Fabrizio Bianchi

Tien-Cuong Dinh

EXPONENTIAL MIXING OF ALL ORDERS AND CLT FOR AUTOMORPHISMS OF COMPACT K ÄHLER MANIFOLDS

Keywords: 2010 Mathematics Subject Classification. 37F80 (primary), 32U05, 32H50, 37A25, 60F05 (secondary) Automorphism, Exponential Mixing of all orders, Central Limit Theorem

We consider the unique measure of maximal entropy of an automorphism of a compact Kähler manifold with simple action on cohomology. We show that it is exponentially mixing of all orders with respect to Hölder observables. It follows that the Central Limit Theorem (CLT) holds for these observables. In particular, our result applies to all automorphisms of compact Kähler surfaces with positive entropy.

Notation. The pairing •, • is used for the integral of a function with respect to a measure or more generally the value of a current at a test form. By (p, p)-currents we mean currents of bi-degree (p, p). The notations and stand for inequalities up to a multiplicative constant. If R and S are two real currents of the same bi-degree, we write |R| ≤ S when S ± R ≥ 0. Observe that this forces S to be positive.

Given a compact Kähler manifold X of dimension k, for every 0 ≤ q ≤ k we will denote by D q (X) (resp. D 0 q (X)) the real space generated by positive closed (resp. dd c -exact) (q, q)currents on X. For S ∈ D q (X), we will denote by {S} the cohomology class of S in H q,q (X, R).

Introduction

Let (X, ω) be a compact Kähler manifold of dimension k and f a holomorphic automorphism of X. We refer to [START_REF] Bedford | Dynamics of Rational Surface Automorphisms: Linear Fractional Recurrences[END_REF][START_REF] Filip | Gaps in the support of canonical currents on projective K3 surfaces[END_REF][START_REF] Mcmullen | Dynamics on K3 surfaces: Salem numbers and Siegel disks[END_REF][START_REF] Mcmullen | Dynamics on blowups of the projective plane[END_REF][START_REF] Oguiso | A remark on Dynamical degrees of automorphisms of compact Hyperkähler manifolds[END_REF][START_REF] Oguiso | Explicit examples of rational and Calabi-Yau threefolds with primitive automorphisms of positive entropy[END_REF] for interesting examples of such maps, and to [START_REF] De Thélin | Dynamics of automorphisms on compact Kähler manifolds[END_REF][START_REF] Dinh | Suites d'applications méromorphes multivaluées et courants laminaires[END_REF][START_REF] Dinh | Green currents for holomorphic automorphisms of compact Kähler manifolds[END_REF][START_REF] Dinh | Super-potentials for currents on compact Kähler manifolds and dynamics of automorphisms[END_REF] for their general properties, see also [START_REF] Cantat | Dynamique des automorphismes des surfaces K3[END_REF][START_REF] Cantat | Automorphisms of surfaces: Kummer rigidity and measure of maximal entropy[END_REF][START_REF] Dang | Dynamical degrees of automorphisms on abelian varieties[END_REF][START_REF] Filip | Kummer rigidity for K3 surface automorphisms via Ricci-flat metrics[END_REF][START_REF] Guedj | Propriétés ergodiques des applications rationnelles, in Quelques aspects des systèmes dynamiques polynomiaux[END_REF][START_REF] Oguiso | Some aspects of explicit birational geometry inspired by complex dynamics[END_REF]. We denote by f n the iterate of order n of f . For 0 ≤ q ≤ k, the dynamical degree d q is defined as the spectral radius of the pull-back operator acting on the cohomology group H q,q (X, R). We have d 0 = d k = 1 and d q (f n ) = d q (f ) n for all n ∈ N.

By a fundamental result of Khovanskii [START_REF] Khovanskii | The geometry of convex polyhedra and algebraic geometry[END_REF], Teissier [START_REF] Teissier | Du théorème de l'index de Hodge aux inégalités isopérimétriques[END_REF], and Gromov [START_REF] Gromov | Convex sets and Kähler manifolds[END_REF], the sequence q → log d q is concave, see also [START_REF] Dinh | The mixed Hodge-Riemann bilinear relations for compact Kähler manifolds[END_REF]. This implies that there exist integers 0 ≤ p ≤ p ≤ k such that 1 = d 0 < d 1 < . . . < d p = . . . = d p > . . . > d k = 1.

We say that f has simple action on cohomology if p = p and if moreover the action of f * on H p,p (X, R) admits a unique eigenvalue of maximal modulus. Such eigenvalue is then necessarily equal to d p . We denote in this case by δ = δ(f ) the maximum between max q =p d q and the moduli of the other eigenvalues for the action of f * on H p,p (X, R). We call d p the main dynamical degree and δ the auxiliary dynamical degree of f .

From now on, we assume that f has simple action on cohomology. It admits a unique probability measure of maximal entropy µ, which is the intersection of a positive closed (p, p)current T + and a positive closed (k -p, k -p)-current T -(the main Green currents of f ), see [START_REF] Dinh | Green currents for holomorphic automorphisms of compact Kähler manifolds[END_REF][START_REF] Dinh | Super-potentials for currents on compact Kähler manifolds and dynamics of automorphisms[END_REF]. Such measure is also called the equilibrium measure of f , and is mixing and hyperbolic. It was shown in [START_REF] Dinh | Exponential mixing for automorphisms on compact Kähler manifolds[END_REF] that such measure is exponentially mixing for Hölder observables, see also [START_REF] Dinh | Decay of correlations for Hénon maps[END_REF][START_REF] Vigny | Exponential decay of correlations for generic birational maps of P k[END_REF][START_REF] Wu | Exponential mixing property for Hénon-Sibony maps of C k[END_REF]. We consider here the following stronger property. Definition 1.1. Let ν be an invariant measure and (E, • E ) a normed space of real functions on X, with • L r (ν)

• E for all 1 ≤ r < ∞. We say that ν is exponentially mixing of all orders for observables in E if for all κ ∈ N * there exist constants C κ > 0 and 0 < θ κ < 1 such that, for all observables g 0 , . . . , g κ ∈ E and integers 0 =:

n 0 ≤ n 1 ≤ • • • ≤ n κ , we have ν, g 0 (g 1 • f n 1 ) . . . (g κ • f nκ ) - κ j=0 ν, g j ≤ C κ • κ j=0 g j E • θ min 0≤j≤κ-1 (n j+1 -n j ) κ
.

The exponential mixing of all orders is one of the strongest ergodic properties for a dynamical system. It implies a number of statistical properties that seem unattainable without such quantitative control, see for instance [START_REF] Björklund | Central Limit Theorems for group actions which are exponentially mixing of all orders[END_REF][START_REF] Dolgopyat | Multiple Borel Cantelli Lemma in dynamics and MultiLog law for recurrence[END_REF]. It is a main open question if this is implied by the exponential mixing of order 1, see for instance [START_REF] Dolgopyat | Exponential mixing implies Bernoulli[END_REF]Question 1.5]. We established in [START_REF] Bianchi | Every complex Hénon maps is exponential mixing of all orders and satisfies the CLT[END_REF] the exponential mixing of all orders for every complex Hénon map. The following is the main result of this paper.

Theorem 1.2. Let f be a holomorphic automorphism of a compact Kähler manifold (X, ω) with simple action on cohomology. Let d p be its main dynamical degree and δ its auxiliary dynamical degree. Then, for every δ < δ < d p and 0 < γ ≤ 2, the equilibrium probability measure µ of f is exponentially mixing of all orders κ ∈ N * for all C γ observables, with θ κ = (d p /δ ) -(γ/2) κ+1 /2 , see Definition 1.1.

In order to prove Theorem 1.2 we rely on delicate estimates coming from pluripotential theory and on the theory of positive closed currents. We partially follow the strategy in [START_REF] Bianchi | Every complex Hénon maps is exponential mixing of all orders and satisfies the CLT[END_REF], but the absence of plurisubharmonic functions on compact Kähler manifolds makes it not possible to employ a key step developed there. Instead, we rely on the theory of super-potentials for positive closed currents, and on quantitative estimates on the convergence of sufficiently regular positive closed currents towards the Green currents. We will give below an overview of our strategy.

As in [START_REF] Bianchi | Every complex Hénon maps is exponential mixing of all orders and satisfies the CLT[END_REF], the following is a consequence of Theorem 1.2 and [START_REF] Björklund | Central Limit Theorems for group actions which are exponentially mixing of all orders[END_REF]. Recall that u satisfies the Central Limit Theorem (CLT) with variance σ 2 ≥ 0 with respect to µ if n -1/2 (S n (u)n µ, u ) → N (0, σ 2 ) in law, where N (0, σ 2 ) denotes the (possibly degenerate, for σ = 0) Gaussian distribution with mean 0 and variance σ 2 , i.e., for any interval I ⊂ R we have

lim n→∞ ν S n (u) -n µ, u √ n ∈ I =    1 when I is of the form I = (-δ, δ) if σ 2 = 0, 1 √ 2πσ 2 I e -t 2 /(2σ 2 ) dt if σ 2 > 0.
Corollary 1.3. Let f be a holomorphic automorphism of a compact Kähler manifold (X, ω) with simple action on cohomology. Then all Hölder observables u : X → R satisfy the Central Limit Theorem with respect to the measure of maximal entropy µ of f with

σ 2 = n∈Z µ, ũ(ũ • f n ) = lim n→∞ 1 n X (ũ + ũ • f + . . . + ũ • f n-1 ) 2 dµ,
where ũ := u -µ, u .

Let now X be a compact Kähler surface and f an automorphism of positive entropy. By [START_REF] Gromov | On the entropy of holomorphic maps[END_REF][START_REF] Yomdin | Volume growth and entropy[END_REF], the topological entropy is equal to log d 1 , see also [START_REF] Dinh | Regularization of currents and entropy[END_REF]. In particular, d 1 is strictly larger than 1. A result by Cantat [START_REF] Cantat | Dynamique des automorphismes des surfaces K3[END_REF] says that all the eigenvalues of the action of f * on H 1,1 (X, R) have modulus 1, except for two eigenvalues d 1 and 1/d 1 , which have multiplicity 1. In particular, every automorphism of positive entropy of a Kähler surface has simple action on cohomology.

Corollary 1.4. Let f be a holomorphic automorphism of positive entropy on a compact Kähler surface X. Then, for every 1 < d < d 1 and 0 < γ ≤ 2, the equilibrium probability measure µ of f is exponentially mixing of all orders κ ∈ N * for all C γ observables, with θ κ = (d ) -(γ/2) κ+1 /2 , see Definition 1.1. Moreover, all Hölder observables satisfy the Central Limit Theorem with respect to the measure of maximal entropy of f . Strategy of the proof of Theorem 1.2. Using the classical theory of interpolation [START_REF] Triebel | Interpolation theory, Function Spaces, Differential Operators[END_REF], it is enough to prove the theorem in the case where γ = 2. Consider the compact Kähler manifold X := X × X. The automorphism F := (f, f -1 ) of X and its inverse have simple action on cohomology, with the largest dynamical degree being that of order k, which is equal to d 2 p . We can define the main Green currents T + and T -for F as

T + := T + ⊗ T -and T -:= T -⊗ T + . They satisfy (F n ) * (T + ) = d 2 p T + and (F n ) * (T -) = d 2 p T -.
Proving the exponential mixing of order κ for the κ + 1 observables g 0 , . . . , g κ can be reduced to proving the estimate

(1.1) | d -n 1 p (F n 1 /2 ) * [∆] -T -, Θ {g j },{n j } | κ j=0 g j C 2 d -min 0≤j≤κ-1 (n j+1 -n j )/2 ,
where [∆] denotes the current of integration on the diagonal ∆ ⊂ X × X, (z, w) the coordinates on X × X, we set

Θ {g j },{n j } := g 0 (w)g 1 (z)(g 2 • f n 2 -n 1 (z)) . . . (g κ • f nκ-n 1 (z))T + ,
and we assumed for simplicity that n 1 is even.

In the case of Hénon maps, we established in [START_REF] Bianchi | Every complex Hénon maps is exponential mixing of all orders and satisfies the CLT[END_REF] the above convergence by proving that Θ {g j },{n j } can be replaced by suitable currents Θ ± with dd c Θ ± ≥ 0, for which the estimate above can be proved thanks to the properties of plurisubharmonic functions. As non-trivial plurisubharmonic functions do not exist on compact Kähler manifolds, that approach cannot work here. Instead, we use more refined estimates on the regularity of the currents involved. We prove that if Θ {g j },{n j } is Hölder continuous in a precise sense (i.e., when seen as a function on the space of positive exact (k, k)-currents, endowed with a suitable metric), then the convergence (1.1) holds. This is done by exploiting the theory of super-potentials for positive closed currents, as developed by Sibony and the second author.

The main task becomes to prove the Hölder continuity of the current Θ {g j },{n j } . Observe that the estimate needs to be uniform in the n j 's and in the g j 's (assuming g j C 2 ≤ 1 for all j), in order for the implicit constant in (1.1) not to depend on such parameters. Observe also that this problem does not exist when just proving the mixing of order κ = 1, see [START_REF] Dinh | Exponential mixing for automorphisms on compact Kähler manifolds[END_REF]. This is the main technical point of the current paper.

By means of a general comparison principle for the super-potentials of positive closed currents [START_REF] Dinh | Super-potentials, densities of currents and number of periodic points for holomorphic maps[END_REF], we show that it is enough to find a positive closed current Ξ with a Hölder continuous super-potential and such that |dd c Θ {g j },{n j } | ≤ Ξ for all n j and all g j with g j C 2 ≤ 1.

Finding such Ξ and establishing such an estimate rely on the gap between d p and the auxiliary dynamical degree of f and on Hölder estimates for the action on f * on (p + 1, p + 1)-currents, that we also develop in this paper.

Acknowledgments. We would like to thank the National University of Singapore, the Institut de Mathématiques de Jussieu-Paris Rive Gauche, IMPAN, Xiaonan Ma, and Feliks Przytycki for the warm welcome and the excellent work conditions. This project has received funding from the French government through the Programme Investissement d'Avenir (LabEx CEMPI /ANR-11-LABX-0007-01, ANR QuaSiDy /ANR-21-CE40-0016, ANR PADAWAN /ANR-21-CE40-0012-01) and the NUS and MOE through the grants A-0004285-00-00 and MOE-T2EP20120-0010. This work is also partially supported by the Simons Foundation Award No. 663281 granted to the Institute of Mathematics of the Polish Academy of Sciences for the years 2021-2023.

Super-potentials of currents on compact Kähler manifolds

We fix in this section a k-dimensional compact Kähler manifold X and a Kähler form ω on X. For 0 ≤ q ≤ k, we denote by D q (X) the real space generated by positive closed (q, q)currents on X, and by D 0 q (X) the subspace of D q (X) given by the currents whose cohomology class in H q,q (X, R) vanishes. By the ∂ ∂-lemma, this is the set of currents in D q (X) which are ∂ ∂-exact. Since X is fixed, for simplicity in this section we will drop the dependence on X for these spaces and denote them by D q and D 0 q . We set h q := dim H q,q (X, R) and fix a collection α q = (α q,1 , . . . , α q,hq ) of real smooth (q, q)-forms on X such that the family of cohomology classes {α q,j } is a basis for H q,q (X, R). We also choose a collection of smooth real closed (k -q, k -q)-forms αq = (α q,1 , . . . , αq,hq ) which represent a dual basis of α q in H k-q,k-q (X, R) for the Poincaré duality.

Recall that the mass of a positive closed current S ∈ H q,q (X, R) is defined as S := S, ω k-q and it depends only on the cohomology class {S} of S in H q,q (X, R). The norm • * is defined for S ∈ D q as S * := min{ S : |S| ≤ S }. When S is dd c -exact the norm S * is equivalent to the norm given by min S + , where the minimum is taken over all decompositions S = S + -S -with S ± positive closed. Observe that, for each such decomposition, we have S + = S -as a consequence of the equality

{S + } = {S -}. We say that a subset of D q is * -bounded if it is bounded for • * .
Definition 2.1. We say that a sequence S n ∈ D q * -converges to S ∈ D q if S n → S in the sense of currents, and S n * is bounded by a constant independent of n.

Remark 2.2. The convergence is Definition 2.1 defines a topology on D q , which is not the one given by • * . We can also define a topology on D 0 q with the same definition. By [START_REF] Dinh | Regularization of currents and entropy[END_REF], smooth forms are dense in D q and D 0 q with respect to the topology of Definition 2.1.

For any 0 < l < ∞, denote by • C l the standard C l norm on the space of differential forms. We consider a norm • C -l and a distance dist l on D q defined by

S C -l := sup Φ C l ≤1 | S, Φ | and dist l (S, S ) := S -S C -l ,
where the supremum in the first definition is on smooth (k -q, k -q)-forms Φ on X. Observe that, by interpolation [START_REF] Triebel | Interpolation theory, Function Spaces, Differential Operators[END_REF], for every 0 < l < l < ∞ and m > 0, we have

(2.1) S C -l ≤ S C -l ≤ c l,l ,m S l/l C -l
for all S such that S * ≤ m, for some positive constant c l,l ,m . In particular, the above inequalities imply that

dist l ≤ dist l ≤ c l,l ,m (dist l ) l/l on {S ∈ D p : S * ≤ m/2}.
We now recall the notion of super-potential for currents in D q , see [START_REF] Dinh | Super-potentials for currents on compact Kähler manifolds and dynamics of automorphisms[END_REF]. This notion generalizes that of potentials for (1, 1)-positive closed currents on X, which are the quasi-plurisubharmonic functions on X. Super-potentials of a current S ∈ D q that we use here are functions, depending linearly on S, and defined on a subset of

D 0 k-q+1 1 . Take R ∈ D 0 k-q+1 . As {R} = 0, we have R = dd c U R for some (k -q, k -q)-current U R
, that we call a potential of R. By adding to U R a suitable combination of the αq,j 's, one can assume that U R is α q -normalized, i.e., that U R , α q,j = 0 for all 1 ≤ j ≤ h q . The α q -normalized super-potential U S of S is defined as

(2.2) U S (R) := S, U R
whenever the RHS of the above expression is well-defined. This is the case for instance when S is smooth or when R is smooth and we choose U R smooth.

Lemma 2.3. Let S, R, U R be as above and such that either R or S is smooth.

(i) The α q -normalized super-potential U S of S does not depend on the choice of the α qnormalized potential U R ; in particular, it does not depend on the choice of the αq,j 's;

(ii) If {S} = 0, then U S does not depend on the choice of α q .

(iii) If S is a linear combination of the α q,j 's, then the α q -normalized super-potential U S of S vanishes identically on D 0 k-q+1 . Proof. (i) Let U R and U R be two α q -normalized potentials of R. As dd c (U R -U R ) = R -R = 0, by Poincaré duality and the ∂ ∂-lemma the class {U R -U R } ∈ H k-q,k-q (X, R) is well defined and S, U R -U R only depends on the cohomology class of S. Since both U R and U R are α q -normalized, the cup-products of such class with all the classes {α q,j } are equal to 0. As the {α q,j }'s form a basis of H q,q (X, R), it follows that S, U R -U R = 0, as required.

(ii) As {S} = 0, we have S = dd c U S for some (q -1, q -1)-current U S . For any potential V R of R, non necessarily α q -normalized, we have

(2.3) S, V R = dd c U S , V R = U S , dd c V R = U S , R .
In particular, the first term of the above expression does not depend on the choice of V R , as this is the case for the last term. The assertion follows.

(iii) By definition, we have U R , α q,j = 0 for all j. As S is a linear combination of the α q,j 's, we have U S (R) = S, U R = 0. The assertion follows.

Remark 2.4. Observe that, on the other hand, the value S, U R and, by consequence, the α q -normalized super-potential U S of S, depend on α q when S is not exact. Definition 2.5. We say that S ∈ D q has a continuous super-potential if U S extends continuously to D 0 k-q+1 , with respect to the topology given by Definition 2.1. We also use the notation U S for the extended super-potential in this case.

Proposition 2.6. Take q, q with q + q ≤ k, S ∈ D q , S ∈ D q , and R ∈ D 0 k-q+1 . (i) If S is smooth, then it has a continuous α q -normalized super-potential for every choice of α q ; (ii) If S has a continuous α q -normalized super-potential, it also has continuous α q -normalized super-potentials for every other normalization α q = (α q,1 , . . . , α q,hq ). (iii) If S belongs to D 0 q and has continuous super-potentials and R is smooth we have

U S (R) = U R (S),
where U R is the super-potential of R (which is also independent of the choice of the normalization); (iv) If S has continuous super-potentials, then the current S ∧S is well defined and depends continuously on S.

In particular, by the third item, for every R ∈ D 0 k-q+1 we can define U R (S) := U S (R).

when S ∈ D 0 q has a continuous super-potential. Proof. (i) This is clear from the definition of the super-potential U S .

(ii) By (i) we can add to S a smooth closed form. This does not change the problem. So, we can assume that {S} = 0. By Lemma 2.3(ii), U S is independent of the normalization. The assertion follows.

(iii) Since R is smooth, the RHS in (2.2) is well defined for every S ∈ D 0 p . As {S} = 0, by Lemma 2.3(ii), U S does not depend on the normalization and (2.3) holds. As the last term of that expression is equal to U R (S), the assertion follows.

(iv) This is a consequence of [START_REF] Dinh | Super-potentials for currents on compact Kähler manifolds and dynamics of automorphisms[END_REF]Theorem 3.3.2]. Definition 2.7. Take S ∈ D q . For l > 0, 0 < λ ≤ 1, and M ≥ 0, we say that a super-potential U S of S is (l, λ, M )-Hölder continuous if it is continuous and we have

|U S (R)| ≤ M R λ C -l for every R ∈ D 0 k-q+1 with R * ≤ 1.
Lemma 2.8. Take S ∈ D q and R, R ∈ D 0 k-q+1 with R * ≤ 1 and R * ≤ 1. (i) If S is smooth, then the α q -normalized super-potential U S of S is (2, 1, M )-Hölder continuous with M ≤ c S C 2 for some constant c > 0 independent of S (but possibly depending on α q ). (ii) If S has an (l, λ, M )-Hölder continuous α q -normalized super-potential U S , then

|U S (R) -U S (R )| ≤ 2 1-λ M R -R λ C -l .
In particular, U S can be seen as a Hölder continuous function on {R ∈ D 0 k-q+1 : R * ≤ 1}. (iii) If S has an (l, λ, M )-Hölder continuous α q -normalized super-potential U S , then it also has (l, λ, M )-Hölder continuous α q -normalized super-potentials for every choice of α q and for some M independent of S.

Proof. (i) By Lemma 2.3(iii), we can add to S a combination of the α q,j 's and assume that {S} = 0. By Lemma 2.3(ii), we have U S (R) = U S , R , where U S is a smooth potential of S.

The assertion is now clear.

(ii) As R -R ∈ D 0 k-q+1 and R -R * ≤ 2, the assertion follows from Definition 2.7 applied with (R -R )/2 instead of R.

(iii) As in Proposition 2.6(ii), by (i) we can assume that {S} = 0. The assertion now follows from Lemma 2.3(ii). Proposition 2.9. Take q, q with q + q ≤ k. Let α q , α q , α q+q be as above and take S, T ∈ D q and S ∈ D q with S * ≤ 1, T * ≤ 1, and S * ≤ 1. Assume that the α q -normalized superpotential U S of S is (l, λ, M )-Hölder continuous and that the α q -normalized super-potential U S of S is (l , λ , M )-Hölder continuous.

(i) For every l 1 > 0, U S is (l 1 , λ 1 , M 1 )-Hölder continuous, for some λ 1 and M 1 depending on l, l 1 , λ, M , and α q , but independent of S. (ii) If S, T are positive and T ≤ S, then any α q -normalized super-potential U T of T is (2, λ 1 , M 1 )-Hölder continuous, for some λ 1 and M 1 depending on l, λ, M , and α q , but independent of S and T . (iii) S ∧ S has a (2, λ 1 , M 1 )-Hölder continuous α q+q -normalized super-potential, for some λ 1 and M 1 depending on l, l , λ, λ , M , M , α q , α q , and α q+q , but independent of S and S .

Proof. The first assertion is a consequence of (2.1). The second one is a consequence of [START_REF] Dinh | Super-potentials, densities of currents and number of periodic points for holomorphic maps[END_REF]Theorem 1.1] and the first one. The third assertion is a consequence of [14, Proposition 3.4.2] and the first one.

Dynamical Green currents

We keep in this section the notations of Section 2.

3.1. Convergence towards Green currents. We fix here an automorphism f of X with simple action on cohomology. We let p be such that d p is the largest dynamical degree of f , and let δ be its auxiliary degree. We also fix δ such that δ < δ < d p . By assumption, the eigenspace associated to the eigenvalue d p of the action of f * on H p,p (X, R) is a real line H + . A Green (p, p)-current T + of f is a non-zero positive closed (p, p)-current invariant under d -1 p f * , i.e., satisfying f * (T + ) = d p T + . The cohomology class {T + } generates H + . Lemma 3.1. Let f , d p , T + , α p be as above.

(i) T + is the unique positive closed (p, p)-current in {T + }, and it has a (2, λ, M )-Hölder continuous α p -normalized super-potential for some λ and M . (ii) For every S ∈ D p we have d -n p (f n ) * S → sT + . Here the constant s depends linearly on {S}. More precisely, s is the constant such that that d -n p (f n ) * {S} → s{T + }.

Proof. (i) The first assertion is a consequence of [14, Theorem 4.2.1] and Proposition 2.9(i).

(ii) As f has simple action on cohomology, there exists a constant s such that d -n p (f n ) * {S} → s{T + }. It is clear that the constant s depends linearly on S. When S ≥ 0 the statement follows from (i). The general case follows by linearity.

As the inverse f -1 of any automorphism with simple action on cohomology satisfies the same property, the above also holds for the automorphism f -1 . Since (f -1 ) * = f * , by Poincaré duality, we have d q (f ) = d k-q (f -1 ) for all 0 ≤ q ≤ k. Hence, the main dynamical degree of f -1 is equal to the one of f and is the dynamical degree of order k -p of f -1 . The eigenspace associated to this eigenvalue for the action of andwe have d -n p (f n ) * S → sT -for every S ∈ D k-p , where the constant s depends linearly on {S}.

f * on H k-p,k-p (X, R) is a real line H -. A Green (k -p, k -p)-current T -of f is a non-zero positive closed (k -p, k -p)-current invariant under d -1 p f * , i.e., satisfying f * (T -) = d p T -. The cohomology class {T -} generates H -, T -is the unique positive closed (k -p, k -p)-current in {T -},
Note that the currents T + and T -are unique up to multiplicative constants. We choose T + and T -such that the positive measure T + ∧ T -is of mass 1. This is then the unique measure of maximal entropy of f , see [START_REF] Dinh | Super-potentials for currents on compact Kähler manifolds and dynamics of automorphisms[END_REF] for details.

The following result gives a quantitative description of the convergences above, see [START_REF] Dinh | Exponential mixing for automorphisms on compact Kähler manifolds[END_REF]Proposition 3.1]. Note that the independence of the constant c from S is given in [15, Proposition 2.1]. Proposition 3.2. Let f , d p , δ be as above and S be a current in D p with S * ≤ 1. Let s be the constant such that d -n p (f n ) * (S) converge to sT + . Let R be a current in D 0 k-p+1 with R * ≤ 1 and whose α k-p+1 -normalized super-potential U R is (2, λ, 1)-Hölder continuous for some λ > 0. Let U T + and U n be the α p -normalized super-potentials of T + and d -n p (f n ) * (S), respectively. Then

|U n (R) -sU T + (R)| ≤ c(d p /δ ) -n
, where c > 0 is a constant independent of R, S, s, and n.

In particular, we will need the following consequence of the above result, see also Taflin [START_REF] Taflin | Dynamique des endomorphismes holomorphes de l'espace projectif[END_REF]Theorem 3.7.1] for a similar result in the case where p = 1.

Corollary 3.3. Let f , d p , δ be as above and S be a current in D p with S * ≤ 1 and such that d -n p (f n ) * (S) → 0. Let ξ be a (k -p, k -p)-current with dd c ξ * ≤ 1 and such that the superpotential U dd c ξ of dd c ξ (which is independent of the normalization) is (2, λ, 1)-Hölder continuous for some λ > 0. Assume that either S or ξ is smooth. Then

| d -n p (f n ) * (S), ξ | ≤ c(d p /δ ) -n
, where c > 0 is a constant independent of R, S, and n.

Observe that, since either S or ξ is smooth, the pairing in the statement is well defined.

Proof. We first consider the case where S is exact. Recall that, in this case, the super-potential of S is independent of the normalization. As we have {(f n ) * (S)} = (f n ) * {S} = 0, the same is true for the super-potential of (f n ) * (S) for all n ∈ N.

Setting S n := d -n p (f n ) * (S), we then have | S n , ξ | = |U Sn (dd c ξ)|.
By the assumptions on ξ, we can apply Proposition 3.2 with dd c ξ instead of R and s = 0. The assertion in this case follows.

Let us now consider the general case. Observe that d -n p (f n ) * {S} → 0. So, the set of the classes {S} of the currents S ∈ D p with this property is an hyperplane H ⊂ H p,p (X, R), which is a complement of the line generated by {T + } and is invariant under the action of f * . For simplicity, denoting by h the dimension of H p,p (X, R), we let α 1 , . . . , α h-1 be real smooth (p, p)forms such that {α 1 }, . . . , {α h-1 } form a basis for H, and α h be a smooth form in the class of T + . We will consider α-normalized super-potentials of currents in D p , with α = (α 1 , . . . , α h ). We also fix α = (α 1 , . . . , αh ) a dual basis of α. As {S} ∈ H, we have

S = a 1 α 1 + • • • + a h-1 α h-1 + S
for some S ∈ D 0 p and a j ∈ R. By the first part of the proof applied to S , it is enough to prove the statement for S := h-1 j=1 a j α j instead of S. Observe, in particular, that S is smooth and that the α-normalized super-potential U S of S satisfies U S = 0 by Lemma 2.3(iii).

Denote S n := d -n p (f n ) * (S ). We have

S n = h j=1 c n,j α j + dd c V n ,
where the c n,j are defined by {S n } = h j=1 c n,j {α j } and the (p -1, p -1)-current V n is chosen so that V n , αj = 0 for all 1 ≤ j ≤ h. Observe that c n,h = 0 for all n because of the invariance of H. It follows that, for any ξ as in the statement, we have

S n , ξ = h-1 j=1 c n,j α j , ξ + dd c V n , ξ = h-1 j=1 c n,j α j , ξ + U S n (dd c ξ),
where U S n is the α-normalized super-potential of S n . By the assumptions on ξ and Proposition 3.2 applied with dd c ξ instead of R and with s = 0, we have

|U S n (dd c ξ)| (d p /δ ) -n . Hence, it is enough to prove that, for all 1 ≤ j ≤ h -1, we have |c n,j | (d p /δ ) -n .
Let A be the (h -1) × (h -1) matrix representing f * |H with respect to the basis {α j } 1≤j≤h-1 , i.e., whose j-th column is given by the coordinates of f * {α j } with respect to the given basis. Denoting c n := (c n,1 , . . . , c n,h-1 ) t and a := (a 1 , . . . , a h-1 ) t , we see that

c n = d -n p A n a.
As the spectral radius of the action of f * on H is smaller than δ , we see that A n = o(δ n ). It follows that |c n,j | (δ /d p ) -n , as desired. The assertion follows.

In our study we will also need the case where q = p + 1. It follows from the definition of d p+1 and the fact that d p+1 < d p that d -n p (f n ) * (S) → 0 as n → ∞ for every S ∈ D p+1 . We will need later a more quantitative version of this convergence, for S with Hölder continuous super-potentials. This is given by the next proposition. Recall that we normalize potentials with respect to a given α p+1 = (α p+1,1 , . . . , α p+1,h p+1 ), where h p+1 is the dimension of H p+1,p+1 (X, R). Proposition 3.4. Let f be as above. Take S ∈ D p+1 with S * ≤ 1.

(i)

d -n p (f n ) * S * → 0 as n → ∞; in particular, d -n p (f n ) * S → 0 as n → ∞. (ii) Assume that S has an (l, λ, M )-Hölder continuous α p+1 -normalized super-potential.
Then there exist λ and M independent of S such that, for every n ≥ 0,

d -n p (f n ) * S has a (2, λ , M )-Hölder continuous α p+1 -normalized super-potential.
Proof. The first assertion is a consequence of the inequality d p+1 < d p and of the fact that, for every > 0, we have (f n ) * (S) * ≤ C(d p+1 + ) n S * , where the constant C is independent of S because the mass of a positive closed current only depends on its cohomology class. It remains to prove the second assertion. By Proposition 2.9(i) we can assume that l = 4. Particular case. We first prove the third assertion assuming that S is exact. Recall that, in this case, the super-potentials U S is independent of the normalization, see Lemma 2.3(ii).

Fix R ∈ D 0 k-p with R * ≤ 1 and set S n := d -n p (f n ) * S.
We need to show that

|U Sn (R)| ≤ M R λ C -4
for some λ and M independent of S and R. The assertion is then a consequence of Proposition 2.9(i).

Fix L ≥ max(2, sup Φ C 4 ≤1 f * Φ C 4 )
where the supremum is on smooth (q -1, q -1)-forms Φ on X. As S and R are both exact, by Lemma 2.3(ii) and the definition of super-potential we have

(3.1) |U Sn (R)| = d -n p U S (f n ) * (R) ≤ M d -n p (f n ) * (R) λ C -4 . By Poincaré duality, d -n p (f n ) * (R) * is bounded independently of n. We also have d -n p (f n ) * (R) C -4 = sup Φ C 4 ≤1 d -n p (f n ) * (R), Φ = sup Φ C 4 ≤1 U d -n p (f n ) * (R) (dd c Φ) θ n
for some 0 < θ < 1, where the last step follows from Corollary 3.3 applied with f -1 , R, and Φ instead of f , S, and ξ. Observe that the assumption on dd c ξ in that corollary is satisfied by dd c Φ by Lemma 2.8(i).

Assume first that n ≥ -(2 log L) -1 log R C -4 . In this case, we have

d -n p (f n ) * (R) C -4 θ n ≤ R 1 2 | log θ| log L C -4
.

Together with (3.1), this implies the assumption.

Assume instead that n ≤ -(2 log L) -1 log R C -4 . Observe that this implies that L n R C -4 ≤ R 1/2 C -4 .
Hence, for all such n, we have

d -n p (f n ) * R C -4 ≤ sup Φ C 4 ≤1 d -n p | R, (f n ) * Φ | ≤ d -n p L n R C -4 ≤ d -n p R 1/2 C -4 ,
which also implies the assertion in this case. The proof in the particular case is complete.

General case. We now consider the general case. By the previous part of the proof, and as H p+1,p+1 (X, R) is finite dimensional, it is enough to prove the assertion for any finite family of smooth forms whose classes generate H p+1,p+1 (X, R). Therefore, it is enough to prove the statement for a fixed smooth form S.

Let 1 ≤ m < dim H p+1,p+1 (X, R) be the minimal integer such that {S}, {f * (S)}, . . . , {(f m ) * (S)} are linearly dependent over R, and define a 0 , . . . , a m-1 by the relation

(3.2) {(f m ) * (S)} = a 0 {S} + • • • + a m-1 {(f m-1 ) * (S)}.
Let E be the subspace of H p+1,p+1 (X, R) spanned by {S}, . . . , {(f m-1 ) * (S)}. Then these m classes form a basis B of E. The action of f * |E with respect to the basis B is represented by the m × m matrix

A E :=        0 0 . . . 0 a 0 1 0 . . . 0 a 1 0 1 . . . 0 a 2 . . . . . . . . . . . . . . . 0 0 . . . 1 a m-1       
.

We denote by B E the transpose of A E . Fix 0 < < d p -d p+1 . By the definition of d p+1 , we have

A n E = B n E (d p+1 + ) n = o(d n p ). It follows from (3.2) that U := (f m ) * (S) - m-1 j=0 a j (f j ) * (S)
is an exact form, i.e., it belongs to D 0 p+1 . Since (f j ) * (S), 0 ≤ j ≤ m -1, and U are smooth, they have (l, λ, N )-Hölder continuous α p+1 -normalized super-potentials for some constant N .

For every n ≥ 1, set

W n :=      (f n ) * (S) (f n+1 ) * (S) . . . (f n+m-1 ) * (S)      and U n :=      0 0 . . . (f n ) * (U )     
, where both W n and U n have m components. As W 1 = B E W 0 + U 0 , we obtain by induction that

d -n p W n = d -n p B n E W 0 + d -n p n-1 j=0 B n-j E U j .
As the first component of W n is equal to (f n ) * (S), we need to consider the α p+1 -normalized super-potential of the first component of the RHS of the above expression.

By the above, the currents (f j ) * (S), 0 ≤ j ≤ m -1, have an (l, λ, N )-Hölder continuous α p+1 -normalized super-potential. As

B n E = o(d n p+1
), it follows that the α p+1 -normalized super-potential of every component of d -n p B n E W 0 is (l, λ, M /2)-Hölder continuous, for some M large enough.

In order to prove the assertion, using again that B n-j E = o(d n-j p ), it is enough to show that there exist l , λ , M such that, for every n ≥ 0, U n := d -n p (f n ) * (U ) has a (l , λ , M /2)-Hölder continuous α p+1 -normalized super-potential. As U is exact, this follows from the particular case considered above. This concludes the proof.

3.2.

Further properties of the Green currents. We prove here two lemmas that we will need in the next section in order to apply Proposition 3.2 and Corollary 3.3. As in the previous section, we let f be an automorphism of X with simple action on cohomology, and d p its largest dynamical degree. In particular, the Green current T + has bi-degree (p, p). We also fix a normalization α p+1 := (α p+1,1 , . . . , α p+1,h p+1 ), where h p+1 is the dimension of H p+1,p+1 (X, R). Lemma 3.5. Let f be as above. Let κ ≥ 1 be an integer and g 0 , . . . , g κ : X → R satisfy g j C 2 ≤ 1. Then there is a positive constant c independent of the g j 's such that for all 0 , . . . , κ ≥ 0 we have

dd c (g 0 • f 0 ) . . . (g κ • f κ ) ∧ T + * ≤ c.
Proof. Set gj := g j • f j and notice that i∂g j ∧ ∂g j = (f j ) * (i∂g j ∧ ∂g j ) for all j. We have We deduce from the above inequalities and the invariance of T + that

(3.3) dd c (g 0 • f 0 ) . . . (g κ • f κ ) ∧ T + κ j=0 f j ) * (ω ∧ T + = κ j=0 (f j ) * (ω) ∧ d -j p (f j ) * (T + ) = κ j=0 d -j p (f j ) * ω ∧ T + .
We now use that the (p + 1, p + 1)-current ω ∧ T + is positive and closed. We have

(f j ) * ω ∧ T + = (f j ) * ω ∧ T + , ω k-p-1 = ω ∧ T + , (f j ) * (ω k-p-1 ) ,
where the last form is positive closed. The last pairing only depends on the cohomology classes of ω, T + , and (f j ) * (ω k-p-1 ). Hence, it is bounded by a constant times the mass of (f j ) * (ω k-p-1 ). Since

(f j ) * (ω k-p-1 ) = ω p+1 , (f j ) * (ω k-p-1 ) = (f j ) * (ω p+1 ), ω k-p-1 ,
for all > 0 this number is also equal to (f j ) * (ω p+1 ) (d p+1 + ) n . As d p+1 < d p by assumption, it follows that each term in the last sum in (3.3) is bounded, which implies that the sum is also bounded. The lemma follows.

Lemma 3.6. Let f be as above. Let κ ≥ 1 be an integer and g 0 , . . . , g κ : X → R satisfy g j C 2 ≤ 1. Then there exist constants 0 < λ ≤ 1 and M > 0, independent of the g j 's, such that for all 0 , . . . , κ ≥ 0 the current

dd c (g 0 • f 0 ) . . . (g κ • f κ ) ∧ T + has a (2, λ, M )-Hölder continuous super-potential.
Observe that the current in the statement is exact, hence its super-potential is independent of the normalization.

Proof. We denote the current in the statement by R. It is a real exact (p + 1, p + 1)-current. Recall that, by Proposition 2.9(i), it is enough to show that R has an (l, λ, M )-Hölder continuous super-potential, for some l > 0, 0 < λ ≤ 1, and M > 0 independent of g 0 , . . . , g κ and 0 , . . . , κ . By Proposition 2.9(ii), it is enough to show that there exists a positive closed current V with a (2, λ , M )-Hölder continuous α p+1 -normalized super-potential such that |R| ≤ V . Indeed, this implies that both V + R and V -R are positive, closed, and bounded by 2V . Hence, they have (2, λ , M )-Hölder continuous α p+1 -normalized super-potentials for some constants 0 ≤ λ ≤ 1 and M > 0. It follows that R has a (2, λ , M + M )-Hölder continuous super-potential.

We have already seen in (3.3) that

|dd c g 0 • f 0 ) . . . (g κ • f κ ) ∧ T + | κ j=0 d -j p (f j ) * ω ∧ T + .
In order to prove the assertion, it is then enough to prove that there exist λ and M such that, for every j ≥ 0, the current d -j p (f j ) * ω ∧ T + has a (2, λ , M )-Hölder continuous α p+1normalized super-potential. For j = 0, this follows from Lemma 3.1(i) and Proposition 2.9(iii). Proposition 3.4(ii) implies that that same holds for all j ∈ N. The proof is complete.

Exponential mixing of all orders and Central Limit Theorem

We prove now Theorem 1.2. By interpolation techniques [START_REF] Triebel | Interpolation theory, Function Spaces, Differential Operators[END_REF] as in [9, pp. 262-263], it is enough to prove the statement in the case γ = 2. The statement is clear for κ = 0. By induction, we can assume that the statement holds for up to κ functions and prove it for κ + 1 ≥ 2 functions. In particular, by the induction assumption, we are allowed to add to the g j 's a constant and assume that µ, g j = 0 for all 0 ≤ j ≤ κ. We can also assume that g j C 2 ≤ 1 for all 0 ≤ j ≤ κ. Then, we need to show that µ, g 0 (g 1 • f n 1 ) . . . (g κ • f nκ ) d -min 0≤j≤κ-1 (n j+1 -n j )/2 .

We can also assume that n 1 is even. Indeed, by the invariance of µ, the case where n 1 is odd can be treated similarly by replacing n j with n j -1 for 1 ≤ j ≤ κ and g 0 with g 0 • f -1 .

Consider the automorphism F of the compact Kähler manifold X := X × X given by F (z, w) := (f (z), f -1 (w)). We first show that F -1 = (f -1 , f ) satisfies the assumptions of Proposition 3.2 and Corollary 3.3. Recall that δ is the auxiliary degree of f and that by assumption we have δ < d p .

Lemma 4.1. The automorphisms F and F -1 of X have simple action on cohomology. More precisely, the dynamical degrees of order k of F and F -1 are equal to d p (f ) 2 , they are eigenvalues of multiplicity one of both F * and F * acting on H k,k (X, C), and all the other dynamical degrees of F and F -1 , as well as the other eigenvalues of F * and F * on H k,k (X, C), have modulus smaller than or equal to d p • δ.

Proof. By Künneth formula [START_REF] Voisin | Théorie de Hodge et géomérie algébrique complexe[END_REF]Theorem 11.38], for every 0 ≤ q ≤ k we have a canonical isomorphism H q,q (X × X, C) = s+r=q H s,r (X, C) ⊗ H r,s (X, C).

The operators F * and F * preserve the above decomposition. By [START_REF] Dinh | Suites d'applications méromorphes multivaluées et courants laminaires[END_REF], the spectral radius of f * on H r,s (X, C) is (equal to the spectral radius of f * on H k-r,k-s (X, C) and) smaller than or equal to √ d r d s . The assertion follows from the fact that f and f -1 have simple actions on cohomology and the definitions of p and δ.

In this section, we will denote by T ± the Green currents of f and by T ± the Green currents of F . More precisely, we fix Green currents T ± for f (they are unique up to a multiplicative constant) such that µ = T + ∧ T -and we set Set h := g 1 (g 2 • f n 2 -n 1 ) . . . (g κ • f nκ-n 1 ). Recalling that µ, g 0 = 0 and that g j C 2 ≤ 1, by the induction hypothesis it is enough to show that µ, g 0 (h

• f n 1 ) d -n 1 /2 p .
We denote by (z, w) the coordinates on X = X × X and we set Ψ(z, w) := g 0 (w)h(z).

Lemma 4.2. The following assertions hold: (i) dd c Ψ ∧ T + * ≤ c;

(ii) dd c Ψ ∧ T + has a (2, λ, M )-Hölder continuous super-potential, where c, λ, and M are positive constants depending on κ, but not on the g j 's and the n j 's.

Proof. The two assertions are consequences of Lemmas 3.5 and 3.6 applied to F : X → X, respectively. Here, we use the functions g0 (z, w) := g 0 (w) and gj (z, w) := g j (z) for j ≥ 1, and the integers 0 := 0, 1 := 0, and j := n j+1 -n j for j ≥ 2.

Remark 4.4. The rate of mixing in Theorem 1.2 can be improved by considering F := (f l , f -m ) instead of F , for suitable positive integers l and m, see for instance [START_REF] Dinh | Exponential mixing for automorphisms on compact Kähler manifolds[END_REF]Remark 4.1].

  Since g j C 2 ≤ 1 we have |g j | ≤ 1 and |dd c g j | ω. Then ) * (ω) and an application of Cauchy-Schwarz inequality gives 0≤j =l≤κ i∂g j ∧ ∂g l m =j,l gm κ j=0 i∂g j ∧ ∂g j κ j=0 (f j ) * (ω).

T

  + = T + ⊗ T -and T -= T -⊗ T + , see [18, Section 4.1.8] for the tensor (or cartesian) product of currents. Since f * (T + ) = d p T + and f * (T -) = d p T -, we have F * (T + ) = d 2 p T + and F * (T -) = d 2 p T -.

Super-potentials can be defined on a subset of D k-q+1 , but it is simpler to use D 0 k-q+1 , and we will only need this definition in this paper.

End of the proof of Theorem 1.2. Recall that we are assuming that n 1 is even. By the invariance of µ, we see that µ, g 0 (h

) . We first transform the above integral on X to an integral on X by means of the map F . Namely, using the invariance of T + , we have

Observe, in particular, that the wedge product [∆] ∧ T + is well defined by Proposition 2.6(iv).

Proof. By Lemma 4.1, F -1 has simple action on cohomology and its main dynamical degree is that of order k, which is equal to

Hence, we only need to show that s = 1. We have

On the other hand, by the invariance of T + , for every n ∈ N we also have

The assertion follows.

We will apply the results of Sections 2 and 3 to the automorphism F -1 of X. In order to do this, for simplicity, we let h be the dimension of H k,k (X, R) and we fix a collection α := (α 1 , . . . , α h ) of real smooth (k, k)-forms on X with the property that {α 1 } = {T -} and {α 2 }, . . . , {α h } are a basis for the F * -invariant hyperplane in H k,k (X, R) transversal to the line generated by {T -}. In the following, we only consider α-normalized super-potentials for currents in D k (X). By Lemma 4.2, the current R := dd c Ψ ∧ T + satisfies R * 1 and has a (2, λ, M )-Hölder continuous potential U R , for some 0 < λ ≤ 1 and M > 0 independent of the g j 's and n j 's. By linearity, up to multiplying g 0 by a constant, we can assume that R * ≤ 1 and that U R is (2, λ, 1)-Hölder continuous, for some 0 < λ ≤ 1. Recall that δ < δ < d p . By Lemmas 4.1 and 4.3, we can apply Corollary 3.3 with F -1 , ΨT + , and n 1 /2 instead of f , ξ, and n, and get that, for all smooth S ∈ D k (X) such that d -2n p (F n ) * (S) → 0, we have

Let now S be a regularization of [∆] -T -, with S → [∆]

-T -as → 0 and S * bounded [START_REF] Dinh | Regularization of currents and entropy[END_REF]. We can also add to S a small smooth form so that {S } = {[∆] -T -}. We apply the above inequality with S instead of S and take → 0. It follows from Proposition 2.6(iv) and Lemma 4.2 that (4.2)

Combining (4.1) and (4.2) and using the invariance of T -we obtain that µ, (g 0

To conclude, it is enough to show that T -, ΨT + = 0. As µ, g 0 = 0, we have

The proof is complete.

CNRS, Univ. Lille, UMR 8524 -Laboratoire Paul Painlevé, F-59000 Lille, France Email address: fabrizio.bianchi@univ-lille.fr

National University of Singapore, Lower Kent Ridge Road 10, Singapore 119076, Singapore Email address: matdtc@nus.edu.sg


Youcef Dahmani
email: dahmaniyoucef@yahoo.fr

Maamar El

Amine Hamri
email: amine.hamri@lis-lab.fr

Nesrine Driouche
email: driouche.nesrine@hotmail.fr

Closure under Coupling of Cellular-DEVS for the Optimization of Memory Resource: Wildfire Spread Case Study

The present work aims to show one of the advantages of using the property of closure under coupling in the DEVS specification. The advantage concerned in this paper attempts to address the need for memory resources during the simulation of systems by cellular-DEVS.

This improvement of performance is based on the usage of the property closure under coupling in the DEVS formalism. With this property and taking account of the iterative behavior of each cellular-DEVS atomic model, we provide simulation of many models simultaneously. The method starts with the specification of the cellular-DEVS coupled model which is then converted into its equivalent DEVS atomic model. Thus, the goal of this conversion is to transform large quantities of atomic models coupled together, which require huge computational resources, into one DEVS atomic model.

A case study is presented at the end of the work on modeling and simulation of forest fire propagation using DEVS and cellular-DEVS. A specification by cellular-DEVS of the forest fire model and its non-modular equivalent DEVS atomic model are presented. Finally, a comparison between both methods is presented in term of consumption of resources.

Introduction

Modeling and Simulation (M&S) have accumulated a large number of successes in a wide and varied range of domains. However, M&S consumes time, effort and resources when you need to resolve class of problems where the analytic or modeling solution is very hard to find. This class of problems includes dynamic systems where there is a wide number of parameters with time and causal dependencies between them, a non-linear behavior of the system, etc.

Discrete event M&S deals with systems whose temporal and spatial behaviors are complex to be treated analytically. The DEVS formalism (Discrete EVent system Specification) is one of the common formalism used in the simulation of dynamic systems [START_REF] Zeigler | Theory of Modeling and Simulation[END_REF][START_REF] Zeigler | Theory of Modeling and Simulation[END_REF]. It is known for its modularity and expressiveness. This formalism offers, compared to others, a general purpose.

However, the DEVS formalism has undergone several extensions to meet specific needs. Many variants on DEVS were adopted by introducing appropriate theories such the parallel-DEVS [START_REF] Chow | Parallel-DEVS: a parallel, hierarchical, modular modeling formalism and its distributed simulator[END_REF] DS-DEVS [START_REF] Barros | Modelling formalisms for dynamic structure system[END_REF], Cell-DEVS [START_REF] Wainer | Application of the cell-DEVS paradigm for cell spaces modelling and simulation[END_REF], etc.

Although these variants exist, it is very difficult to simulate complex systems, especially those represented by large-scale cellular-DEVS models. The need for computational resources becomes vital in order to ensure reliable and fast simulations. Many techniques were adopted with their advantages and disadvantages [START_REF] Shiginah | A new cell space DEVS specification: Reviewing the parallel-DEVS formalism seeking fast cell space simulations[END_REF].

In this work, we take advantage of the property, closure under coupling of DEVS which allows us to reduce computational resources without changing the bases of DEVS specification.

A case study of the Wildfire Spread Simulation is modeled by the cellular-DEVS coupled model and its equivalent DEVS atomic model. Simulation results for both techniques are given and a comparison between these two simulations is described.

The remainder of this paper is outlined as follows: Section 2 resumes the basic concepts on the DEVS formalism. Section 3 is divided into two subsections, in the first one, a literature review on optimization methods to improve the performances of simulation within the DEVS formalism is reported, and in the second subsection, an overview on forest fire spread simulations is presented. Section 4 depicts our approach by converting the cellular-DEVS into its equivalent DEVS atomic model using the property closure under coupling applied to wildfire spread. Section 5 illustrates simulation experiments on our case study on forest fire spread, we specify both models: cellular-DEVS coupled model versus DEVS atomic model. At the end, conclusion and perspectives are given.

DEVS Formalism

Background

Being based on system theory, the DEVS formalism is one of the means theoretically wellgrounded to express discrete event systems in a hierarchical and modular manner. A discrete event system is a dynamic system whose behavior is directed by the appearance of discrete events [START_REF] Lee | Simulation speedup for DEVS models by composition-based compilation[END_REF].

The use of discrete event-based simulation, rather than time-driven simulation has been proven to reduce computation time in many applications. DEVS formalism can be used to specify systems whose input, state and output trajectories are piecewise constant [START_REF] Zeigler | DEVS framework for modeling, simulation, analysis, and design of hybrid systems[END_REF]. In addition, DEVS can give highest performance for simulation of continuous systems typified by spatial and temporal heterogeneity [START_REF] Hu | A high performance simulation engine for large-scale cellular DEVS models[END_REF][START_REF] Nutaro | Discrete event solution of gas dynamics within the DEVS framework: Exploiting spatio-temporal heterogeneity[END_REF].

Theoretically, each system is characterized by two features: functional (behavioral) and structural aspects. At the lowest level, a basic part called DEVS atomic, describes the autonomous behavior of a discrete event system. At the highest level, DEVS coupled describes a system as modular and hierarchical structure [START_REF] Zeigler | Theory of Modeling and Simulation[END_REF][START_REF] Zeigler | Theory of Modeling and Simulation[END_REF].

A DEVS atomic model is based on continuous time, inputs, outputs, states and functions (output, internal and external transitions, life states). Formally, a DEVS atomic model is described by the following seven-tuple:

AM devs = (X, S, Y, δ int , δ ext , λ, t a )
Where X, S and Y are the sets of input events, states, and output events, respectively, δ int : S → S the internal transition function, describes the state changes, that occurs when the elapsed time reaches the lifetime of the state, δ ext : Q × S → S is the external transition function, where Q = {(s, e)|s ∈ S, e ∈ R + , 0 ≤ e ≤ t a (s)} is the total state set and e describes the elapsed time since the last transition of the current state s, λ : S → Y : when the elapsed time reaches the lifetime of the state, this function generates an output event, and t a : S → R + ∪ ∞: time advance function, which is the lifespan of a state.

The DEVS coupled model allows formalizing the modeled system in a set of inter-connected and reused components. A DEVS coupled model is defined as an eight-tuple:

CM devs = (X self , Y self , D, {M d |d ∈ D}, EIC, EOC, IC, Select)
Where X self : set of inputs events, Y self : set of outputs events, D : is the name set of sub-components, {M d |d ∈ D}: set of sub-components of DEVS models, EIC: set of External Input Coupling, EOC: set of External Output Coupling, IC: defines the Internal Coupling, and Select : 2 D → D: defines a priority between simultaneous events.

Cellular-DEVS

Several DEVS formalisms have evolved compared to the classic DEVS to fit specific needs. Cellular-DEVS models have been created to model and simulate several phenomena such as fire propagation, traffic control, etc. [START_REF] Hu | A high performance simulation engine for large-scale cellular DEVS models[END_REF].

Cellular-DEVS originates from the cellular automata formalism. The latter is based on discretetime simulation which consumes the computation power to update all cells at each time-step. A decrease in the time-step used in the models would increase accuracy, but would also result in a longer simulation time and requires large computational resources. In fact, in many cases, there are few cells that are concerned to be updated at each time-step, which makes this manner inefficient. In order to overcome this problem, cellular-DEVS was adopted to offer computational resources to the active cells that really execute state transitions and therefore avoid useless computation on inactive cells.

The cellular-DEVS formalism divides the spatial space into a set of identical cells where computations are done locally. A cell is considered as a DEVS atomic model which executes the local computations based on its own state as well as its neighbor states. The spatial space is implemented as a DEVS coupled model where the internal couplings between cells are given by neighborhood rules [START_REF] Shiginah | A new cell space DEVS specification: Reviewing the parallel-DEVS formalism seeking fast cell space simulations[END_REF].

Closure under Coupling

The property closure under coupling, in DEVS and parallel-DEVS, reports that every coupled model has its own equivalent atomic model. Therefore a DEVS coupled model regroups several DEVS models, which can be regarded as another DEVS atomic model.

The transition from the parallel-DEVS coupled model into its equivalent parallel-DEVS atomic model is described as follows [START_REF] Shiginah | Multi-Layer Cellular DEVS Formalism for Faster Model Development and Simulation Efficiency[END_REF][START_REF] Shiginah | A new cell space DEVS specification: Reviewing the parallel-DEVS formalism seeking fast cell space simulations[END_REF]:

The state set S of the subsequent model will be the Cartesian product of the total state sets of all the DEVS atomic models. Thus, the time advance t a (s) defines the time remaining to the next event in component d. if e = t a (s) and

S = × d∈D Q d t a (s) =
x b = φ Where δ ext : × d∈D Q d × X → × d∈D Q d δ conf : × d∈D Q d × X → × d∈D Q d δ int : × d∈D Q d → × d∈D Q d λ : × d∈D Q d → Y
DEVS coupled model can be reduced to a behaviorally equivalent DEVS atomic one. Thus, the concept of closure under coupling ensures that the coupled model results in a model of the same class which has a basic specification.

3 Related Works

Optimization Techniques in DEVS Simulation

In many situations, simulate complex systems with DEVS requires machines with very high performance. To overcome this crucial need, various techniques have been used and can be classified in two categories: those that modify the DEVS formalism by adding either specific functions or variables to manage in a dynamic way the structure of the models, which require demonstrating again some of the properties of DEVS (closure under coupling, hierarchy, etc.), and those that preserve the DEVS formalism but which integrate information concerning the behavioral model structure, which decrease the modularity, flexibility and its reuse.

For both categories, different techniques have been developed in the DEVS community such as quantization [START_REF] Kofman | Quantized-state systems: A DEVS approach for continuous system simulation[END_REF], which helps in enhancing performance simulation by decreasing the number of state transitions and messages. However, [START_REF] Shiginah | A new cell space DEVS specification: Reviewing the parallel-DEVS formalism seeking fast cell space simulations[END_REF] have noted that quantization is an approximation method that satisfies the tolerance requirement, i.e, the simulation should stay within acceptable error.

The parallel and distributed DEVS simulations need more hardware as well as extra work for parallelization of prevailing sequential models [START_REF] Wang | Extending the DEVS formalism for massively parallel simulation[END_REF][START_REF] Park | Hierarchical Model Partitioning for Distributed Simulation of Hierarchical and Modular DEVS Models[END_REF][START_REF] Lee | Simulation speedup for DEVS models by composition-based compilation[END_REF]. However, [START_REF] Wainer | Standardizing DEVS Model Representation[END_REF] have noted that, the overhead of check pointing and rollback operations may result in unstable and degraded performance.

DEVS components within variable structure [START_REF] Hu | Variable structure in DEVS component-based modeling and simulation[END_REF], permit models and couplings to be dynamically added and/or removed during simulations run. This main advantage is loosen when an important number of update (add and/or remove) of models and couplings are done during a simulation cycle [START_REF] Sun | Performance measurement of dynamic structure DEVS for large-scale cellular space models[END_REF].

However, there are few works based on converting DEVS coupled into DEVS atomic to increase performance [START_REF] Shiginah | A new cell space DEVS specification: Reviewing the parallel-DEVS formalism seeking fast cell space simulations[END_REF]. Among these works, [START_REF] Lee | Simulation speedup for DEVS models by composition-based compilation[END_REF] presented a formal approach of this property with addition of scheduling mechanism, they introduced a composition-based method that converts a DEVS coupled model into its equivalent DEVS atomic model at compile time in order to speed up the simulation by accounting events and messages at compile time. The main advantage is that the resultant atomic model should keep track of all the functions of these internal models. However, this technique results in overhead as the number of the internal models grows up, and the conversion process will be more complex to be done properly. [START_REF] Beltrame | Design and Development of a Dymola/Modelica Library for Discrete Eventoriented Systems Using DEVS Methodology[END_REF] followed the same approach by converting a coupled model into atomic one in order to eliminate the message overhead based on Modelica parallel variable update. Unfortunately, this approach shows slow simulations because of the large number of variables used. Thereby, these two techniques proceed at compile time to allow the conversion.

In [START_REF] Shiginah | A new cell space DEVS specification: Reviewing the parallel-DEVS formalism seeking fast cell space simulations[END_REF], the idea is to convert a set of cellular-DEVS atomic model into its single DEVS atomic model while keeping the same accuracy and remaining an error-free approach. Thus, instead of considering each cell as an atomic model, with the closure under coupling and at the specification and development stage, the method will transform a group of cells into a non-hierarchal and non-modular model which leads to increase the performance simulations run.

Although the resulting model is not hierarchical, but this does not present any problem to the user since they are identical sub-models. Therefore, the user will bypass the hierarchy due to the iterative and repetitive behavior and structure of cells.

The present work is closely related to [START_REF] Shiginah | A new cell space DEVS specification: Reviewing the parallel-DEVS formalism seeking fast cell space simulations[END_REF]. Our work focuses mainly on the usage of the memory resource. We apply the property closure under coupling in the modeling of forest fire to obtain reliable simulations by decreasing the use of memory resource; and as a collateral profit, we speed up the simulation run and we increase considerably the number of concurrent simulated cells.

Forest fire spread

Wildfires continue to cause considerable losses of human lives, wildlife and houses every year and are costly to contain. In United States of America alone, about one billion dollars is spent annually on wildfire suppression and containment [START_REF] Ntaimo | DEVS-FIRE: Towards an integrated simulation environment for surface wildfire spread and containment[END_REF][START_REF] Hu | DEVS-FIRE: design and application of formal discrete event wildfire spread and suppression models[END_REF].

Fire spread is a propagation process and needs to build simulation models that relate the system evolution as accurately as possible to plan scenarios in order to save lives and money. However, simulating wildfire spread remains a challenging problem due to the complexity of wildfire behavior.

Such far, no model has tried to combine the different features of fire behavior (vegetation, weather, topography, etc.) that are already established separately because of the important factors that influence wildfire behavior and the interaction between them [START_REF] Finney | FARSITE: Fire Area Simulator-Model Development and Evaluation[END_REF][START_REF] Ntaimo | DEVS-FIRE: Towards an integrated simulation environment for surface wildfire spread and containment[END_REF], that why there are many models in the literature.

The most common approaches to simulate fire spread are based either on vector or wave approach, or on the cellular models [START_REF] Finney | FARSITE: Fire Area Simulator-Model Development and Evaluation[END_REF]. For the former, the model produces vector fire perimeters at determined time intervals. For the latter, the cellular models simulate fire spread as a discrete process of ignitions across a regular division of the space in cells. Successive computations are carried out on each cell to ignite its neighbors.

The majority of fire models in use today are principally based on fire propagation relations developed by [START_REF] Rothermel | A mathematical model for predicting fire spread in wildland fuels[END_REF], [START_REF] Albini | Estimating wildfire behavior and effects[END_REF] and [START_REF] Anderson | Predicting wind-driven wildland fire size and shape[END_REF]. The popular Rothermel model was in particular chosen because of its robustness and stability which have been extensively tested and proven [START_REF] D'ambrosio | A model for the simulation of forest fire dynamics using cellular automata[END_REF][START_REF] Ntaimo | DEVS-FIRE: Towards an integrated simulation environment for surface wildfire spread and containment[END_REF].

Both categories of models (vector, cellular) are simulated either in discrete or continuous time [START_REF] Ntaimo | Two-dimensional fire spread decomposition in cellular DEVS models[END_REF]. The following non-exhaustive list of examples uses the Rothermel model. HFIRE [START_REF] Morais | Comparing Spatially Explicit Models of Fire Spread through Chaparral Fuels: A New Model Based Upon the Rothermel Fire Spread Equation[END_REF] includes discrete time model, FARSITE [START_REF] Finney | FARSITE: Fire Area Simulator-Model Development and Evaluation[END_REF] where the fire growth is based on Huygens principle of wave propagation, and BEHAVE [START_REF] Burgan | BEHAVE: Fire Behavior Prediction and Fuel Modeling System-FUEL Subsystem[END_REF] uses continuous simulation for the vector models.

The forest fire simulation models include cellular space, cellular automata [START_REF] D'ambrosio | A model for the simulation of forest fire dynamics using cellular automata[END_REF], cellular-DEVS models [START_REF] Vasconcelos | Simulation of fire growth in GIS using discrete event hierarchical modular models[END_REF], DEVS-Fire [START_REF]Forest fire spread and suppression in DEVS[END_REF], and cell-DEVS which uses heat transfer partial differential equations to calculate fire spread in each cell [ [START_REF] Ameghino | Models of complex physical systems using cell-DEVS[END_REF][START_REF] Muzy | Comparing simulation methods for fire spreading across a fuel bed[END_REF]].

Cellular-DEVS and Cell-DEVS simulation results were compared and the analysis concluded the validation of both models [START_REF] Muzy | Comparing simulation methods for fire spreading across a fuel bed[END_REF]. However, the present cellular implementation knows a simulation issue whenever all cells in the cell space are created simultaneously which causes simulation performance degradation. This degradation is attributed to the initialization of the cells and the functions needed, and the memory that is required to run the simulation [START_REF] Ntaimo | DEVS-FIRE: Towards an integrated simulation environment for surface wildfire spread and containment[END_REF].

A dynamic structure was proposed to overcome this issue [START_REF] Ntaimo | DEVS-FIRE: Towards an integrated simulation environment for surface wildfire spread and containment[END_REF][START_REF] Hu | DEVS-FIRE: design and application of formal discrete event wildfire spread and suppression models[END_REF]. This approach keeps track of active cells along the fire front as the simulation proceeds. The cells are created/deleted dynamically; however this implementation has computational overhead at runtime when we proceed with a large-scale cell space.

Other implementations were proposed to overcome these concerns. Indeed, besides using high performance computers, researchers still develop other techniques including innovative algorithms that allow handling only active cells or research on how the non-modularity/non-hierarchy in modeling can improve simulations run [START_REF] Shiginah | A new cell space DEVS specification: Reviewing the parallel-DEVS formalism seeking fast cell space simulations[END_REF][START_REF] Hamri | A non-modular modeling and simulation approach based on DEVS for the forest fire spread[END_REF].

Formal Specification and Design Issue

The M&S of forest fire spread is confronted to size of the map in a geographic information system (GIS). So, obtaining reliable and accurate simulations needs huge computation performance. The aim of this section is to show the benefit of applying the closure under coupling property of DEVS to enhance the use of memory space and increase the size of the map on GIS during simulation runs.

Problem Description

The forest is represented as a 2D cell space of square cells whose dimensions depend on the resolution of the GIS. Each cell has eight neighbors and carried out its local computation of the rate of fire spread and direction based on its local conditions (parameters). The literature classifies the parameters which set the fire spread ratio into three groups: vegetation type (caloric content, density, etc.), fuel properties (vegetation size) and environmental parameters (wind speed, humidity and slope). The flaming fire evolves principally according to the wind speed and its direction [START_REF] Scot | Standard fire behavior fuel models: a comprehensive set for use with Rothermel's surface fire spread model[END_REF]. We assume, in this work, that fuel, topography and weather conditions are uniform for each cell.

This work is closely related to [START_REF] Ntaimo | DEVS-FIRE: Towards an integrated simulation environment for surface wildfire spread and containment[END_REF][START_REF] Hu | DEVS-FIRE: design and application of formal discrete event wildfire spread and suppression models[END_REF]. The rate of fire spread of each cell relies on Rothermel model [START_REF] Rothermel | A mathematical model for predicting fire spread in wildland fuels[END_REF] which is decomposed thereafter into eight spreading directions according to the cell neighborhood and schedules ignition of its neighboring cells accordingly.

The ignition process consists on spreading fire from a burning cell to its neighbors using Moore neighborhood. A cell space is considered as a DEVS atomic model, therefore, the forest cell space is a coupled model composed of a set of cell atomic models.

We consider that each cell can be in one of the following possible states:

• Nonflammable (N): It can be a road, a surface of water or just an empty surface.

• Unburned (U): Passive state; it represents any fuel which is not consumed yet by fire.

• Burning (B): represents a consuming fire.

• Ash (A): It is afterburning state, it is the final combustion process state.

Functional Architecture

The overall system proposed in this work is composed of (Figure 1): • Forest cell space, in which each forest cell is an atomic model.

• Geographic information system interface which provides the different data that influence the spread rate. It consists of:

-Environmental parameters interface: provides weather and topography data such as wind speed, its direction, relative humidity, slope, aspect, etc., -Fuel properties: provide size of vegetation, fuel model, etc., and -Vegetation type: provides density of vegetation, its calorific content, etc.

• Forest cell igniter: ignites an initial set of cells to start the simulation, and • Rothermel model: computes fire spread and its direction.

In its basic form, Forest cell space is modeled as a grid composed of n rows and m columns, which depends on the data resolution. The dynamic system of the flaming front propagation speed is given by the simulator. It is based on the current cell position and its own variables, each of which is given by an appropriate model. The forest cell igniter is a DEVS atomic model; it is coupled to all forest cells to ignite cells. The Environmental interface and Fuel and Vegetation interface provide two kinds of values: spatial-temporal and environmental data. These data are fed into the forest cells which in turn send these data to Rothermel model to compute the rate of fire spread and its main direction.

Formal Specification and Implementation

In this section, we are going to describe the overall system design. As seen above, the fire spread model is composed of forest cell space coupled to the igniter model, Rothermel model, and GIS interface (environmental, vegetation and fuel models).

Forest cell DEVS atomic model

For each cell O = (a, b), its Moore neighborhood is given by the set:

V = {O, N, N E, E, SE, S, SW, W, N W }
Each neighbor is reported as one of eight compass points (N, N E, E, SE, S, SW, W, N W ). In Figure 2, the fire is propagated from the center of the cell to the center of the neighbor cells. Each forest cell DEVS atomic model has four states: Nonflammable (N), Unburned (U), Burning (B) and Ash (A). All cells are assumed to be initially in unburned state, except those defined by the user in the passive state Nonflammable.

Once a cell is ignited it computes the ignition delays for its eight neighbors, and sends out the ignition message for each of them. Therefore, each cell is ignited either by its neighbor or by the igniter model. The ignited cell, in its turn, ignites its neighbors. Thus, an additional input port is coupled with each forest cell DEVS atomic model. The ignition delays are calculated by the Rothermel model. Each cell sends out its state variables (fuel model, size of vegetation, wind speed, wind direction, etc.) obtained from the GIS interface toward the Rothermel model. The latter sends out as a result the rate of fire spread and its main direction to the concerned cell. Therefore the forest cell interacts with the weather and fuel interface, and the Rothermel model via these additional input/output ports.

Consequently, each forest cell has 11 input ports and 9 output ports by which it reacts and acts on its environment (Figure 3).

The dynamics of fire spread is described as follows: once a cell is ignited, it sends out ignition messages one by one in each of the eight directions. The decomposition of fire spread in each direction is based on the model that defines the fire shape as an ellipse [START_REF] Hu | DEVS-FIRE: design and application of formal discrete event wildfire spread and suppression models[END_REF][START_REF] Finney | FARSITE: Fire Area Simulator-Model Development and Evaluation[END_REF]. The contagion process of fire spread across a cell considered in our case study is center-to-center as in [START_REF] Ntaimo | Two-dimensional fire spread decomposition in cellular DEVS models[END_REF]. Consequently, the burning cell remains in this state at least until all the eight messages will be sent out. Let us consider:

• T td = {t 1 , t 2 , t 3 , t 4 , t 5 , t 6 , t 7 , t 8 }, the set of these eight different time delays sorted respectively from the smallest to the largest. • ∆T td = {ρ 1 , ρ 2 , ρ 3 , ρ 4 , ρ 5 , ρ 6 , ρ 7 , ρ 8 }, the set of the time interval between two adjacent ignition messages.

ρ 1 = t 1 and ρ i = (t i -t i-1 )|i = 2..8.
The first message is sent at t + t 1 and the last one at t + t 8 , at this moment, the forest cell transitions from the burning state to the ash one (burned) after sending out its last ignition message. The set T td can be affected by the parameters change (wind flow, speed, direction) and consequently the set ∆T td . In this case, the forest cell space model notifies the Rothermel model which updates the rate of fire spread and subsequently the spread in each direction. The rate of fire spread in each direction is depicted on Figure 4. The formal specification of the forest cell DEVS atomic model (Figure 5) is:

OutNW

F orestCellAM = (X,
= R 9 × R d × R 2
Where dimension 9 represents the nine ignition messages (8 from neighbors and one from igniter model), d is the number of parameters of each forest cell (weather, wind, humidity, etc.), and the number 2 is the couple, rate of fire spread and its main direction.

OutP orts = {OutN, OutN E, OutE, OutSE, OutS, OutSW, OutW, OutN W, OutP ar} Y OutP orts = R 8 × R d
We have 8 output ignition messages coupled respectively to the 8 neighbor cells; and d is a dimension, that represents the cell parameters, its output is coupled with Rothermel model.

X = {(in, x)|in ∈ InP orts, x ∈ XInP orts} S = {(N onf lammable, ∞), (U nburned, ∞), (Burning, ρ), (Ash, ∞)}|ρ ∈ R + Y = {(out, y)|out ∈ OutP orts, y ∈ Y OutP orts} δ int (Burning, ρ) = (Burning, ρ i ) if(∆T td = ∅) ∆T td = ∆T td -ρ ρ i = min(∆T td ) δ int (Burning, ρ) = (Ash, ∞)
if|∆T td | = 1(i.e. ∆T td has a cardinality of 1) δ ext (U nburned, ρ, (In?x)) = (Burning, ρ 1 ) δ ext (Burning, ρ, (InP ar?x)) = (Burning, ρ i ) δ conf (s, ρ, x) = δ ext (δ int (s), 0, x) λ(Burning) = Out!ignition t a (phase, ρ) = ρ

The initial state of this model is (U nburned, ∞).

Where In ∈ InP orts -{InP ar, InRot}, ρ i is computed by the Rothermel model whenever parameters change, Out ∈ OutP orts -{OutP ar}, ignition ∈ R. In Figure 5, the solid line describes the occurrence of external event, while the dashed line depicts the internal state changes, and occurs when the elapsed time reaches the lifetime of the state.

In?x Out!ignition

Forest cell atomic model

Forest Fire Spread and Decomposition

The Rothermel mathematical model [START_REF] Rothermel | A mathematical model for predicting fire spread in wildland fuels[END_REF] computes a one-dimensi-onal maximum fire spread rate and its direction. Fire geometry has been determined empirically [START_REF] Finney | FARSITE: Fire Area Simulator-Model Development and Evaluation[END_REF]. However, in the literature, the fire shape is considered as an ellipse which is widely used [START_REF] Anderson | Predicting wind-driven wildland fire size and shape[END_REF]. Therefore the fire spread is inferred in all directions from the 1-D maximum rate using the mathematical properties of the ellipse and the equations defined in [START_REF] Finney | FARSITE: Fire Area Simulator-Model Development and Evaluation[END_REF][START_REF] D'ambrosio | A model for the simulation of forest fire dynamics using cellular automata[END_REF]. The spread rate in an arbitrary direction θ (Figure 6) is obtained by:

R(θ) = R max 1 - 1 -cos θ (1) = √ lw 2 -1 lw (2) lw = 0.936e 0.2566v + 0.461e -0.1548v -0.397 (3)
Where R max is maximum rate of spread, lw is the ellipse ratio of the semi-major over semiminor (Length to breath ratio) and v is the midflame wind speed (effective wind speed).

Once the rate of spread is known in all directions, we compute the time delays to reach the center of the neighbors by this equation:

t i = d i R i |i = 1..8 (4)
Where t i is the time to reach the neighbor i, d i is the distance from the center of the burning cell to the center of its neighbor i and R i is the fire spread rate in this direction. In case of parameters change (wind speed, direction, humidity, etc.), the new R new i in direction of the neighbor i, is inferred from Rothermel model if the fire hasn't yet reached it. The new remaining time delay is: Where ∆d i is the remaining distance to reach the neighbor i and t new i is the new time delay. As the fire propagates, the equation 5 allows the model to be dynamic and sensitive to weather changes.

t new i = ∆d i R new i ( 5

System Formal Design by Closure under Coupling

In the forest cellular approach, the forest fire spread is usually presented as a set of arranged cells whose dimensions depend on the resolution of GIS [START_REF] Hu | DEVS-FIRE: design and application of formal discrete event wildfire spread and suppression models[END_REF]. Thus, forest cell space coupled model is a discrete dynamical system formed by coupling a finite number of forest cells. These cells are arranged uniformly in a two-dimensional space composed of n rows and m columns. Each forest cell atomic model is coupled to 8 neighbors as described in Figure 7. The fire spread model is described in Section 4.2. It is composed of forest cell space (a grid of forest cells), igniter model, GIS interface and Rothermel model. However, simulate such a model will be affected by large-scale forest cell model and particularly during the simulation run and involves more memory usage. A huge memory is needed to simulate the overall forest and if you lack memory, you simulate just a part of it. In [START_REF] Ntaimo | DEVS-FIRE: Towards an integrated simulation environment for surface wildfire spread and containment[END_REF], memory required for some implementations is illustrated.

To overcome to this issue, we propose to use the closure under coupling. This property of the DEVS formalism consists to transform a coupled model into its equivalent atomic model. Thus, the forest cell space coupled model will be converted into an atomic model.

Each cell becomes a state variable of this atomic model and accedes to its neighbor state directly by removing the ports from the atomic models that is resulted by small volume of inter-component messages during simulation run. In fact, the communication between cells is done directly inside the model, except outdoor events [START_REF] Shiginah | A new cell space DEVS specification: Reviewing the parallel-DEVS formalism seeking fast cell space simulations[END_REF].

Based on the Forest Cell DEVS atomic model described in Section 4.3.1, the forest cell space DEVS coupled model is transformed into forest cell space DEVS atomic model and would benefit from the property of closure under coupling. It is described as follows:

F orestCellSpaceAM = (X, S, Y, δ int , δ ext , δ conf , λ, t a )

• X = {(ignite, list), (inP aram, parameters)} -list is a list of initial ignited cells

• Y = ∅, there is no output event to send out.

• S = Cell × inP aram = {. . . , (phase i,j , ρ i,j ), . . . } × R d |i = 1..n j = 1..m
Where Cell is the set of forest cell. Each cell is identified by its position (row, column) which can be in one of the four states (Nonflammable, Unburned, Burning, Ash) and d is the number of parameters provided by GIS interface. Each cell keeps its current state and according to external and internal events state changes will occur to update the state of the concerning cell. It is pointed out that each cell has eight neighbors except those situated on the bound of the forest. Environmental parameters are assumed to be uniform at each cell except the wind which can change its speed and direction over time. Thus, the fire spread direction can change dynamically. Thus, we get a non-modular modeling structure (see Figure 8). With a modular modeling, when a cell ignites its neighbors, it invokes internal state change to provoke the execution of the output function λ(), which sends out the event ignition to all neighbors. On the other hand, the non-modular modeling propagates the output event ignition directly to neighboring cells without calling the simulator for dispatching events. Such a communication optimizes the DEVS M&S structure and decreases the number of exchanged messages between cells. Consequently, the fire spread model in its non-modular structure is comprised of four atomic models: Igniter atomic model, Forest cell space atomic model, GIS interface and Rothermel model as shown on Figure 9. It is identical to the structure depicted in Figure 1 except the Forest Cell Space which is converted to an atomic model.

Note that this architecture remains an open and modular one which allows enhancing the functionalities of the GIS and updating the Rothermel package.

Simulation Experiments

In this section some output results that measure the simulation performance of the two models are provided. The package Behave was used to compute the rate of fire spread given by the Rothermel model. The experiments were carried out on laptop Dell T2300@1.66GHz with Intel CPU, 1.66 GHz processor, 2.50 GB of RAM and Windows XP c operating system and as a simulator, we have used our own simulator developed in Java and respecting DEVS specifications.

Rothermel Model

Igniter Model

GIS interface (Fuel, vegetation and environment )

Forest Cell Space Atomic model

Comparison Results

As mentioned by [START_REF] Ntaimo | DEVS-FIRE: Towards an integrated simulation environment for surface wildfire spread and containment[END_REF], an assessment of memory usage based on the implementation in [START_REF] Devsjava | DEVS M&S tool[END_REF] shows that each cell needs about 35 kB memory space. This fact is time consuming and resource intensive and therefore not very effective for large-scale cells simulation. We were tempted to see the limits of our simulator for these two specifications of forest fire spread. The ForestCellSpace DEVS coupled model versus ForestCellSpace DEVS atomic model.

Table 1 summarizes some important results. The model ForestCellSpace coupled model uses a conventional architecture, where a grid of cells is used. While ForestCellSpace atomic model uses the closure under coupling to get a unique atomic model of the forest cell space. The experiment revealed that only about 70×70 cells of the overall forest space has been simulated in coupled model, whereas the other specification, i.e., forest cell atomic model, a limit of 2682×2682 cells was reached therefore, we get a ratio close of 1 over 39 for each cell.

Another advantage of the forest cell space atomic model is the simulation time; in fact, the latter is 8.15 times faster than the forest cell space coupled model.

In order to get more results, we have used additional free software Java Virtual Machine Monitor (JVM Monitor) to profile our Java application. The snapshot of the Figure 10(a Table 2 gives the performance indicators of both Forest Cell Space simulations coupled and atomic at some critical times which are extracted from curves of Figure 10 in case of 60 × 60 cells.

As mentioned before, despite the different implementations of forest fire spread, it is essential to reminder that both models have the same behavior, whereas their simulation performances are different (see Figure 11 for a cell space of 60×60).

The front flame is depicted in color yellow while the black color is dedicated to the ash area. The colors pink, orange, red and light gray represent the different phases of the burning state (flanking and backing spread).

Verification and validation

The validation of fire spread model is not an easy task, and few works describe validations based on real fire [START_REF] D'ambrosio | A model for the simulation of forest fire dynamics using cellular automata[END_REF]. Nevertheless, we are going to do a partial validation by comparing the results of our approach by those of FARSITE. The latter is the most generally distributed and accepted fire behavior predictive model used in forestry [START_REF] Ntaimo | DEVS-FIRE: Towards an integrated simulation environment for surface wildfire spread and containment[END_REF].

On one hand, we are going to simulate our model described in Section 4.2 by adopting this functional architecture, on the other hand, we are going to simulate by FARSITE software the same scenario and compare the different results. Therefore, the equivalence of the two forms is verified using simulation methods.

In these simulations, we suppose that:

• Virtual forest is constructed as a grid of 260×260 cells where each cell represents an area of 1×1 m 2 (m: meter).

• Fire spreading is on each of the eight compass points.

• Starting ignited point is the cell(130,30).

We assume that uniform parameters characterize the cell space, i.e., each cell has uniform fuel, terrain, weather and fire behavior along the forest fire area. The fire shape is assumed elliptical and the decomposition scheme considered is center-tocenter. We use the Rothermel model to compute the maximum fire spread rate. Fire spread in all other directions is inferred from the spread rate using the mathematical properties of the ellipse (equation 1).

We consider the following values to get the rate of spread by the Rothermel model:

The model of fuel considered in this experiment is Northern Fire Forest Laboratory (NFFL) Fuel Model 1: short grass (1 feet (ft)) with its standard characteristics.

Besides that, we assume:

- Length to breath ratio lw is obtained via equation 3, its value for theses parameters is:

-lw = 1.50 The spreading rate in each direction is given by equation 1 based on the ratio lw. Thus, we obtain the following spreading rates according to each direction as shown on Table 3.

According to these results, we simulate our forest fire model. Figure 12 shows the fire spread taken at different simulation times and gives us the effective burned area [hectare (ha)].

On the other hand, we have done the simulation on FARSITE with the same parameters and we obtained output results shown on Figure 13. A comparison was done on the progression of fire area over time (see Figure 14). The result shows the likeness between both methods. Of course, the goal is not to mimic FARSITE software behavior, it is possible to fine-tune or parameterize our model so that it mimics FARSITE and obtain the same graph. However, the real purpose is to show the vitality of our approach. As you can see, the same scenario can't be simulated with the forest cell space coupled model and we will get a message as Out of memory error. A solution suggested is, either we divide this forest space into small areas or decrease the cell resolution (increase cell dimension) which may result in incorrect results and degrade accuracy. As cited previously, this is a partial and preliminary validation. More validation is definitely required because of the many potential sources of error which can confuse the comparisons.

These tests have underlined the necessity to consider errors associated with:

• The spatial and temporal resolution of the inputs,

• The model input data (fuels and environmental parameters) and

• The nature of fire growth projections used for comparison in both methods (Huygens principle of wave propagation in FARSITE versus Rothermel spread equation in Behave package).

Logically, a wildfire spread simulation should be most accurate when using accurate data at high spatial and temporal resolutions.

The acceptability of this statement is dependent on the spatial resolution required by the user and the resolution specified for the simulation. Thus, the sensitivity of fire simulation to the resolution, fire spread patterns generated (Huygens principle) and model of different fire input parameters remain to be verified.

However, it should be noted that even FARSITE needs to be validated [START_REF] Finney | FARSITE: Fire Area Simulator-Model Development and Evaluation[END_REF], the most important result of FARSITE tests to date has been proven that spread rates for all fuel models tended to be over predicted by the Rothermel spread equation. Different adjustments have been added in order to correct the observed error with real forest fire, but there is no way to guide settings for these adjustments. Consequently, future work needs to focus more on validating our approach using observed fire growth patterns in several conditions of weather and vegetation.

Conclusion and Future Works

This work attempts to foster the integration between two concepts: closure under coupling in cellular-DEVS and the forest fire spread to gain memory resource during simulation run.

The study of the simulation of forest fire has increased considerably, it requires building simulation models that allows for system evolution in both time and space.

The most common approaches to simulate fire spread are based either on vector approach, or on the cellular models. Cellular models simulate fire spread as a discrete process of ignitions across a regular division of the space in cells. However, dealing with high cell resolutions naturally defies efficient computer simulation.

In this work, we tried to gain some performance in simulation run. This was achieved by converting the cell space into atomic model in order to eliminate inter-cell messages. This approach was initially deduced from the property closure under coupling of the DEVS formalism. Some relevant results are presented and a comparison between our approach and the simulation by FARISTE software is given.

However, this implementation needs an additional work in verification and validation (which we crossed an important step) to confirm more again the vitality of this approach. The functional architecture needs to take into account more parameters since the fire spread process might become complicated. A model for these parameters must be coupled to our architecture in order to get more realistic results. Also we need to validate our approach by using observed fire growth patterns in several conditions of weather and vegetation. An issue that we hope explore in the near future.

  minimum{ρ d |d ∈ D} Where s ∈ S, s = (. . . , (s d , e d ), . . .) for all d ∈ D and ρ d = t a (s d ) -e d The overall transition function is defined as: δ(s, e, x b ) =    δ ext (s, e, x b ) if 0 ≤ e < t a (s) and x b = φ δ conf (s, e, x b ) if e = t a (s) and x b = φ δ int (s)

Figure 1 :

 1 Figure 1: Functional architecture of fire spread simulation.

Figure 2 :

 2 Figure 2: Cell center-to-center fire spread.

Figure 3 :

 3 Figure 3: Forest cell DEVS atomic model.

Figure 4 :

 4 Figure 4: Rate of spread calculation in eight directions.

Figure 5 :

 5 Figure 5: Forest cell DEVS behavior.

Figure 6 :

 6 Figure 6: Rate of spread according to an arbitrary direction.

Figure 7 :

 7 Figure 7: Forest cell space DEVS coupled model.

  δext(s, e, x) c, c´: Cell if (x = ignite){ for each c ∈ ignite list sc = δext (sc, e, ignite)} if (x = inParam) update parameters recompute the lifetime for each active cell c δ int (s) for each c ∈ Cell{ if (lifetime(c) = lifetime(Cell)){ if (burning(sc)) for each c´∈ neighbor(c) s c = δext(s c , lifetime(Cell), ignite) sc = δint(sc)} else lifetime(c) = lifetime(c) -lifetime(Cell) } δ conf (s, e, x) δext(δint(s), 0, x) lifetime(s) return min {lifetime(c) | c ∈ Cell}

Figure 8 :

 8 Figure 8: DEVS atomic model functions of forest fire spread

Figure 9 :

 9 Figure 9: The forest fire spread model via closure under coupling.

  ) resumes some characteristics of simulation of ForestCellSpace coupled model; whereas Figure 10(b) gives the values of these characteristics of ForestCellSpace atomic model. Through these Figures 10(a) & 10(b) and by looking inside execution time windows of each simulation, we can see the advantages of the usage of closure under coupling property. We have less amount of used heap memory. The maximum values reached with Forest Cell Space Coupled Model are greater in overall cases compared to Forest Cell Space Atomic Model.

  (a) ForestCellSpace coupled model: grid of cells. (b) ForestCellSpace atomic model: closure under coupling.

Figure 10 :

 10 Figure 10: Performances of forest fire spread simulation.

  (a) DEVS coupled: time = 900 sec. (b) DEVS atomic: time = 900 sec. (c) DEVS coupled: time = 1600 sec. (d) DEVS atomic: time =1600 sec.

Figure 11 :

 11 Figure 11: ForestCellSpace coupled model vs. ForestCellSpace atomic model.

  wind speed [meter/second (m/s)] wsp = 1.0 wind direction [degree( • )] wdr = 0.0 slope [ • ] slp = 16.7 aspect [ • ] asp = 0.0 The Rothermel model produces the following results: rate of spread [m/s] ros = 0.26 direction of maximum spread [ • ] sdr = 180 effective wind speed [m/s] efw = 1.37

Figure 12 :

 12 Figure 12: Fire spread progression (260×260 cells)

  (a) Fire perimeter. (b) Burned area curve.

Figure 13 :

 13 Figure 13: Fire spread progression on FARSITE

Figure 14 :

 14 Figure 14: Fire area: FARSITE vs. DEVS.

Rothermel Model Igniter Model Vegetation Type Fuel Properties Environmental Parameters Slope Wind Humidity density Veget. size Forest Cell Space GIS Caloric content

  S, Y, δ ext , δ int , δ conf , λ, t a ) where:

	InP orts = {InN, InN E, InE, InSE, InS, InSW, InW, InN W, InIgn,
	InP ar, InRot}
	XInP orts

Table 1 :

 1 Comparison results.

	Results	ForestCellSpace	ForestCellSpace
		Coupled Model	Atomic Model
	Maximum number of simulated cells	70×70	2682×2682
	[cells]
	Used Memory [MegaBytes (MB)] for	8.74	1.23
	60×60 cells
	Used Memory [MB] for 600×600 cells Out of Memory Er-	12
		ror
	Simulation Execution Time [second	109,131	13,390
	(sec)] for 60×60 cells

Table 2 :

 2 Sample of simulation performances of ForestCellSpace: Coupled vs. Atomic

	Indicator	ForestCellSpace	ForestCellSpace
		Coupled Model	Atomic Model
	Simulation Execution Time	109,131	13,390
	[sec.]
	Used Heap Memory [MB]	[2, 20]	[2, 8]
	Thread Count	[9, 3500]	[9, 14]
	CPU Usage [%]	100	100

Table 3 :

 3 Spreading rate in the eight directions

	θ	R(θ)	% ros
	0	0.26	100
	∓ 45	0.14	53.77
	∓ 90	0.066 25.41
	∓ 135 0.04	16.63
	180	0.038 14.55