Nicolas Augier

Ugo Boscain ‡ ¶

Mario Sigalotti

† Cmap

École Polytechnique

Universit É Paris-Saclay

ADIABATIC ENSEMBLE CONTROL OF A CONTINUUM OF QUANTUM SYSTEMS *

In this article we discuss how to control a parameter-dependent family of quantum systems. Our technique is based on adiabatic approximation theory and on the presence of curves of conical eigenvalue intersections of the controlled Hamiltonian. As particular cases, we recover chirped pulses for two-level quantum systems and counter-intuitive solutions for three-level stimulated Raman adiabatic passage (STIRAP). The proposed technique works for systems evolving both in finitedimensional and infinite-dimensional Hilbert spaces. We show that the assumptions guaranteeing ensemble controllability are structurally stable with respect to perturbations of the parametrized family of systems.

1. Introduction. The problem of controlling quantum mechanical systems has undergone a huge development in recent years, motivated by its applications in manipulation of single spins, photons or atoms, optical spectroscopy, photochemistry, nuclear magnetic resonance, and quantum information processing (see, for instance, [START_REF] Altafini | Modeling and control of quantum systems: an introduction[END_REF][START_REF] Boscain | A weak spectral condition for the controllability of the bilinear Schrödinger equation with application to the control of a rotating planar molecule[END_REF][START_REF] Brif | Control of quantum phenomena: past, present and future[END_REF][START_REF]Introduction to quantum control and dynamics[END_REF][START_REF] Dong | Quantum control theory and applications: a survey[END_REF][START_REF] Glaser | Training Schrödinger's cat: quantum optimal control. Strategic report on current status, visions and goals for research in Europe[END_REF] and references therein).

In this paper we are concerned with the problem of ensemble controllability, namely the problem of controlling using finitely many scalar controls (typically few of them, e.g., two or three) a family of systems having slightly different parameters. Equivalently, the problem is to control a quantum system for which certain parameters are unknown. Such a problem arises very often in practical applications. For instance, the dynamics of different spins in a nuclear magnetic resonance sample are characterized by different sets of parameters.

Notice that in such a problem the assumption that the unknown parameter belongs to a continuous set plays an important role. If the parameter belonged to a discrete set, then the classical techniques of controllability could be applied to the system written in a higher-dimensional space (see, for instance, [START_REF] Belhadj | Ensemble controllability and discrimination of perturbed bilinear control systems on connected, simple, compact Lie groups[END_REF][START_REF] Dirr | Ensemble controllability of bilinear systems[END_REF]).

The ensemble controllability problem is studied in quantum mechanics since a long time. In particular, it is known that the so-called chirped pulses for two-level systems have good robustness properties with respect to parameter uncertainties [START_REF] Chelkowski | Efficient molecular dissociation by a chirped ultrashort infrared laser pulse[END_REF][START_REF] Shore | The theory of coherent atomic excitation[END_REF]. Similarly, the so-called counterintuitive pulses for STIRAP processes in three-level quantum systems are known to realize a population transfer even if certain parameters are unknown [START_REF] Gaubatz | Population transfer between molecular vibrational levels by stimulated Raman scattering with partially overlapping laser fields. a new concept and experimental results[END_REF][START_REF] Shapiro | Complete transfer of populations from a single state to a preselected superposition of states using piecewise adiabatic passage: Theory[END_REF][START_REF] Shore | The theory of coherent atomic excitation[END_REF]. Ensemble controllability is often tackled in experimental situations via direct optimization methods (see, for instance, [START_REF] Chen | Sampling-based learning control of inhomogeneous quantum ensembles[END_REF][START_REF] Skinner | Application of optimal control theory to the design of broadband excitation pulses for high-resolution NMR[END_REF]). The mathematical aspects of the ensemble controllability problem have been addressed, in particular, in [START_REF] Beauchard | Controllability issues for continuous-spectrum systems and ensemble controllability of Bloch equations[END_REF][START_REF] Chittaro | Asymptotic ensemble stabilizability of the Bloch equation[END_REF][START_REF] Li | Control of inhomogeneous quantum ensembles[END_REF][START_REF] Li | Ensemble control of Bloch equations[END_REF] for a spin- 1 2 system whose Larmor frequency is not known precisely. In [START_REF] Leghtas | Adiabatic passage and ensemble control of quantum systems[END_REF] the effect of chirped pulses for a class of many-level systems is analyzed using adiabatic techniques.

In this paper we present a technique based on adiabatic approximation and on the presence of conical intersections between eigenvalues (see Figure 2.1). We are mainly considering the situation in which the unknown parameters vary in a one-dimensional set (although in several practical applications, as in the use of chirped pulses or in the STIRAP process, we admit the unknown parameters to vary in higherdimensional spaces). We recall that eigenvalue intersections are generically conical for finite-dimensional systems having a real-symmetric (respectively, Hermitian) Hamiltonian in presence of 2 (respectively, 3) controls and that such a genericity property also extends to systems evolving in infinite-dimensional spaces (see Section 6 for more details).

Our technique is based on the simple idea that if two levels of a quantum system are connected by a conical intersection, then generically a one-parameter perturbation results in a one-dimensional manifold of conical intersections, each one corresponding to a value of the parameter. Then an adiabatic path in the space of controls that passes through such a one-dimensional manifold realizes a transfer for all systems (see Figure 2.4).

It is interesting to notice that such an idea explains and generalizes what happens for chirped pulses. The fact that chirped pulses exploit the presence of curves of conical intersections was not noticed before, because of the special symmetries of the system (see Section 4).

Our main result can be roughly resumed in a statement like this: assume that for a given parameter the spectrum of the systems is conically connected (i.e., every energy level is connected to the adjacent ones by conical eigenvalue intersections). Assume that by modifying the parameter the eigenvalue intersections remain conical and describe disjoint 1-d manifolds in the set of admissible controls. Assume, in addition, that the space of controls minus the support of such curves is pathwise connected. Then the system is approximately ensemble controllable. Such a result proves the robustness of adiabatic transitions with respect to variations of the parameters of the system. We recall that the conically connectedness hypothesis implies the exact controllability of each individual system [START_REF] Boscain | Approximate controllability, exact controllability, and conical eigenvalue intersections for quantum mechanical systems[END_REF].

Our result permits to obtain only transitions between eigenstates of the controlled Hamiltonian, and hence it is less general than the one in [START_REF] Beauchard | Controllability issues for continuous-spectrum systems and ensemble controllability of Bloch equations[END_REF] (where initial and final conditions can depend on the parameter). However, it can be applied to a much more general class of systems (in particular, it is not restricted to two-level systems). Actually, our result permits to establish a little more than controllability between eigenstates, namely it states that it is possible to induce any arbitrary permutation within a basis of eigenstates of the controlled Hamiltonian. In this sense it generalizes [START_REF] Leghtas | Adiabatic passage and ensemble control of quantum systems[END_REF], where a result of this kind was presented for a specific class of systems. As it always happens when using adiabatic theory to prove controllability properties, the result is constructive in the sense that it provides a simple algorithmic procedure for obtaining the required control laws.

It should be noticed that the technique that we present here is the only one that applies to the ensemble control of quantum systems evolving in an infinite-dimensional Hilbert space.

The key tool on which we base our reasoning is a uniform adiabatic theorem for systems having conical eigenvalues intersections. A crucial role is played by a technical result guaranteeing that a Hamiltonian that depends C k on a parameter and whose eigenvalues, seen as functions of such a parameter, are conical, admits an orthonormal family of eigenstates depending C k-1 on the parameter. The regularity of eigenvalues and eigenvectors is well established in the analytic setting [START_REF] Kato | Perturbation theory for linear operators[END_REF] and has been studied in the C ∞ case [START_REF] Gauthier | On the codimension one motion planning problem[END_REF]. We prefer to work in the C k setting because of its generality and since the analytic setting is less natural when studying structural stability and genericity of the assumptions on eigenvalue intersections. Our result is in the same spirit as a recent result on the regularity of projectors obtained in [START_REF] Chittaro | Approximate controllability via adiabatic techniques for the three-inputs controlled Schrödinger equation[END_REF], of which we became aware while preparing the final version of this article. The regularity of eigenpairs is a useful complement to the classical adiabatic theory in presence of conical intersections, where it is usually assumed that eigenpairs are C 2 with respect to the parameter. With our result one can replace such a regularity assumption with the more easily verifiable hypothesis on the C 3 regularity of the Hamiltonian itself.

It should be mentioned that the ensemble controllability problem is not restricted to quantum systems. In particular, in [START_REF] Helmke | Uniform ensemble controllability for one-parameter families of time-invariant linear systems[END_REF][START_REF] Li | Ensemble control of finite-dimensional time-varying linear systems[END_REF][START_REF] Li | Ensemble control of time-invariant linear systems with linear parameter variation[END_REF] such a problem has been addressed for linear systems. The case of driftless nonlinear systems has been studied in [START_REF] Agrachev | Ensemble controllability by Lie algebraic methods[END_REF], thanks to an infinite-dimensional version of the Rashevsky-Chow theorem.

The paper is organized as follows: In Section 2 we set the ensemble controllability problem and we state our sufficient conditions for approximate ensemble controllability in finite-dimensional spaces. The proof of the sufficiency of such conditions in given in Section 3. The proof of the technical result on the regularity of eigenpairs is postponed to the appendix. In Section 4 we apply the general result to the case of two-level systems, recovering the classical results on chirped pulses. The extension to permutations within a basis of eigenvectors is studied in Section 5. The genericity of the conditions appearing in the sufficient conditions for approximate ensemble controllability is discussed in Section 6. In Section 7 we consider the case in which the Hamiltonian depends on more than one parameter and we illustrate our results on the STIRAP process. Finally, in Section 8, we discuss the extensions to the case of infinite-dimensional spaces, presenting as an example a version of the classical Eberly-Law model. We consider the controlled Schrödinger equation in C N , N ∈ N,

i ψ = H α (u(t))ψ (2.1)
where u : R → U is a L ∞ map. Here α is a time-independent parameter belonging to a compact interval [α 0 , α 1 ] ⊂ R. Each matrix H α (u) belongs to the set Herm(N ) of N × N Hermitian matrices. The map (α, u) → H α (u) is sufficiently regular, as it will be specified later on.

Denote the spectrum of H α (u) by (λ α j (u)) N j=1 where j → λ α j (u) is the nondecreasing sequence of eigenvalues of H α (u) repeated according to their multiplicities. We also write (φ α 1 (u), . . . , φ α N (u)) to denote an orthonormal basis of associated eigenvectors.

Definition 2.1. We say that system (2.1) is ensemble approximately controllable between eigenstates if for every ε > 0, j, k ∈ 1, N and u 0 , u 1 ∈ U such that λ α j (u 0 ) and λ α k (u 1 ) are simple for every α ∈ [α 0 , α 1 ], there exists a control u(•) : [0, T ] → U such that for every α ∈ [α 0 , α 1 ] the solution of (2.1) with initial condition ψ α (0) = φ α j (u 0 ) satisfies ψ α (T ) -e iθ φ α k (u 1 ) < ε for some θ ∈ R (possibly depending on α and ε).

Remark 2.2. The typical case of interest is when u = 0 ∈ U represents an isolated (i.e., uncontrolled) system (the so-called drift Hamiltonian) and one seeks to steer φ α j (0) towards φ α k (0) for some j, k ∈ 1, N . Definition 2.3. Let us fix α ∈ [α 0 , α 1 ]. We say that ū ∈ U is a conical intersection between λ α j and λ α j+1 if λ α j-1 (ū) < λ α j (ū) = λ α j+1 (ū) < λ α j+2 (ū) and there exists c > 0 such that for every v in a neighborhood of ū in U, we have

λ α j (v) -λ α j+1 (v) ≥ c ū -v .
(As example of conical intersection is shown in Figure 2.1.) Let us denote by γ j the projection of the set {(u, α)

u 1 u 2 λ 1 (u) λ 2 (u)
∈ U × [α 0 , α 1 ] | λ α j (u) = λ α j+1 (u)} onto the u-component, that is, γ 0 = ∅, γ j = {u ∈ U | ∃α ∈ [α 0 , α 1 ] such that λ α j (u) = λ α j+1 (u)}, j ∈ 1, N -1 , γ N = ∅.
Assumption A j . There exist a connected component γj of γ j and a map β j : [α 0 , α 1 ] → U such that β j is a C 3 embedding and

• γj = β j ([α 0 , α 1 ]) • γj is contained in U \ (γ j-1 ∪ γ j+1 )
• For every α ∈ [α 0 , α 1 ], λ α j and λ α j+1 have a unique intersection on γj , which is conical and occurs at β j (α). Moreover the set U \ (γ j-1 ∪ γ j ∪ γ j+1 ) is pathwise connected.

Remark 2.4. The assumption that β j is an embedding between the set of parameters [α 0 , α 1 ] and γj is used here for simplicity and can be relaxed. This will be done in the general context of multi-dimensional sets of parameters in Section 7 (see Assumption A * j ). The structural stability and the genericity of Assumption A j are discussed in Section 6. In the following theorem we show that the curve γj appearing in Assumption

γj γj γj+1 γj-1 γj+2 U Fig. 2.2. The component γj is such that Assumption A j is satisfied γj u(t 1 ) = β j (α 1 ) u(t 0 ) = β j (α 0 ) u(0) = u 0 u(1) = u 1 Fig. 2.3. A control u(•)
as in the statement of Theorem 2.5.

A j can be used to induce an adiabatic transition between the j-th and the (j + 1)-th eigenstate for every α.

Theorem 2.5. Consider a C 3 map [α 0 , α 1 ] × U (α, u) → H α (u) ∈ Herm(N ). Let j ∈ 1, N -1 be such that Assumption A j is satisfied. Take u 0 , u 1 ∈ U \ (γ j-1 ∪ γ j ∪ γ j+1 ) and consider a C 3 path u(•) : [0, 1] → U satisfying u(0) = u 0 , u(1) = u 1 , and such that u| [t0,t1] is a reparameterization of β j for some

0 < t 0 < t 1 < 1 such that u(t) / ∈ γ j-1 ∪ γ j ∪ γ j+1 for every t ∈ [0, 1] \ [t 0 , t 1 ]. (See Figure 2.3.) Assume, moreover, that u(t) = 0 for every t ∈ [0, 1].
Then there exists a constant C > 0 such that for every α ∈ [α 0 , α 1 ] and ε > 0 the solutions

ψ α,± ε of i ψα,± ε (t) = H α (u(εt))ψ α,± ε (t)
with initial conditions

(2.2) ψ α,+ ε (0) = φ α j (u 0 ), ψ α,- ε (0) = φ α j+1 (u 0 ) satisfy (2.3) ψ α,+ ε (1/ε) -e iθ + φ α j+1 (u 1 ) ≤ C √ ε, ψ α,- ε (1/ε) -e iθ - φ α j (u 1 ) ≤ C √ ε,
for some θ ± ∈ R.

Remark 2.6. By unitarity of the evolution, if in the statement of Theorem 2.5 we replace the initial conditions (2.2) by

ψ α,+ ε (0) -φ α j (u 0 ) ≤ c, ψ α,- ε (0) -φ α j+1 (u 0 ) ≤ c, u 0 = u 1 U γ3 γ2 γ1 Fig. 2.4.
A control realizing an ensemble transition between φ α 1 (u 0 ) and φ α 4 (u 0 ).

then the conclusion (2.3) becomes

ψ α,+ ε (1/ε) -e iθ + φ α j+1 (u 1 ) ≤ C √ ε + c, ψ α,- ε (1/ε) -e iθ - φ α j (u 1 ) ≤ C √ ε + c. Corollary 2.7. Consider a C 3 map [α 0 , α 1 ] × U (α, u) → H α (u) ∈ Herm(N ). Let Assumption A j be satisfied for every j ∈ 1, N -1 . Then (2.

1) is ensemble approximately controllable between eigenstates.

The corollary follows from an iterated application of Theorem 2.5 and its proof works by constructing a C 3 control such that u(0) = u 0 and u(T ) = u 1 (where u 0 and u 1 are as in Definition 2.1) and going through the curves γj as in Figure 2.4. Starting from the second iteration of the application of Theorem 2.5, we use Remark 2.6 with c of order √ ε to guarantee that the adiabatic approximation after each iteration remains of order √ ε.

Remark 2.8. The proof of Theorem 2.5 is based on a uniform adiabatic theorem, which is recalled in next section (Theorem 3.1). Actually, under the additional hypothesis that the curves γj are non-mixing for every α ∈ [α 0 , α 1 ], in the sense of [9, Section V], one can replace the factor √ ε in (2.3) and (3.1) by ε.

3. Proof of the ensemble controllability result in the finite-dimensional case.

Theorem 3.1. Consider a C 3 map [α 0 , α 1 ] × U (α, u) → H α (u) ∈ Herm(N ). Let u : [0, 1] → U be a C 2 control. For every α ∈ [α 0 , α 1 ] and t ∈ [0, 1], let Λ α 1 (t), . . . , Λ α N (t) be the eigenvalues of H α (u(t)
) repeated according to their multiplicities and denote by (Φ α 1 (t), . . . , Φ α N (t)) an orthonormal basis of associated eigenvectors. Assume that for every α ∈ [α 0 , α 1 ] and every j ∈ 1, N ,

(Λ α j , Φ α j ) ∈ C 2 ([0, 1], R × C N ). Let j ∈ 1, N -1
and assume that for every α ∈ [α 0 , α 1 ], there exists τ ∈ (0, 1) such that Λ α j (t) and Λ α j+1 (t) are simple for every t ∈ [0, 1] \ {τ } and

Λ α j (τ ) = Λ α j+1 (τ ) = Λ α m (τ ) for m = j, j + 1, d dt t=τ Λ α j (t) = d dt t=τ Λ α j+1 (t).
Then there exists C > 0 such that for every α ∈ [α 0 , α 1 ], and every ε > 0 the solution

ψ = ψ α ε of the equation i ψ(t) = H α (u(εt))ψ(t)
with initial condition ψ(0) = Φ α j (0) satisfies

(3.1) ψ(1/ε) -e iθ Φ α j (1) ≤ C √ ε for some θ ∈ R.
The proof of this theorem can be obtained by applying the classical adiabatic theorem for every α, using the continuity of the corresponding constant C(α) and the compactness of the interval [α 0 , α 1 ] (see for instance [START_REF] Jansen | Bounds for the adiabatic approximation with applications to quantum computation[END_REF]Theorem 4] and [START_REF] Teufel | Adiabatic perturbation theory in quantum dynamics[END_REF]Theorem 1.2]).

In order to guarantee the regularity of eigenvalues and eigenvectors along a regular but not necessarily analytic path in the domain of admissible controls, we are going to apply the following result, whose proof is given in the appendix. (For a result in the same spirit in the infinite-dimensional setting, see [START_REF] Chittaro | Approximate controllability via adiabatic techniques for the three-inputs controlled Schrödinger equation[END_REF].) Lemma 3.2. Let I be an interval in R and take

H ∈ C k+1 (I, Herm(N )), k ≥ 0. For every t ∈ I, let λ 1 (t) ≤ • • • ≤ λ N (t)
be the eigenvalues of H(t), repeated according to their multiplicities. Assume that for every t ∈ I and every j ∈ 1, N -1 such that λ j ( t) = λ j+1 ( t), then λ h ( t) = λ j ( t) for h = j, j + 1 and there exist c > 0 and a neighborhood Ī of t in I such that

(3.2) λ j+1 (t) -λ j (t) ≥ c|t -t|, for every t ∈ Ī.
Then there exist Λ 1 , . . . ,

Λ N ∈ C k+1 (I, R) and Φ 1 , . . . , Φ N ∈ C k (I, C N ) such that, for every t ∈ I, Λ 1 (t), . . . , Λ N (t)
are the eigenvalues of H(t) repeated according to their multiplicities and (Φ 1 (t), . . . , Φ N (t)) is an orthonormal basis of corresponding eigenvectors. Moreover, Φ 1 , . . . , Φ n are C k+1 (I, C N ) if all eigenvalues of H are simple along I.

Proof of Theorem 2.5. We are going to apply Theorem 3.1 on the path u(•). Notice that for every α the curve u(•) passes through exactly one point ūα such that λ α j (ū α ) = λ α j+1 (ū α ) and through no point v ∈ U where either

λ α j-1 (v) = λ α j (v) or λ α j+1 (v) = λ α j+2 (v)
. Denote by t α the time such that u(t α ) = ūα . By Lemma 3.2 applied to the map t → H(u(t)), we can assume that

[0, 1] t → (Λ α j (t), Φ α j (t)) = (λ α j (u(t)), φ α j (u(t))) if t < t α (λ α j+1 (u(t)), φ α j+1 (u(t))) if t ≥ t α is C 2 for every j ∈ 1, N and every α ∈ [α 0 , α 1 ]
. The application of Theorem 3.1 then concludes the proof of Theorem 2.5.

Remark 3.3. In Theorem 2.5 the control u(•) is assumed to be C 3 . The C 3 regularity could actually be relaxed. The same proof would work, for instance, by assuming u(•) to be continuous, C 3 in a neighborhood of γj and piecewise C 2 elsewhere.

4. Example 1: Two-level system driven by a chirped pulse. A classical example of ensemble control by adiabatic evolution is the famous chirped pulse used in two-level systems. It is used, for instance, for controlling via magnetic fields an ensemble of spin systems with slightly different parameters. We illustrate chirped pulses on the following model

i d dt ψ α 1 ψ α 2 = E + α Ω(t) Ω * (t) -E -α ψ α 1 ψ α 2 , (4.1)
where E > 0 is fixed, Ω(•) ∈ L ∞ ([0, T ], C) is the control, Ω * (t) denotes its complex conjugate, and α is a real parameter that can vary in a fixed range α ∈ [α 0 , α 1 ] ⊂ (-E, ∞). The quantity E + α is called the proper frequency of system (4.1).

Chirped pulses are well studied in the literature (see [START_REF] Chelkowski | Efficient molecular dissociation by a chirped ultrashort infrared laser pulse[END_REF][START_REF] Shore | The theory of coherent atomic excitation[END_REF][START_REF] Leghtas | Adiabatic passage and ensemble control of quantum systems[END_REF]). We show here below how they naturally show up in the general framework proposed in this paper. Consider the following ensemble approximate controllability problem for (4.1): we want to construct explicitly a control steering, for every value of α ∈ [α 0 , α 1 ], the eigenstate φ α 1 (0) = (1, 0) to φ α 2 (0) = (0, 1) of the Hamiltonian in (4.1) corresponding to Ω = 0.

Theorem 2.5 and Corollary 2.7 do not directly apply to system (4.1) since the eigenvalues of the Hamiltonian are simple for every value of α ∈ [α 0 , α 1 ] and Ω ∈ C.

Hence we perform a suitable time-dependent change of variables (to recast the system in interaction picture). In order to do so, we consider controls having the form

(4.2) Ω(t) = u 1 (t)e -i(2Et+∆(t)) ,
where u 1 (•) and ∆(•) are real-valued, and we set

h α (t) = E + α u 1 (t)e -i(2Et+∆(t)) u 1 (t)e i(2Et+∆(t))
-E -α .

Let us apply the time-dependent unitary change of variable ψ α (t) = U (t)Φ α (t) with

U (t) = e -iEt 0 0 e i(Et+∆(t)) .
Notice that the transformation U (t) preserves (up to phases) the two eigenstates φ α 1 (0) = (1, 0) and φ α 2 (0) = (0, 1). Then Φ α satisfies the equation

i d dt Φ α = H α (t)Φ α with H α (t) = U -1 (t)h α (t)U (t) -iU -1 (t) dU dt (t) = α u 1 (t) u 1 (t) -α + d∆ dt (t)
.

Setting u 2 (t) = d∆ dt (t), we are then left to control the family of systems

i d dt Φ α 1 Φ α 2 = α u 1 (t) u 1 (t) -α + u 2 (t) Φ α 1 Φ α 2 . (4.3)
We apply now Theorem 2.5 to system (4.3). To this purpose we have to check that Assumption A 1 is satisfied. Set

H α (u 1 , u 2 ) = α u 1 u 1 -α + u 2 .
The eigenvalues of H α (u 1 , u 2 ) are

λ α 1 (u 1 , u 2 ) = u 2 -(2α -u 2 ) 2 + 4u 2 1 2 , λ α 2 (u 1 , u 2 ) = u 2 + (2α -u 2 ) 2 + 4u 2 1 2 .
For any fixed α ∈ [α 0 , α 1 ], H α has a unique eigenvalue intersection, which is conical and takes place at (u 1 , u 2 ) = (0, 2α). In other words for each point of the set

u 2 u 1 2α 1 2α 0 γ 1 (u 1 (t), u 2 (t)) u 21 u 20
γ 1 = {(0, 2α) ∈ R 2 | α ∈ [α 0 , α 1 ]} = {0} × [2α 0 , 2α 1 ]
there is exactly one system having an eigenvalue intersection at such a point, which is, moreover, conical. Assumption A 1 is then satisfied.

By applying Corollary 2.7 to system (4.3) we deduce the following result. The controls used to achieve the controllability stated in Proposition 4.1 can be taken continuous, C 3 in a neighborhood of γ 1 and piecewise C 2 elsewhere (see Remark 3.3). Notice that the same regularity holds for Ω, according to (4.2).

As a consequence of Proposition 4.1 we can steer (up to phases) system (4.3) (simultaneously with respect to α ∈ [α 0 , α 1 ]) from (1, 0) to an arbitrary neighborhood of (0, 1), which are eigenstates of H α (0, u 20 ) for every u 20 ∈ R \ [2α 0 , 2α 1 ]. Now, since U (t) preserves (up to phases) (1, 0) and (0, 1), we can conclude that system (4.1) can be steered simultaneously by adiabatic control from (1, 0) to (0, 1) for every α ∈ [α 0 , α 1 ]. Notice that the control u = (0, u 20 ) corresponds to Ω = 0 in the original system, independently of u 20 .

We now construct explicitly the adiabatic control realizing the transition. We start by doing this for (4.3) and then we translate the result for (4.1).

Let u 21 be in the connected component of R \ [2α 0 , 2α 1 ] not containing u 20 (see Figure 4.1). Let u : [0, 1/2] → R 2 be a path following a curve contained in the right half-plane connecting (0, u 20 ) to (0, u 21 ). Complete the path u by letting u : [1/2, 1] → R 2 follow a straight segment from (0, u 21 ) to (0, u 20 ). The path u(•) is such that t → u(εt) yields the desired ensemble approximate transition when ε → 0.

In terms of the original system, the control Ω obtained from u as in (4.2) is identically equal to zero on [1/2, 1]. Hence, the evolution corresponding to this portion

|ψ2(εt)| 2 t α = 1 α = 0 t |ψ1(εt)| 2 α = -1 u2(t) u1(t) 2α -2α
Re(Ω(εt))

Im(Ω(εt)) of the control u has no effect on the dynamics and can be ignored. The original system is then controlled by Ω(εt) = u 1 (εt)e -i(2Et+

1 ε εt 0 u2(s)ds)
with t ∈ [0, 1 2ε ]. Notice that t → Ω(εt) is a complex function with slow-varying modulus and a phase oscillating with a frequency given by

d dt 2Et + 1 ε εt 0 u 2 (s)ds = 2E + u 2 (εt).
Since u 2 (0) = u 20 and u 2 (1/2) = u 21 , such a frequency slowly varies between two values, one smaller and one larger than every proper frequency E + α of system (4.1) (see Figure 4.2 for an example).

Let us conclude by a remark on the precision of the adiabatic estimate. According to Theorem 2.5, the precision of the transition of system (4.3) corresponding to u :

[0, 1] → R 2 is of order √ ε.
Taking into account, moreover, Remark 2.8, the precision is of order ε. For what concerns system (4.1), we are just interested, as we have seen, to the restriction of u on the interval [0, 1/2] and its corresponding control Ω. Hence, since u crosses no eigenvalue intersection on [0, 1/2], one can conclude that the precision of the adiabatic transition is of order ε even without recalling Remark 2.8. [START_REF] Leghtas | Adiabatic passage and ensemble control of quantum systems[END_REF], we refine in this section our approach in order to select adiabatic controls performing a prescribed permutation of the eigenvectors of the controlled Hamiltonian.

γ 1 \ γ1 U γ1 γ2 γ3 γ 2 \ γ2

Permutations. Inspired by

The set of assumptions A j , j ∈ 1, N -1 , is replaced by the slightly stronger assumption P below (see Figure 5.1 for an illustration).

Assumption P. For every j ∈ 1, N -1 there exist a connected component γj of γ j and a map β j : [α 0 , α 1 ] → U such that β j is a C 3 embedding and

• γj = β j ([α 0 , α 1 ]). • γj is contained in U \ (∪ k∈ 1,N k =j γ k )
• For every α ∈ [α 0 , α 1 ], λ α j and λ α j+1 have a unique intersection on γj , which is conical and occurs at β j (α). Moreover the set U \ (∪ k∈ 1,N γ k ) is pathwise connected.

Under Assumption P, Corollary 2.7 can be refined as follows.

Theorem 5.1. Assume that P is satisfied. Then for every u 0 , u

1 ∈ U\(∪ k∈ 1,N γ k ), every p 1 , . . . , p N ∈ C such that |p 1 | 2 + • • • + |p N | 2 = 1 and every permutation σ : 1, N → 1, N , there exists a C 3 path u(•) : [0, 1] → U satisfying u(0) = u 0 and u(1) = u 1 and a constant C > 0 such that for every α ∈ [α 0 , α 1 ] and ε > 0 the solution ψ α ε of i ψα ε = H α (u(εt))ψ α ε with initial condition ψ α ε (0) = N j=1 p j φ α j (u 0 ), satisfies ψ α ε (1/ε) - N j=1 p j e iθj φ α σ(j) (u 1 ) ≤ C √ ε,
for some θ 1 , . . . , θ N ∈ R.

Proof. The path u(•) is constructively obtained by requiring it to pass through the curves γj in a suitable order. Such an order is identified by the algorithm described below.

U f 1 f 3 f 2 t 3 t 1 t 2 γ1 γ2 u 1 u 0 Fig. 5.2.
On the left, the functions f j and the times t k . On the right, the control u(•). In this case N = 3, µ = 3, τ (1) = 1, τ (2) = 2, τ (3) = 1. Hence, u(•) passes through γ1 , γ2 , and again γ1 while connecting u 0 to u 1 .

Let h ∈ 1, N . We say that the functions f 1 , . . . , f h : [0, 1] → R satisfy property (Π h ) if they are continuous, piecewise affine, such that f j (0) = j, f j (1) = σ(j) for j ∈ 1, h and, moreover, if for every t ∈ (0, 1) and j = k, j, k ∈ 1, h such that

f j (t) = f k (t) we have • f l (t) = f m (t) for every l ∈ 1, h \ {j, k}, m ∈ 1, h \ {l}, • (f j (t + ε) -f k (t + ε))(f j (t -ε) -f k (t -ε))
< 0 for every ε > 0 small enough. We now construct recursively a set of functions f 1 , . . . , f N satisfying (Π N ).

Let f 1 and f 2 be the affine functions uniquely determined by

f 1 (0) = 1, f 2 (0) = 2, f 1 (1) = σ(1), f 2 (1) = σ(2). Notice that they satisfy (Π 2 ).
By induction, assume to have selected f 1 , . . . , f h satisfying (Π h ).

Let f h+1 be the affine function satisfying f h+1 (0) = h + 1, f h+1 (1) = σ(h + 1). If (Π h+1 ) is satisfied then the recursion step is complete. Otherwise modify f h+1 into a continuous function that is constantly equal to h + 1 in an interval [0, η] and affine on [η, 1], with the same boundary conditions. For every positive small enough η one has that f 1 , . . . , f h+1 satisfy (Π h+1 ) and this concludes the induction step.

For every t ∈ [0, 1] let σ t : 1, N → 1, N be a permutation such that

f σt(1) (t) ≤ f σt(2) (t) ≤ • • • ≤ f σt(N ) (t).
Notice that t → σ t is piecewise constant and σ 0 = Id, σ 1 = σ. Let t 1 < t 2 < • • • < t µ be the values in (0, 1) at which the graphs of the functions f 1 , . . . , f n intersect (see Figure 5.2). Then t 1 , . . . , t µ are the discontinuity points of t → σ t . For every j ∈ 1, µ , let τ (j) ∈ 1, N -1 be defined by

f σt j (τ (j)) (t j ) = f σt j (τ (j)+1) (t j ).
The control u(•) is constructed in such a way that it passes through γτ(1) , . . . , γτ(µ) . More precisely let s 1 , . . . , s µ be such that 0 =:

t 0 < s 0 < t 1 < s 1 < t 2 < • • • < s µ-1 < t µ < s µ := 1. Let u(•) : [0, 1] → U be such that u(t) = 0 for every t ∈ [0, 1], u(•) ∈ C 3 , u(0) = u 0 , u(1) = u 1 .
Moreover for every j = 0, . . . µ, the restriction u| (tj ,sj ) has values in U \ (∪ k∈ 1,N γ k ) and for j = 1, . . . µ the curve u| [sj-1,tj ] is a reparameterization of β τ (j) (see Figure 5.2).

By construction and by a repeated application of Theorem 2.5 and Remark 2.6, the solution of i ψα ε = H α (u(εt))ψ α ε with initial condition ψ α ε (0) = φ α j (u 0 ), satisfies for every t ∈ ∪ µ k=0 (t k , s k )

ψ α ε (t/ε) -e iθj φ α σt(j) (u(εt)) ≤ C √ ε,
for some θ j ∈ R. The proof of Theorem 5.1 is then concluded arguing by linearity.

Remark 5.2. Let us comment on the difference between Assumption P and the set of hypotheses A 1 , . . . , A N appearing in the statement of Corollary 2.7. Assumption P is stronger in the sense that each γj is required to have empty intersection not only with γ j-1 and γ j+1 , but also all with γ k for k = j. Assumption P guarantees that while inducing a transition between the levels j and j + 1 the other energy levels are untouched.

6. Genericity. We discuss in this section the genericity of Assumption A j which appears in Theorem 2.5 and Corollary 2.7. Recall that, by the Wigner-von Neumann theorem [START_REF] Neumann | Überdas Verhalten von Eigenwerten bei Adiabatischen Prozessen[END_REF], the set of N × N Hermitian matrices of rank equal to N -1 (i.e., with one degenerate eigenvalue of multiplicity 2 and all other eigenvalues simple) is the finite union of submanifolds of codimension 3. More generally the set Υ of all N ×N Hermitian matrices with degenerate eigenvalues is a Whitney stratified set (see for instance [25, Section 1.2]) of codimension 3. Similarly, symmetric real matrices with degenerate eigenvalues form a Whitney stratified set of codimension 2 of the space of all symmetric matrices. Let d = 3, H : U → Herm(N ) be a C 3 map and h = H(u) ∈ Herm(N ) have rank equal to N -1. Then H(u) intersects Υ transversally at h = H(u) if and only if u is a conical intersection. By standard transversality arguments (see, for instance, [1, Proposition 19.1] and [25, Section 1.3.2]), it follows that there exists a residual set R in C 3 (U, Herm(N )) such that for every H ∈ R, all intersections between the eigenvalues of H are conical. (In particular for every H ∈ R and every u ∈ U such that H(u) ∈ Υ, the rank of H(u) is N -1, since the strata of Υ corresponding to matrices of lower rank are of codimension larger than 3.) Moreover, a conical intersection is structurally stable in the sense that, if u is a conical intersection for H ∈ C 3 (U, Herm(N )), then any small perturbation of H has a conical intersection near u. Similar results hold for d = 2 in the case of real symmetric Hamiltonians.

The following two results, whose proof can be directly derived from the above considerations, establish that a small one-parameter perturbation of a Hamiltonian with conically connected spectrum satisfies the assumptions of Theorem 2.5 and Corollary 2.7. Here we impose U to be bounded and the Hamiltonian H to be C 3 on U, meaning that it admits a C 3 extension on a neighborhood of U. We also require H not to have eigenvalue intersections on ∂U. This prevents the occurrence of a sequence of eigenvalue intersections converging to ∂U. Theorem 6.1. Let U be an open, connected and bounded subset of R 3 . Let H : U → Herm(N ) be a C 3 map and (λ j (u)) j∈ 1,N be the increasing sequence of eigenvalues of H(u), u ∈ U, repeated according to their multiplicities. Fix j ∈ 1, N -1 . Assume that the levels λ j and λ j+1 intersect and that all intersections between them are conical and correspond to controls u ∈ U. If j > 1 (respectively, j < N -1), assume, moreover, that all intersections between the levels λ j-1 and λ j (respectively, λ j+1 and λ j+2 ) are conical and correspond to controls u ∈ U.

Let us define

Ξ = {h ∈ C 3 (U × R, Herm(N )) | h(•, 0) = H(•)}
endowed with the C 3 Whitney topology induced by C 3 (U × R, Herm(N )).

Then there exists an open and dense subset Ξ of Ξ, such that for every h ∈ Ξ, the restriction h : U × [-δ, δ] → Herm(N ) satisfies Assumption A j for some δ > 0. Corollary 6.2. Let U be an open, connected and bounded subset of R 3 . Let H : U → Herm(N ) be a C 3 map and (λ j (u)) j∈ 1,N be the increasing sequence of eigenvalues of H(u), u ∈ U, repeated according to their multiplicities. Assume that all eigenvalue intersections are conical and correspond to controls u ∈ U. Assume moreover that, for every j ∈ 1, N -1 , λ j and λ j+1 intersect.

Let Ξ be defined as in Theorem 6.1. Then there exists a open and dense subset Ξ of Ξ, such that for every h ∈ Ξ, the restriction h : U × [-δ, δ] → Herm(N ) satisfies Assumption P for some δ > 0 (and, in particular, Assumption A j every j ∈ 1, N ).

As a consequence of Corollary 2.7 and Theorem 5.1, every h in the set Ξ of parameter-dependent Hamiltonians appearing in the statement of Corollary 6.2 defines, when restricted to U × [-δ, δ], a system which is ensemble approximately controllable between eigenstates and which satisfies the conclusion of Theorem 5.1.

The same conclusions as those of Theorem 6.1 and Corollary 6.2 hold when d = 2 (i.e., U is a subset of R 2 ) and the Hamiltonians H and h take values in the set of symmetric N × N real matrices.

7. Multidimensional set of parameters. We consider in this section the situation in which the parameter on which the Hamiltonian depends varies in a set of dimension larger than one. Our technique still applies when the set of eigenvalue intersections projects onto a one-dimensional curve in the space of controls. Even if this situation is not generic, it however shows up in several physical examples, as we illustrate in Sections 7.1 and 7.2.

Let Σ be the set of parameters, contained in R m for some m ∈ N. We consider the ensemble controllability problem for the Schrödinger equation on C N , N ∈ N,

i ψ = H σ (u(t))ψ, σ ∈ Σ. (7.1)
Denote by (λ σ j (u)) N j=1 the nondecreasing sequence of eigenvalues of H σ (u) repeated according to their multiplicities and by (φ σ 1 (u), . . . , φ σ N (u)) an orthonormal basis of associated eigenvectors. Let us define

γ 0 = ∅, γ j = {u ∈ U | ∃σ ∈ Σ such that λ σ j (u) = λ σ j+1 (u)}, j ∈ 1, N -1 , γ N = ∅. Assumption A * j .
There exists a connected component γj of γ j such that • There exists an interval

[α 0 , α 1 ] ⊂ R and a C 3 embedding β j : [α 0 , α 1 ] → U such that γj = β j ([α 0 , α 1 ]); • γj is contained in U \ (γ j-1 ∪ γ j+1 );
• For every σ ∈ Σ, there exists a unique u ∈ γj such that λ σ j and λ σ j+1 have an intersection at u, which is conical. Moreover the set U \ (γ j-1 ∪ γ j ∪ γ j+1 ) is pathwise connected.

Theorem 2.5 and Corollary 2.7 hold true by replacing [α

0 , α 1 ] × U (α, u) → H α (u) ∈ Herm(N ) by Σ × U (σ, u) → H σ (u) ∈ Herm(N ) and A j by A * j .
The same proof works without major modifications.

7.1. Chirped pulses for two-level systems with two parameters. We consider here below an extension of the example studied in Section 4, in which we add an uncertain parameter also in the coupling between the control and the system. Consider the controlled equation

i d dt ψ σ 1 ψ σ 2 = E + α βΩ(t) βΩ * (t) -E -α ψ σ 1 ψ σ 2 , (7.2) where σ = (α, β) ∈ [α 0 , α 1 ] × [β 0 , β 1 ] ⊂ R × (0, +∞).
As before, E > 0 is fixed and we want to steer the system from φ σ 1 (0) = (1, 0) to φ σ 2 (0) = (0, 1), up to phases. The same change of variables as in Section 4 and the control trasformation

Ω(t) = u 1 (t)e -i(2Et+∆(t)) , u 2 (t) = d∆ dt (t), yield the equivalent system i d dt Φ σ = H σ (t)Φ σ with H σ (t) = α βu 1 (t) βu 1 (t) -α + u 2 (t)
.

Since the eigenvalue intersections of H σ only depend on α and not on β, the entire discussion of Section 4 still applies in the presence of the parameter β. This underlines once more the robustness of the chirped pulse strategy for two-level systems. The eigenvalue intersections of H σ (u 1 , u 2 ) occur on the axes u 1 and u 2 . More precisely,

Example 2: STIRAP. Consider the three-level system with controlled Hamiltonian

H (α1,α3,β1,β2) (u 1 , u 2 ) =   E 1 + α 1 β 1 u 1 0 β 1 u 1 E 2 β 2 u 2 0 β 2 u 2 E 3 + α 3   with α j ∈ [α j0 , α j1 ], j = 1, 3, β j ∈ [β j0 , β j1 ], j = 1,
γ 1 = 0, - (E 1 + α 1 -E 3 -α 3 )(E 1 + α 1 -E 2 ) β 2 | σ ∈ Σ ∪ 0, (E 1 + α 1 -E 3 -α 3 )(E 1 + α 1 -E 2 ) β 2 | σ ∈ Σ , γ 2 = - (E 1 + α 1 -E 3 -α 3 )(E 2 -E 3 -α 3 ) β 1 , 0 | σ ∈ Σ ∪ (E 1 + α 1 -E 3 -α 3 )(E 2 -E 3 -α 3 ) β 1 , 0 | σ ∈ Σ .
In particular, γ 1 and γ 2 are both unions of two segments (see Figure 7.1), hence they are 1-dimensional smooth embedded manifolds. All hypotheses of Corollary 2. (in the multi-dimensional parameter extension discussed above) are satisfied and we conclude that the system is ensemble approximately controllable between eigenstates. Moreover, according to Remark 2.8, the precision of the transition described in Theorem 2.5 can be made of order ε, where ε is the velocity at which we follow the adiabatic path.

For instance, in order to steer system (7.1) from φ σ 1 = (1, 0, 0) to φ σ 3 = (0, 0, 1) we can follow the path in Figure 7.2a.

In Figure 7.2b we plot the two components of the control realising the transition. Notice that such controls are in the celebrated counter-intuitive order, meaning that, in order to go from state 1 to state 3, one first activates the control u 2 , responsible for the transition 2 → 3, and then the control u 1 , responsible for the transition 1 → 2 (creating the so-called dark state). The approach presented in this paper gives a complete mathematical explanation of why the counter-intuitive order works and why it is so robust with respect to parameter fluctuations.

8. Extension to the infinite-dimensional case. The results of the previous sections extend, under some suitable regularity assumptions, to the case where C N is replaced by an infinite-dimensional complex separable Hilbert space H.

In order to avoid excessive technicalities, we present this extension in the case where the Hamiltonian H depends affinely on the controls and where the controlled Hamiltonians are bounded. (For the general nonlinear case, one could follow the approach in [START_REF] Chittaro | Approximate controllability by adiabatic methods of the Schrödinger equation with nonlinear Hamiltonian[END_REF].)

We then consider a Hamiltonian of the type

H σ (u) = H σ 0 + d j=1 u j H σ j ,
wih the parameter σ belonging to Σ ⊂ R m for some m ∈ N, and with H σ j , j = 0, . . . , d, satisfying the following assumption. (H ∞ ) Fix σ 0 ∈ Σ and assume that:

• H σ0 0 is self-adjoint and bounded from below; • H σ 0 -H σ0 0 , H σ 1 , . . . , H σ d are bounded for all σ ∈ Σ; • the map (σ, u) → H σ0 0 -H σ (u) is C 3 from Σ × U to the Banach space of bounded self-adjoint operators endowed with the operator norm; Let P σ,u : H → H be the orthogonal projector onto the sum of the eigenspaces corresponding to λ σ 1 (u), . . . , λ σ N (u). Without loss of generality this space is of complex dimension N . Then (see, for instance, [START_REF] Reed | Methods of modern mathematical physics. IV. Analysis of operators[END_REF]), for every (σ, ū) ∈ Σ × U and every (σ, u)

• for all (σ, u) ∈ Σ × U the eigenvalues λ σ 1 (u) ≤ • • • ≤ λ σ N (u) of H σ (u) are such that • spectrum(H σ (u)) ∩ [λ σ 1 (u), λ σ N (u)] = {λ σ 1 (u), . . . , λ σ N (u)} and λ σ j (u) has finite multiplicity for every j ∈ 1, N , • dist(spectrum(H σ (u)) \ [λ σ 1 (u), λ σ N (u)], {λ σ 1 (u), λ σ N (u)}) ≥ Γ, with Γ > 0 independent of (σ, u) (see
λ σ 1 (u) λ σ 1 (u) λ σ 2 (u) λ σ 2 (u) λ σ 3 (u) (σ, u) λ σ 3 (u) λ σ 2 (u) Fig. 8.1. The eigenvalues λ σ 1 (u) ≤ λ σ 2 (u) ≤ λ σ 3 (u) of H σ (
u) separated from the rest of the spectrum (which is contained in the shaded regions).

in a neighborhood of (σ, ū),

P σ,u (ψ) = - 1 2πi γ (H σ (u) -c) -1 dc, where γ is a Jordan curve in C separating λ σ 1 (ū), . . . , λ σ N (ū) and spectrum(H σ (ū)) \ [λ σ 1 (ū), λ σ N (ū)]
. Hence (σ, u) → P σ,u is C 3 as a map from Σ × U to the Banach space of bounded operators on H endowed with the operator norm.

Let I σ,u : C N → H be a linear map such that I σ,u is unitary between C N and the image of P σ,u .

Assume for now that (σ, u) → I σ,u is globally C 3 on Σ × U. We denote by (I σ,u ) -1 the inverse of I σ,u on the image of P σ,u . The Hamiltonian (σ, u) → H σ (u) = (I σ,u ) -1 H σ (u)I σ,u ∈ Herm(N ) is C 3 and its eigenvalues are λ σ 1 (u), . . . , λ σ N (u). A general adiabatic theorem (see, for instance, [43, Theorem 2.2]) states that, given a C 2 path u : [0, 1] → U and ψ0 ∈ C N , the solutions ψ(•) and ψ(•) of, respectively, i ψ(t) = H σ (u(εt))ψ(t), ψ(0) = (I σ,u(0) ) -1 ψ0 , and i ψ(t) = H σ (u(εt)) ψ(t), ψ(0) = ψ0 , are such that ψ(1/ε) is ε-close, up to phases, to (I σ,u(1) ) -1 ψ(1/ε). When (σ, u) → I σ,u cannot be globally defined as a C 3 map, the same arguments can be applied in local charts leading to the following generalization of Theorem 2.5 and Corollary 2.7

As in the previous section, the sets γ j are defined as

γ 0 = ∅, γ j = {u ∈ U | ∃σ ∈ Σ such that λ σ j (u) = λ σ j+1 (u)}, j ∈ 1, N -1 , γ N = ∅. Theorem 8.1. Assume that (H ∞ ) holds true. Let j ∈ 1, N -1 be such that Assumption A * j of Section 7 is satisfied. Take u 0 , u 1 ∈ U \ (γ j-1 ∪ γ j ∪ γ j+1 ) and consider a C 3 path u(•) : [0, 1] → U satisfying u(0) = u 0 , u(1) = u 1 , and such that u| [t0,t1] is a reparameterization of β j for some 0 < t 0 < t 1 < 1 such that u(t) / ∈ γ j-1 ∪ γ j ∪ γ j+1 for every t ∈ [0, 1] \ [t 0 , t 1 ]. Assume, moreover, that u(t) = 0 for every t ∈ [0, 1].
Then there exists a constant C > 0 such that for every σ ∈ Σ and ε > 0 the solutions ψ

σ,± ε of i ψσ,± ε (t) = H σ (u(εt))ψ σ,± ε (t)
with initial conditions

ψ σ,+ ε (0) = φ σ j (u 0 ), ψ σ,- ε (0) = φ σ j+1 (u 0 ) satisfy ψ σ,+ ε (1/ε) -e iθ + φ σ j+1 (u 1 ) ≤ C √ ε, ψ σ,- ε (1/ε) -e iθ - φ σ j (u 1 ) ≤ C √ ε,
for some θ ± ∈ R.

Corollary 8.2. Assume that (H ∞ ) holds true. Let Assumption A * j be satisfied for every j ∈ 1, N -1 . Then (8.1) is ensemble approximately controllable between the eigenstates {φ σ 1 (u), . . . , φ σ N (u)} in the sense that for every ε > 0, j, k ∈ 1, N and u 0 , u 1 ∈ U such that λ σ j (u 0 ) and λ σ k (u 1 ) are simple for every σ ∈ Σ, there exists a control u(•) : [0, T ] → U such that for every σ ∈ Σ the solution of (8.1) with initial condition ψ σ (0) = φ σ j (u 0 ) satisfies ψ σ (T ) -e iθ φ σ k (u 1 ) < ε for some θ ∈ R. Remark 8.3. When Σ is one-dimensional, genericity results similar to those in Section 6 can be obtained. For related structural stability and genericity considerations, see [START_REF] Boscain | Adiabatic control of the Schroedinger equation via conical intersections of the eigenvalues[END_REF].

8.1. Example 3: Eberly-Law-like models. We study in this section a model for coupled spin-oscillator dynamics (see [START_REF] Law | Arbitrary control of a quantum electro-magnetic field[END_REF][START_REF] Yuan | Controllability of the coupled spin-1 2 harmonic oscillator system[END_REF][START_REF] Bloch | Finite controllability of infinite-dimensional quantum systems[END_REF]). For other spin-boson models, see [START_REF] Boscain | On the control of spin-boson systems[END_REF][START_REF] Keyl | Controlling Several Atoms in a Cavity[END_REF][START_REF] Paduro | Approximate controllability of the two trapped ions system[END_REF] and references therein. The state space is the tensor product of the state spaces of an harmonic oscillator and of a spin- 1 2 particle. The system has two control parameters that we assume to be real, the first one, u 1 , coupling the two levels of the spin system and the other one, u 2 , producing simultaneous spin transitions and vibrational phonon excitations.

We let ω be the gap between the levels of the harmonic oscillator and δ be the internal gap of the spin system, α (respectively, β) the coupling strength between u 1 (respectively, u 2 ) and the system. The vector σ = (α, β, ω, δ) will be considered as a set of parameters whose uncertainty will be tackled by the technique proposed in the previous sections.

The model can then be represented by the infinite-dimensional controlled quantum system i ψ = H σ (u)ψ, ψ ∈ 2 , Let J be a neighborhood of t in I such that for every t ∈ J \ { t} both j (H(t)) and j+1 (H(t)) are simple. For every t ∈ J, let P t : C N → C N be the orthogonal projector on the sum of the eigenspaces corresponding to j (H(t)) and j+1 (H(t)).

where u = (u 1 , u 2 ) ∈ R 2 , σ ∈ [α 0 , α 1 ] × [β 0 , β 1 ] × [ω 0 , ω 1 ] × [δ 0 , δ 1 ] = Σ, and
H σ (u) =              0 αu 1 0 0 0 • • •
Then (see, for instance, [START_REF] Reed | Methods of modern mathematical physics. IV. Analysis of operators[END_REF]),

P t = - 1 2πi γ (H(t) -c) -1 dc
where γ is a Jordan curve in C separating j (H(t)), j+1 (H(t)) and spectrum(H(t)) \ { j (H(t)), j+1 (H(t))}.

Up to taking a smaller J if necessary, the curve γ can be taken independently of t ∈ J. Hence t → P t is C k+1 as a map from J to the space of linear operators on C N . Then there exists a linear map I t : C 2 → C N , depending C k+1 on t, such that I t is unitary between C 2 and the image of P t . We denote by (I t ) -1 the inverse of I t on the image of P t .

The Hamiltonian h(t) = (I t ) -1 H(t)I t : C 2 → C 2 is well defined on J, depends C k+1 on t, and has j (H(t)) and j+1 (H(t)) as eigenvalues. If, moreover, v is an eigenvector of h(t), then I t v is an eigenvector of H(t). We are therefore left to prove the result for the eigenpairs of h(t).

Without loss of generality we can consider the case where the trace of h(t) is zero for every t ∈ J. Hence, h(t) has the form Hypothesis (3.2) guarantees that α(0) = 0 or β(0) = 0 or γ(0) = 0. Notice that, up to applying a unitary change of variables in C 2 and restricting J, we can assume that β(t) = 0 for every t ∈ J.

The eigenvalues of h(t) are ± a(t)

2 .

 2 Basic definitions and statement of the main results in the finitedimensional case. For every n ∈ N , let 1, n denote the set of integers {1, . . . , n}. Let U be an open connected subset of R d , d ≥ 2.

Fig. 2 . 1 .

 21 Fig. 2.1. An example of conical intersection in the case d = N = 2.

Fig. 4 . 1 .

 41 Fig. 4.1. The control u(•).

Proposition 4 . 1 .

 41 System (4.3) is ensemble approximately controllable between eigenstates.

Fig. 4 . 2 . 3 2 ( 1 -

 4231 Fig. 4.2. Two-level system driven by a chirped pulse. Here E = 2, α ∈ [-1, 1], u 1 (t) = 3 2 (1 -cos(4πt)), u 2 (t) = -3 cos(2πt), for t ∈ [0, 1 2 ] and ε = 0.004. The frequency of the pulse is 2E + u 2 (εt) = 4 -3 cos(2πεt) which varies monotonically between 1 and 7.

Fig. 5 . 1 .

 51 Fig. 5.1. A situation in which Assumption P is verified.

2 ,

 2 β 10 , β 20 > 0 and α j0 , α j1 such that E 1 + α 1 < E 2 < E 3 + α 3 . (One could clearly add a further parameter uncertainty in the level E 2 , which is not relevant, since the trace of the matrix can always be shifted without modifying the dynamical properties of the system.) The controls u 1 and u 2 are real-valued.Denote σ = (α 1 , α 3 , β 1 , β 2 ) and Σ = [α 10 , α 11 ] × [α 30 , α 31 ] × [β 10 , β 11 ] × [β 20 , β 21 ].

7

 7

1 Fig. 7 . 1 .

 171 Fig. 7.1. The sets γ 1 and γ 2 for the STIRAP process.

2 tFig. 7 . 2 .

 272 Fig. 7.2. Example of STIRAP process withE 1 = -1, E 2 = 0, E 3 = 1, α 1 , α 3 ∈ [-0.1, 0.1], β 1 , β 2 ∈ [0.8, 1.2].In (a) we give the parametric plot of the control, in (b) the shape of its components as functions of time and in (c) the components of the wave function for the three choices of parameters (α 1 , α 3 , β 1 , β 2 ) = (0, 0, 1, 1), (0.1, -0.1, 0.8, 1.2), (-0.1, 0.1, 1.2, 0.8). For this simulation we used ε = 0.05.

Figure 8 .

 8 1).Under the hypothesis (H ∞ ), for every continuous control u : [0, T ] → U and every initial condition, the equationi ψ = H σ (u(t))ψ (8.1) admits a unique solution [43, Proposition 2.1].

  h(t) = a(t) b(t) + ic(t) b(t) -ic(t) -a(t) ,where a(•), b(•), c(•) are real-valued C k+1 functions on J. Moreover, since h( t) has a double eigenvalue, we have a( t) = b( t) = c( t) = 0. Without loss of generality t = 0.Let α, β, γ : J → R be C k functions such that a(t) = tα(t), b(t) = tβ(t), c(t) = tγ(t), t ∈ J.

  2 + b(t) 2 + c(t) 2 = ±|t| α(t) 2 + β(t) 2 + γ(t) 2 . The function t → -a(t) 2 + b(t) 2 + c(t) 2 if t < 0 a(t) 2 + b(t) 2 + c(t) 2 if t ≥ 0 is then equal to t α(t) 2 + β(t) 2 + γ(t)2 which is C k+1 on J. Indeed, the only term of the (k + 1)-th derivative of t α(t) 2 + β(t) 2 + γ(t) 2 involving the (k + 1)-th derivative of α, β , and γ ist(α (k+1) (t)α(t) + β (k+1) (t)β(t) + γ (k+1) (t)γ(t)) α(t) 2 + β(t) 2 + γ(t) 2 .Now, since a(t) = tα(t) is C k+1 , the term tα (k+1) (t) is continuous. The same argument holds for β and γ and the C k+1 regularity of t α(t) 2 + β(t) 2 + γ(t) 2 is proved.For what concerns the unit eigenvectors, a simple calculation shows that, up to phases and scaling, they are equal to-a(t) ± a(t) 2 + b(t) 2 + c(t) 2 b(t) -ic(t) , 1 = -α(t) ± sign(t) α(t) 2 + β(t) 2 + γ(t) 2 β(t) -iγ(t) ,1for t = 0. As above, connecting suitably the two branches, we can rewrite the eigenvectors as -α(t) ± α(t) 2 + β(t) 2 + γ(t) 2 β(t) -iγ(t) , 1 , which are C k on J.

Acknowledgements: This work was supported by the project DISQUO of the DEFI InFIniTI 2017 by CNRS, by the ANR project SRGI ANR-15-CE40-0018, by the ANR project Quaco ANR-17-CE40-0007-01.

We assume that α 0 , β 0 , ω 0 , δ 0 > 0. For simplicity, we also assume that ω 0 > δ 1 , that is, the diagonal of H σ (u) is a strictly increasing sequence. Further conditions on α j , β j , ω j , δ j , j = 0, 1, will be imposed below (see (8.3)).

The Hamiltonian H σ (u) is self-adjoint, has purely discrete spectrum, and, if both u 1 and u 2 are different from zero, then all eigenvalues of H σ (u) are non-degenerate (see [START_REF] Adami | Controllability of the Schrödinger equation via intersection of eigenvalues[END_REF]).

In order to apply our general strategy, let us describe the eigenvalue intersections of H σ (u). This is quite simple, since, for u 1 = 0 or u 2 = 0, the matrix describing

Let us first consider intersections along the axis u 2 = 0. The eigenvalues of H σ (u 1 , 0) as a function of u 1 are shown in Figure 8.2. A simple computation shows that the smallest value of

and the second-smallest value is ū1 =

. We reason similarly along the axis u 1 = 0 and we get that the smallest value of

and the second-smallest value is ū2 =

. The smallest value of |u 2 | for which 0 is a degenerate eigenvalue for

We assume in the following that

In particular, all eigenvalue intersections in γj , j ≥ 1, are conical. Condition (8.3) also implies that, for every n ∈ N, Assumption (H ∞ ) is satisfied by the first 2n + 1 eigenvalues with U = U ∩ {u 2 < u 1 + η}, by the first 2 eigenvalues λ σ 1 , λ σ 2 with U = U ∩ {u 1 < u 2 + η} and by the eigenvalues λ σ 3 , . . . , λ σ 2n with U = U ∩ {u 1 < u 2 + η, u 2 < max σ∈Σ u * 2 + η}. Henceforth, ensemble approximate controllability between eigenvectors holds true. Figure 8.3 shows a path in the space of controls leading to an ensemble transfer from φ σ 1 (0) to φ σ 4 (0). The path is a loop starting and ending at u = 0 that goes through γ1 , γ2 , and γ3 . min ū2 Appendix A. Regularity of eigenvalues and eigenvectors. In this section we prove Lemma 3.2, which states that a time-dependent Hamiltonian H ∈ C k+1 (I, Herm(N )) whose eigenvalues intersections are all conical has C k+1 eigenvalues and C k eigenvectors.

Proof of Lemma 3.2. For each j ∈ 1, N , consider the function j : Herm(N ) → R that associates with a Hermitian matrix its j-th eigenvalue, where eigenvalues are counted according to their multiplicities, in such a way that 1 ≤ • • • ≤ N . By Rellich theorem [START_REF] Rellich | Perturbation theory of eigenvalue problems[END_REF], each map j is analytic on

Moreover, for every h ∈ Herm j (N ), there exist a neighborhood V h of h in Herm j (N ) and an analytic function ϕ j : V h → C N such that ϕ j (g) is a norm 1 eigenvector of g of eigenvalue j (g) for every g ∈ V h . Hence, for every j ∈ 1, N , the j-th eigenvalue and a choice of a corresponding eigenvector of norm 1 are C k+1 on {t ∈ I | j (H(t)) is a simple eigenvalue of H(t)}.

We are left to asses the regularity of eigenpairs in a neighborhood of a time t ∈ I such that H( t) / ∈Herm j (N ).


Pierre Aucouturier

Jean-Paul Maillochon

Joseph-Théodore Guy

Françoise Duarte

Jean-Louis Preud'homme

A Filter Affinity Transfer Method for the Analysis of Rheumatoid Factors

Keywords: anti-lgG antibodies, monoclonal immunoglobulins, Western blotting, affinity blot ting, rheumatoid arthritis

Transfer of serum proteins separated by thin layer agarose electrophoresis onto nitro cellulose sheets precoated with purified hu man polyclonal lgG followed by revelation with enzyme-coupled anti-µ or anti-a antisera resulted in the specific detection of rheuma toid factors (RF) belonging to the lgM or lgA classes. Mono-or polyclonality of such RF can be evaluated from the patterns of the blots (sharp bands). ln addition, their light chain type can be determined using affinity filters coated with a 'Y heavy chain disease protein or with lgG Fe fragments. This sim ple and rapid procedure allows an easy char acterization of monoclonal RF, even if they are present in minute amount amongst poly clonal RF as in certain sera from rheumatoid arthritis patients.

IN T RODUCTION

High level rheumatoid factors (RF ) are present in the serum of patients with various autoimmune diseases, especiall y rheu matoid arthritis (RA), the Sjôgren's syndrome and idiopathie mixed cryoglobulinemia. On the other hand, a strikingly large proportion of serum monoclonal immunoglobulins (lg) dis plays a RF activity. Such monoclonal RF may belong to any of the three main lg classes, predominantly IgM. They may result in clinical manifestations such as vasculitis, hyper gammaglobulinemic purpura or symptoms due to type II cryoglobulinemia (1,2). lt is therefore of clinical interest to characterize serum monoclonal RF. Severa! procedures have been developed to analyse RF charge heterogeneity and light chain restriction. These methods are based upon the isola tion of RF by affinity chromatography, followed by conven tional immunochemical analysis or on the revelation by conjugated aggregated IgG of blots of zone electrophoresis or isoelectric focusing of serum proteins (2-5). Such meth ods are technically complex and time consuming and they can hardly be used in large-scale studies. Filter affinity trans fer (FAT), which consists in a specific transfer of electropho retically separated proteins onto ligands (antigens, antibodies, lectins) bound to a solid phase (6) offers an elegant alterna tive. It was previously used to detect small amounts of serum monoclonal Ig using anti-Ig antibody-coated filters (7) or to characterize, with antigen-coated filters, monoclonal Ig reac tive with viral, proteic or gangliosidic antigens (8-10).

We developed a rapid and easy method for the analysis of lgM and IgA RF by FAT after thin layer agarose electropho-resis. This inexpensive method is applicable to large-scale studies of the clonality of RF and to screening of monoclonal Ig for RF specificity.

MATERIALS AND METHODS

The principle of the method is to separate serum proteins by high resolution electrophoresis and to transfer them on an uncoated nitrocellulose sheet to characterize serum lg using conjugated anti-Ig antibodies, as previously described (11,12), and on IgG-coated nitrocellulose to specifically detect RF. Preliminary experiments allowed to determine technical con ditions under which the same electrophoresis can be efficiently successively blotted onto uncoated and lgG-coated blots.

Sera

RF containing sera included 17 sera from RA patients and 4 sera which contained monoclonal lgM or IgA of known RF activity (3). As controls, we used a normal serum pool and 37 monoclonal IgM containing sera and a monoclonal IgA containing serum from patients with Waldenstrôm's macroglobulinemia (WM), lymphoma or chronic lymphocytic leukemia (CLL) with no known or an irrevelant (cold agglu tinin, anti-myelin associated glycoprotein, MAG) antibody activity of the monoclonal Ig.

for 10 minutes on uncoated nitrocellulose, and revelation by conjugates specific for the various Ig heavy and light chains.

Preparation of Affinity Filters

Human polyclonal IgG were prepared from Cohn fraction I1 (Sigma, St. Louis, MO) by DEAE-trisacryl (IBF, Villeneuve La Garenne, France) chromatography in 10 mM Tris-HC1 pH 7.5 buffer. The y heavy chain disease (HCD) protein Riv, which is a short y chain made up of the hinge region and of the two last constant domains of the y l chain (13) and IgG Fc fragments prepared by papain digestion of a monoclonal IgGlX, were kind gifts of Dr. E. Mihaesco (Paris). Nitrocellulose sheets (HAHY 304, Millipore, Bedford, MA) were incubated for 1 hr at room temperature with solutions of polyclonal IgG at 10 mg/ml in 0.01 M phosphate, 0.15 M NaCl, pH 7.4, buffer (PBS) or of the y HCD protein or Fc fragments at 5 mg/ml in PBS and rinsed once with PBS. IgG filters were saturated by incubation for 1 hr in 5% skimmed milk in PBS and rinsed in PBS. Saturation of y HCD or Fc fragments filters required an additional step of incubation with a 10 mg/ml solution of bovine serum albumin in PBS (PBS-BSA). Such affinity filters can be stored at -20°C for at least three months with no noticeable loss of antigenic activity.

Filter Affinity Transfer

Thin layer (0.4 mm) agarose electrophoresis of 2 p l of serum diluted 1/20 in saline was performed using the Paragon'" kit (Beckman, Gagny, France). A first short blotting onto uncoated nitrocellulose was performed under pressure (15 g per cm2 for 10 seconds). Then the IgG affinity filter was applied to the same agarose gel under the same pressure for 7 minutes. The uncoated nitrocellulose was saturated with 5% skimmed milk in PBS. Then, both types of membranes were incubated for one hour with alkaline phosphatase-conjugated anti-p or anti-a antibodies (Biosys, Compiegne, France) diluted 1/500 and 1/5,000 in PBS-BSA, respectively, washed six times with PBS containing 0.05% Tween 20 (Merck, Darmstadt, Germany) (PBS-Tween). Bound antibodies were revealed with 0.4 mM tetrazolium nitroblue (Sigma), 0.4 mM 5-bromo-4chloro-3-indolylphosphate (p-toluidine salt, Sigma) in 0.1 M Tris, 0.1 M NaCl, 50 mM MgC12, pH 9.5, buffer. The reaction was stopped in distilled water. Light chains were similarly detected on uncoated nitrocellulose or on yHCD or Fc fragments FAT filters using conjugated anti-x and anti-A antibodies (Sigma) or the monoclonal antibodies HP 6023 (anti-x) or HP 6024 (anti-A) (both kind gifts of Dr. C. Reimer, Atlanta, GA) followed by revelation with alkaline phosphatasecoupled anti-mouse IgG antibodies extensively absorbed on human IgG (12), as above.

Standard lrnmunoblots

The sera were also analysed under our standard conditions of Western blotting (1 1,12): electrophoresis of sera diluted 1/50 to 111 ,OOO according to serum Ig levels, pressure blotting

RESULTS

RF were undetectable by FAT in normal serum pools (not shown). In contrast, FAT allowed an easy detection of RF in RA sera with RF titers as low as 1/40 by the IgG-latex agglutination test and the Waaler-Rose reaction (Fig. 1) and in the sera containing monoclonal IgM or IgA with known RF activity (Fig. 2). It is worth noting that several RA sera (at least Nos. 3 to 6 on Fig. 1) clearly contained small amounts of homogeneous RF (narrow bands on the IgG-coated nitrocellulose). These narrow bands were undetectable by electrophoresis nor on standard immunoblots. As for monoclonal RF containing sera, comparison of the blots of the same electrophoresis onto uncoated and IgG-coated nitrocellulose allowed an easy identification of monoclonal RF. Monoclonal Ig lacking RF activity were consistently negative. In some of the latter sera, polyclonal RF or discrete narrow bands of monoclonal RF which were clearly distinct from the major monoclonal Ig were detectable, generally together with a weak RF activity by agglutination tests. The internal controls in the experiment shown on Fig. 1 are sera containing monoclonal IgM (lanes 1) or IgA (lanes 3) without RF activity. As other examples of specificity controls of the FAT, Figure 3 shows a CLL serum with no detectable RF by the latex agglutination test and Waaler-Rose reaction (this serum contained small amounts' of a monoclonal RF which was otherwise undetectable) and Figure 4 the serum from a patient with WM and a peripheral neuropathy and an anti-MAG activity of the monoclonal IgM (latex agglutination test and Waaler-Rose reaction titers were 1/360 and 1/40, respectively).

In addition to their charge homogeneity, monoclonal RF may also be identified on the basis of light chain restriction using the yHCD protein or IgG Fc fragments as antigen in the FAT. However, a correct saturation of the blots (and hence specificity) was more difficult to obtain than with IgG coated nitrocellulose and the specific staining of bound RF was not as strong.

DISCUSSION

We describe a new method for the characterization of RF which appears to be especially suitable for an evaluation of their monoclonality on the basis of charge homogeneity and light chain restriction. This method is simple and rapid and it does not need any sophisticated material or reagent, when applied to the detection of RF belonging to Ig classes other than IgG. It can be used on a large scale to screen monoclonal IgM or IgA for RF activity or to evaluate the mono-or oligoclonality of RF in inflammatory diseases. Conceivably, FAT could also be used to detect IgG RF, using Fc fragments as antigen in the FAT and antibodies specific for IgG Fab fragments as revelating antibodies, for instance. RF display a stronger affinity for polymeric than for monomeric IgG. IgG bound to a solid phase are therefore well suited for their detection. Even IgG-RF complexes preexisting in vivo appear to be displaced enough for RF binding to the affinity filter to occur. Monoclonal IgM with known RF specificity yielded narrow bands by FAT whereas the monoclonal IgA from the patient with hypergammaglobulinemic purpura did not (Fig. 2B). This IgA was electrophoretically homogeneous when isolated but was diffuse in the whole serum due to high level of binding to polyclonal IgG (3). Residual complexes bound to the affinity filter likely explain the charge heterogeneity of this monoclonal IgA by FAT. This example illustrates a limitation of the method for evidencing the monoclonality of certain RF. The difference between this IgA RF and IgM RF is possibly due to a stronger affinity of the IgA for monomeric IgG which might reflect differences in the variable regions orland polymerization of the monoclonal IgA ( 14).

In such cases, other methods, such as affinity chromatography isolation of RF after immune complex dissociation (3), appear to be required. Under the present technical conditions, FAT is specific, as shown by the negative internal controls with irrevelant WM sera. This specificity is dependent on the efficacy of the saturation of the coated nitrocellulose. Filters were easily and regularly properly saturated after coating with polyclonal IgG. After coating with the y HCD protein or with Fc fragments, an additional saturation step was required. There is, hence, clearly a need to control the binding specificity of every newly prepared affinity filter. In fact, we include an internal control using a WM serum with no RF activity of the monoclonal IgM in every experiment. FAT detects polyclonal RF in RA sera with RF titers as low as 1/40 by agglutination tests. Being a solid phase assay, it is likely to detect monomeric IgM RF which might be present in certain RA sera and do not yield positive agglutination reactions [START_REF] Carson | Rheumatoid factors and immune networks[END_REF]. It is not sensitive enough to detect RF in normal sera, which is rather an advantage in clinical terms. FAT is highly resolutive, and narrow bands of IgM with RF activ-ity can be detected in the serum of patients with immunoproliferative disorders or even with autoimmune disease. Such oligoclonal RF are likely present in minute amounts since they were undetected by standard immunoblotting which allows the identification of monoclonal Ig in concentrations as low as 25 pgiml (11). The finding of homogeneous RF amongst polyclonal RF in autoimmune processes was not unexpected but its significance remains to be established (1,2,4,5). In the present study, we merely examined technical parameters but in the near future we intend to apply FAT to the screening of a large number of sera from autoimmune patients to evaluate the precise incidence of oligoclonal RF under such conditions.

Fig. 1 .Fig. 2 .

 12 Fig. 1. FAT study of nine individual RA sera (revelation with anti-p conjugate). The electrophoresis of the first RA serum is shown for comparison and the RF titers of the RA sera are indicated on the right part of the figure.

Fig. 3 .EPFig. 4 .

 34 Fig. 3. Electrophoresis (EP), standard Western blots (A) and FAT (B, N, normal uncoated nitrocelllose; IgG, polyclonal IgG-coated nitrocellulose) of a CLL serum containing oligoclonal Ig (IgMX, IgMh, IgGK, 1gGX) without RF activity by FAT and a narrow band of IgM anti-IgG that is undetectable by electrophoresis and on standard blots.


Marie-Andrée Jacob

Da Col

Loïc Mazo
email: mazo>@unistra.fr

nD Quasi-Affine Transformations

A Quasi-Affine Transformation (QAT) is a transformation on Z n which corresponds to the composition of a rational affine transformation and an integer part function. The aim of this paper is twofold. Firstly, it brings new insight into the periodic structures involved by a QAT. Secondly, some new results in nD are presented specifically about the behavior under iteration of a QAT.

1

Introduction.

 to model them by Quasi-Affine Transformations (QAT) which are actual affine transformations on Q n followed by a floor function. We stress that this model slightly differs from the model used for instance by Nouvel et al. or Ngo et al. who consider Euclidean affine transformations (in R n ) followed by some rounding function.

At this point, we need to introduce a few notations. Points of R n , mixed up with vectors, are denoted with a bold font and, for any c ∈ R, c denotes the vector whose coordinates are all equal to c (in particular, 0 is the null vector). Binary relations between vectors (e.g. v ≤ w, or v ≤ 0), are to be understood coordinatewise. The floor function is denoted by ⌊•⌋ and, ω being a positive integer, we write ⌊x⌋ ω for 1 ω x . If g : Z n → Z n is an affine transformation of the n-dimensional Z-module, we note g 0 the linear part of g and v its constant part: g = g 0 + v. The Quasi-Affine Transformation (QAT) associated with g and ω is the transformation ⌊g⌋ ω : Z n → Z n defined by ⌊g⌋ ω (x) = ⌊ 1 ω g(x)⌋. We can now explain how a QAT ⌊g⌋ ω can be derived from an Euclidean affine transformation t. To fix ideas, let t be the rotation of π/4 radians such that t(0) = (1/3, 1/7) and u be the Euclidean vector with coordinates (π, 7/3).

The first way to obtain a QAT from the Euclidean rotation t is to approximate the real coefficients of (the matrix of) t by rationals whose greatest common denominator is ω leading to a rational affine transformation t ′ : Q n → Q n . Then the application of t ′ on the integer vectors ⌊x⌋, x ∈ R n , followed by the floor function leads to the QAT ⌊g⌋ ω : Z n → Z n where g = ω t ′ is an affine transformation on Z n . For instance, one can set ω = 1000 and g(x) = g 0 (x) + v where g 0 is defined by its matrix 707 -708 707 707 and v is the vector 333 142 . Then, t(u) is computed as .

t(u) = ⌊g⌋ ω (⌊u⌋) =
We have proposed in [START_REF] Andres | Les omega-aqa : Représentation discrète des applications affines[END_REF] another way to discretize an affine transformation in order to introduce the multigrid convergence scheme in the framework of QATs. Firstly, each real number is 'projected' on the integer line by a scaling operation followed by a rounding as in π → ⌊ω π⌋ = 3141 taking ω = 1000 as the scaling factor 1 . Then, any multiplication must be followed by an integer division. Indeed, consider for instance the calculus √ 2 × π + 6/7 with the precision ω = 1000. This calculation can be done by ⌊1414 × 3141⌋ 1000 + 857 = 4441 + 857 = 5298. Then, a "back-projection" on the real line gives the final result 5.298. In this setting, the Euclidean transformation t corresponds to the integer affine transformation ⌊g⌋ ω : x → ⌊g 0 (x)⌋ ω + v, that is to the QAT ⌊g 0 + ω v⌋ ω and t(u) is computed as t(u) = 1 ω ⌊g⌋ ω (⌊ωu⌋) = .

However, here it doesn't matter what the discretization scheme is. Indeed, the present paper is only interested in the properties of the QATs by themselves. More precisely, it is devoted to the understanding of the periodic structure created by a QAT, which basically expresses the lack of bijectivity of the transformation, and on the behavior under iteration, which brings out the loss of precision. We present new results about these two topics in the general case and also give a new perspective on the links with the lattice group theory. The article is organized as follows. In Sect. 2, we study tilings generated by QATs: a tile is the set of the inverse images of a given point. Indeed, as explained above, a QAT derived from a bijective affine transformation in the Euclidean world will generally not be bijective and fibers can be empty, contain one or several points. Then, if we only focus on the difference between the affine transformation and the quasi-affine transformation, tiles yield a lattice structure in the discrete space. In 2D and 3D, explicit formulas have been proposed to compute a minimal basis of the QAT periodic structure [START_REF] Col | Applications quasi-affines et pavages du plan discret[END_REF][START_REF] Coeurjolly | Quasi-affine transformation in 3-d: Theory and algorithms[END_REF]. In the n-dimensional case, an upper bound on the number of distinct tiles is given in [START_REF] Col | Quasi-linear transformations and discrete tilings[END_REF]. We now give a closed formula for the period in any dimension. In Sect. 3, we study the behavior under iterations of a contracting QAT. If g is a contracting affine transformation of R n then g has a unique fixed point and for each x ∈ R n the sequence g n (X) tends toward this fixed point. But the corresponding QAT has not necessarily a unique

O (1,1) (0,0) (1,3) (4,2) (4,0) (5,5 (2,4) (5,3) (2,2) (3,1) (0,2) (1,5) (4,4) (1,3) (4,2) (5,5) (5,1) (0,0) (3,5) T (0,0) T T T T (1,0) (2,0) (1,1) (2,1)
Fig. 1: Example of tiles and remainders. A point of Z 2 is represented by a unit square whose bottom-left corner corresponds to the represented point. For each point in a tile we provide its corresponding remainder. Tiles T (2,1) and T (0,0) are arithmetically equivalent. The tiles T (1,0) and T (1,1) have the same shapes but they are not equivalent.

fixed point. The behavior under iteration of the 2D contracting QATs has been studied in [START_REF] Jacob | Applications quasi-affines[END_REF], [START_REF] Nehlig | Applications Affines Discrètes et Antialiassage[END_REF] and [START_REF] Nehlig | Applications quasi-affines : pavages par images réciproques[END_REF]. In this paper, we study the nD case and prove that a QAT has a unique fixed point if and only if there are no cycle among the points of norm 1.

Periodicity of the tiles

In order to obtain efficient computations for the affine transformation of discrete images, we can use some properties of QATs. In [START_REF] Jacob | Gaussian numeration systems[END_REF] the 2D case has been treated and extended to 3D in [START_REF] Coeurjolly | Quasi-affine transformation in 3-d: Theory and algorithms[END_REF]. More precisely, we have seen in these papers that the periodicity properties of the tilings associated to QATs improve considerably the computation of the discrete affine transformations. This periodicity has been studied in the n-dimensional case in [START_REF] Col | Quasi-linear transformations and discrete tilings[END_REF] and [START_REF] Blot | Quasi-affine transformation in higher dimension[END_REF]. Nevertheless, no closed formula for the tile period was given in these papers. We begin by recalling some definitions about QATs and their tiles. A nonsingular QAT, that is a QAT derived from a bijective affine transformation, is generally not bijective and for a given QAT ⌊g⌋ ω , the preimages ⌊g⌋ ω -1 (y) can have either none, one or several elements. As these preimages define a tessellation of the space Z n , we call them tiles. In the sequel we assume a QAT ⌊g⌋ ω = 1 ω (g 0 + v) and we note T y the tile of y for this QAT. The remainder of a tile T y , noted r y , is the set of the remainders modulo ω of the vectors g(x), x ∈ T y , that is

r y = {g(x) -ω y | x ∈ T y } .
Observe that the sets T y and r y have the same cardinal since g is nonsingular.

Two tiles T y and T y ′ are said to be (arithmetically) equivalent when their remainders are the same. We then write T y ≡ T y ′ . Two vectors y and y ′ of Z n are equivalent modulo ⌊g⌋ ω -we write y ≡ y ′ -if the tiles T y and T y ′ are equivalent. Eventually, we write p ∧ q, resp. p ∨ q, for the gcd, resp. the lcm, of the integers p and q. Two equivalent tiles of a QAT ⌊g⌋ ω share the same shape but the converse is false. Figure 1 exhibits some tiles of the QAT defined by the integer matrix 3 1

-1 3 , the vector v = 0 and the integer ω = 6.

The next proposition and its corollary describes the periodic structures induced by the equivalence modulo ⌊g⌋ ω and the corresponding tile equivalence.

Proposition 1. Let ⌊g⌋ ω be a non singular QAT. The equivalence modulo ⌊g⌋ ω has a translational symmetry whose lattice group is

G = g 0 ( 1 ω Z n ) ∩ Z n .
In particular, two non empty tiles T y and T y ′ are equivalent iff yy ′ ∈ 1 ω g 0 (Z n ). Moreover, the multi-valued function defined by q → T ⌊q⌋ induces a surjection from the quotient space

1 ω g(Z n ) ∪ Z n /G onto the tile classe set {T y } y∈Z n / ≡. Proof. Let y, y ′ in Z n such that y ′ -y = 1
ω g 0 (z) for some z ∈ Z n . We shall prove that the tiles T y and T y ′ have the same remainders. Let x ∈ T y , if such an integer vector exists. The remainder of g(x

) modulo ω is r(x) = g(x) -ω y. We observe that 1 ω g(x + z) -y ′ = 1 ω g(x) + 1 ω g 0 (z) -y ′ = 1 ω g(x) -y = 1 ω r(x)
by definition of z and r(x). Thus, the integer vector x ′ = x + z lies in T y ′ and the remainder of g(x ′ ) modulo ω is equal to the remainder of g(x) modulo ω. Thereby, we proved that either r y = ∅ or ∅ = r y ⊆ r y ′ . It is plain that, by the same reasoning, we can also prove r y ′ = ∅ or ∅ = r y ′ ⊆ r y . Hence, either

T y = T y ′ = ∅ or r y = r y ′ , that is, in both cases, T y ≡ T y ′ .
Conversely, if two non empty tiles T y and T y ′ are equivalent, there exist two vectors x ∈ T y and

x ′ ∈ T y ′ such that g(x) ≡ g(x ′ ) (mod ω). Thus g 0 (x ′ -x) ≡ 0 (mod ω) so z = 1 ω g 0 (x ′ -x) ∈ Z n . Then, ⌊g(x ′ )⌋ ω = 1 ω g(x) + g 0 (x ′ -x) = 1 ω g(x) + z = 1 ω g(x) + z = y + z where z ∈ 1 ω g 0 (Z n ).
We cannot assert that for two empty tiles T y and T y ′ there exists z ∈ 1 ω g 0 (Z n ) such that y ′ = y + z. Nevertheless, this is not necessary to obtain the result since the tiles cannot all be empty. Incidentally, we have shown that, for any x, x ′ in Z n , if g(x) ≡ g(x ′ ) (mod ω), then ⌊g(x ′ )⌋ ω = ⌊g(x)⌋ ω + z where z ∈ 1 ω g 0 (Z n ) which, from the first part of the proof, implies that

T ⌊g(x ′ )⌋ ω ≡ T ⌊g(x)⌋ ω . ⊓ ⊔
Note that, from a practical point of view, the last assertion of Prop. 1 means that one can prove that two tiles are equivalent by just exhibiting two integer points in these tiles whose images under 1 ω g have the same fractional part. Corollary 1. Let ⌊g⌋ ω be a non singular QAT. The tiles of ⌊g⌋ ω have a translational symmetry whose lattice group is g -1 0 (ωZ n ) ∩ Z n . Proof. Let y, z ∈ Z n and t the translation of vector g -1 0 (ωz). Assume that the tiles T y and T y+z are equivalent. Then r y = r y+z and

T y+z = {g -1 (ω (y + z) + r) | r ∈ r y+z } = {g -1 (ω (y + z) + r) | r ∈ r y } = {g -1 (ω y + r) + g -1 0 (ωz) | r ∈ r y } = t(T y ) .
In particular, g -1 0 (ωz) is an integer vector. Conversely, let y, u ∈ Z n and let t be the translation of vector u. Assume that there exists z ∈ Z n such that u = g -1 0 (ωz). Since z = 1 ω g 0 (u), we derive from Prop. 1 that T y+z ≡ T y . Then, from the first part of the proof, t(T y ) = T y+z . Thus, t(T y ) is a tile and this tile is equivalent to T y . ⊓ ⊔

In [START_REF] Col | Quasi-linear transformations and discrete tilings[END_REF], the authors give upper and lower bounds for the number of tile equivalent classes. Thanks to the Smith normal form of integer matrices, we give hereafter the exact number of classes. We also provide the cardinal of any (maximal) union of representative tiles. Firstly, we give two lemmas that describe the role of unimodular transformations on the lattices associated to a QAT.

Lemma 1 ([5]

). Let u be an unimodular transformation. The QAT ⌊g⌋ ω and ⌊g • u⌋ ω have the same tile remainders. In particular, they share the same tile equivalence and the same empty tiles.

Proof. We note T u y , resp. r u y , the tile of y, resp. the remainder of T u y , for the QAT ⌊g • u⌋ ω while T y and r y are the tile of y, resp. the remainder of T y , for the QAT ⌊g⌋ ω . We have x ∈ T u y ⇐⇒ u(x) ∈ T y , thus T y = u(T u y ). Furthermore,

r u y = {g • u(x) -ω y | x ∈ T u y } = {g(z) -ω y | z ∈ T y } = r y . ⊓ ⊔ Lemma 2.
Let u be an unimodular transformation. The cardinals of the fundamental domains of the equivalences modulo ⌊g⌋ ω and modulo ⌊u • g⌋ ω are equal.

Proof. The lattice groups of the equivalences modulo ⌊g⌋ ω and modulo ⌊u •

g⌋ ω are L 1 = Z n ∩ 1 ω g 0 (Z n ) and L 2 = Z n ∩ 1 ω u • g 0 (Z n ). Note that L 2 = u(L 1 ) since u(Z n ) = Z n
and u is bijective (u is unimodular). As |det(u)| = 1, we get that the volumes of the parallelepipeds wrapping L 1 and L 2 are equal. Then, from Picks's theorem, we derive that the numbers of integer vectors in these domains are identical.

⊓ ⊔

The following lemma describes the lattice groups of QATs whose matrix is diagonal.

Lemma 3. Let ⌊g⌋ ω be a nonsingular QAT whose matrix (d i,j ) is diagonal. Then, -the lattice group of the equivalence modulo ⌊g⌋ ω is n i=1 di,i ω∧di,i Z; -the lattice group of the non empty tiles is

n i=1 ω ω∧di,i Z.
Proof.

-From Prop. 1, the lattice group of the equivalence modulo

⌊g⌋ ω is Z n ∩ g 0 1 ω Z n = 1 ω (ω Z n ∩ g 0 (Z n )) and g 0 (Z n ) = n i=1 (d i,i Z) for g 0 is diagonal. Then, the relation ω ∨ d i,i = (ω d i,i )/(ω ∧ d i,i
) yields the first part of the result.

-From Corollary. 1, the lattice group of the non empty tiles is g

-1 0 (ω Z n )∩Z n = n i=1 1 di,i (ω ∨ d i,i )Z = n i=1 ω ω∧di,i Z.

⊓ ⊔

Thanks to the Smith normal form of the affine transformations of Z n , we now derive from the three preceding lemmas the cardinal of the fundamental domains of the lattice structures involved in a QAT.

Theorem 1. Let ⌊g⌋ ω be a nonsingular QAT and D be a fundamental domain of the equivalence modulo ⌊g⌋ ω . Noting s = (s i,j ) the Smith normal form of the matrix of g 0 and δ its determinant, one has

#D = δ n i=1 ω ∧ s i,i and y∈D #T y = ω n n i=1 ω ∧ s i,i
.

Proof.

-From Prop. 1, the cardinal of D only depends on g 0 . Let s = u • g 0 • v be the Smith normal form of g 0 . From Lemmas 1 and 2, we derive that #D = #D ′ where D ′ is the fundamental domain of the equivalence modulo ⌊s⌋ ω . Then Lemma 3 gives the first result. -From Corollary 1, y∈D #T y is equal to the cardinal of the fundamental domain of g -1 0 (ω Z n ) ∩ Z n . Then, thanks to the properties of unimodular transformations, we have

g -1 0 (ω Z n ) ∩ Z n = v • s -1 • u (ω Z n ) ∩ Z n = v • s -1 (ω Z n ) ∩ Z n = v s -1 (ω Z n ) ∩ v -1 (Z n ) = v s -1 (ω Z n ) ∩ Z n .
Hence, the lattice of the non empty tiles of g is the image by an unimodular transformation of the non empty tile lattice of s. We conclude straightforwardly by invoking the second part of Lemma 3.

⊓ ⊔

Example : Figure 2 illustrates a QAT in Z 2 (g 0 : 12 -11 18 36 , v = 0, ω = 84) whose fundamental domain contains 15 tiles (the tiles with same color are equivalent). In this example, for all i, j ∈ N, T (i+5,j) ≡ T (i,j) and T (i+2,j-3) ≡ T (i,j) . The fundamental domain of this QAT is the set T (i,j) | i = 0, 1, 2, 3, 4, j = 0, 1, 2 . The Smith normal form of the QAT is 1 0 0 630 . From Th. 1, we derive that there is no empty tile (#D = 15) and there are 168 integer points in 0≤i≤4,0≤j≤2 T (i,j) .

In [START_REF] Coeurjolly | Quasi-affine transformation in 3-d: Theory and algorithms[END_REF] we used the periodicity to improve the transformation of a 3D image by a linear transformation. In order to use this periodicity we need to determine a basis of the equivalence modulo ⌊g⌋ ω lattice group such that, for any m ≤ n, (y i ) m i=1 is a basis of Z m × {0} n-m . Proposition 2. Let ⌊g⌋ ω be a non singular QAT. There exists a basis (y i ) n i=1 of the equivalence modulo ⌊g⌋ ω lattice group such that, for any m ≤ n, (y i ) m a basis of Z m (more specifically, Z m × {0} n-m that we identify with Z m ). Each vector y i , 1 ≤ i ≤ n, is defined by y i = h i (λ i ) where h i is the restriction to Z i of the Hermite normal form of g 0 and λ i = (λ i,1 , . . . , λ i,i ) is the solution of h i (λ i ) = 0 (mod ω) such that λ i,i is minimal.

Proof. The modulo ⌊g⌋ ω equivalence lattice group is g 0 1 ω Z n ∩ Z n . Let h be the Hermite normal form of g 0 . As h -1 • g 0 is unimodular, the group g 0 (Z n ) is generated by the column vectors of h. Let c j = (h i,j ) n i=1 be the j-th column vector of h. Note that, since h is triangular, c j ∈ Z j × {0} n-j . Then finding the basis (y i ) amounts to find the integer tuples (λ i,1 , . . . , λ i,i ) such that

y i = i j=1 λ i,j c j ∈ ω Z n and λ i,i is minimal. ⊓ ⊔
Proposition 2 shows that we can compute the basis (y i ) n i=1 by solving triangular systems of i linear equations in the modules (Z/ωZ) i , 1 ≤ i ≤ n, which can be done by Gaussian elimination in polynomial time.

Behavior under iteration of a quasi-linear transformation

In this section we consider affine transformations g whose vector v is such that 0 ≤ v < ω. Then, the associated QAT can be seen as the composition of a linear transformation with some rounding operator. For this reason, we say that such a QAT is a quasi linear transformation (QLT). Since we are interested in fixed points, or more generally in cycles, we restrict our study to non expansive transformations for the infinite norm, which is defined on a linear transformation

f whose matrix is A = (a i,j ) by f ∞ = A ∞ = max i j |a i,j | . Thus we have for any vector x, f (x) ∞ ≤ f ∞ × x ∞ .
It is well known that if f is a contracting linear transformation of R n then f has the origin as unique fixed point and for each x ∈ R n the sequence f n (x) tends toward this fixed point. But a QLT built from a non expansive linear transformation has not necessarily a unique fixed point. Consider the sequence y n = (⌊g⌋ ω (y n-1 )) n≥0 = (⌊g⌋ ω n (y 0 )) n≥0 with y 0 ∈ Z n . Depending on y 0 we can obtain a cycle (y p = y q for some integer pair p < q), a fixed point (y n = y n+1 for some n ∈ N), or leaves The first results concern the localisation of the cycles. Firstly, the following lemma gives a bound on the difference between a QLT and the corresponding linear transformation. Lemma 4. Let ⌊g⌋ ω be a QLT, d = ⌊g⌋ ω -1 ω g 0 and s = max( ||v||∞ ω , 1 -||v||∞ ω ). Then, for any integer vector x, d(x) ∞ ≤ s, the inequality being strict if s > 1 2 .

Proof.

Consider x ∈ Z n , y = ⌊g⌋ ω (x) ∈ Z n , y ′ = 1 ω g 0 (x) ∈ Q n and d(x) = y -y ′ . We have d(x) = 1 ω (g 0 (x) + v) - 1 ω g 0 (x) = 1 ω (g 0 (x) + v) -r - 1 ω g 0 (x) with 0 ≤ r < 1 = 1 ω v -r with 0 ≤ r < 1 It follows that 1 ω v -1 < d(x) ≤ 1 ω v.
As we assume in the current section that

0 ≤ v < ω , we get d(x) ∞ ≤ s, the inequality being strict if s = 1 -1 ω v ∞ , that is to say if v ∞ < ω 2 . ⊓ ⊔
As a consequence of Lemma 4, we show that a QLT corresponding to a non expansive linear transformation is itself non expansive from the origin.

Corollary 2. If ⌊g⌋ ω is a QLT and 1 ω g 0 ∞ ≤ 1, then ⌊g⌋ ω (x) ∞ ≤ x ∞ for any x ∈ Z n .
Proof. Let ⌊g⌋ ω be a QLT such that 1 ω g 0 ∞ ≤ 1. With the notations of Lemma 4, we have:

⌊g⌋ ω (x) ∞ = 1 ω g 0 (x) + d(x) ∞ ≤ 1 ω g 0 (x) ∞ + d(x) ∞ ≤ 1 ω g 0 ∞ x ∞ + d(x) ∞ ≤ x ∞ + d(x) ∞ < x ∞ + 1 (from Lemma 4) ≤ x ∞ for ⌊g⌋ ω (x) ∞ and x ∞ are integers . ⊓ ⊔ Remark 1.
From Corollary 2, we derive that, if ⌊g⌋ ω is a QLT, 1 ω g 0 is non expansive and x belongs to a cycle, then ⌊g⌋ ω (x) ∞ = x ∞ .

Thanks to Lemma 4, we can also prove that the cycles of non expansive QLTs are 'not too far' from the origin. Thereby, we extend to the general case a result that was obtained in [START_REF] Jacob | Applications quasi-affines[END_REF] for the 2D space.

Theorem 2. Consider a QLT ⌊g⌋ ω such that 1 ω g 0 is contracting and note s = max(

v ∞ ω , 1 - v ∞ ω ). If x belongs to a cycle then x ∞ ≤ s 1-g0 ∞ /ω , the inequality being strict if s > 1 2 . Proof. Let g 1 = (1/ω) g 0 . From lemma 4, inductively we get ⌊g⌋ ω k (x) = g k 1 (x) + g k-1 1 (d 1 (x)) + g k-2 1 (d 2 (x)) + • • • + g 1 (d k-1 (x)) + d k (x) with d i (x) ∞ ≤ s for i = 1, 2, . . . , k each inequality being strict if s > 1 2 . It follows that ⌊g⌋ ω k (x) ∞ ≤ g k 1 (x) ∞ + k i=1 g k-i 1 (d i (x)) ∞ ≤ g 1 k ∞ x ∞ + k i=1 g 1 k-i ∞ d i (x) ∞ ≤ g 1 k ∞ x ∞ + s k i=1 g 1 k-i ∞ ≤ g 1 k ∞ x ∞ + s 1 -g 1 k ∞ 1 -g 1 ∞ .
If x belongs to a cycle, it exits k ∈ N such that ⌊g⌋ ω k (x) = x and so

x ∞ (1 -g 1 k ∞ ) ≤ s 1 -g 1 k ∞ 1 -g 1 ∞ .
Finally, x ∞ ≤ s/(1g 1 ∞ ), the inequality being strict if s > 1 2 .

⊓ ⊔

As noted in Remark 1, the points of a cycle of a non expansive QLT share the same infinite norm. Then, the following lemma shows that, for contracting QLTs with null constant part, those points cannot lie in the first hyperoctant.

Lemma 5. Let ⌊g⌋ ω a QLT such that 1 ω g 0 ∞ < 1 and v = 0. Consider y = ⌊g⌋ ω (x) where x ∈ Z n . If y ∞ = x ∞ , then the coordinates of y with maximal absolute value are negative.

Proof. From the hypotheses, we have

1 ω g 0 (x) ∞ ≤ 1 ω g 0 ∞ x ∞ < y ∞ .
Thus, any positive coordinate of 1 ω g 0 (x) is less than y ∞ which prove the result. ⊓ ⊔ Corollary 3 applies Theorem 2 to two common particular cases: v ∞ = ω/2 (rounding half up) and v = 0 (rounding down). In this corollary and in the following theorem we denote by S n-1 the unit sphere {x | x ∞ = 1}. Corollary 3. Let ⌊g⌋ ω be a QLT.

-If v ∞ = ω 2 and g 0 ∞ < 3 4 ω then any cycle is included in the unit sphere S n-1 . -If v = 0 and g 0 ∞ < 1 2 ω, then any cycle is included in the unit cube {-1, 0} n .
Proof. The first assertion is a direct consequence of Theorem 2. The second assertion is a consequence of Lemma 5 and Theorem 2.

⊓ ⊔

Eventually with the next theorem we show that not only the cycles of a non expansive QLT are not too far from the origin but moreover one of these cycles (if such a cycle exists) lies in the unit sphere S n-1 . Thereby, we solve a conjecture that was stated in [START_REF] Jacob | Applications quasi-affines[END_REF] for the 2D case. Theorem 3. Any QLT ⌊g⌋ ω derived from a non expansive linear transformation (1/ω) g 0 has a cycle in the unit sphere S n-1 or has no cycle.

In order to prove the theorem we need the following technical lemma.

Lemma 6. Let ⌊g⌋ ω be a QLT such that 1 ω g 0 ∞ ≤ 1. Let x ∈ Z n \ {0}, p ∈ S n-1 be such that x -p ∞ is minimal. If y = ⌊g⌋ ω (x)
has the same norm as x, then q = ⌊g⌋ ω (p) is in S n-1 and yq ∞ is minimal.

Proof. As p ∈ S n-1 , Corollary 2 induces that q ∈ S n-1 ∪ {0}. Moreover, again from Corollary 2, we derive that:

y -q ∞ = ⌊g(x)⌋ ω -⌊g(p)⌋ ω ∞ = ⌊g 0 (x -p) + g(p)⌋ ω -⌊g(p)⌋ ω ∞ = ⌊g 0 (x -p) + u⌋ ω ∞ where u = g(p) -ω ⌊g(p)⌋ ω ≤ x -p ∞ for x → g 0 (x) + u is a non expansive QLT. Since x -p ∞ is minimal and p ∈ S n-1 , one has x -p ∞ = x ∞ -1 = y ∞ -1. Then, y -q ∞ < y ∞ .
We conclude that q = 0 so that q ∈ S n-1 and yq ∞ is minimal. ⊓ ⊔ Proof (Theorem 3). Consider x a point of a cycle and a point p defined as in lemma 6 and consider the sequences x i = ⌊g⌋ ω i (x) and p i = ⌊g⌋ ω i (p). For each i ∈ N the points x i and p i verify the condition of lemma 6. The sequence p i is then an infinite sequence of points in S n-1 . But it exits a finite number of points in S n-1 so there is a cycle among these points.

⊓ ⊔ Remark 2. If v = 0 and g 0 ∞ < ω, from Lemma 5 and Theorem 3, we derive that there exists a cycle in the unit cube {-1, 0} n .

Conclusion

In this paper we completed and improved some theoretical results abouts QATs. We gave a closed formula for the period of the tiles generated by QATs and proved a conjecture on the localization of their cycles. We plan to use these results in a multigrid and multiprecision framework that we are developing in a joint work between Poitiers University and Strasbourg University [START_REF] Andres | Les omega-aqa : Représentation discrète des applications affines[END_REF]. The goal of this project is to build a formal theory in which the consequences of the change of universe between affine and quasi-affine transforms, that is between real numbers and integers, is precisely described so that one could choose a calculus precision and an image resolution to ensure a set of properties up to a tolerance threshold. For instance, at any precision and resolution, almost all quasi-affine rotations have non-singular tiles. Nevertheless, it would be desirable firstly to prove that though the size in pixels of the tiles grows with the precision, their 'true' sizes, taking into account the space resolution, tend toward zero and secondly to bound the convergence speed. Alike, we have to ensure for example that, despite the increasing complexity of cycles of quasi-affine rotations as the precision goes to infinity, they converge, together with fixed points, toward the center of the rotation when the pair precision -resolution tends toward infinity.

Fig. 2 :

 2 Fig. 2: Periodicity of thes tiles of a 2D QAT (see text)

Fig. 3 : 3 - 1 1 - 1 - 1 x

 331111 Fig. 3: (a) Example of cycle, fixed point and leaf. (b) QLT with many cycles. The points of a cycle and those that reach this cycle have the same colours. (c) QLT with a unique fixed point. The colour of a point is determined by the number of iterations necessary to reach the fixed point.

One can use two distinct scale factors for the digitization of the space and the quantification of the transformation.