F Leyvraz

R A Méndez-Sánchez

M Lombardi

T H Seligman

Multichannel quantum defect theory: a quantum Poincaré map

Keywords: 05.45.Mt, 33.80.Rv, 03.65.Sq

The multichannel quantum defect theory (MQDT) can be reinterpreted as a quantum Poincaré map in representation of angular momentum. This has two important implications: we have a paradigm of a true quantum Poincaré map without semiclassical input and we get an entirely new insight into the significance of MQDT.

In recent years there has been a rapidly growing interest in the quantum Poincaré map (QPM) [START_REF] Bogomolny | Chaos 2 (1992) 5; Comments At[END_REF]2,3,4,5,6,7,8,9,10], i.e. the quantization of a classical Poincaré map, for a time independent Hamiltonian system. Bogomolny [START_REF] Bogomolny | Chaos 2 (1992) 5; Comments At[END_REF] started out with a semi-classical formulation. Among other things he shows that unitarity of the representation is reached in the limit → 0, while this is only approximate for finite [8]. Prosen [4] gives an elegant general solution to the unitarity problem at the expense of obtaining an infinite matrix for the QPM. The semi-classical approach common to most discussions causes a number of problems that make the use of this new and powerful tool a little obscure. With other words, the quantum Poincaré section implicitly defined by Bogomolny, lacks a paradigmatic example where a quantum treatment can be performed properly throughout and leads to a finite unitary matrix.

Multichannel quantum defect theory (MQDT) [11,12,13,14,15] and its classical limit [16,17,18] will be shown to provide the framework for such a paradigm. Indeed we shall see that a simplified model of the Rydberg molecule allows to construct a classical Poincaré map on the unit sphere, whose exact quantization is provided by MQDT. Thus the result is necessarily entirely quantal, exactly unitary and for finite given in terms of a finite matrix. We shall show that the results commonly derived for MQDT are directly properties of the unitary representation of this classical map as obtained by MQDT.

After a short description of the model for a Rydberg molecule and the simplification introduced in Ref. [16], we proceed to give the quantum map for this case explicitly. We illustrate the two important aspects of our result by two applications. First the new interpretation allows modifications of the MQDT method, that prove particularly effective in near integrable systems. Second we proceed to show by way of examples that the properties of this map are relevant to the study of chaos and order in this system.

Simplifying to the most basic case, these molecules can be viewed as a rotating system with positive charge and cylindrical symmetry that binds one electron in an orbit that is at large distances hydrogenic. The classical limit of the MQDT is the following classical model [16]: The motion is composed of two consecutive steps. (i) when the electron is far from the molecular core (i.e. most of the time for a Rydberg electron) it feels only the Coulomb part (-1/r) of the potential. Its orbit is hydrogenic and its angular momentum L is fixed in the laboratory reference frame. Meanwhile the core rotates freely with an angular momentum N which is also fixed in the laboratory frame. The total angular momentum J = L + N is always conserved. In the molecular reference frame, the OZ axis is the cylindrical symmetry axis of the core. The core angular momentum N points in a perpendicular direction, taken as the OX axis. The angles θ and ϕ are the polar and azimuthal angles respectively of the electronic angular momentum L in this frame. During this step, L rotates freely in the molecular frame clockwise around the OX axis, at a speed ω N = N/I (I is the moment of inertia of the core), during the time of a Coulomb revolution: T e = 2π (-2E e ) -3/2 (E e is the electron energy, all is in atomic units e = = m = 1). (ii) during the so called "collision" step, the electron senses also the cylindrically symmetric short range part of the potential of the core. Aside from the energy and J, the projection of L onto the core axis Λ = L cos θ is conserved due to the cylindrical symmetry of the core. We will add an extra, simplifying, hypothesis, namely that the magnitude L of L remains constant [16]. This is justified for Rydberg molecules at least for small L's, but the classical and quantum map with this approximation exist for all L. Thus the collision can be described by a θ-dependent rotation of L around the core axis. The simplest form of this rotation compatible with the symmetry is [16]: δϕ = K cos θ, where K is a coupling constant. This simplification is not essential. Notice further that the conservation of the total angular momentum J implies that the molecular core feels a simultaneous recoil which changes both the direction and magnitude of N . N precesses around the core axis OZ by an angle δϕ ′ . These changes of N are computed by writing the conservation of J in the initial (before collision) reference frame:

L sin(θ 1 ) cos(ϕ 1 ) + N 1 = L sin(θ 2 ) cos(ϕ 1 + δϕ) + N 2 cos(δϕ ′ ) L sin(θ 1 ) sin(ϕ 1 ) = L sin(θ 2 ) sin(ϕ 1 + δϕ) + N 2 sin(δϕ ′ ) L cos(θ 1 ) = L cos(θ 2 ), (1)
where subscript 1 (2) labels parameters before (after) the collision. These three equations are solved for three unknowns θ 2 , δϕ ′ and N 2 . The change of the magnitude of N in turn entails a change of the rotational energy

E N = N 2 /2I
of the core and because of conservation of total energy a change of the energy E e of the electron. This exchange of energy makes this model much richer than the kicked spin model [19] (which is its limit when L ≪ J, where this recoil can be neglected). In particular the energy of the electron may become positive after the collision, allowing to treat on equal footing bound and unbound (ionized) states. Possibility of chaotic motion comes from the conflict of these two steps, which consist of two rotations around distinct axes with different laws. The classically chaotic case can be obtained by increasing the coupling K. Near-integrable cases can be obtained for small coupling or at resonance, i.e. when the period of the electron is a multiple of half the period of rotation of the core.

The quantum problem is solved by using the MQDT. The configuration space is divided by a sphere of radius r 0 in a collision (r < r 0 ) and an asymptotic region (r > r 0 ) for the motion of the electron. r 0 is of the order of the core size and is chosen such that in the asymptotic region the potential acting on the electron is only Coulombic, whereas in the collision region it feels both Coulomb and cylindrical potential. The wave functions for both regions are joined appropriately at r = r 0 . The conflict between the two motions is expressed in quantum mechanics by the existence for the wave function of two bases with different good quantum numbers (in addition to J, J z , L and total energy E). At short distance the Born Oppenheimer basis is appropriate. The Rydberg electron is strongly bound to the core, thus quantized in the molecular reference frame and the additional good quantum number is Λ. At long distances the collision basis is appropriate. Here the electron is uncoupled from the core and the angular momentum N of the core remains a good quantum number. The collision is described by phase shifts πµ Λ , which are identical to usual collision phase shifts (relative to pure Coulomb phase shifts) if the total energy is positive enough for all channels to be open, and which are related to the classical K parameter by the relation δϕ = -2π∂µ Λ /∂Λ (see the derivation of the classical limit of MQDT in Ref. [16], Appendix A), i.e. in our simple case

µ Λ = µ 0 -(K/4πL)Λ 2 .
We focus on the completely bound situation, when total energy is low enough for the electron energy to be always negative whatever the value of N within the allowed range [J -L, J + L]. Demanding that the electron wave function goes to zero when r → ∞ leads to demanding that the following determinant vanishes [15], i.e.

det S = det {U N Λ sin(π(µ Λ + ν N ))} = 0, (2)
where the unitary U matrix given by

U N,Λ =< L, -Λ, J, Λ|N, 0 > (-1) J-N +Λ (2 -δ Λ , 0) 1/2 . ( 3
)
relates the two conflicting bases through a Clebsh Gordan coefficient. The principal (non integer) action ν N (E) of the Coulomb electron, is related to the electron energy through

E = E N + E e = N (N + 1)/(2I) -1/(2ν 2 N )
, where I is the moment of inertia of the core, and we use atomic units (e = m = = 1). Corresponding wave functions are the eigenkets of S for the eigenvalue zero, labeled by Λ in the Born Oppenheimer basis:

S|A Λ = 0. ( 4
)
Similarly the eigenbras B N | of S (eigenkets of S t ), labeled by N describe the corresponding wave function in the collision basis.

To proceed notice first that S is the imaginary part of a complex unitary matrix

E = C + iS = {U N Λ exp(iπ(µ Λ + ν N ))} . (5)
This non symmetrical matrix maps the N basis on the Λ basis: it is half the QPM we look for, and describes the motion between apogee and perigee. From it we can construct two QPM, which, by construction, turn out to be symmetrical unitary complex matrices.

E t E operates in Born Oppenheimer Λ space, and is the exact quantization of a classical Poincaré map. The latter is nearly the classical map used in refs. [16,17]. That map on the unit sphere described the position of L in the molecular frame immediately after the collision, whereas the present map describes the position of L in the middle of the collision (perigee). That this matrix is the T matrix defined by Bogomolny [START_REF] Bogomolny | Chaos 2 (1992) 5; Comments At[END_REF] to quantize a Poincaré map will be shown by proving that the eigenvalues and eigenfunctions of the entire system result from

det(1 -T (E n )) = 0, (6)
i.e. Bogomolny's equation for the quantized energy E n . Indeed at quantized energies given by Eq. ( 2)

E t E|A Λ = 1|A Λ , (7)
which implies (6). To prove this key point first notice that unitarity of E, namely

E † E = (C t -iS t )(C + iS) = I, leads to C t C + S t S = I and C t S = S t C. Then E t E = (C t + iS t )(C + iS) = I -2S t S + 2iC t S, so that if S|A Λ = 0 then E t E|A Λ = I|A Λ .
Conversely EE t operates in collision N space and corresponds to a classical map between apogee and apogee in the laboratory frame. The rationale is that the matrix E is the product of three unitary matrices, namely Diag{exp(iπµ Λ )}, {U N Λ } and Diag{exp(iπν N )}, which describe respectively half a collision in molecular frame, change between molecular and laboratory frame, and half a free Coulomb orbit in laboratory frame. We have thoroughly checked this point by comparing the time evolution of a given swarm of classical trajectories and the corresponding quantum Husimi distribution [20].

We will now compare the traditional way of solving MQDT to the one implied by our QPM. The traditional way is to look for zeros of the determinant of the non-symmetric matrix S of Eq. ( 2), all of whose elements depend in a complex way on energy through ν N (E). It is computed by a LU or a SVD method followed by a root searching algorithm [21,16]. The present method is to look for eigenphases of E t E or EE t . They are computed by the diagonalization of a symmetrical unitary matrix. This is efficient and unproblematic because it diagonalizes in an orthonormal basis. Finally we search the zeros of the resulting eigenphases.

The situation is sketched in Fig. 1. The search for zeros of eigenphases which vary nearly linearly with energy is obviously much simpler than the search of zeros of a determinant which is sometimes nearly tangent to the horizontal axis. Moreover, in this case of near tangency we had problems to converge the wave functions because the eigenfunction switches between two nearly orthogonal values in a very narrow energy range [17], requiring the use of the more efficient but slower SVD algorithm. Such a situation occurs frequently in nearly integrable cases, due to the lack of level repulsion. The diagonalization, on the contrary, gives always correctly the orthogonal eigenfunctions even if eigenvalues are very close.

The implications of this procedure for the study of quantum chaos is of great interest. Figure 2 displays a chaotic situation. Comparing eigenphases in the near integrable (Fig. 1) and chaotic (Fig. 2) situations, we see that in the first case the phases as a function of energy display avoided crossings of straight lines running at different angles while for the second case these lines are prac- tically parallel. This shows that the drift of eigenphases as a function of energy displays presence or absence of level repulsion or spectral rigidity more obviously than the energy levels themselves. This consideration is important in relation with the theory put forward by two of us [START_REF] Leyvraz | [END_REF], which relates Random Matrix Properties of eigenvalues of a quantum system to properties of invari-ance under canonical transformations of the structure of the corresponding classical system (structural invariance). This theory was developed for maps such as the scattering map, the stroboscopic map or the Poincaré map leading to results about their unitary representations, i.e. about eigenphases. To transfer statistical properties of eigenphases of the QPM to energy eigenvalues, it is necessary that the drift of eigenphases as a function of energy be nearly parallel for all phases. Analytic evidence that this must be true for chaotic systems is given in [START_REF] Leyvraz | [END_REF][START_REF] Leyvraz | [END_REF], in agreement with the numerical results shown in Fig. 2. To put this on a more quantitative basis we display on Fig. 3 histograms of the distributions of the velocities of the eigenphases curves. The chaotic case has a narrow distribution (approximately gaussian when viewed in an horizontally expanded scale), while the integrable one shows long tails. The present study is much more convincing in that respect than another one for billiards using directly Bogomolny theory [START_REF] Leyvraz | [END_REF], which is garbled by the non unitarity of the T matrix for finite . The difference seen in Fig. 3 for the near integrable and the chaotic case is remarkable and it is thus tempting to consider the eigenphase velocity distri-bution as another signature of classical chaos in quantum mechanics. While the narrow distribution for chaotic systems is typical, the tails for the integrable systems are not generic: they depend on the details of the partition of phase space by separatrices. In other examples [START_REF] Leyvraz | [END_REF] tails have different shapes.

Summarizing: in this paper we proposed to interpret the multichannel quantum defect theory as a quantum Poincaré map. This was implemented in detail in the approximation that the absolute value of the electron angular momentum is conserved, but it is quite clear that it is true in general. The new interpretation allows for a more stable and efficient way to find solutions to MQDT, particularly for near-degenerate levels. Beyond that, the approximate system has been used to exemplify quantum features of classically chaotic systems. This can be now extended to use MQDT as a paradigm for QPM. Indeed a study of the velocity distribution of eigenphases confirms properties expected from other studies.

This work was partially supported by DGAPA (UNAM) project IN112998 and the CONACYT grant 25192-E

Fig. 1 .Fig. 2 .

 12 Fig. 1. Comparison of Determinant and Eigenphases methods to solve the MQDT problem. Abscissa axis is energy on a scale ν N (E) for the middle N = J value.

Present address: Fachbereich 7, Physik, Universität G. H. Essen, 45117 Essen

Corresponding author; Fax: +33 476 514 544; e-mail: Maurice.Lombardi@ujfgrenoble.fr


Rata Suwantong
email: rata.suwantong@onera.fr

Sylvain Bertrand
email: sylvain.bertrand@onera.fr

Didier Dumur
email: didier.dumur@supelec.fr

Dominique Beauvois
email: dominique.beauvois@supelec.fr

Stability of a Nonlinear Moving Horizon Estimator with Pre-Estimation

In this paper, a Moving Horizon Estimator with pre-estimation (MHE-PE) is proposed for discrete-time nonlinear systems under bounded noise. While the classical Moving Horizon Estimator (MHE) compensates for model errors by estimating the process noise sequence over the horizon via optimization, the MHE-PE does it using an auxiliary estimator. The MHE-PE is shown to require significantly less computation time compared to the MHE, while providing the same order of magnitude of estimation errors. The stability of the estimation errors of the MHE-PE is also proven and an upper bound on its estimation errors is derived. Performances of the MHE-PE is illustrated via a simulation example of pressure estimation in a gas-phase reversible reaction.

I. INTRODUCTION

A Moving Horizon Estimator (MHE) computes an estimate at the current instant by solving an optimization problem based on information from a fixed-number of latest measurements collected over a finite horizon. In this problem, the cost function to be minimized is traditionally described by the norm of the difference between real and predicted measurements over the horizon, a norm of the process noise over the horizon and a norm of the difference between an estimate at the beginning of the horizon and an a priori one. The latter term of the cost function is usually referred to as an "arrival cost" [START_REF] Haseltine | Critical evaluation of extended Kalman filtering and moving-horizon estimation[END_REF][6] [START_REF] Zavala | A fast moving horizon estimation algorithm based on nonlinear programming sensitivity[END_REF]. Once a new measurement is available, the oldest measurement is discarded and the horizon is moved forward. The main advantages of the MHE are that it allows to handle nonlinear systems without linearisation and to incorporate constraints directly during the optimization. The MHE has also been shown to be more robust against poor initialization than the Extended Kalman Filter (EKF) [START_REF] Haseltine | Critical evaluation of extended Kalman filtering and moving-horizon estimation[END_REF], the Unscented Kalman Filter (UKF) and the particle filter [START_REF] Suwantong | Robustness analysis of a moving horizon estimator for space debris tracking during atmospheric reentry[END_REF]. The convergence of the estimation errors of the MHE has been demonstrated in [START_REF] Alessandri | Advances in moving horizon estimation for nonlinear systems[END_REF] provided that a quadratic arrival cost is adopted and its weight matrix is adequately chosen.

However, the computation time of the MHE can be very large due to its high number of optimization parameters which are not only the estimate at the beginning of the horizon but also the process noise sequence over the horizon. To decrease this computation time, fast optimization techniques for the MHE have been proposed in [START_REF] Ferreau | High-speed moving horizon estimation based on automatic code generation[END_REF] [START_REF] Zavala | A fast moving horizon estimation algorithm based on nonlinear programming sensitivity[END_REF]. Another strategy to reduce computation time without changing the optimization method consists in replacing the state equation in the MHE by an auxiliary estimator, called pre-estimating estimator. This pre-estimating estimator allows the MHE-PE to compensate for model errors without searching for the optimal process noise sequence over the horizon via optimization like the MHE. Such strategy has been proposed in [START_REF] Sui | Linear moving horizon estimation with pre-estimating observer[END_REF] for discrete-time linear systems where a deterministic observer is chosen as pre-estimating estimator. It has been shown to be robust against model errors and require smaller computation time than the MHE since the process noise sequence is no longer included in the optimization parameters.

In this paper, an MHE with pre-estimation (MHE-PE) is proposed for discrete-time nonlinear systems under bounded noise. The novelties of this paper compared to [START_REF] Sui | Linear moving horizon estimation with pre-estimating observer[END_REF] are the extension of the method to nonlinear systems and the possibility to use a bounded-error estimator instead of a deterministic observer in the pre-estimation part. The stability of the proposed MHE-PE is also analysed and an upper bound on the estimation errors is derived. Performance of the MHE-PE is illustrated via numerical simulations in terms of accuracy of the estimates and computation time. Note that an attempt to incorporate an observer in the MHE strategy for nonlinear systems under bounded noise has been proposed in [START_REF] Liu | Moving horizon state estimation for nonlinear systems with bounded uncertainties[END_REF]. However, in their strategy, the observer is only used to impose an additional constraint in the MHE and the process noise sequence is still included in the optimization parameters which induces large computation time.

The paper is organized as follows. In section II, some notations, definitions and general assumptions are introduced regarding the pre-estimating estimator and the MHE-PE formulation. In section III, an upper bound on the estimation errors provided by the pre-estimator is derived. In section IV, the stability of the estimation error dynamics of the MHE-PE is studied. In section V, the numerical studies are presented and the paper ends with concluding remarks.

II. PRELIMINARIES

A. Notations and Definitions

v denotes the Euclidean norm of a vector v. A function f (x) is said to be locally Lipschitz with respect to its argument x if there exists a positive constant

L x f such that f (x ′ ) -f (x ′′ ) ≤ L x f x ′ -x ′′
, for all x ′ and x ′′ in a given region of x and L x f is the associated Lipschitz constant. A continuous function φ : [0, a) → [0, ∞) is said to belong to class K if it is strictly increasing and satisfies φ (0) = 0. A function β (r, s) is said to be a class KL-function if, for each fixed s, β (r, s) is a class K function with respect to r and, for each fixed r, β (r, s) is decreasing with respect to s and β (r, s) → 0 as s → ∞.

B. System, Estimator and Associated Assumptions

Consider a nonlinear discrete-time system with additive noise of the form

x t+1 = f (x t ) + w t y t = h(x t ) + v t (1)
where

x t ∈ X ⊂ R n is the state, y t ∈ Y ⊂ R m is the measure- ment, w t ∈ W ⊂ R n is the process noise, v t ∈ V ⊂ R m
is the measurement noise and t ∈ N is the time index. The problem addressed in this paper consists in the estimation of the state vector x t from the measurements y k (k ≤ t). The system (1) will be referred to as the real system. The following assumptions are assumed to be satisfied: (A1) X is a convex compact set, W and V are compact sets with 0 ∈ W and 0 ∈ V. Define

r w max w∈W w , r v max v∈V v (2)
(A2) The initial state x 0 is such that, for any possible sequence of process noise {w t }, the system trajectory {x t } lies in the convex compact set X, ∀t (A3) f and h are C 2 functions with respect to x on X.

Consequently, f and h are also locally Lipschitz. Define their Lipschitz constants as L x f and L x h respectively Define the nominal system as the "noise-free" part of the real system and denote n x t ∈ X the nominal state and n y t ∈ R m the noise free output. The nominal system is defined as:

n x t+1 = f ( n x t ) n y t = h( n x t ) (3)
Let us define an estimator map g(•, •): Z × R m → Z, Z ⊂ R n a convex compact set, by:

z t+1 = g(z t , γ t ) (4)
where z ∈ Z is the estimate given by g and γ t ∈ R m is the input of g. When y t is used as input (γ t = y t ), i.e. when the estimator is applied to the real system (1), the state z t will be denoted as z t . In the same way, when n y t is used as input (γ t = n y t ), i.e. when the estimator is applied to the nominal system (3), the state z t will be denoted as n z t . The estimator g is assumed to satisfy: (A4) g is locally Lipschitz with respect to its arguments, with the associated Lipschitz constants L z g and L y g (A5) ∀z 0 ∈ Z, ∀x 0 ∈ X, there exists a class K function ψ such that n z t -n x t 2 ≤ ψ( n z 0 -n x 0 2 ), ∀t, i.e. the estimation error in a noise-free case is equal to zero if g is initialized at the real state, or (A5a) g provides bounded estimation errors under bounded noise An example of g that verifies (A5) is a discrete-time deterministic observer, i.e. g is designed such that lim t→∞ n z t = n x t , ∀ n x 0 ∈ X and ∀ n z 0 ∈ Z. In this case, there exists a KLfunction β such that

n z t -n x t ≤ β ( n z 0 -n x 0 ,t) (5)
In section III, an upper bound on the estimation errors provided by g satisfying (A4) and (A5) will be derived.

C. Formulation of the MHE-PE

Assume that the state vector x t has to be estimated at instant t ≥ N using the latest N + 1 measurements collected within the "sliding horizon" [t -N,t]. Let us denote z t-N,t an estimate of x t-N computed at instant t, y t t-N = y t-N , y t-N+1 , . . . , y t the measurement collection over the horizon and zo t-N,t , an a priori value of z t-N,t .

The MHE-PE is formulated as follows:

ẑo t-N,t = arg min z t-N,t∈Z J t (z t-N,t , zo t-N,t , y t t-N ) (6a)
J t = µ z t-N,t -zo t-N,t 2 + N ∑ i=0 y t-N+i -h(z t-N+i,t ) 2 (6b) z t-N+i+1,t = g(z t-N+i,t , y t-N+i ), ∀i ∈ [0, N -1] ( 6c
)
where µ is a positive scalar representing the confidence in the a priori value zo t-N,t with respect to the observation model. The estimate ẑ• t,t of x t at time t provided by the MHE-PE is computed using (6c) for i = 1, . . . , N from ẑo t-N,t , and zo

t-N,t is determined from zo t-N,t f (ẑ • t-N-1,t-1
). Note that in the MHE-PE the evolution of the state at each instant over the horizon is subject to the equation of the estimator g instead of the state equation as in the classical MHE. In fact, the MHE-PE locally implements the estimator g which is re-initialized at each instant at the beginning of the horizon by solving the optimization problem [START_REF] Liu | Moving horizon state estimation for nonlinear systems with bounded uncertainties[END_REF].

III. UPPER BOUND ON THE ESTIMATION ERRORS OF AN ESTIMATOR VERIFYING (A4) AND (A5)

In this section, an upper bound on the estimation errors provided by an estimator g verifying (A4) and (A5) is derived in case of bounded noise for system [START_REF] Alessandri | Moving horizon state estimation for nonlinear discrete-time systems: New stability results and approximation schemes[END_REF]. The work is inspired by the method proposed in [START_REF] Liu | Moving horizon state estimation for nonlinear systems with bounded uncertainties[END_REF] where an upper bound on the estimation errors of a nonlinear continuoustime deterministic observer is derived. Consider the real system starting from the initial state x t evolving as in (1) and the nominal system starting from the same initial condition n x t = x t evolving as in [START_REF] Ferreau | High-speed moving horizon estimation based on automatic code generation[END_REF]. ∀i ∈ N * , denote z t+i the estimate provided by the estimator g receiving real (noisy) measurements y t+i t starting from the initial estimate z t . Denote n z t+i the estimate provided by g receiving noise-free measurements n y t+i t and starting from the same initial estimate n z t = z t .

We would like to find an upper bound on the square norm of the estimation error z t+ix t+i 2 at time t + i. Thanks to the triangle inequality, we remark that

z t+i -x t+i 2 ≤ 3 z t+i -n z t+i 2 + 3 n z t+i -n x t+i 2 +3 n x t+i -x t+i 2 (7)
The second term of the r.h.s. of [START_REF] Rao | Constrained state estimation for nonlinear discrete-time systems: stability and moving horizon approximations[END_REF] verifies

n z t+i -n x t+i 2 ≤ ψ( n z t -n x t 2 ) = ψ( z t -x t 2
). For the third term of the r.h.s. of (7), using (1), ( 2), [START_REF] Ferreau | High-speed moving horizon estimation based on automatic code generation[END_REF], assumption (A3) and the triangle inequality, the following propositions are derived by recurrence, ∀i ∈ N * . Proposition 3.1: The square norm of the difference at t +i between the real state x t+i of (1) starting from x t , and the nominal state n x t+i of (3) starting from n x t = x t is bounded

as x t+i -n x t+i 2 ≤ ( i-1 ∑ j=0 {2(L x f ) 2 } j )2r 2 w Proposition 3.2:
The square norm of the difference at t +i between the real measurement y t+i of (1) starting from x t and the nominal measurement n y t+i of the nominal state of (3) starting from n x t = x t is bounded as

y t+i -n y t+i 2 ≤ 2(L x h ) 2 ( i-1 ∑ j=0 {2(L x f ) 2 } j )2r 2 w + 2r 2 v
For the first term of the r.h.s. of [START_REF] Rao | Constrained state estimation for nonlinear discrete-time systems: stability and moving horizon approximations[END_REF], by using ( 1), ( 2), (3), assumptions (A3) and (A4) along with the triangle inequality and by defining α

t+ j = j-1 ∑ l=0 {2(L x f ) 2 } l , ∀ j ∈ N + and α t = 1,
the following proposition is derived. Proposition 3.3: Consider system (1) starting from x t and system (3) starting from n x t = x t . ∀i ∈ N * , consider at t + i the estimate z t+i of the real state x t+i given by g using y t+i t initialized at z t . ∀i ∈ N * , consider at t + i the estimate n z t+i of the nominal state n x t+i given by g using n y t+i t initialized at n z t = z t . We have:

z t+i -n z t+i 2 ≤ 2(L y g ) 2 × . . . 4(L x h ) 2 r 2 w i-1 ∑ j=0 {2(L z g ) 2 } i-1-j α t+ j + 2r 2 v i-1 ∑ j=0 {2(L z g ) 2 } j Proof: Let us start by considering when i = 1. Since n z t = z t , it yields z t+1 -n z t+1 2 = g(z t , y t ) -g( n z t , n y t ) 2 ≤ 2(L y g ) 2 y t -n y t 2
Using proposition 3.2, the proposition 3.3 holds for i = 1. Now suppose that the proposition holds for i N * , let us prove that it is also the case for i + 1.

z t+i+1 -n z t+i+1 2 = g(z t+i , y t+i ) -g( n z t+i , n y t+i ) 2 ≤ 2(L z g ) 2 z t+i -n z t+i 2 + 2(L y g ) 2 y t -n y t 2 ≤ 4(L y g ) 2 r 2 v i ∑ j=0 {2(L z g ) 2 } j + 2(L x h r w ) 2 i ∑ j=0 {2(L z g ) 2 } i-j α t+ j
Theorem 3.4: The square norm of the difference between the estimate z t+i given by an estimator g verifying (A4) and (A5) receiving measurements y t+i t initialized at time t by z t and the state x t+i of the real system, ∀i ∈ N * , is bounded as:

z t+i -x t+i 2 ≤ 3ψ( z t -x t 2 ) + c w,i r 2 w + c v,i r 2 v where c w,i = 6 i-1 ∑ j=0 {2(L x f ) 2 } j + 24(L y g L x h ) 2 α t+ j c v,i = 12(L y g ) 2 i-1 ∑ j=0 {2(L z g ) 2 } j (8)
Proof: The proof is straightforward by combining (8) and proposition 3.3 in [START_REF] Rao | Constrained state estimation for nonlinear discrete-time systems: stability and moving horizon approximations[END_REF], recalling that n z t = z t and n x t = x t . Now that an upper bound on the estimation errors provided by an estimator verifying (A4) and (A5), applied to a system under bounded noise, has been determined, this result will be used to analyse the error dynamics of the MHE-PE when such pre-estimating estimator is chosen. The following study is also verified for g verifying (A5a) instead of (A5) with different values of associated constants.

IV. ERROR DYNAMICS OF THE MHE-PE

To determine the dynamics of the estimation error of the MHE-PE, upper and lower bounds on the optimal cost, noted by J o t J t (ẑ o t-N,t , zo t-N,t , y t t-N ) where ẑo t-N,t is the optimal solution of the problem (6), are calculated. The approach is inspired by [START_REF] Alessandri | Moving horizon state estimation for nonlinear discrete-time systems: New stability results and approximation schemes[END_REF] where the error dynamics of the MHE (without pre-estimation) is derived.

A. Upper Bound on the Optimal Cost

Denote x t-N , the state of (1) at instant t -N. By optimality of ẑo t-N,t , we get J o t ≤ J t (x t-N , zo t-N,t , y t t-N ), i.e. 9)

J o t ≤ µ x t-N -zo t-N,t 2 + N ∑ i=0 y t-N+i -h(z t-N+i,t ) 2 (
with z t-N+i+1,t = g(z t-N+i,t , y t-N+i ), ∀i ∈ [0, N -1] z t-N,t = x t-N
Let us find an upper bound on the second term of the r.h.s. of [START_REF] Suwantong | Robustness analysis of a moving horizon estimator for space debris tracking during atmospheric reentry[END_REF]. Using the triangle inequality, we have

y t-N+i -h(z t-N+i,t ) 2 ≤ 2 y t-N+i -h(x t-N+i ) 2 + 2 h(x t-N+i ) -h(z t-N+i,t ) 2 ≤ 2r 2 v + 2(L x h ) 2 x t-N+i -z t-N+i,t 2 Therefore, N ∑ i=0 y t-N+i -h(z t-N+i,t ) 2 ≤ 2 N ∑ i=0 r 2 v + 2(L x h ) 2 N ∑ i=0 x t-N+i -z t-N+i,t 2 (10)
To bound the second term of the r.h.s. of (10), we use theorem 3.4 recalling that ψ( z t-N,t -

x t-N 2 ) = 0 since z t-N,t = x t-N . N ∑ i=0 y t-N+i -h(z t-N+i,t ) 2 ≤ l w,N r 2 w + l v,N r 2 v (11a) l w,N = 2(L x h ) 2 N ∑ i=1 c w,i (11b) l v,N = 2(L x h ) 2 N ∑ i=1 c v,i + 2(N + 1) (11c)
where c w,i and c v,i are defined in [START_REF] Sui | Linear moving horizon estimation with pre-estimating observer[END_REF].

Hence, an upper bound on the optimal cost J • t is given by1

J o t ≤ µ x t-N -zo t-N,t 2 + c 2 N ( 12
)
where

c 2 N = l w,N r 2 w + l v,N r 2 v ( 13
)
Now that we have an upper bound on the optimal cost, let us pursue with the computation of a lower bound.

B. Lower Bound on the Optimal Cost

We start by determining a lower bound on any common cost J t in (6), so z t-N,t will be replaced by z t-N to refer to any value of the estimate at the beginning of the horizon in general. After that, a lower bound on the optimal cost J • t will be derived by replacing z t-N by ẑ• t-N,t . Let us define first the observation maps of the real system in (1) and of the pre-estimating estimator g initialized at t -N by z t-N receiving the real measurements y t t-N respectively by

F(x t-N )      h(x t-N ) h( f (x t-N ))
. . .

h( f N (x t-N ))      (14)
and

G(z t-N , y t-1 t-n )      h(z t-N ) h • g(z t-N , y t-N ) . . . h(g N (z t-N , y t-1 t-N ))      (15)
where for i ≥ 1

g i (z t-N , y t-N+i-1 t-N ) g(g . . . g i times (z t-N , y t-N ), . . . , y t-N+i-1 )
Therefore, the second term of the r.h.s. of (6b) becomes

N ∑ i=0 y t-N+i -h(z t-N+i ) 2 = y t t-N -G(z t-N , y t-1 t-N ) 2 . Denote n y t-1
t-N the noise free measurements provided by system (3) initialized at time t -N with n x t-N = x t-N . Thanks to the triangle inequality, we deduce:

y t t-N -G(z t-N , y t-1 t-N ) 2 ≥ 1 4 G(z t-N , n y t-1 t-N ) -G(x t-N , n y t-1 t-N ) 2 -y t t-N -F(x t-N ) 2 -F(x t-N ) -G(x t-N , n y t-1 t-N ) 2 -G(z t-N , y t-1 t-N ) -G(z t-N , n y t-1 t-N ) 2 (16)
To calculate a lower bound on the first term of the r.h.s. of ( 16), consider the system of the estimator g in ( 4) as a nonlinear system as defined in [START_REF] Hanba | Further results on the uniform observability of discretetime nonlinear systems[END_REF] where measurements are considered as the input of the system. It is shown in [START_REF] Hanba | Further results on the uniform observability of discretetime nonlinear systems[END_REF] that a lower bound on the term of interest exists if the following assumptions are satisfied: (A6) g is K-uniformly observable on Z with respect to all admissible measurements, i.e. ∃N > 0,

∀(z ′ , z ′′ ) ∈ Z 2 , ∀y ∈ Y, there exists a K-function φ (•) such that φ ( z ′ -z ′′ 2 ) ≤ G(z ′ , y t-1 t-N ) -G(z ′′ , y t-1 t-N ) 2 (17) (A7)
The observation map G(•, •) has a finite sensitivity to the estimate, i.e. the K-function φ (•) in (17) satisfies:

δ = inf (z 1 ,z 2 )∈Z 2 ,z 1 =z 2 φ ( z 1 -z 2 2 ) z 1 -z 2 2 > 0 (18)
Thanks to ( 17) and ( 18), we obtain a lower bound on the first term of the r.h.s. of ( 16)

G(x t-N , n y t-1 t-N ) -G(z t-N , n y t-1 t-N ) 2 ≥ δ z t-N -x t-N 2 (19)
Consider now the second term of the r.h.s. of ( 16), similarly to [START_REF] Alessandri | Moving horizon state estimation for nonlinear discrete-time systems: New stability results and approximation schemes[END_REF], we have

y t t-N -F(x t-N ) 2 ≤ c 2 Ale,N (20) c Ale,N ∆ w √ Nr w + √ N + 1r v + k 2 N(N + 1)(2N + 1) 6 r 2 w
where ∆ w and k characterize the model sensitivity to state noise, see [START_REF] Alessandri | Moving horizon state estimation for nonlinear discrete-time systems: New stability results and approximation schemes[END_REF]. The third term of the r.h.s. of ( 16) is equal to zero since g satisfies (A5) 2 Now, consider the forth term of the r.h.s of ( 16)

G(z t-N , y t-1 t-N ) -G(z t-N , n y t-1 t-N ) 2 ≤ (L x h ) 2 N ∑ i=1 G i ( 21
)
where

G i g i (z t-N , y t-N+i-1 t-N ) -g i (z t-N , n y t-N+i-1 t-N ) 2 .
By recurrence, it can be shown that

G i ≤ 2(L z g ) 2 G i-1 for i≥1 +2(L y g ) 2 y t-N+i-1 -n y t-N+i-1 2 (22)
To calculate an upper bound on G i , consider the following proposition: Proposition 4.1: The square norm of the difference at time t -N + i between the estimate provided by the estimator g receiving the collection of real measurements y t-N+i-1 t-N and initialized at t -N with z t-N and the estimate provided by the estimator g initialized at t -N with n z t-N = z t-N but receiving the collection of the noise-free measurements n y t-N+i-1 t-N associated to the nominal system (3) initialized at t -N with n x t-N = x t-N , is bounded as

G i ≤ 2 L y g 2 × . . .       2r 2 v i-1 ∑ j=0 2 L z g 2 j for i≥1 + 2 (L x h ) 2 2r 2 w i-2 ∑ j=0 2(L x f ) 2 + 2(L z g ) 2 j for i≥2      
Proof: It is easy to verify that the proposition 4.1 holds for i = 1 and i = 2 using (22) and proposition 3.2, where t -N is considered as the first instant of the window. Now suppose that the proposition 4.1 holds for i and prove that it is also the case for i + 1. Equation ( 22) and proposition 4.1 give

G i+1 ≤ 2(L y g ) 2 × 2r 2 v i ∑ j=1 2(L z g ) 2 j + 4(L x h L z g ) 2 2r 2 w i-2 ∑ j=0 2(L x f ) 2 + 2(L z g ) 2 j +2(L y g ) 2 2(L x h ) 2 i-1 ∑ j=0 2(L x f ) 2 j 2r 2 w + 2(L y g ) 2 2r 2 v G i+1 ≤ 2(L y g ) 2 2r 2 v i ∑ j=0 2(L z g ) 2 j + 2(L x h ) 2 2(L y g ) 2 × 2(L z g ) 2 i-2 ∑ j=0 2(L x f ) 2 + 2(L z g ) 2 j + i-1 ∑ j=0 2(L x f ) 2 j 2r 2 w
Using lemma 6.1 in appendix, proposition 4.1 holds for i+1, which completes the proof by recurrence. Reconsider (21). Proposition 4.1 leads to

G(z t-N , y t-1 t-N ) -G(x t-N , n y t-1 t-N ) 2 ≤ (L x h ) 2 λ 2 N ( 23
)
where 19), (20), and (23) in ( 16), we finally obtain a lower bound on the second term of the r.h.s. of (6b)3

λ 2 N = 2(L y g ) 2 × {2r v 2 N ∑ i=1 i-1 ∑ j=0 2(L z g ) 2 j (24) +2(L x h ) 2 2r 2 w N ∑ i=2 i-2 ∑ j=0 2(L x f ) 2 + 2(L z g ) 2 j } Replacing (
y t t-N -G(z t-N , y t t-N ) 2 ≥ 1 4 δ z t-N -x t-N 2 -(L x h ) 2 λ 2 N -c 2 Ale,N (25)
Let us continue with a lower bound on the first term of (6b) with the general notation z t-N introduced at the beginning of the section:

µ z t-N -zo t-N 2 .
First of all, we remark that 25) to (26) and replace the general notation z t-N by the solution ẑ• t-N,t . By defining e • t-N ẑ• t-N,tx t-N , a lower bound on the optimal cost J • t is derived as follows:

z t-N -z• t-N,t 2 ≥ 1 2 x t-N -z t-N 2 -x t-N -z• t-N,t 2 (26) Combine (
J • t ≥ 1 2 µ + 1 4 δ e • t-N 2 -µ x t-N -z• t-N 2 -(L x h ) 2 λ 2 N -c 2 Ale,N (27)
Now that a lower bound on the optimal cost J • t has been calculated, it will be combined with the upper bound on J • t computed in the previous section to analyse the dynamics of the estimation error of the MHE-PE.

C. Dynamics of the Estimation Errors

Theorem 4.2: Consider a discrete-time nonlinear system verifying assumptions (A1)-(A3). If the pre-estimating estimator of the MHE-PE verifies (A4)-(A7), then the square norm of the estimation error of the MHE-PE is bounded as e

• t-N 2 ≤ ζ t-N
where {ζ t } is a sequence generated by

ζ 0 = β 0 (28) ζ t = α N ζ t-1 + β N , ∀t ∈ N * with α N = 8µ(L x f ) 2 µ + δ 2 β N = 2 µ + δ 2 (4µr 2 w + C 2 N ) β 0 = 2 µ + δ 2 (2µd 2 x + C 2 N )
where

C 2 N (L x h ) 2 λ 2 N +c 2 Ale,N +c 2 N 4 and d x max (x,x ′ )∈X 2 x -x ′ . Moreover, if µ is selected s.t. 8µ(L x f ) 2 /(µ + δ 2 ) < 1 (29)
The sequence {ζ t } has the following properties: (a) {ζ t } converges exponentially to the asymptotic value e

• ∞ (µ) β N /(1 -α N ) (b) if ζ t > e • ∞ (µ), then ζ t < ζ t-1
Proof: Combine the upper bound on the optimal cost J • t in (12) with its lower bound in (27) to get:

1 2 µ + 1 2 δ e • t-N 2 ≤ 2µ x t-N -z• t-N,t 2 + C 2 N If z• t-N,t f (ẑ • t-N-1,t-1 ), therefore x t-N -z• t-N,t 2 = f (x t-N-1 ) + w t-N-1 -f (ẑ • t-N-1,t-1 ) 2 ≤ 2(L x f ) 2 e • t-N-1 2 + 2r 2 w ( 30
)
Thanks to (30), we have

1 2 µ + 1 2 δ e • t-N 2 ≤ 4µ(L x f ) 2 e • t-N-1 2 + 4µr 2 w + C 2 N
(31) Applying condition (29) to (31) completes the proof. Thanks to theorem 4.2, the convergence of the estimates provided by MHE-PE can be guaranteed by choosing an appropriate µ satisfying condition (29). This is easy to do for any value of L x f , once δ is calculated using (18) assuming that the reachable values of the estimates can be considered as equal to those of the real system.

In the case of a noise-free system (r w = r v = 0), the asymptotic convergence of the MHE-PE is guaranteed by corollary 4.

V. NUMERICAL EXAMPLE

The performances of the MHE-PE and of a classical MHE whose formulation can be found in [START_REF] Rao | Constrained state estimation for nonlinear discrete-time systems: stability and moving horizon approximations[END_REF] are compared through a pressure estimation problem of a gas-phase, reversible reaction as defined in [START_REF] Haseltine | Critical evaluation of extended Kalman filtering and moving-horizon estimation[END_REF]. The state is defined as the partial pressures of the system x t = x 1,t x 2,t T . Define kg = 0.16 (atm • s) -1 and the sampling time T s = 0.1 s. The dynamics of the real system is:

x t+1 = f (x t ) + w t =     x 1,t 2 kg T s x 1,t + 1 x 2,t + kg T s x 2 1,t 2 kg T s x 1,t + 1     + w t y t = 1 1 x t + v t , x 0 = 5 1 T (32)
The noise w t and v t are supposed to be independent random variables, uniformly distributed in the intervals process noise estimates given by the MHE are imposed to be in the interval [-0.3, 0.3] 2 . A deterministic observer for the noise-free system associated to system (32) is

z t+1 = g(z t , y t ) = f (z t ) + L(y t -1 1 z t ) (33)
where L = 0.0026 0.7046 T has been computed using proposition 3 in [START_REF] Zhang | A note on observers for discrete-time Lipschitz nonlinear systems[END_REF]. The values of L x f = 1, δ = 1, µ = 5•10 -4 are chosen. This value of µ gives α = 0.008. N = 5 is chosen for both MHE and MHE-PE and N = 50 is chosen in addition for MHE-PE. Results for MHE with N = 50 are not included due to its too large computation time. All available measurements are used for t < N for every MHE.

The accuracy of the estimates are studied in terms of Root Mean Square Error (RMSE) and Asymptotic Root Mean Square Error (ARMSE) as defined in [START_REF] Alessandri | Moving horizon state estimation for nonlinear discrete-time systems: New stability results and approximation schemes[END_REF]. The evaluation batch size for ARMSE calculation is set to 50. The RMSEs provided by the MHE, the MHE-PE and the nonlinear deterministic observer defined in (33) are shown in figure 1. To finish, the ARMSE and the mean computation times per iteration are presented in table 1 It can be observed from figure 1 that the deterministic observer gives high RMSE and converges slowly and that a too small horizon can lead MHE-PE to have large estimation errors. However, increasing the horizon size does not affect much on MHE-PE's computation time since the number of the optimization parameters does not change. Moreover, when N is sufficiently high, MHE-PE gives estimation errors closed to those of MHE but uses much smaller computation time. According to table 1, MHE-PE can be used in real-time applications for this example since its computation time per iteration is lower than the sampling period.

To conclude, MHE-PE leads to a good trade-off between accuracy and small computation time making it an interesting alternative MHE strategy for discrete-time nonlinear systems under bounded noise, for which an estimator verifying (A4)-(A7) can be designed. VI. CONCLUSIONS A Moving Horizon Estimator with Pre-Estimation (MHE-PE) has been proposed for a nonlinear discrete-time system under bounded noise. The stability of the estimation error of the MHE-PE has been demonstrated and its upper bound has been derived. It has been shown via numerical simulations that the performance of the proposed MHE-PE offers a good trade-off between accuracy of the estimates and small computation time. Therefore, the MHE-PE can be considered as an alternative estimator to save computation time which opens the possibilities of real-time applications of Moving Horizon strategies.

i ≤ b a + b ⇒ b i (a + b) i-1 ≤ b ⇒ a + b i (a + b) i-1 ≤ a + b ⇒ a(a + b) i-1 + b i ≤ (a + b) i

3 . 4 . 3 :

 343 Corollary Suppose that assumptions (A1)-(A7) hold. If r w = r v = 0, e • t-N 2 ≤ α t-N N β 0 , ∀t ≥ N. Moreover, if µ satisfies (29), lim

  [-0.06, 0.06] × [-0.3, 0.3] and [-0.3, 0.3], respectively. 100 Monte Carlo simulations of the system have been simulated for t ∈ [0, 10] s using MATLAB on a standard PC. The a priori value z• 0 = 2 4.5 T and the constraint Z = [0, 5] 2 are imposed to every MHE and MHE-PE for every run. The

Fig. 1 :

 1 Fig. 1: (Left) Example of the real system state and measurement from one Monte Carlo run. (Right) RMSE given by each tested estimator

APPENDIX Lemma 6 . 1 :

 61 If a > 0 and b > 0, for i ≥ 2 we have a It is evident that the lemma holds for i = 2. Now suppose that it holds for i, it holds fort = i + 1 ⇔ a i-2 ∑ j=0 (a + b) j + a(a + b) i-1 + i-1 ∑ j=0 b j + b i ≤ i-1 ∑ j=0 (a + b) j + (a + b) iThanks to (34), there is only a(a + b) i-1 + b i ≤ (a + b) i left to be verified. Since a > 0 and b > 0, we have b a + b

Table 1 :

 1 . ARMSE and mean computation time per iteration of MHE-PE for N=5 and N=50 and MHE for N=5.

	MHE	ARMSE	N=5 mean time/	ARMSE	N=50 mean time/
			iteration (s)		iteration (s)
	MHE-PE	1.4971	0.0526	0.3997	0.0822
	MHE	0.2231	0.4831	×	×

R. Suwantong and S. Bertrand are with the Department of System Design and Performance Evaluation, ONERA-The French Aerospace-

Note that for g satisfying (A5a), only the value of the constant c N will change.

This term is equal to a constant if g verifies (A5a) instead.

For g satisfying (A5a), another constant will be added on the r.h.s.

C N takes another constant value for g verifying (A5a).

ACKNOWLEDGMENT

The authors would like to express our sincerest thanks to Damien Jacquemart of ONERA, without whom this study would not have been completed, for the proof of lemma 6.1.


Benoit Berard

The "South-Dominica" archaeological mission: the Soufrière site

Keywords: Archaeology, Caribbean, Dominica, Soufrière, Saladoid Archéologie, Caraïbe, Dominique, Soufrière, Saladoïde

The "South-Dominica" archaeological mission of the French ministry of foreign affairs has started work in 2005. Designed to enlighten the archaeological terra incognita dominicana, this research has essentially been dedicated to the early ceramic occupation of the island. Our work has been based on the analysis of the few collections already existing, on systematic survey campaigns and on an important test pit program. This lecture will be concentrating on the presentation of the Soufrière site excavation. The Soufriere site, found in the seventies, is composed of an important early cedrosan saladoid occupation covered by volcanic deposits. The data obtained during this excavation, associated with other results of the "South Dominica" archaeological mission, incites us to think about a revision of the territories associated to the complexes in the Lesser Antilles.

RESUME

La mission archéologique "Sud-Dominique" du ministère français des affaires étrangères a débuté ses travaux en 2005. Destinées à en finir avec la terra incognita dominicana 1 Maître de conférence, Université des Antilles et de la Guyane, EA 929 "Archéologie Industrielle, Histoire et Patrimoine de la Caraïbe". archéologique, ces recherches sont essentiellement consacrées à l'étude du premier peuplement agro-céramique de l'île. Le travail de la mission archéologique repose sur l'étude de séries anciennes, des campagnes de prospections systématiques et un important programme de sondages.

Notre présentation se concentrera sur la fouille que nous avons menée du site de Soufrière dans le sud de la Dominique. Ce gisement identifié dans les années 1970 correspond à une importante occupation saladoïde cedrosane ancienne qui a été recouverte par des dépôts volcaniques. Les données obtenues lors de cette fouille associées à l'ensemble des résultats obtenus en 3 années par la mission archéologique Sud-Dominique nous incitent à nous interroger sur la nécessité d'une révision des territoires associés à la notion de complexe dans les Petites Antilles.

RESUMEN

La misión arqueológica " Sud-Dominique " del ministerio francés de los Asuntos Exteriores empezó sus trabajos en 2005. Destinando a acabarlo con la tierra incógnita dominicana arqueológica, estas búsquedas son esencialmente dedicadas al estudio de la primera población agro-cerámica de la isla. El trabajo de la misión arqueológica reposa en el estudio de series antiguas, campañas de prospecciones sistemáticas y un importante programa de sondeos.

Nuestra presentación se concentrará sobre la excavación del sitio de Soufrière al sur de Dominique. Este yacimiento identificado en los años 1970 corresponde a una ocupación importante saladoïda cedrosana antigua que ha sido recubierta por depósitos volcánicos.

Los datos obtenidos en el momento de esta excavación asociadas con conjunto de los resultados obtenidos en 3 años por la misión arqueológica Sud-Dominique nos incitan a interrogarnos sobre la necesidad de una revisión de los territorios asociados con la noción de complejo en las Pequeñas Antillas.

Introduction

The pre-Columbian occupation of Dominica is one of the poorly documented in the Lesser Antilles. Before the beginning of our work on the island only seven papers had been published concerning that subject, and they are now all more than 30 years old [START_REF] Honychurch | Carib to Creole: A History of Contact and Culture Exchange[END_REF]. Walter Fewkes [START_REF] Fewkes | A Prehistoric Island Culture Area of America. A Report on Prehistoric Objects from the West Indies in the Museum of the American Indian (Heye Foundation) in 1914[END_REF] who staid few hours in the island and only took a look at the stone artifacts store at the Roseau Public Library, published the first one. After that, Marschall McKusick did the first test pits in the island, on the site of Vieille Case [START_REF] Mckusick | Distribution of Ceramic Styles in the Lesser Antilles, West Indies[END_REF].

But the most influencal effect on the archaeology of Dominica had a paper presented by Clifford Evans during the second congress for the study of the pre-Columbian cultures of the Lesser Antilles. In that paper, entitled "The lack of archaeology in Dominica", he explained that the poor quantity of archaeological sites he had been able to discover was a precise mirror of pre-Columbian reality. This was a result, so Evans explained, of the ecological limitations of Dominica which did not allow for the development of an important population [START_REF] Evans | The Lack of Archeology on Dominica[END_REF]. This paper, even if a few researchers like for example Robert Myers had tried to prove that Evans' statement was incorrect [START_REF] Myers | Ethnohistorical vs.ecological considerations : the case of Dominica's Amerindians[END_REF], was for some time the burial of Dominican pre-Columbian archaeology.

We have to be grateful to Henry Petitjean Roget to have been since then the only archaeologist to have proven in the field the poverty of Evans' assessments. Petitjean Roget went two times to Dominica, in 1976 and[START_REF] Myers | Ethnohistorical vs.ecological considerations : the case of Dominica's Amerindians[END_REF], and did two important things: a general survey of the island, finding 24 new sites, and a small excavation of the early saladoid site of Soufrière [START_REF] Roget | Reconnaissance archéologique à l'île de la Dominique (West Indies)[END_REF]. During his work he found a few other early saladoid sites in the south of the island. These findings are at the origin of the research program of the South Dominica archaeological mission of the French ministry of foreign affairs.

The South Dominica archaeological mission.

The South Dominica archaeological mission, funded essentially by the French ministry of foreign affairs and conducted by myself, consists of researchers from: l'Université des Antilles et de la Guyane, le Ministère français de la culture: Jean-Pierre Giraud, the University of West Indies : Lennox Honychurch, the University of Vermont: James Petersen and Florida University : Josh Toney.

The origin of this program is linked to important research conducted by a part of this team between 1996 and 2003, concerning the early saladoid occupation of Martinique [START_REF] Bérard | Les premières occupations agricoles de l'Arc Antillais, Migrations et insularité[END_REF]. All the sites associated to that chronological phase are located in the northeastern part of Martinique, opposite the south coast of Dominica, which is where H. Petitjean Roget found contemporary sites during is survey. Our main objective was to develop a research program studying those south shore sites in Dominica in a similar way to those recently done in Martinique, and also similar to recent work carried out in other Lesser Antillean islands to the north, especially Montserrat.

Our basic idea was that the comparison of archaeological results obtained in Dominica with those from Martinique and Montserrat, among other islands, should be able to provide us with a basis for discussion about the nature of social and economical mechanisms underlying the origin of the agro-ceramic pioneers in the Caribbean, thus helping us to identify the internal structure of the cultural pattern related to the migration.

On a theoretical level, we hope that our results will be useful to review the concept of archaeological "complex," generally associated, following Irvin Rouse in the Lesser Antilles with single islands [START_REF] Rouse | The Tainos: Rise and Decline of the People Who Greeted Colombus[END_REF], since it has never been fully validated. In fact, our hypothesis was that the term 'complex' may better be applicable to terrestrial and maritime spaces including nearby islands, in this case the North of Martinique and the South of Dominica and the channel between them.

We tried to test this hypothesis during our three years research program in Dominica. We hope that our work will enable us to determine if the sea provided a link or a frontier for the early Amerindian pioneers. Indeed, we strongly suspect that modern perceptions of the Lesser Antilles, dominated by important inter-island divisions that came into being with colonization and, more recently, decolonization probably have little or no relations to how the pre-Columbian people perceived their world. The idea is to leave the terrestrial island approach to develop an archipelagic maritime one following the idea first developed by Irvin Rouse and David Watters [START_REF] Watters | Environmental Diversity and Maritime Adaptations in the Caribbean Area[END_REF].

Our three years of research in Dominica had included a review of all the collections already existing linked with early ceramic sites, a systematic survey of the island, which enabled us to increase by 40% the number of pre-Columbian sites known, the digging of tests pits in four early ceramic sites and a larger excavation of the Soufrière site.

It is the preliminary results of that excavation, we are going to present you now.

The Soufrière site.

Localization

The site of Soufrière is found in the south of the west coast of Dominica. It is located 10 m up the actual river level, on the south side of the sulphur spring valley, near to its mouth. The valley's stretch is broader at the east of the site where it constitutes a circus limited by the mornes: Acouma, Plat Pays and Patates. The land inside that circus, made up by young volcanic soils, seems to be perfectly suitable for agriculture [START_REF] Lang | Soils and land use surveys[END_REF]. The water of the sulphur spring river is not drinkable, but a spring water source is available near to the site on the north side of the valley.

History of the research

Nowadays, the early saladoid site of Soufriere is located just beneath the actual village of Soufrière. It has been discovered in 1976 during the preparatory works linked to an estate project by Carl Winston, a Bulldozer driver. Following the exceptional findings of Mr.

Winston, H. Petitjean Roget was contacted and he came to Soufrière to conduct the excavation of a 3x1m test pit. He was able to discover an early saladoid layer, well preserved under a pumice level. The collection linked to that excavation is nowadays stored at the Roseau Public Library and partially exhibited in the Roseau Historical Museum.

The field works

The south Dominica archaeological mission worked in Soufrière in 2005 and 2006. We have excavated in four different loci in the village.

In Locus 1, the place where Henry Petitjean Roget had worked in 1976, we dug two new 2x1 test pits and a trench in 2005. Locus two is near to the river level, where we did three test pits with a backhoe. Locus 3 is located in a lot just beneath locus 1; there, we conducted a 16-m2 excavation. Locus 4 did not correspond to a controlled archaeological excavation. In that lot, located around 50 m in the east of the loci 1 and 3, construction work for the building of a house was in progress in 2006. We were only able to screen the sediment coming from the excavation of the construction site.

Stratigraphy

We faced a number of problems during the excavation of the Soufrière site: First, it was only possible for us to work on the unbuilt lots; second, the paleotopography of the site is complex; and third, the stratigraphy of the site is complex.

The site is cut by paleoditches running down the slope of the Morne Patates and the early saladoid occupation is covered by an important level of volcanic deposits, accumulated in the paleoditches. In locus 2, for example, we have never been able, even with a backhoe, to reach the early saladoid layer located under more than two meters of volcanic deposits.

Broadly the stratigraphy of the site can be divided in four complexes:

The base complex is composed by a series of alluvial layers linked with the activity of the sulphur spring river. A gray/yellow sand layer including some reddish gravel lentils composes the D complex. It is in the D complex that we have found the artifacts linked with the earliest Amerindian occupation of the site. These layers have been partially destroyed by paleoditches. The C complex is associated with a black/brown anthropic layer. In the C complex we have found the remains of a denser occupation of the site. Volcanic deposits constitute the B complex. The A complex is linked with the reoccupation of the site after the volcanic eruption. Only the structures associated to that occupation dug inside the B complex have been preserved. Historical layers cover them.

The two earliest Amerindian layers (complex D and C) are associated with the early cedrosan saladoid phase. The C complex has been dated by C14 to 1800±40 BP, 150-250 AD calibrated, the volcanic eruption has been dated by vulcanologists and ourselves to cedrosan saladoid phase.

The artefacts

We will now concentrate on the early saladoid occupations of the site.

The remains coming from the excavation of the Soufrière site are essentially composed of ceramic elements. The high acidity level of the sediments did not permit the preservation of the faunal remains and only few lithic artifacts have been found.

Included in the earliest collection coming from the H. Petitjean Roget test pit, we have analyzed near to 2500 sherds linked with the early saladoid occupation of the Soufrière site.

The interpretation of the results of this analysis is still in progress. But some results can already be presented.

First, the early ceramic occupation of Soufrière is without doubt linked to the cedrosan saladoid sub-series. Second, the first comparison of the ceramic material coming from Soufrière with the collection of Vivé, Fond Brulé and Moulin l'Etang in the north of Martinique [START_REF] Bérard | Les premières occupations agricoles de l'Arc Antillais, Migrations et insularité[END_REF] demonstrates the great similarity of the two assemblages. This similarity is visible at different levels: Sizes and shapes of the vessels, relative proportion of the different decoration techniques and designs produced by those techniques.

We are highly aware that the great similarity of the early ceramic productions coming from the north of Martinique and South of Dominica is not sufficient to demonstrate definitively the existence of a small cultural sphere linked with that geographical space. However, the preliminary results of the 2007 field works in the north of Dominica seem to confirm that idea. Only two early ceramic sites have been identified. Those sites are clearly geographically distinct from the south dominica sites set. Furthermore, a part of the ceramic remains founded in those north sites seems to have no equivalent in the south dominica and north Martinique early ceramic collections.

We need now to compare our data with early ceramic collections from Marie-Galante and Guadeloupe in the north and St Lucia in the south to ensure that we have been able to locate a small cultural sphere linked to the internal diversity of the first agro-ceramic pioneer groups in the Lesser Antilles.

Figure 2 :

 2 Figure 2: Location of the Soufrière Loci.

Figure 3 :

 3 Figure 3: Soufrière Locus 1 and Locus 3 -Map of the excavations.

Figure 4 :

 4 Figure 4: exemple of great proximity between the early cedrosan saladoid ceramic productions of Martinique and Dominica. a. Vivé (Martinique), b. Soufrière (Dominica).

Figure 5 :

 5 Figure 5: exemple of great proximity between the early cedrosan saladoid ceramic productions of Martinique and Dominica. a. Soufrière (Dominica), b. Vivé (Martinique).


Benoît Bérard

Pre-Columbian Antillean civilizations in an archipelagic perspective

Keywords: Caribbean, Early Ceramic, Martinique, Dominica, Navigation techniques, Territories

This lecture main goal will be to present a synthesis of the researches we have conducted during the last 15 years about the pre-Columbian occupation of the West Indies between 300 BC and 400 AD. Those researches have been realized in Martinique, Dominica, Marie-Galante, Guadeloupe, Antigua and Barbuda. They have been enriched by the work of others archaeologists in Grenada, St. Vincent, Montserrat, Nevis, St Kitts and St. Martin. This huge set of data give us the opportunity to analysis the West Indian Amerindian civilizations in a maritime and archipelagic perspective, falling out by this way with the traditional approach focused on the insularity concept. The last part of this research have been an experimental archaeology program about Amerindian navigation techniques. Thus, between 2008 and 2010 we have navigated on a 60 feet long traditional dugout canoes from island to island, from Grenada to Antigua following the road of ours ancestors.

Introduction

Reading this paper title you may believe that it will not directly concern Garifuna people that are the fruit of a post-Columbian ethno-genesis. That is far to be the reality. The Garifuna history is maybe, in the West Indies, the most spectacular proof that cultural heritage is not a matter of genetic, blood or skin color. Therefore, a large part of Garifuna heritage, history and identity is linked to the Kalinago ones. Then, the six thousand years of Amerindian pre-Columbian occupation of the West Indies is also part of the Garifuna heritage.

The main goal of this presentation is to analyze the geographical dynamics of the West Indies Amerindian occupation in terms of territories. This analyze will be essentially based on the researches we have conducted since near to 20 years on the early ceramic occupation of the Antillean archipelago and on the results of the experimental archaeology program on Amerindian navigation techniques we have conducted with Karisko association between 2007 and 2010.

The West Indies early ceramic occupation is the result of a migration of people coming from Orinoco basin in actual Venezuela. They entered in the Antilles during the second half of the first millennium B.C. and they have quickly occupied a large area going from north South America lowlands to Puerto Rico. Those groups are associated in two cultural entities called early Cedrosan Saladoid and Huecan Saladoid by the archaeologists.

Several major archaeological programs developed since 30 years in Grenada, Martinique, Dominica, Guadeloupe, Montserrat, Saint-Martin, Vieques and Puerto Rico offer us a large data set on this Antillean history specific phase. Based on those investigations we now have strong foundations that make us able to characterize those earliest Saladoid cultural expressions in the West Indies and go relatively deeply in the analysis of the geographical dynamics set up by those groups.

The early Cedrosan Saladoid general presentation

Early Cedrosan Saladoid groups can be defined as pioneer societies. They are characterized by a predetermined economic system and a strong cultural identity2 . This early Cedrosan Saladoid economic system is characterized by slash-and-burn horticulture associated with collecting, fishing and hunting. More than that, it is defined by the introduction of plants and (wild) animals from the mainland 3 ; an important long-distance exchange network incorporating the entire Cedrosan Saladoid expansion sphere from the mainland to Puerto Rico 4 ; the definition of specific environmental characteristics for their village locations 5 ; a centralized way of managing island resources 6 ; and finally, a possible village mobility within and between islands that is linked, for a part, to the practice of slashand-burn horticulture 7 .

Our knowledge of their tools and handicrafts is based on the recovery and analysis of objects constructed from a wide variety of raw materials. The most commonly preserved raw materials are ceramics, lithics, and mollusk shells, which were used to create cutting and grinding tools, vessels, beads and pendants. Cutting and grinding tools in stone and shell are common to a large number of archaeological cultures in the Amazonian-West Indian space.

The most typical saladoid technique is the bipolar débitage technique used to produce numerous small flakes 8 but there is also evidence for freehand flaking, and the pecking and grinding of stone axes. Despite their sophisticated knowledge of manipulating stone, their cutting and grinding tools production show a low level of technical and social investment ("expedient" tools). The situation is totally different concerning ceramic vessels, and stone and shell beads and pendants.

Early Cedrosan Saladoid ceramics have been described in detail. Their decorations are characterized by painted polychromic designs especially white-and-red and white-on-red, incised motifs including the typical zoned-incised-crosshatched pattern and modeled elements like adornos and rim punctuated pellets. An exceptional collection of more than 300 complete vessels from Martinican sites has provided the basis for a comprehensive typology of ceramic production and decoration 9 . A high technical quality and very high degree of decoration characterize early Cedrosan Saladoid ceramics. In general more than 30% of the shards are decorated. The motifs demonstrate the significant social and symbolic value of this production 10 . The importance of ceramics as a symbolic media is reinforced by the great diversity of shapes and designs. This diversity is not the result of complete independence and freedom among the craftspeople. The collection conforms to the rigorous use of a complex and very codified system 11 . This rigorous code expresses a cultural standard that structured the topological organization of symbolic content in a three-dimensional space (decoration techniques and design motifs), the use of colors, vessel shapes, and the link between vessels shapes and the decorative technique and message.

The beads and pendants were made from semi-precious stones and shells. This production is very homogenous across the entire early Cedrosan Saladoid sphere. Beads and pendants also provide testimony to long-distance exchanges inside and outside this sphere. Finished products, but also semi-precious stones as raw materials, circulated between the Lesser Antilles, the Greater Antilles and the mainland. They show the importance of the links maintained between those groups dispersed over a large continental and archipelagic space.

Therefore, the early Cedrosan Saladoid is characterized by a standardized and predetermined economic system, by a very strong cultural identity visible in the ceramic, bead and pendant production, and by a broad long-distance exchange network. Those three elements may have represented the starting point and success of their pioneer project.

Martinique/Dominica, re-evaluate pre-Columbian territories

In Martinique all the early Cedrosan Saladoid sites are located in the north Atlantic coast facing Dominica. Therefore, after near to 10 years of excavation in Martinique we have decided in 2004 to cross the channel. Our desire was to evaluate the validity of the classic territorial model used by the archaeologist for the Lesser Antilles pre-Columbian occupation. This model developed by I. Rouse12 is centered on the "complex" concept that is his minimal entity corresponding to a level of perfect cultural equivalence of all the groups (archaeological sites) related to a same complex. Geographically, a complex is associated in the Lesser Antilles with single islands. Thus, in this model each island appears to be a perennial valid cultural territory. The underlying element of this vision is that each channel constitutes a geographical boundary and that those supposed "natural" boundaries are stronger than any social or cultural considerations.

In fact, our hypothesis was that the term 'complex' may better be applicable to terrestrial and maritime spaces including nearby islands, in this case the North of Martinique and the South of Dominica and the channel between them.

We tried to test this hypothesis during our three years research program in Dominica. We hoped that our work will enable us to determine if the sea provided a link or a frontier for those early Amerindian pioneers. Indeed, we strongly suspected that modern perceptions of the Lesser Antilles, dominated by important inter-island divisions that came into being with colonization and, more recently, decolonization probably have little or no relations with how the pre-Columbian people perceived their world. The idea was to leave the terrestrial island approach to develop an archipelagic and maritime perspective.

Our three years of research in Dominica had included a review of all the collections already existing linked with early ceramic sites, a systematic survey of the island, which enabled us to increase by 40% the number of pre-Columbian sites known, the digging of tests pits in four early ceramic sites and a larger excavation of the Soufrière site.

Early ceramic sites in Dominica are spatially divided in two sets separated by an important gap. The first set is composed by 3 sites located in the north part of the Atlantic coast facing Marie-Galante, the second set is composed by seven sites located in the south of Dominica.

Two types of analysis have been conducted concerning those sites. First, we have analyzed the environmental characteristics of each of them (distance to the sea, distance to fresh water, soil agricultural quality, vegetation, annual quantity of rain and distance to the nearer landing point suitable for a canoes). Second, we have realized a typological and iconographical analysis of the ceramic remains. Finally, we have compared the results of those works with the ones we obtained earlier in Martinique.

After those researches, we have been able to clearly demonstrated 13 that 2000 years ago the peoples living in the south of Dominica have shared more cultural affinities with the contemporaneous groups living in the north of Martinique than with the groups living at the same time in the north of Dominica. Therefore, if we have to delimit a minimal cultural territory for that period, it will integrate north Martinique, the Martinique channel as space of marine territories 14 . Thus, it is necessary to completely change our Lesser Antilles Amerindian territories vision adopting a marine and archipelagic point of view. This new perspective may have multiple consequences on the archaeological practice. However, it seemed to us that it was first necessary to evaluate the inter-insular relationship technical conditions before to be able to obtain satisfying results in this new frame and thus to produce solid data about pre-Columbian navigation techniques in the West Indies. It was the scientific motivation of the Kytangomingo Ema program 15 we set up with Karisko association.

Kytangomingo Ema program

Until then, our knowledge was based on very rare archaeological data, European descriptions from contact period and the offshore dugout canoe building tradition maintain in the central Lesser Antilles, especially in the Dominica Carib Territory, and also by mainland by the Kali'na Amerindian groups. Richard Callaghan exploited those data 16 to create digital simulation. With Karisko association we decided to follow another way developing an experimental archaeology program.

Therefore, in 2007 we have asked to Felix Brinkman, a Kali'na offshore dugout canoe building master from Surinam to conduct in French Guyana the production of two sixty feet long canoes more or less identical to the Carib kanawa described in the 16 th and 17 th European accounts. This is one of those kanawa baptized Akayouman 17 we used for our expeditions. Harbour, Antigua, 2011. pp.577-589. 16 Callaghan R., Mainland origins of the preceramic cultures of the greater antillles. PhD dissertation, University of Calgary, 1990, UMI dissertation services, Ann Arbor. 17 L'esprit du grand père serpent en caraïbe insulaire. 2010. Those expeditions were associated to hundreds of coastal navigation hours. The scientific program was articulated around three axes: The ethno-archaeological study of the kanawa building process including all the technical, symbolic and religious aspects, the Akayouman 3D digital modelization for his hydrostatic and hydrodynamic analysis by the of the Ecole Navale de Brest Hydrodynamic Laboratory18 and finally the experimental navigation data exploitation.

The first major information we have obtained is a valid evaluation of kanawa transportation capacity. The Akayouman hydrostatic analysis has clearly demonstrated that those boats are transportation ones. Even with a 27 persons complete crew, it is necessary to add an important ballast quantity to make it stable during navigation. Thus, we have transported 500 kg of ballast each time we have crossed a channel and we have done coastal navigations with one tons without any problems. This is a crucial element when you want to evaluate the interinsular material exchanges Amerindian capacity and this way to evaluate the contact intensity through the non-local row materials identification in the archaeological sites.

More than that we have discovered during our navigation all the marine qualities of that embarkation type. Therefore, even if we stay an inexperienced crew regarding the Amerindian multisecular tradition, we have been able to run Akayouman in rough sea with a 2 m high average swell and 3 m for the higher waves. Our average surface speed was around 3-3,5 knots. Therefore, theoretically (excluding currents, wind and swell influence) the kanawa speed is equivalent to the one of a fast walker in an open and flat landscape. After hundreds of navigation hours, we start also to have an intuitive appreciation of the distances you can do in one day. A travel inferior to 10-12 nm in one day does not represent real effort and must be considered as ordinary. A day travel between 12 and 20 nm start to be consequent and at least represent a real time investment (around between 4 and 6 hours). Finally, over 20 nm by day the journey can be considered as a real expedition (it start to be difficult to go and back the same day), especially if you have to cross a channel (the maximum distance we have did in one day during our expeditions was 37nm corresponding to a 11h 30mn navigation).

Moreover, the Kytangomingo Ema program has highly contributed to change our perspective.

We have been able this way to move from scientific theoretical positions to a live experience.

Sometimes the intelligence and the understanding run from the hands to the brain. During the experience we have intimately (in our bodies) take conscience that paddling between the Antillean islands must not be looked as something exceptional but as one of the Amerindian groups everyday life elements. This is those frequent travels, contacts and exchanges that have been the social and historical dynamics basement that authorized the Amerindians to make the Antillean archipelago a civilization space.

Finally, it is important to point out that on the side of scientific aspects, Karisko association Kytangomingo Ema program also integrated strong social, cultural and symbolic elements.

Re-opening ancestral maritime roads Akayaouman have been the media of a reappropriation by the Lesser Antillean populations of a shared heritage, a shared space and a shared identity.

Conclusion

The lack of "given" territories for the Antilles Amerindian occupation analysis, has made indispensable a specific modelization effort. We are deeply convinced that the vision produced may also be useful to the Antillean colonial history understanding. That is the way roots, totally emerging from the center and produced by it, inconstant even without signification and interest out of the colonial relationship" 21 . The fourth is based on the insularity notion and the contemporaneous geo-political context. Historians working at a larger scale generally consider it pejoratively as "local history". All those approaches are legitimism and useful to write Lesser Antilles history but they are not all the story, several others geographical scales have to be evaluated to make us able to analyze the internal factors role and importance in the Caribbean historical dynamics. Micro local level and the nonimperial inter-insular relationship (interlope exchanges, short-sea shipping, Antillean internal contacts and population moves…) for example need to be taken into account. Those internal dynamics were numerous and have had a major impact a several levels. If, Saint Vincent is the Garifuna birth land, the Antillean archipelago is also the good scale to understand their history, theirs tide relations with Victor Hugues in Guadeloupe during the last Carib war is a good example of this reality. 21 Ibidem.

  contact and fishing place and south Dominica. Researches now conducted about the early ceramic occupation of others lesser Antillean spaces (North Dominica/Marie Galante and south Basse-Terre, Antigua and Barbuda) seems to produce the same type of results and other archaeologists working on other chronological phases are also defining multi-insular and 13 Bérard B., Le phénomène pionnier agro-céramiste antillais : vers une vision archipélique. Les Nouvelles de l'Archéologie, N°108/109, Maison des Sciences de l'Homme, Paris, 2007. and Bérard B., La mission archéologique française en Dominique. Les Nouvelles de l'Archéologie, N°111/112, Paris, Maison des Sciences de l'Homme, 2008. pp. 95-100.

  After several preparation and training month, we first crossed Martinique channel in May 2008, then linked Martinique to Antigua in May 2009 and Grenada to Martinique in May 14 Hofman C., A. Bright et M. Hoogland. Archipelagic ressource procurement and mobility in the Northern Lesser Antilles: the view from a 3000-years-old tropical forest campsite on Saba. Journal of Island and Coastal Archaeology, Volume 1 Issue 2, 2006, pp. 145-164 and Hofman C. et al., Island rhythms: the web of social relationships and interaction networks in the Lesser Antillean archipelago between 400 B.C. and A.D. 1492. Latin American Antiquity, Vol.13, Number 3, September 2007, Washington, Society for American Archaeology, 2007. 15 Bérard B., Billard J.-Y. et Ramstein B., Ioumoúlicou "Koumoúlicou nhányem amonchéentium oúbao" (Les Caraïbes qui viennent des autres îles sont gens de notre nation), In Rebrovich S. (Ed.) The Proceedings of the XXIII Congress of the International Association for Caribbean Archaeology, Antigua, 29 june-3 july 2009. Dockyard Museum, English

  we have tried to follow developing a specific collective research program called Les territoires de l'histoire antillaise and conducted by J. Dumont, J.-P. Sainton and the author 19 . "Caribbean history has been the victim of a triple peripheralization: geographical, typological and epistemological" 20 . Our goal was to develop a self-centered vision by a systematic evaluation of external models and concepts used or adapted (if not adopted) by Caribbean historians and also to produce our own conceptual or geographical categories. If we analyze Lesser Antillean history as a mosaic of secant various scale geographical territories, four are well conceptualized and used by contemporaneous historians: the new world plantations space, the imperial perspective (F.W.I, B.W.I, Spanish America…), more recently the Atlantic world and finally the insular level. The three first, considered as global or connected history approaches, are focusing on the links produced by the colonial process and therefore on the external factors of the Antillean historical dynamics. Thus, the Antilles are seen too often as "peripheral societies with no 19 Bérard B., J. Dumont & J.-P. Sainton (dir.), Les territoires de l'histoire antillaise, Outre-Mers, Revue d'Histoire, n°378-379, SFHOM, Paris, 2013 20 Sainton J.-P., Territoires de l'histoire antillaise et dynamiques des sociétés in Bérard B., J. Dumont & J.-P. Sainton (dir.), Les territoires … (op.cit.), Translation from the french by the author.

  Archaeology of Trants, Montserrat. Part 4. Flaked Stone and Stone Bead Industries. Annals of Carnegie Museum 67: 197-224, 1998 ; Bérard Benoit, Lithic technology, one of the ways to complexity for Caribbean archaeology, In Hofman, Corinne L. Hoogland, Menno L. P. Van Gijn, A. L. (eds.), Crossing the Borders: New Methods and Techniques in the Study of Archaeology Materials from the Caribbean. Tuscaloosa : University Alabama Press, 2008 9 Bérard B., Les premières occupations agricoles de l'Arc Antillais, Migrations et insularité. op. cit. 10 Petitjean Roget, Henry, Contribution à l'étude de la préhistoire des Petites Antilles, Ph.D. dissertation, Ecole Pratique des Hautes Etudes, Paris, 1975 ; Roe Peter, A grammatical analysis of cedrosan saladoïd vessel form categories and surface decoration : aesthetic and technical styles in early antillean ceramics. In P.E. Siegel (ed.), Early Ceramic Populations Lifeways and Adaptive Strategies in the Caribeean, pp. 267-282. BAR International Series 506, Oxford, 1989 ; Waldron Lawrence, Like Turtles, Islands Float Away: Emergent Distinctions in the Zoomorphic Iconography of Saladoid Ceramics of the Lesser Antilles, 250 BCE TO 650 CE. Ph.D. dissertation, Central University of New York, 2010.

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