h (v)(η) 2 (1 + ξ 2 ) 2 (1 + η 2 ) 2 dηdξ ≤ h Å R |F 1 1 χ(ξ)| 2 (1 + ξ 2 ) 2 dξ ã Å R |F 1 h (η)| 2 (1 + η 2 ) 3/2 dη ã ≤ Ch||v|| 2 H 3/2 h (R)
We then assume that for all v ∈ H 3/2 h (∂O j ), supp(T h j v) ⊂ supp χ. Then, we claim that

H j (z) = 1 R 2 \Oj T h j -R j,θ (z)1 R 2 \Oj P θ (z)T h j
where R j,θ (z) is the resolvent of the complex scaled Dirichlet realization of -h 2 ∆ -1 on R 2 \ O j . Indeed, the boundary condition on ∂O j is satisfied since Ran(R j,θ ) ⊂ H 1 0 (R 2 \ O j ), and by definition,

P θ R j,θ w = w in R 2 \ O j , for w ∈ L 2 (R 2 \ O j ).
As a consequence, is suffices to show that χR j,θ (z)χ is holomorphic in D(0, Kh) with the bound

χR j,θ (z)χ = O(h -1 ) : L 2 (R 2 \ O j ) → L 2 (R 2 \ O j )
This is a rather standard non-trapping estimates (here, when there is a single obstacle, the billiard flow is non-trapping). As explained in [START_REF] Nonnenmacher | Fractal weyl law for open quantum chaotic maps[END_REF] in the proof of Lemma 6.1, such an estimate relies on propagation of singularities concerning the wave propagator : one can check that an abstract non-trapping condition for black box Hamiltonian is satisfied (see for instance [START_REF] Dyatlov | Mathematical Study of Scattering Resonances[END_REF], Definition 4.42). This implies that the required statement ([DZ19], Theorem 4.43) holds.

Finally, we recall the crucial relation between R Ω,θ (z) and M(z) (see [START_REF] Nonnenmacher | Fractal weyl law for open quantum chaotic maps[END_REF], formula 6.11 and the references given there). Assume that z ∈ D(0, Kh) and that Id -M(z) is invertible. Then, so is R Ω,θ (z) and we have

(2.18) R Ω,θ (z) = 1 Ω R θ (z) -1 Ω H(z)(Id -M(z)) -1 γR θ (z)
In particular, we see that if we have a bound for

||(Id -M(z)) -1 || H 3/2 h →H 3/2 h
we find a resolvent bound for R Ω,θ (z). In fact, as explained in [START_REF] Nonnenmacher | Fractal weyl law for open quantum chaotic maps[END_REF], it is sufficient to work on L 2 (∂O) in virtue of the following result :

Lemma 2.3. ([NSZ14], Lemma 6.5) For j = 1, . . . , J, let B * ∂O j := {(y, η), y ∈ ∂O j , |η| ≤ 1} and consider χ j ∈ C ∞ c (T * ∂O j ) such that χ j = 1 near B * ∂O j . Then, by denoting Op h (χ j ) a quantization of χ j and by D the diagonal operator-valued matrix Diag(Op h (χ 1 ), . . . , Op h (χ J )), we have

(Id -D)(Id -M(z)) = O(h ∞ ) L 2 (∂O)→C ∞ (∂O) (Id -M(z))(Id -D) = O(h ∞ ) L 2 (∂O)→C ∞ (∂O)
As a consequence of this lemma, Id -M(z) extends to an operator L 2 (∂O) → L 2 (∂O) and as soon as Id -M(z) is invertible and z ∈ D(0, Kh)

||(Id -M(z)) -1 || H 3/2 h →H 3/2 h ≤ C 1 ||(Id -M(z)) -1 || L 2 →L 2 (with a constant C 1 independent of z).
Microlocal properties of M(z) and reduction to a simpler problem. We recall the main microlocal properties of M(z) and reduce the invertibility of Id -M(z) to a nicer Fourier integral operator, as explained in [START_REF] Nonnenmacher | Fractal weyl law for open quantum chaotic maps[END_REF] (Section 6). To do so, let us introduce the following notations.

For j ∈ {1, . . . , J}, let B * ∂O j be the co-ball bundle of ∂O j , S * ∂Oj be the restriction of S * Ω to ∂O j , π j : S * ∂Oj → B * ∂O j the natural projection and ν j (x) be the outward normal vector at x ∈ ∂O j (see Figure 4a).

For i = j, let B ± ij : B * ∂O j → B * ∂O i be the symplectic open maps defined by

ρ = B ± ij (ρ ) ⇐⇒ ∃t > 0 , ∃ξ ∈ S 1 , ∃x ∈ ∂O j (2.19) x + tξ ∈ ∂O i , ν j (x), ξ > 0 , ± ν i (x + tξ) , ξ < 0, (2.20) π j (x, ξ) = ρ , π i (x + tξ, ξ) = ρ (2.21) B +
ij is the billiard map, whereas B - ij is a shadow map (see Figure 4b and4c).These maps are open. (see Figure 4d). Note that due to our definition of these maps, the glancing rays (that is the rays associated with a point ρ = (x, ξ) ∈ B * ∂O with |η| = 1) are not in the set of definition of B ± ij . Moreover, due to Ikawa's condition, if a point ρ ∈ B * ∂O j has an image by B ± ij , it cannot have one by B ± kj for k = i. Let A ij be the closure of the arrival set of the billiard map, that is

A ij = {ρ ∈ B * ∂O i , ∃ρ ∈ B * ∂O j , ρ = B + ij (ρ )} Similarly, let D ij
be the closure of the departure set of the billiard map, that is

D ij = {ρ ∈ B * ∂O j , ∃ρ ∈ B * ∂O i , ρ = B + ij (ρ )} We also note A i = j =i A ij ; D i = j =i D ji
Finally, we introduce the arrival and departure glancing regions :

A G i = A i ∩ S * ∂O i ; ‹ D G i = D i ∩ S * ∂O i
We recall the main facts proved in [START_REF] Nonnenmacher | Fractal weyl law for open quantum chaotic maps[END_REF] concerning these relations and their link with M(z) :

Lemma 2.4. (See Figure 5). Assuming that the obstacles satisfy the no-eclipse condition, the following holds :

let i = j = k. Let (ρ 1 , ρ 1 ) ∈ Gr(B - ji ) and (ρ 2 , ρ 2 ) ∈ Gr(B ± kj ). Then, ρ 1 = ρ 2 Figure 5
. The no-eclipse condition prevents such situation. The points ρ 1 , ρ 2 , ρ 1 are represented on the form ρ = (y, η) with a blue point for y ∈ ∂O and a blue arrow for η ∈ B * y ∂O. The limit situations, where the dotted line would be tangent to one of the obstacle are also excluded. 6a : it means that the ray continues to infinity) 6b : it means that the ray comes from infinity)

In particular, it is possible to consider open neighborhoods U

A i and U D i of A G i and ‹ D G i re- spectively, such that (see Figure 6), by noting π R (resp. π L ) the projection (ρ , ρ) → ρ (resp. (ρ , ρ) → ρ ) ρ ∈ U A i =⇒ ρ ∈ k =i π R Gr(B ± ki ) (see Figure
ρ ∈ U D i =⇒ ρ ∈ k =i π L Gr(B ± ik ) (see Figure
Let us fix cut-off functions χ D i (resp.

χ A i ) such that χ D i = 1 near D G i (resp. χ A i = 1 near A G i ) and supp χ D i ⊂ U D i (resp. supp χ A i ⊂ U A i ).
We gather the results of Proposition 6.7 in [START_REF] Nonnenmacher | Fractal weyl law for open quantum chaotic maps[END_REF] and some of its consequence in the following proposition. It is based on the microlocal analysis of the operators involved.

Proposition 2.3. For i = j ,

• uniformly in z ∈ D(0, Kh),

M ij (z) Op h (χ A j ) = O(h ∞ ) L 2 →C ∞ ; Op h (χ D i )M ij (z) = O(h ∞ ) L 2 →C ∞ (a)
Due to the no-eclipse condition, if a trajectory coming from an obstacle becomes glancing, then it goes on to infinity without hitting another obstacle. This holds in a neighborhood of the glancing ray and allows to define U A i .

(b) Due to the no-eclipse condition, if a glancing trajectory hits an obstacle, it means that the ray comes from infinity. This holds in a neighborhood of the glancing ray and allows to define U D i .

Figure 6. The sets U A i and U D i are built by using the properties of the glancing rays. The dotted lines correspond to glancing rays, the broken lines represent trajectories close to the glancing region.

• By excluding the glancing region on the left and on the right, we have

(1 -Op h (χ D i ))M ij (z)(1 -Op h (χ A j )) ∈ I 0 (∂O i × ∂O j , Gr Ä B + ij ä ) + I 0 (∂O i × ∂O j , Gr( Ä B - ij ä ) so let us write (1 -Op h (χ D i ))M ij (z)(1 -Op h (χ A j )) = M + ij (z) + M - ij (z) with M ± ij (z) ∈ I 0 (∂O i × ∂O j , Gr Ä B ± ij ä ).
Only compact parts of the interior of the graphs of B ± ij are involved in the definition of the class I 0 (∂O i × ∂O j , Gr

Ä B ± ij ä
), depending on the support of χ D i and χ A j (see A.2.2 in the appendix, for a description of this class). • The operators M ± ij (z) have amplitude α ± ij (z) satisfying, for z ∈ D(0, Kh) and for some

C 1 , τ > 0, α ± ij (z) ≤ C 1 e -τ Im z h • Finally, in virtue of Lemma 2.4, M ± ij (z) • M - jk (z) = O(h ∞ ) L 2 →C ∞ uniformly for z ∈ D(0, Kh).
Let us note M ± (z) the matrix of operators with

M ± (z) ij = ß M ± ij (z) if i = j 0 if i = j
Then, we observe that

(2.22) (Id -M(z)) (Id +M -(z)) = Id -M + (z) + O(h ∞ ) L 2 →C ∞
Since we are interested in invertibility in strips, let's note :

Ω(γ, K, h) = D(0, Kh) ∩ {Im z ≥ -γh}
We have the rather obvious lemma :

Lemma 2.5. Assume that for z ∈ Ω(γ, K, h), Id -M + (z) is invertible and satisfies the bound

||(Id -M + (z)) -1 || L 2 →L 2 ≤ a(z, h)
with a(z, h) ≤ h -N for some N independent of z ∈ Ω(γ, K, h). Then, there exists h 0 > 0 and C > 0, such that for 0 < h ≤ h 0 , and for all z ∈ Ω(γ, K, h), Id -M(z) is invertible and satisfies

||(Id -M(z)) -1 || L 2 →L 2 ≤ Ca(z, h)
Proof. Assuming the invertibility of Id -M + (z), it suffices to write

(Id -M(z)) (Id +M -(z))(Id -M + (z)) -1 = Id +R(z, h) with R(z, h) = (Id -M + (z)) -1 O(h ∞ ) L 2 →C ∞ = O(h ∞ ) L 2 →L 2 uniformly in z.
We conclude by a Neumann series argument to invert the right hand side Id +R(z, h) and use the bound on the amplitude of M -(z) given in Proposition 2.3, which gives a uniform bound for M -(z) in D(0, Kh).

It is then enough to prove the invertibility of Id -M + (z) with polynomial resolvent bounds, where M + (z) is associated with the billiard map.

Conjugation by an escape function. The operator M + (z) satisfies almost all the assumptions of Proposition 2.1 for the relation F = B + , except that it is not very small outside a fixed compact neighborhood of T 2 . To fix this problem, following [NSZ14] (Section 6.3), we can introduce a smooth escape function g 0 . Recall that T is the trapped set for F = B + and let W 1 W 2 W 3 be subsets of i ‹ D i such that T ⊂ W 1 and such that W 3 is large enough so that M + (z) Op h (φ) = O(h ∞ ) L 2 →C ∞ for any smooth function φ such that supp(φ) ∩ W 3 = ∅. This is possible in virtue of the third point in Proposition 2.3. Concerning W 2 , it can be an arbitrarily small neighborhood of T . Then, one can construct g 0 such that (see Lemma 4.5 in [NSZ14]),

g 0 = 0 in W 1 g 0 • F -g 0 ≥ 0 in W 3 g 0 • F -g 0 ≥ 1 in W 3 \ W 2
Then, we set g = T log(1/h)g 0 for some T > 0 fixed and large enough, so that e ± Op h (g) are pseudodifferential operators and satisfies

e ± Op h (g) = O h -T C0 ; C 0 = ||g 0 || ∞ (Note that Op h (g) is a diagonal matrix-valued operator on L 2 (∂O) = L 2 (∂O i ))
, and in virtue of Egorov's theorem, the operator M + g := e -Op h (g) M + (z)e Op h g is O(h L ) for some L > 0, microlocally outside a neighborhood of T , which can be made as small as necessary if W 2 is small enough.

End of proof. We can now apply Proposition 2.1 and then, Proposition 2.2, to M (h) = M + g (z) for z ∈ D(0, Kh) with Im z ≥ -γh. To control the amplitude α h (z) of M + g (z), we simply need a bound in a small neighborhood of T in which M + g is not O(h L ). In virtue of Egorov's theorem, the amplitude of M + g is smaller than the amplitude of M + . We now claim that there exists τ > 0 such that the amplitude satisfies :

||α h (z)|| ∞ ≤ e -τ Im z h
In fact, as explained in [START_REF] Nonnenmacher | Spectral problems in open quantum chaos[END_REF] (Theorem 6), microlocally near the trapped set, it is possible to write

M + (z) = M + (0) Op h (a h,z ) + O(h 1-ε ) ; a h,z (ρ) = exp Å izt(ρ) h ã
where t(ρ) is the time needed for a ray emanating from ρ to hit another obstacle. This fact is a consequence of the microlocal analysis performed in [START_REF] Gérard | Asymptotique des pôles de la matrice de scattering pour deux obstacles strictement convexes[END_REF] (see Appendix II for the construction of a parametrix and III.2 for precise computations near the unique trapped ray for two obstacles, see also [START_REF] Stefanov | Distribution of resonances for the Neumann problem in linear elasticity outside a strictly convex body[END_REF]). In particular, τ in the estimate above is a maximal return time for the billiard flow, in a small neighborhood of T . Now, let γ, δ be the constants given by Proposition 2.1, depending, in this context, on the dynamics of the billiard map. Let us introduce the following threshold

γ lim = 1 2τ γ δ so that z ∈ Ω(γ lim , K, h) =⇒ ||α h (z)|| ∞ ≤ e γ/2δ < e γ/δ
2 strictly speaking, ∂O is a not a disjoint union of intervals, but since we work with the relation B + , we can use microlocal cut-offs to restrict to the relevant part of the obstacles, which is included in a disjoint union of open intervals Proposition 2.2 now gives for z ∈ Ω(γ lim , K, h),

Id -M + g (z) -1 ≤ 2δ| log h|h -δ log A
where A := max(1, τ γ lim ) Indeed, A ≥ 1 and it allows to have ||α h (z)|| ∞ ≤ A for z ∈ Ω(γ lim , K, h). Going back to M + (z), we get that

Id -M + (z) -1 ≤ C| log h|h -δ log A-2C0T
where the extra h -2C0T comes from the norm of e ± Op h (g) . We conclude with Lemma 2.5 and the formula (2.18), using the estimates of Lemma 2.1 and 2.2. This gives for h small enough and z ∈ Ω(γ lim , K, h),

(2.23) ||R θ (z)|| L 2 →L 2 ≤ C| log h|h -1-2C0T -δ log A 2.3.
Proof in the case of scattering by a potential. The treatment of scattering by a potential is different and relies on a reduction to Poincaré sections of the Hamiltonian flow, under the assumption that the trapped set is totally disconnected.

2.3.1. Assumptions. We refer the reader to [NSZ11] (Section 2.1) for more general assumptions.

Here, we simply consider a smooth compactly supported potential V ∈ C ∞ c (R 2 ) and work with the semiclassical differential operator -h 2 ∆ + V . We fix an energy E 0 > 0 and consider

P h = -h 2 ∆ + V -E 0
We note p(x, ξ) = ξ 2 + V -E 0 and we assume that 0 is not a critical energy of p, that is

dp = 0 on p -1 (0)
Let's note H p the Hamiltonian vector field associated with p and Φ t = exp(tH p ) the corresponding Hamiltonian flow. The trapped set at energy 0 is the set

K 0 = {(x, ξ) ∈ p -1 (0), ∃R > 0, ∀t ∈ R, Φ t (x, ξ) ∈ B(0, R)}
It is a compact subset of p -1 (0). Here are the two crucial assumptions :

(i) Φ t is hyperbolic on K 0 ;

(ii) K 0 is topologically one dimensional.

The reduction of [NSZ11]

. We recall the main ingredients of the reduction to open quantum maps performed in [START_REF] Nonnenmacher | From open quantum systems to open quantum maps[END_REF]. The aim of the following lines is to explain their crucial Theorem 5. Let us note

R(η, M 0 , h) = {z ∈ C, | Re z| ≤ η, | Im z| ≤ M 0 h}
Here, M 0 is fixed (but large). As in the case of obstacle scattering, we fix once and for all the cut-off function χ ∈ C ∞ c (R 2 ) (with χ = 1 in a neighborhood of supp(V )) and we consider a complex scaled version of P (h), P θ (h) whose eigenvalues coincide with the resonances in R(η, M 0 , h) and such that χ(P θ -z) -1 χ = χ(P -z) -1 χ for z ∈ R(δ, M 0 , h). Note that the parameter θ chosen in [START_REF] Nonnenmacher | From open quantum systems to open quantum maps[END_REF] depends on h.

Here are now the crucial ingredients of the reduction.

• Poincaré sections. There exist finitely many smooth contractible hypersurfaces Σ i ⊂ p -1 (0), i = 1, . . . , J with smooth boundary and such that

∂Σ i ∩ K 0 = ∅ ; Σ i ∩ Σ k = ∅, k = i
H p is transversal to Σ i uniformly up to the boundary Moreover, for 1 ≤ i ≤ J and for every ρ ∈ K 0 , there exists t -(ρ) < 0 and i -(ρ) (resp. t + (ρ) > 0 and i + (ρ)) such that

Φ t±(ρ) (ρ) ∈ K 0 ∩ Σ i±(ρ) Σ ∩ {Φ t (ρ), t -(ρ) < t < t + (ρ), t ∈ R * } = ∅
where we note The maps t ± (ρ) are uniformly bounded on K 0 and can be smoothly extended in a neighborhood of K 0 . For convenience, it is also assumed that i + (ρ) = i for all ρ ∈ K 0 ∩ Σ i . This can be achieved by taking smaller and more Poincaré sections. Finally, there exist Σ i R 2 and symplectic diffeomorphisms

Σ = J i=1 Σ i
κ i : Σ i → Σ i
smooth up to the boundary.

• Poincaré return map. For 1 ≤ i, j ≤ J, the map ρ → Φ t+(ρ) (ρ), initially defined on K 0 , extends smoothly to a symplectic diffeomorphism

F ij : D ij → A ij by taking the intersection of the flow of a point ρ ∈ D ij with Σ i where D ij (resp. A ij ) is a neighborhood of {ρ ∈ T ∩ Σ j , i + (ρ) = i} (resp. {ρ ∈ T ∩ Σ i , i -(ρ) = j}) in Σ j (resp. Σ i ).
The map F ij is called the Poincaré return map. By writing it in the charts κ i and κ j , we can consider the following map between open sets of

T * R ‹ F ij = κ i • F ij • κ -1 j : ‹ D ij ⊂ Σ j → A ij ⊂ Σ i
Using the continuity of the flow, the same objects can be defined on energy shells p -1 (z) for z ∈ [-δ, δ] with δ small enough and we will note these objects

Σ (z) i , F (z)
ij , etc. In fact, it is possible to use the same open sets Σ i and define,

‹ F (z) ij = κ i • F (z) ij • κ -1 j : ‹ D (z) ij ⊂ Σ j → A (z) ij ⊂ Σ i
The hyperbolicity of the flow implies the hyperbolicity of these open maps.

• Open quantum maps. The notion of open quantum hyperbolic map associated with F has been given in Definition 2.1. Since Σ j ⊂ T * R, we will simply say that it is an operator-

valued matrix T = (T ij ) 1≤i,j≤J : L 2 (R) J → L 2 (R) J with T ij ∈ I δ (R × R, Gr Ä ‹ F ij ä ).
In [START_REF] Nonnenmacher | From open quantum systems to open quantum maps[END_REF], the authors construct a particular family of open quantum hyperbolic maps, called M(z), where M(z) is associated with ‹ F (Re z) . This family is first microlocally defined near the trapped set and satisfies uniformly in R(η, M 0 , h) and microlocally in a fixed neighborhood of the trapped set :

M ij (z) = M ij (Re z) Op h (a z ) + O(h log h) ; a z = exp i (Im z)t (z) + h
This particular family is built to solve a microlocal Grushin problem (see section 4 in [START_REF] Nonnenmacher | From open quantum systems to open quantum maps[END_REF]). • Escape functions. To perform a global study (i.e. no more microlocal) and to make the amplitude of M(z) very small outside a fixed neighborhood of the trapped set, the authors introduce an escape function for the flow

Φ t , denoted G 0 ∈ C ∞ c (T * R 2 ) (see Lemma 5.3 in [NSZ11]
, where it is chosen independent of the energy variable near Σ). Let us note

G = M h log(1/h)G 0 , g j = G| Σj • κ -1 j and g : J j=1 Σ j → R, ρ ∈ Σ j → g j (ρ)
g is an escape function for the map ‹ F . Each g j can be extended as an element of C ∞ c (R 2 ) (see equation 5.2 and below in [START_REF] Nonnenmacher | From open quantum systems to open quantum maps[END_REF]).

• Conjugated operators. As in the case of obstacle scattering, we can consider the operators

e ± Op h (G) : L 2 (R 2 ) → L 2 (R 2 ) ; e ± Op h (g) : L 2 (R) J → L 2 (R) J
Again, their norm is bounded by O(h -K G ) for some K G > 0 depending on G 0 and M . We now introduce the following conjugated operators :

P θ,G = e -Op h (G) P θ e Op h (G) ; M ij,g (z) = e -Op h (gi) M ij (z)e Op h (gj ) M g (z) = (M ij,g (z)) 1≤i,j≤J
The escape function G 0 is built so that M g (z) :

L 2 (R) J → L 2 (R) J is O(h K0
) for some (large) K 0 microlocally outside a small neighborhood of the trapped set T . In particular, M g (z) satisfies the assumptions of the propositions 2.1 and 2.2. • A finite dimensional space. For practical and technical reasons,3 the authors choose to work with a finite dimensional version of the open quantum map M g (z). To do so, they introduce finite rank projections

Π j : L 2 (R) → L 2 (R) and the finite dimensional subspace of L 2 (R) J H = Ran Π 1 × • • • × Ran Π J
The Π j 's are built so that the projector Π = Diag(Π 1 , . . . , Π J ) satisfies the very important relation

(2.24) ΠM(z)Π = M(z)Π + O(h K1 )
for some large K 1 (in particular K 1 K G so that the same relation holds after conjugation by e Op h g with K 1 replaced by K 1 -2K G ). We will note Π g = e -Op h (g) Πe Op h (g) .

• The Grushin problem. To obtain a global Grushin problem (see section 5 in [START_REF] Nonnenmacher | From open quantum systems to open quantum maps[END_REF]) the authors construct global operators R + (z) :

H 2 h (R 2 ) → H, R -(z) : H → H 2 h (R 2 ) which depend holomorphically on z ∈ R(η, M 0 , h). The Grushin problem concerns (2.25) P g (z) = Å P θ,G -z R -(z) R + (z) 0 ã : H 2 h (R 2 ) × H → L 2 (R 2 ) × H
The goal of such a Grushin problem is to transform the eigenvalue equation P θ,G u = zu into an equation on a simpler operator E ± (z). This transformation is possible when P g (z) is invertible. Indeed, in virtue of the so-called Schur complement formula, if P g (z) is invertible with inverse

E(z) = Å E(z) E + (z) E -(z) E ± (z) ã : L 2 (R 2 ) × H → H 2 h (R 2 ) × H, then P θ,G -z is invertible if and only if E ± (z
) is and we have

(P θ,G -z) -1 = E(z) -E + (z)E ± (z) -1 E -(z)
The authors prove the following result :

Theorem 4 ([NSZ11], Theorem 5). The Grushin problem (2.25) is invertible for all z ∈ R(η, M 0 , h).

If we note

E(z) = Å E(z) E + (z) E -(z) E ± (z) ã : L 2 (R 2 ) × H → H 2 h (R 2 ) × H the inverse of P g (z), then • 4 ||E||, ||E + ||, ||E -||, ||E ± || = O(h -1 ) uniformly in R(η, M 0 , h).
• The operator E ± (z) takes the form, for some L 2 > 0

E ± (z) = Id -M g (z, h) + O(h L2 ) ; M g (z, h) := Π g M g (z)Π g
Remark. As explained after Theorem 2 in [START_REF] Nonnenmacher | From open quantum systems to open quantum maps[END_REF], L 2 = c M for some c , where M is the one in the definition of the escape function G. In particular, M can be chosen arbitrarily large, independently of c , so that L 2 can be made as large as necessary.

2.3.3. End of proof. To rigorously apply Proposition 2.1 to M g (z), we fix η 0 ∈ [-η, η] for η small enough and consider z ∈ D(η 0 , Kh) for some fixed 0 > K < M 0 . For such z, M(z) is an open quantum map associated with the Poincaré return map between the Poincaré sections Σ(η 0 ) = 1≤j≤J Σ j (η 0 ) inside the energy shell p -1 (η 0 ). Since M g (z) satisfies the assumption of Proposition 2.1 for z ∈ D(η 0 , Kh) ∩ R(η, M 0 , h), it also satisfies its conclusion :

(2.26)

∃h 0 > 0, δ, γ > 0 ; ∀0 < h ≤ h 0 , ||M(z) N (h) g || L 2 →L 2 ≤ h γ ||α h (z)|| N (h) ∞
with N (h) ∼ δ| log h|. A priori, h 0 , δ, γ depend on η 0 . Nevertheless, as explained after Proposition 2.1, δ and γ depend only on the properties of the Poincaré return map F (η0) and h 0 depends on semi-norms of α h (z) = e i z h t (η 0 ) +

. As explained in Section 4.1.1 in [START_REF] Nonnenmacher | From open quantum systems to open quantum maps[END_REF], this dynamics depends continuously on η 0 in a neighborhood of 0 : that is, the departure sets, the arrival sets, the Poincaré maps and the return time function depend continuously on η 0 . As a consequence, we can find η and constants δ, γ, h 0 such that (2.26) holds for z ∈ Ω(η, γ, h) := {| Re z| ≤ η, -γh ≤ Im z ≤ 0}. From this, we see that for 0 < h ≤ h 0 and for z ∈ Ω(η, γ, h),

||M(z) N (h) g || L 2 →L 2 ≤ h γ e -N (h)τ Im z h ; τ = sup |η0|≤η ||t (η0) ∞ || ∞
From (2.24), we see that for N (h) ∼ δ| log h|,

M g (z) N (h) = Π g M N (h) g Π g + O((log h)h K1 )
so that we deduce that M g (z) satisfies also the conclusion of Proposition 2.1 and hence, of Proposition 2.2. As a direct consequence, we obtain that for h small enough, E ± (z) is invertible for all z ∈ Ω(η, γ, h) and it satisfies for some β > 0 :

||E ± (z) -1 || H→H ≤ h -β
We now conclude the proof as in the case of obstacle scattering, essentially replacing the formula (2.18) by the standard Schur complement formula for the Grushin problem above : E ± (z) is invertible if and only if P θ -z is and

(P θ,G -z) -1 = E(z) -E + (z)E ± (z) -1 E -(z)
Then, for h small enough and for z ∈ Ω(η, γ, h),

||(P θ,G -z) -1 || L 2 →H 2 h = O(h -1 ) + O(h -β-2 ) = O(h -β-2 ) which gives ||(P θ -z) -1 || L 2 →H 2 h = O(h -β-2-2K G )
where c 0 depends on G 0 .

Application to the local energy decay for the wave equation

We present an application of the resolvent estimate obtained in the case of obstacle scattering to the decay of the local energy for the wave equation outside the obstacles. In this note, we follow the main arguments of [START_REF] Burq | Décroissance de l'énergie locale de l'équation des ondes pour le problème extérieur et absence de résonance au voisinage du réel[END_REF] to prove Theorem 3.

3.1. Resolvent estimates. Let us rewrite the resolvent estimate of Theorem 2 in term of λ : there exists γ > 0, λ 0 > 0 and β > 0 such that for any χ ∈ C ∞ c (R 2 ) equal to one in a neighborhood of O, there exists C χ > 0 such that for all λ ∈ C,

(3.1) |λ| ≥ λ 0 , Im λ ≥ -γ =⇒ ||χR(λ)χ|| L 2 →L 2 ≤ C χ |λ| β
Recalling that for f ∈ L 2 comp , with g = R(λ)f it holds that χg ∈ H 1 0 (Ω) and g satisfies -∆g+λ 2 g = f , it is not hard to see that the above estimate implies that

(3.2) ||χR(λ)χ|| L 2 →H 1 0 ≤ C χ |λ| β+1
for |λ| ≥ λ 0 and Im λ ≥ -γ (see for instance the proof of Proposition 2.5 in [START_REF] Burq | On nonlinear Schrödinger equations in exterior domains[END_REF]). This gives resolvent estimates for large λ. We will also need to control the resolvent for small λ, in angular neighborhoods of the logarithmic singularity at 0. For this purpose, we state a consequence of a result proved in [START_REF] Burq | Décroissance de l'énergie locale de l'équation des ondes pour le problème extérieur et absence de résonance au voisinage du réel[END_REF] (Appendix B.2) :

Lemma 3.1. For ε > 0, let S ε = {λ ∈ C * , |λ| ≤ ε, arg λ ∈ [-π/4, 5π/4]}.
There exists ε 0 > 0 such that there is no resonance in S ε0 and for any χ ∈ C ∞ c (R 2 ) equal to one in a neighborhood of O, there exists C χ > 0 such that for all λ ∈ S ε0 , (3.3) ||χR(λ)χ|| L 2 →H 1 0 ≤ C χ Finally, we also mention the following result, proved in [START_REF] Burq | Décroissance de l'énergie locale de l'équation des ondes pour le problème extérieur et absence de résonance au voisinage du réel[END_REF] (Appendix B.1), which will be used below : Lemma 3.2. There are no real resonances (that is with arg(z) = 0 or π).

A = Å 0 Id ∆ 0 ã with domain D(A) = (H ∩ H 2 (Ω)) ⊕ H 1 0 (Ω).
A is maximal dissipative, so that Hille-Yosida theory allows to define the propagator e tA and for (u 0 , u 1 ) ∈ H, the first component u(t) of t → e tA (u 0 , u 1 ) is the unique solution of the following Cauchy problem

   ∂ 2 t u -∆u = 0 u| t=0 = u 0 ∂ t u| t=0 = u 1
Note also that since A is maximal dissipative, for ξ with Re(ξ) > 0, A -ξ is invertible and

(3.4) ||(A -ξ) -1 || H→H ≤ | Re ξ| -1
The global energy of the solution is defined as

E(t) = 1 2 ||(u(t), ∂ t u(t))|| 2 H = 1 2 Ω |∇u(t)| 2 + |∂ t u(t)| 2
It is conserved. If K R 2 , we also define the local energy in K as

E K (t) = 1 2 K∩Ω |∇u(t)| 2 + |∂ t u(t)| 2 Note that, by Poincaré inequality, if B ⊂ Ω is bounded and if χ ∈ C ∞ c (R 2 ) is equal to one in a neighborhood of O and is supported in B, then for f ∈ H(Ω), ||χf || H ∼ ||χf || H 1 (B) ∼ ||χf || L 2 (B) + ||∇(χf )|| L 2 (B) If χ ∈ C ∞ c (R 2
) is equal to one in a neighborhood of O, by abuse we note χ the bounded operator of (u, v) ∈ H → (χu, χv) ∈ H.

A short computation shows that for λ ∈ C, (u 0 , u 1 ) ∈ D(A) and (v 0 , v 1 ) ∈ H,

(A + iλ) Å u 0 u 1 ã = Å v 0 v 1 ã ⇐⇒ ß (-∆ -λ 2 )u 0 = iλv 0 -v 1 u 1 = v 0 -iλu 0
This relation and the remark above for bounded sets B show that for any χ ∈ C ∞ c (R 2 ), the cut-off resolvent χ(A + iλ) -1 χ, well defined for Im λ > 0 extends to the logarithmic cover Λ of C and we have for λ ∈ Λ We deduce that χ(A + iλ) -1 χ has no real resonance and satisfies the following resolvent estimates, for some constant C χ ,

(3.5) χ(A + iλ) -1 χ = Å iλχR(λ)χ -χR(λ)χ χ 2 + λ 2 χR(λ)χ iλχR(λ)χ
λ ∈ S ε0 =⇒ χ(A + iλ) -1 χ H→H ≤ C χ (3.6) |λ| ≥ λ 0 , Im λ ≥ -γ =⇒ χ(A + iλ) -1 χ H→H ≤ C χ |λ| β+2 (3.7)
3.3. Proof of the local energy decay. Let us fix R > 0 such that O B(0, R). We want to estimate the local energy in B(0, R) for solutions with initial data supported in B(0, R), and sufficiently regular, that is in D(A k ) for a sufficiently large k. As we will see, the decay will hold for data in D(A k ) with k ≥ β + 4, where β is the one appearing in (3.7). For this purpose, let us

fix χ ∈ C ∞ c (R 2 ) such that χ = 1 in B(0, R). Let U 0 ∈ D(A k ) with supp(U 0 )
B(0, R). We want to estimate the energy of χe tA U 0 , or equivalently, we want to control χe tA U 0 H . Let us write U

= (I -A) k U 0 ∈ H, so that ||U 0 || D(A k ) = ||U || H .
It is clear that we have supp(U ) ⊂ B(0, R), so that U = χU . With this notation, we want to show that there exists C R > 0 such that for all t ≥ 1,

I(t) := χe tA (I -A) -k χU H ≤ C t ||U || H
The starting point of the proof is the following formula :

Lemma 3.3. Assume that k ≥ 2. For t ≥ 0 and for U ∈ H, we have

(3.8) e tA (I -A) -k U = -1 2π λ∈ i 2 +R e -itλ 1 (1 + iλ) k (A + iλ) -1 U dλ
Proof. First remark that the integral I(t) in the right hand side is absolutely convergent in virtue of (3.4) and since k ≥ 2. Differentiating the right hand side with respect to t, we find that

(∂ t -A)I(t) = -1 2π λ∈ i 2 +R e -itλ 1 (1 + iλ) k (-iλ -A)(A + iλ) -1 U dλ = 1 2π λ∈ i 2 +R e -itλ 1 (1 + iλ) k U dλ = 0
(To see that the last integral is equal to zero, one can for instance perform a contour deformation from Im(λ) = 1/2 to Im(λ) = -ρ and let ρ tend to +∞. ) Finally, we need to check that I(0) = (Id -A) -k U . We have

I(0) = -1 2π λ∈ i 2 +R 1 (1 + iλ) k (A + iλ) -1 U dλ
We perform a contour deformation. Let r > 1 and let Γ r be rectangle joining the points i/2 + r, r(1 + i), r(i -1), i/2 -r. We also note γ

r = Γ r \ [-r + i/2, r + i/2]. The function g k : z → -(1 + iz) -k (iz Id +A) -1 U is meromorphic in Im z > 0, with a unique pole at z = i. As a consequence, we find that 1 2iπ Γr g k (z)dz = Res z=i g k = -1 i k (k -1)! ∂ k-1 z ((iz Id +A) -1 )U | z=i = i(Id -A) -k U
Hence, we have

I(0) = lim r→+∞ -i 2iπ i/2+r i/2-r g k (λ)dλ = lim r→+∞ -i Ç i(Id -A) -k U - γr g k (z)dz å = (Id -A) -k U
Indeed, it is not hard to see that the contribution on γ r tends to 0 as r → +∞.

End of proof of Theorem 3. The proof relies on a contour deformation below the real axis in the integral of (3.8) where the cut-off χ is inserted, but we need to get around the logaritmic singularity at 0 : it is possible due to (3.6).

We fix t > 0. In the estimates below, the constants denoted by C (or C k ) do not depend on t. We know that the map λ → χ(A + iλ) -1 χU is meromorphic in C \ iR -, with no poles in {Im λ > 0} ∪ S ε0 ∪ {Im λ ≥ -γ, |λ| ≥ λ 0 }. By taking ε 0 smaller if necessary, we may assume that 2 -1/2 ε 0 ≤ γ. Let K be the union of the rectangles K + and K -where Since there is only a finite number of resonance in K and since there are no resonances on K ∩ {Im λ = 0}, we can find δ > 0 such that there is no resonance in K ∩ {Im λ ≥ -δ} and since this region is compact, we can find C such that for λ

K ± = {λ ∈ C, Re λ ∈ [±2 -1/2 ε 0 , ±λ 0 ], Im λ ∈ [-2 -1/2 ε 0 , 0]} γ + r l r l + r l - r γ - r C ε γ - ε γ + ε
∈ K ∩ {Im λ ≥ -δ}, ||χ(A + iλ) -1 χ|| H→H ≤ C Let's note z + (resp. z -) the unique point in {|z| = ε 0 } ∩ {Im z = -δ} ∩ {± Re z > 0}.
Fix r 1 and 0 < ε < ε 0 and let's note z ± ε the point of the segment [0, z ± ] with norm ε and let's introduce the following paths, oriented from the left point to the right point :

γ + r = [z + , r -iδ] ; γ - r = [-r -iδ, z -] γ + ε = [z + ε , z + ] ; γ + ε = [z -, z - ε ] l + r = [r -iδ, r + i/2] ; l - r = [-r -iδ, -r + i/2] l r = [-r + i/2, r + i/2]
and C ε be the arc of the circle {|z| = ε} from z - ε to z + ε . (See Figure 8). With f k (z) = -1 2π e -itz (1+iz) -k χ(A+iz) -1 χU , we have χe tA (Id -A) -k χU = lim r→+∞ lr f k (z)dz and since f k (z) is holomorphic in a neighborhood of the compact set surrounded by the above contours, we have The case of γ - ε is treated similarly. On γ ± r , the following holds :

||χ(A + iλ) -1 χ|| H→H ≤ C|λ| β+2
Indeed, this is true for |λ| ≥ λ 0 and there is no resonance on γ ± r . As a consequence, for λ = -iδ +ξ, |ξ| ≥ Re(z + ), we have ||χe -itλ (1 + iλ) -k (A + iλ) -1 χ|| H→H ≤ Ce -tδ |ξ| β+2-k Hence, we assume here that k ≥ β + 4 so that We review the most important notions of semiclassical analysis needed in this note. .

A.1. Pseudodifferential operators and Weyl quantization. We recall some basic notions and properties of the Weyl quantization on R n . We refer the reader to [START_REF] Zworski | Semiclassical Analysis[END_REF] for the proofs of the statements and further considerations on semiclassical analysis and quantizations. We start by defining classes of h-dependent symbols.

Definition A.1. Let 0 ≤ δ ≤ 1 2 . We say that an h-dependent family a := (a(•; h)) 0<h 1 is in the class S δ (T * R n ) (or simply S δ if there is no ambiguity) if for every α ∈ N 2n , there exists C α > 0 such that :

∀0 < h ≤ 1, sup (x,ξ)∈R n |∂ α a(x, ξ; h)| ≤ C α h -δ|α|
In this paper, we will mostly be concerned with δ < 1/2. We will also use the notation S 0 + = δ>0 S δ . We write a = O(h N ) S δ to mean that for every α ∈ N 2n , there exists C α,N such that ∀0 < h ≤ 1, sup For a given symbol a ∈ S δ (T * R n ), we say that a has a compact essential support if there exists a compact set K such :

∀χ ∈ C ∞ c (R n ), supp χ ∩ K = ∅ =⇒ χa = O(h ∞
) S(T * R n ) (here S stands for the Schwartz space). We note ess supp a ⊂ K and say that a belongs to the class S comp δ (T * R n ). The essential support of a is then the intersection of all such compact K's. In particular, the class S comp δ contains all the symbols in S δ supported in a h-independent compact set and these symbols correspond, modulo O(h ∞ ) S(T * R) , to all symbols of S comp δ . For this reason, we will adopt the following notation : for an open set Ω ⊂ R n , a ∈ S comp δ (Ω) ⇐⇒ ess supp a Ω. For a symbol a ∈ S δ (T * R n ), we will quantize it using Weyl's quantization procedure. It is written as :

(Op h (a)u)(x) = (a W (x, hD x )u)(x) = 1 (2πh) n We say that a family u = u(h) ∈ D (R n ) is h-tempered if for every χ ∈ C ∞ c (R n ), there exist C > 0 and N ∈ N such that ||χu|| H -N h ≤ Ch -N . For a h-tempered family u, we say that a point ρ ∈ T * R n does not belong to the wavefront set of u if there exists a ∈ S comp (T * R n ) such that a(ρ) = 0 and Op h (a)u = O(h ∞ ) S . We note WF h (u) the wavefront set of u.

  Spectral gap and resolvent estimates. Let (O j ) 1≤j≤J be open, strictly convex obstacles in R 2 having smooth boundary and satisfying the Ikawa condition of no-eclipse: fori = j = k, O i does not intersect the convex hull of O j ∪ O k . Let O = J j=1 O j ; Ω = R 2 \ O.

Figure 1 .

 1 Figure 1. Scattering by three obstacles in the plane

Figure 2 .

 2 Figure 2. A schematic example with J = 3 in a case where D ii = ∅ for i = 1, 2, 3.

( a )

 a The notations used to define the billiard map and the shadow map.(b) The billiard map.B + ij (yj, ηj) = (yi, ηi).(c) The shadow map. B - ij (yj, ηj) = (yi, ηi). (d) These maps are open. In this figure, the point (yj, ηj) has no image.

Figure 4 .

 4 Figure 4. Description of billiard map and the shadow map.

Figure 7 .

 7 Figure 7. Schematic representation of Poincaré sections for the flow Φ t on an energy shell. The energy shell has dimension 3, so that the Poincaré section are 2-dimensional.

3. 2 .

 2 The wave equation generator. Let H be the Hilbert spaceH(Ω) ⊕ L 2 (Ω), where H is the completion of C ∞ c (Ω) with respect to the norm ||f || H = ||∇f || L 2 (Ω)5 and let A be the operator

ã5

  This choice of Hilbert space makes the wave propagator unitary on H, since the energy of a solution of the wave equation is its norm in H; see[START_REF] Taylor | Partial Differential Equations II: Qualitative Studies of Linear Equations[END_REF], Chapter 9, Section 4

Figure 8 .

 8 Figure 8. The contour used to deform the integral. z -(resp. z + ) is the blue (resp. red) point on the figure.

e

  Note thatC ε ∪ γ + ε ∪ γ - ε ⊂ S ε0. As a consequence, we have Cε f k (z)dz H ≤ C k ε||U || H → ε→0 0 and, with θ = arg z + , ts sin θ ||U || H ds ≤ C k ||U || H t| sin θ| e tε sin θ -e tε0 sin θ ≤ C k t ||U || H

r

  Re(z + ) e -tδ |ξ| β+2-k ||U || H dξ ≤ Ce -tδ +∞ Re(z + ) |ξ| -2 ||U || H dξ ≤ Ce -tδ ||U || HFinally, we treat the vertical segments l ± r .l ± r f k (z)dz H ≤ C sup y∈[-δ,1/2]

  (x,ξ)∈R n |∂ α a(x, ξ; h)| ≤ C α,N h -δ|α| h N If a = O(h N ) S δ for all N ∈ N, we'll write a = O(h ∞ ) S δ .

  (y)e i (x-y)•ξ h dydξ We will note Ψ δ (R n ) the corresponding classes of pseudodifferential operators. By definition, the wavefront set of A = Op h (a) is WF h (A) = ess supp a.

  For y ∈ [-δ, 1/2], we have||f k (±r + iy)|| H ≤ Ce ty r -k ||χ(A + i(r + iy)) -1 χ|| H→H × ||U || H ≤ Ce t/2 r -k ||χ(A + i(r + iy)) -1 χ|| H→H × ||U || HUsing (3.7), we find that for y ∈ [-δ, 1/2], ||χ(A + i(r + iy)) -1 χ|| H→H ≤ Cr β+2 . As a consequence, one finds that for y ∈ [-δ, 1/2],||f k (±r + iy)|| H ≤ Ce t/2 r β+2-k ||U || H ≤ Ce t/2 r -2 ||U || H Ce -tδ + C k ε + Ce t/2 r -2By letting ε tending to 0 and r to +∞, we conclude that ||χe tA (Id -A) -k χU || H ≤

	As a consequence,
	lr	f k (z)dz	H	≤	Å C k t	+ ã	||U || H
						Å C k t	+ Ce -tδ	ã	||U || H ≤	C k t	||U || H
	which gives the required result.

||f k (±r + iy)|| H Appendix A. Tools of semiclassical analysis

We use notations similar to the ones in[START_REF] Nonnenmacher | Fractal weyl law for open quantum chaotic maps[END_REF] but beware that we do not use the exact same conventions.

Mainly, to ensure the existence of determinants without discussion

the norms are associated with the spaces mentionned above. For instance, ||E -|| H→H 2 h (R 2 ) .

We say that a family of operators B = B(h) :

The wavefront set of B, denoted WF h (B) is defined as

The Calderon-Vaillancourt Theorem asserts that pseudodifferential in Ψ δ are bounded on L 2 and as a consequence of the sharp Gärding inequality (see [START_REF] Zworski | Semiclassical Analysis[END_REF], Theorem 4.32), we also have a precise estimate of L 2 norms of pseudodifferential operator, Proposition A.1. Assume that a ∈ S δ (R 2n ). Then, there exists C a depending on a finite number of semi-norms of a such that :

We recall that the Weyl quantizations of real symbols are self-adjoint in L 2 . The composition of two pseudodifferential operators in Ψ δ is still a pseudodifferential operator. More precisely (see [START_REF] Zworski | Semiclassical Analysis[END_REF], Theorem 4.11 and 4.18 D) is a Fourier multiplier acting on functions on R 4n and, writing ρ i = (x i , ξ i ),

A.2. Fourier Integral Operators. We review some aspects of the theory of Fourier integral operators. We follow [START_REF] Zworski | Semiclassical Analysis[END_REF], Chapter 11 and [START_REF] Nonnenmacher | Fractal weyl law for open quantum chaotic maps[END_REF]. We refer the reader to [START_REF] Guillemin | Semiclassical Analysis[END_REF] for further details or to [START_REF] Alexandrova | Semi-classical wavefront set and fourier integral operators[END_REF]. Finally, we will give the precise definition needed to understand the definition 2.1.

A.2.1. Local symplectomorphisms and their quantization. Let us note K the set of symplectomorphisms κ : T * R n → T * R n such that the following holds : there exist continuous and piecewise smooth families of smooth functions (κ t ) t∈[0,1] , (q t ) t∈[0,1] such that : [START_REF] Zworski | Semiclassical Analysis[END_REF], Lemma 11.4, which asserts that local symplectomorphisms can be seen as elements of K, as soon as we have some geometric freedom.

Lemma A.1. Let U 0 , U 1 be open and precompact subsets of T * R n . Assume that κ : U 0 → U 1 is a local symplectomorphism that extends to V 0 U 0 an open star-shaped set. Then, there exists κ ∈ K such that κ| U0 = κ.

If κ ∈ K and if (q t ) denotes the family of smooth functions associated with κ in its definition, we note Q(t) = Op h (q t ). It is a continuous and piecewise smooth family of operators. Then the Cauchy problem (A.1)

is globally well-posed.

From now on, we restrict to the case n = 1. Following [NSZ14], Definition 3.9, we adopt the definition : Definition A.2. Let κ ∈ K and let us note C = Gr (κ) = {(x, ξ, y, -η), (x, ξ) = κ(y, η)} the twisted graph of κ. Fix δ ∈ [0, 1/2). We say that U ∈ I δ (R × R; C) if there exists a ∈ S δ (T * R) and a path (κ t ) from Id to κ satisfying the above assumptions such that U = Op h (a)U (1), where t → U (t) is the solution of the Cauchy problem (A.1).

The class

It is a standard result, known as Egorov's theorem (see [START_REF] Zworski | Semiclassical Analysis[END_REF], Theorem 11.1) that if U (t) solves the Cauchy problem (A.1) and if a ∈ S δ , then

Remark. Applying Egorov's theorem and Beals's theorem, it is possible to show that if (κ t ) is a closed path from Id to Id, and U (t) solves (A.1), then U (1) ∈ Ψ 0 (R n ). In other words,

But the other inclusion is trivial. Hence, this in an equality :

The notation I(R × R, C) comes from the fact that the Schwartz kernels of such operators are Lagrangian distributions associated to C, and in particular have wavefront sets included in C. As a consequence, if

We also recall that the composition of two Fourier integral operators is still a Fourier integral operator : if κ 1 , κ 2 ∈ K and

Quantization of open symplectic maps. As in Section 2, we consider a symplectic map F which is the union of local open symplectic

are open sets. We keep the same notations. In particular, T is the trapped set and the full arrival (resp. departure) set is A (resp. ‹ D). We fix a compact set W ⊂ Ã containing some neighborhood of T . Our definition will depend on W and, is not, in some sense, canonical. Following [START_REF] Nonnenmacher | Fractal weyl law for open quantum chaotic maps[END_REF] (Section 3.4.2), we now focus on the definition of the elements of I δ (Y × Y ; Gr(F ) ). An element

• Fix some small ε > 0 and two open covers of U j , U j ⊂ L l=1 Ω l , Ω l Ω l , with Ω l starshaped and having diameter smaller than ε. We note L the sets of indices l such that Ω l ⊂ ‹ D ij ⊂ U j and we require (this is possible if ε is small enough)

• Introduce a smooth partition of unity associated to the cover (Ω l ), (χ l ) 1≤l≤L ∈ C ∞ c (Ω l , [0, 1]), supp χ l ⊂ Ω l , l χ l = 1 in a neighborhood of U j .

• For each l ∈ L, we denote F l the restriction to Ω l of F ij . By Lemma A.1, there exists κ l ∈ K which coincides with F l on Ω l . • We consider T l = Op h (α l )U l (1) where U l (t) is the solution of the Cauchy problem (A.1) associated to κ l and α

T R is a globally defined Fourier integral operator. We will note

) such that Ψ i ≡ 1 on π(U i ) and Ψ j ≡ 1 on π(U j )(here, π : (x, ξ) ∈ T * Y • → x ∈ Y • is the natural projection) and we adopt the following definitions : Definition A.3. We say that T : D (Y j ) → C ∞ (Y i ) is a Fourier integral operator in the class I δ (Y i × Y j , Gr(F ij ) ) if there exists T R ∈ I δ (R × R, Gr(F ij ) ) as constructed above such that

For U j ⊂ U j and U i = F (U j ) ⊂ U i , we say that T (or T R ) is microlocally unitary in U i × U j if T T * = Id microlocally in U i and T * T = Id microlocally in U j .


Edgardo Antonio Vigo

Dr Paula Klein
email: paulaklein.t@gmail.com

Keywords: Vigo, project, archives, everyday-life, instructions, documentation

The notions of the archive and of documentation occupy a central role in the multiple stages of production, dissemination and conservation of the work of the Argentinian artist Edgardo Antonio Vigo. Such consideration of the conservation and the posterity of the artist's work contrasts with the forms that the work itself utilizes: collective and anonymous performances, fleeting installations, "instructions" for how to see everyday urban space with new perspective, to name only a few. This study aims to highlight Vigo's commitment to what we will call an "art project", in which the techniques of documentation, recording and the construction of a personal archive play a major role.

stands out as a foundational moment for the group. Secondly, we have the CAyC itself, founded in 1969 and headed by Jorge Glusberg, a center aimed at continuing the experience of an "arte de acción" ["action art"] that had been developed and featured at the Centro de Artes Visuales of the Di Tella Institute until its closure in 1970. i The members of the CAyC were pioneers of an "art of the media", which questioned the production of art as "media event", that is to say, art whose very existence depends on the media's construction of it. The CAyC is indispensable to thinking about the decline of Argentinian conceptualism and informalism. ii Moreover, the CAyC and the Grupo de los 13 were in dialogue with other Argentinean artist groups in France and other parts of Europe also seeking to articulate the capacities of the media and the necessity to transform our anesthetized view of the quotidian. Amongst these collectives of artists was the group formed by Raul Escari, Roberto Jacoby et Eduardo Costa, which released the manifesto "Un arte de los medios de comunicación". The manifesto proposed the creation of a false happening, or event, that never took place, through the proliferation of information about this 'event' by the press. Also of note was the Groupe de recherche d'Art Visuel, or GRAV, a collective of Argentinean artists in Paris that included Horacio Garcia Rossi, Julio Le Parc, François Morellet, Francisco Sobrino, Joël Stein and Jean-Pierre Yvaral. This group produced a series of urban "micro-actions" involving the active participation of spectators through playful interference with public space. The project "One day at the street" ["Une journée dans la rue"], which took place in Paris in 1966, figures amongst the most famous experiences generated by this group. The GRAV was aimed at interrogating spectators about the connections they might be able to establish with a piece of art while taking part as active participants in a series of urban performances.

By transforming the city through a series of small artistic and aesthetic interventions, these artists focused on the "network of repetitive facts encountered during an average day in Paris" ["réseau des faits répétés et retrouvés d'une journée à Paris"] iii . Daily routines thus become the trigger which allows artists to challenge the passivity of passersby; the suggestion becomes that the street is "a woven network of daily habits" ["est tramée d'un réseau d'habitudes et d'actes chaque jour retrouvés"] iv capable nonetheless of developing creative "ways of proceeding" [« manières de faire »] v .

The particularity of these artists is that they did not so much aspire to produce artworks as to develop practices, contexts and situations defined by their ephemeral quality -transient and precarious. Much like the Russian formalist theoretician Victor Chklovski's notion of ostranénie, the object of such works is to provoke an effect of de-alienation or "strangement", which destabilizes our manner of interacting in and observing everyday spaces.

The Argentinian 1960's: towards a politicization of the artistic field

In Latin America, this artistic interest in everyday life is inextricable from a desire for the confluence of the avant-guard, on the aesthetic level, and the revolutionary, on the political level. The appropriation of the streets, "subversive" practices in public space, should be considered here as alternative modes of resistance to the violent military dictatorship known as the "Revolución Argentina", which took place between June 1966 and 1973 and whose principal leaders where the dictators Juan Carlos Onganía (1966-1970), Roberto Marcelo Levingston (1970-1971) et Alejandro Agustín Lanusse (1971[START_REF] Lippard | Escape attempts » in Six years: the dematerialization of the art object from 1966 to 1972 ; a cross-reference book of information on some esthetic boundaries[END_REF].

In the late 1960s, the steady processes of state militarization lead to a radicalization of protest movements. vi For the artistic avant-guard, this phenomenon found expression in the progressive politicization of artists, in more and more direct ways. This increasingly pronounced politicization of artistic practices sometimes resulted in the abandonment of artistic activity for the pursuit of direct armed protest.

Vigo's work accounts for this escalation of state violence reflected in the development of subversive practices on the part of civil society. If Vigo's "señalamientos" or "signals", of the late 1960s aspired to show the quotidian from a playfully disruptive angle, his pieces from the 1970s show increasingly politicized slogans such as those used in his adaptation of "arte correo", or "mail art". It is in the context of this progressive state militarization that we should consider Vigo's notion of "SUBVERSION" ["REVULSION"], vii which implies a re-appropriation, in aesthetic terms, of everyday spaces through the transformation of real life as an objective, through the fusion of art and life as it is purported by the historical avantguard. To cite Vigo: "REVULSIONAR" es la palabra para designar la actitud limite del arte actual". La "obra" debe perimir "para dar paso a otro elemento LA ACCIÓN" ["REVOLT is the word designating the ultimate attitude of contemporary art". The "work of art" should "give way to another element, ACTION"]. Here Vigo rejects the creation of aesthetic objects. The work of art transforms into the very process of the spectators' active observation of everyday reality, and then into the artist's simultaneous documentation of the experience.

Before taking on two material artworks in particular, we would like to point out two characteristics fundamental to our understanding of how the practices that derive from the archive undergo production. Firstly, we will consider the idea of an arte pobre [poor art]. On a statement on this subject by Edgardo Antonio Vigo, delivered to Ángel Osvaldo Nessi on January 3rd, 1969, we read:

Creado en Europa, el ARTE POBRE, reacciona contra la técnica y vuelve su mirada a la ecología. En LATINOAMERICA el "ARTE POBRE", en consecuencia de la falta de medios, surge por carencia y no por exceso como en Europa. viii [Created in Europe, POOR ART is a counter-reaction to technology; it shifts the focus back to ecology. In LATIN AMERICA , "POOR ART", the result of a general lack of resources, emerges from shortage rather than from excess as is the case with art in Europe].

Amongst the multiple procedures, techniques and original materials used for this "Latin American arte pobre", we can identify: "mail art", "rubber stamps", "postcards and postal tapes", distribution by means of 8mm films, "photocopy art" and even the publication of avant-guard low-cost journals and performances in unconventional spaces (ranging from the street to vacant lots). These procedures should be considered within the framework of a political response to the economic shortages that exist in Latin America. Another central point is Vigo's commitment to what we will call an "arte del proyecto", ix or project art, in which the techniques of documentation, recording, and the construction of a personal archive play a major role.

A Project Art

Establishing the principals of new procedural art, the North American Sol LeWitt elaborates a kind of manifesto of the conceptual work of art, according to which "the plan would design the work". x According to this principle, the plan should precede the artwork; the idea or the concept determines the final result. Thus, the finished artistic product takes on a supplementary, or secondary, role at the expense of the concept or project that precedes it. The art that is conceived as a "Project" then posits the idea of the artwork not so much as finished product, but rather as the corollary of the collective and anonymous carrying out of a set of protocols, rules and pre-fixed procedures. Focusing on the performative aspect of the project, we will see that the artist's subjectivity is submitted to a work in progress; it remains thus unfinished and subjected to a state of constant improvisation.

The connections between an art of the project and the field of contemporary art are highlight by Lucy Lippard, who emphasizes the secondary role of the materiality of contemporary art, its dematerialized and ephemeral qualities. Lippard says: "conceptual art, for me, means work in which the material form is secondary, lightweight, ephemeral, cheap, unpretentious and/or 'dematerialized'". xi Contrary to the notions of the design, or plan, the project relies more on a mental premise than on the idea of its own accomplishment or achievement. The process thus becomes more important than the result or the final product and, by doing so, an art of the project interrogates the criteria of finality and productivity in the artistic field. The work therefore appears to be the report or the recording of an experience and it's idea of success or possible failure becomes secondary in relation to the whole process of its creation.

Vigo appears to subscribe completely to this idea of art as a "Project to Realize", or to carry out. Vigo does not refer, to this effect, to "the artist", but rather to the "projectista", or, "the designer". xii An excerpt from a draft of Vigo's text, "La calle: escenario del arte actual" ["The Street: the stage for contemporary art"], published in 1972 in the journal Hexágono '71, reads as follows:

La función del 'proyectista' será la de indicar simples elementos que permitan un 'hacer' posterior legando al receptor las mayores posibilidades de desarrollo (…) se deberá PROPONER mas que HACER. (…) "Proponemos el término "PROYECTISTA" pues el mismo es una derivación directa (de algo a realizar) del "PROYECTO". xiii [The function of the 'designer' will be to indicate simple elements that permit a subsequent 'act', or 'doing', granting the receiver the maximum possibility for development (…), to PROPOSE rather than to DO. (…) "We propose the term "DESIGNER" because it is a direct derivation (of something to be done) of the "PROJECT"].

From this perspective, the artist is assimilated, on the one hand, into a conceptual inventor of a series of projects to be realized. On the other hand, the artist is considered a simple operator who establishes the protocols and the "minimal instructions" so that a group of active spectators can carry out, or realize, said experiences.

These two principles of arte pobre and the work of art as a "project" having been established, we can address the central question of: How does Vigo's work relate to the archive? How do these CAyC artists propose an original approach to documentary and archival aesthetics?

The artist as an everyday life archivist

There are four 'c's that make up the major practices associated with the work of the archivist as proposed by Vigo: collection, classification, conservation and communication. Before analyzing the artworks in question, I will define what it is that I mean by archive and by documentation. According to Paul Ricoeur, the archive is defined as a collection of documents or recordings that, independently of their form or their material format, never ceases to grow. xiv The document, meanwhile, is considered, according to a broad definition, as any 'written item providing information or evidence '. xv According to this approach, the document constitutes a clipping of 'the real' before the processes of selection, classification and conservation, which are associated with the archive. It is thanks to these two techniques of photographic documentation and to the collection of materials related to the "mise en route" of his experiences that Vigo's work appears as a true archive of an artist's existence.

We will first concentrate on the "Personal Archive" of Vigo, accessible online on the Centro de Arte Experimental Vigo-Fundación Artes Visuales website, the headquarters of which are located in the city of La Plata, Argentina. Integrating the documentary aspect of the Vigo archive, we find a series of "Personal Documents" made up of 37 boxes, collected, cataloged and conserved by the artist between 1953 and 1997. xvi According to the information provided on the website, the boxes: contienen hojas sueltas y el soporte que prevalece es papel, cartulina y cartón. Desde la caja 6 hasta la 37 fueron denominadas por el artista con el nombre 'Biopsia'. Todas estas cajas son de cartón corrugado y contienen hojas con detalles de la actividad del artista año por año xvii .

[contain loose sheets of paper -paper, poster-board and cardboard being the predominant media. Boxes 6 to 37 were designated by the artist as 'Biopsy'. All of these boxes are made from corrugated cardboard and contain sheets of paper with details about the artist's activity each year].

As pointed out by Ana Bugnone of the University of La Plata, who has thoroughly studied the question of Vigo's personal archive, the author has ordered the collection chronologically until his death. Each box of the archive is also accompanied by a CV, later called a "Biopsia", which Vigo updates annually and which serves as a sort of index of the archive. xviii Departing from the idea that the daily activity of an artist leaves written traces that are linked to their work, these Biopsias constitute the product of the patient act of collecting and sorting what Daniel Fabre would call "écritures ordinaires" ["ordinary writings"], dating back to the beginning of Vigo's career. Following the description of Fabre, this kind of writings: s'opposent nettement à l'univers prestigieux des écrits qui distinguent la volonté de faire oeuvre, la signature authentifiante de l'auteur, la consécration de l'imprimé (…) la plupart de ces écritures-là (…) semblent vouées à une unique fonction qui les absorbe et les uniformise : laisser trace xix .

[are the exact opposite of the prestigious universe of writings which distinguish the will of becoming a piece of artwork, the author's identifiable signature, the recognition of the printed (…) most of these works (…) seem to be oriented towards a unique function which absorbs and makes them uniform: to leave a mark]

The artist also creates a documentary record of his production and of all of those documents which provide an account of his "artist's life. In a way, this conservation of personal "ephemera" confronts us with the artwork as a "work in progress", as a process rather than and end product. At the same time, it leads us to consider the work from the temporality of its "past future", that is to say, seeing the work from the perspective of its posterity.

We can see to what extent the notion of the Archive traverses not only the artwork, but also Vigo's life, from its beginning. From the perspective of the construction of the personal Archive of the artist, each artwork appears to be conceived as part of a potential future archive.

Biopsia's content

The artist also creates a documentary record of his production and of all documents which provide an account of his "artist's life". This archive is mostly composed of: recortes periodísticos sobre exposiciones, reportajes, ensayos escritos por él, críticas y comentarios a sus obras, catálogos, folletos, afiches, invitaciones y convocatorias a exposiciones nacionales e internacionales de grabado, arte conceptual, poesía visual y artecorreo, actuaciones como jurado, comunicaciones vía postal, fotografías, dibujos y grabados originales, pruebas, diseños, correspondencia, manuscritos. xx [newspaper clippings about exhibitions, reports, essays written by him, reviews and commentaries on his works, catalogs, brochures, posters, invitations to and notices of national and international printmaking exhibitions, conceptual art, visual poetry and mail-art, jury performances, communication by mail, photographs, drawings and original prints, proofs, designs, correspondence, manuscripts].

In a way, this conservation of personal "ephemera" confronts us with the artwork as a "work in progress", as a process or an activity rather than and end product. At the same time, it makes us consider the work from the temporality of its "past future", that is to say, seeing the work from the perspective of its posterity.

Furthermore, this project of documentation potentially manages to circumvent the threat of erasure of the subject which, integrated into the logic of the collection and of the archive, transforms itself into a conserved object. We can see, thus, to what extent the notion of the Archive traverses not only the artwork, but also the life of Vigo from its beginnings.

In effect, from the perspective of the construction of the personal Archive of the artist, each artwork appears, in itself, to be conceived as part of a potential future archive. According to this point of view, we see that each of Vigo's pieces is made up of multiple documents and elements, namely: the artwork itself, whether it is a conceptual object or a collective performance that develops from this object; xxi the photographic documentation of said performance-piece or "do-it-yourself project"; and, lastly, a diverse collection of texts (manifestos, newspaper clippings, personal reflections written by the author) created either before or after the presentation of the art piece, which describe and conceptualize the action in question.

We will, at last, focus on three precise cases of signals, documentation and recording of everyday life in Vigo's work.

Señalamientos [Signals] (1968-1975)

The signals were performance pieces, which the artist either proposed or carried out himself, in order to identify different everyday objects, events or spaces. xxii The complete list is as follows: These protocols include a series of rules or instructions to carry out, or realize, a more complex action, and attempt to control the vicissitudes involved in the experience. Furthermore, by playing with legal lingo Vigo proposes, through the photographic record, to "certify" the realization of the performance. What remains of the artwork is then the text of instructions for carrying out, or the "methods of use".

Conceptual art thus becomes literature and, more specifically, a summary of the experience carried out.

Ultimately, we will analyze a work of art that vests in the artist the role of investigator-archivist, a work entitled "Acción de investigar una acción. Documento"

["Action of Investigating an Action. Document"] (September 1973).

The artist as investigator

In the Biopsia box from September 1973, the following explanatory document can be found preceding the images of the piece itself: Vigo chose an envelope he had received from his correspondence (that he himself had previously created and sent to several correspondents) and he started taking out all the stamps and little labels that were glued or patched to the envelope in order to analyze them, as a doctor would do with a corpse in a biopsy. While performing this procedure he asked Juan José Esteves to photograph and document the whole process. He could then later say that the object had become an "investigated object".

Conclusion

Vigo's work interrogates both the production of the work of art as a document and the modalities of conservation and transmission, which determine the transformation of these documents into an autobiographical archive. His work shines light on the "fabrication" of an artist's archive. The analysis of these three specific works highlights the multiple ways through which Vigo imitates the procedures of the archivist. His purpose is to produce everyday archives in order to renew our perception of quotidian spaces as potential sources of aesthetic experience. By collecting, classifying, preserving and communicating the documents that result from his ephemeral performances, Vigo emphasizes their importance in the whole process of making visible the minor, transient elements of everyday life. L., 1993, p. 11. xx Ibid. xxi This statement is made bearing in mind that the decade of the 1970s is a period of reinforcement of the performance as a privileged artistic practice. Cf. RoseLee Goldsberg, Performance Art, from futurism to the present, [1988], London, Thames and Hudson, 1990, p. 7. xxii For a more detailed analysis of the "señalizaciones" procedure, see: Fernando

Vigo's texts:

Edgardo Antonio Vigo, "Manifiesto Primera No-Manifestación Blanca. Manojo de semáforos". "La calle: escenario del arte actual"; "Sellado a mano" in "Edgardo Antonio Vigo: Textos recobrados", Ramona, revista de artes visuales, Buenos Aires, November 2007, p. 14. [On line] URL: https://docplayer.es/9225283-Ramonaedgardo-antonio-vigo-textos-recobrados.html Critical bibliography: AA.VV., "Une journée dans la rue" [START_REF] Cf | Sistemas, Acciones y Procesos. 1965-1975[END_REF], pamphlet du Groupe de recherche d 'Art Visuel, Paris 1964-1966. Intervenants : Garcia Rossi, Le Parc, Morellet, Sobrino, Stein, Yvaral, Archive Julio Le Parc, Paris.

  de semáforos [Bundle of Traffic Lights] 1968 II, Remember Grupo Sí [Remember the Yes Group] 1968 III, No va más!!! [No more!!!] 1968 IV, Poema demagógico [Demagogical Poem], 1969 V, Un paseo visual a la plaza Rubén Darío [A Visual Tour of Plaza Rubén Darío], 1970 VI, 5' de filmación de un monumento en el monumento de B. Mitre [5 minute Film of a Monument at the B. Mitre Monument], 1971 VII, De tu mano [By Your Hand], 1971 VIII, Devolución del agua del '70 [Return of the Water From '70], 1971 IX, Enterramiento y desenterramiento de un taco de madera de cedro [Burial and Exhumation of a Piece of Cedar Wood], 1971-1972 IX bis, Tres actos interconectados '71 [Three Interconnected Acts], 1971 X, Llamado del limonero [The Call of the Lemon Tree], 1972 XI, Souvenir del dolor [Souvenir of Pain], 1972 XII, Almácigo de arena [Sand Seedbed], 1972 XIII, Los puntos cardinales [The Cardinal Points], 1974 XIV, Llamado del silencio [The Call of the Silence], 1974 XV, De doble acción [On Dual Action], 1974 XVI, The Oval House Gallery, 1974. XVII, Acción gratuita [Gratuitous Act], 1974 XVIII, Tres formas de negar la libertad y una propuesta de rescate simbólico [Three Ways of Denying Freedom and a Proposal for a Symbolic Rescue], 1975 S/N° [XIX], Transplante y almácigo de arenas [Transplant and Sand Seedbed], 1992 The signals reveal elements of everyday space as aesthetic objects. According to Vigo, the purpose is: "No construir más imágenes alienantes sino SEÑALAR aquellas que no teniendo intencionalidad estética como fin, la posibiliten" xxiii ["Not to construct more alienating images, but rather to INDICATE those that, not having aesthetic intentionality, do suggest it"]. By graphic and radial means, Vigo participated in an act of contemplation at the intersection of Avenues 1 and 60 of La Plata, Argentina. In the box labeled Biopsia for the month of October, 1968, no. 27, we read: cotidiano. Se citó a las 20 horas al publico pero no me hice presente en esa cita. Una declaración y una utilización de los medios de comunicación, periodístico y radial fomentaron durante veinte días la duda de lo que iba a pasar. [CLUSTER OF TRAFFIC LIGHTS -aesthetic experience with enjoyment of the everyday element. The public was summoned at 8 o'clock [P.M.] but I was not present for the appointment. A statement and the use of the means of communication of radio and newspaper fomented the question of what was going to happen for 20 days]. Through these signals, Vigo expresses his rejection of the creation of aesthetic objects. The work of art transforms into the very process of active observation of everyday reality on the part of the spectators, and of the simultaneous documentation on the part of the artist. Let's consider a second signal : "Señalamiento V. Un paseo visual a la plaza Rubén Darío (Noviembre 1970)" ["Signal V. A Visual Stroll to the Rubén Darío Plaza"]. This piece essentially consists of a series of PROTOCOLS, or "MinimalCodes" -brief instructions that Vigo distributes in his "do-it-yourself projects" as early as 1970. The one from "Un paseo visual a la plaza Rubén Darío", for reads:INDICACIONEStomar una tiza, y marcar con ella una cruz o el límite de una o varias baldosas, o entice una superficie por ud. determinada. Colocarse dentro de la zona demarcada y hacer un giro sobre sí mismo en 360 º ( 1) grabe en ud. lo visto, saque sus conclusiones, en definitiva ud. ha realizado" ( 1) VARIANTE: puede agregar a este movimiento el de PERSPECTIVA POR ALTURA desde la posición de puntas de pié o viceversa, llegar a la posición de cuclillas o of chalk, and mark with it a cross or the limit of one or of various tiles, or any one surface determined by yourself. Place yourself inside the demarcated area and rotate around yourself 360 º (1) record what you have in your line of sight, draw the conclusions you, in essence, have made" (1) VARIATION: you can add to this movement that of PERSPECTIVE FROM HEIGHT from a tip-toe position or vice-versa, take a squatting position or one of horizontal elongation NOTE: claim THE BRAND OF VIGO This will certify YOUR PASSAGE]

  Davies, "Prácticas 'revulsivas'. Edgardo Antonio Vigo en el escena crítica del conceptualismo", Lecture presented at the Workshop Vivid [Radical] Memory. Radical Conceptual Art revisited: A social and political perspective from the East and the South , Stuttgart, Württembergischer Kunstverein, September 2007 [unpublished] [On line]. And also: Ana Bugnone, « El espacio público en la poética de Edgardo Antonio Vigo: los señalamientos », Revista digital de la escuela de historia, UNR, año 5, n° 8, Rosario, 2013. xxiii Edgardo Antonio Vigo, "Manifiesto Primera No-Manifestación Blanca. Manojo de semáforos". Text reproduced in "Edgardo Antonio Vigo: Textos recobrados", art. cit., p. 14. [On line] URL: https://docplayer.es/9225283-Ramona-edgardo-antonio-vigo-textos-recobrados.html

  Lucy Lippard, "Escape attempts" in Six years: the dematerialization of the art object from 1966 to 1972 ; a cross-reference book of information on some esthetic boundaries, New York, Praeger. 1973, p. vii. xii Together with artists Clemente Padín et Elena Pelli, Vigo organizes in 1971 in the CAyC the Expo/Internacional de proposiciones a realizar [Exhi/International of proposals to achieve]. The artist highlights that art no longer aims to be the result of a project but it becomes instead a series of "projects to achieve". xiii Cf. Edgardo Antonio Vigo, "La calle: escenario del arte actual", Hexágono '71, La Plata, 1972. Quoted from the article "Un arte a realizar", published in RITMO revue, La Plata, 1969, N° 3. Quoted from: "Edgardo Antonio Vigo: Textos recobrados", Ramona, revista de artes visuales, Buenos Aires, November 2007, p. 15-16 [On line]. xiv I draw here upon the definition of Jacques André in "De la preuve à l'Histoire. Les archives en France", Traversée, 1986. Cited by Arlette Farge in Le Goût de l'archive, Paris, Seuil, 1989, p. 11. xv Definition of the document taken from Larousse online. xvi Cf. [Unidentified author], "Descripción archivîstica del Fondo Archivo Edgardo Antonio Vigo. Serie: Documentos personales". [On line]. URL: http://www.caev.com.ar/ xvii Ibid. xviii For a detailed analysis of archival practices in Vigo's work, see: Ana Bugnone, "Edgardo Antonio Vigo: Archivo / vanguardia", VI Jornadas de Filología y Lingüística, 7 al 9 de agosto de 2013, La /www.memoria.fahce.unlp.edu.ar/trab_eventos/ev.3844/ev.3844.pdf xix Daniel Fabre (et al.), Écritures ordinaires, Paris, Bibliothèque publique d'information, Centre Georges-Pompidou, P.O.

	Plata,	Argentina.	In	Memoria	Académica.	[On	line].	URL :
	http:/

i For further information about the CAyC, see: Graciela Sarti, Grupo CAyC [en línea], Buenos Aires, xi