B Lods

M Mokhtar-Kharroubi

M Sbihi

SPECTRAL PROPERTIES OF GENERAL ADVECTION OPERATORS AND WEIGHTED TRANSLATION SEMIGROUPS

Keywords: Spectral Mapping Theorem, weighted shift, Annular Hull Theorem, transport equation. 1

We investigate the spectral properties of a class of weighted shift semigroups (U(t)) t 0 associated to abstract transport equations with a Lipschitz-continuous vector field F and noreentry boundary conditions. Generalizing the results of [25], we prove that the semigroup (U(t)) t 0 admits a canonical decomposition into three C0-semigroups (U1(t)) t 0 , (U2(t)) t 0 and (U3(t)) t 0 with independent dynamics. A complete description of the spectra of the semigroups (Ui(t)) t 0 and their generators Ti, i = 1, 2 is given. In particular, we prove that the spectrum of Ti is a left-half plane and that the Spectral Mapping Theorem holds: S(Ui(t)) = exp {tS(Ti)}, i = 1, 2. Moreover, the semigroup (U3(t)) t 0 extends to a C0-group and its spectral properties are investigated by means of abstract results from positive semigroups theory. The properties of the flow associated to F are particularly relevant here and we investigate separately the cases of periodic and aperiodic flows. In particular, we show that, for periodic flow, the Spectral Mapping Theorem fails in general but (U3(t)) t 0 and its generator T3 satisfy the so-called Annular Hull Theorem. We illustrate our results with various examples taken from collisionless kinetic theory.

INTRODUCTION AND PRELIMINARIES

We develop in the present paper a systematic approach to the spectral analysis in L p -spaces (1 p < ∞) of a class of weighted shift semigroups arising in kinetic theory

U(t) : f -→ U(t)f (x) = exp - t 0 ν(Φ(x, -s))ds f (Φ(x, -t))χ {t<τ -(x)} (x) (1.1)
where the flow Φ(x, t) is associated to a globally Lipschitz transport field F and ν(x) is given by

ν(x) = h(x) + div(F )(x), x ∈ Ω
where the functions h and F satisfy the following:

Assumption 1.1. The field F : Ω → R N is Lipschitz-continuous with Lipschitz constant κ > 0.

Moreover, its divergence div(F ) is a bounded function on Ω. The absorption function h(•) is measurable and bounded below.

Before explaining more in details the contents of this work, we have to explicit a bit the first properties of the different terms arising in Eq. (1.1). Since F is Lipschitz over Ω (with constant κ > 0), it is known from Kirszbraun's extension theorem [17, p. 201], that F can be extended as a Lipschitz function (with the same Lipschitz constant κ > 0) over the whole space R N . We shall still denote this extension by F . In Eq. (1.1), Φ(x, t) is the unique maximal solution of the characteristic equation d dt X(t) = F (X(t)), (t ∈ R);

X(0) = x, (1.2)
which is well-defined since the (extended) field F is globally Lipschitz. In (1.1), τ -(x) denotes the stay time in Ω of the characteristic curves t > 0 → Φ(x, -t) starting from x ∈ Ω :

τ ± (x) = inf{s > 0 ; Φ(x, ±s) / ∈ Ω}, (1.3)
with the convention that inf ∅ = ∞. In other words, given x ∈ Ω, I x = (-τ -(x), τ + (x)) is the maximal time interval for which the solution X(t) lies in Ω for any t ∈ I x . We shall denote by τ (x) := τ + (x) + τ -(x) the length of the maximal interval I x .

The general strategy we adopt to describe the spectral properties of the semigroup (U(t)) t 0 consists in a canonical decomposition of the semigroup (U(t)) t 0 into three semigroups (U i (t)) t 0 , (i = 1, 2, 3) with independent dynamics, the third one (U 3 (t)) t 0 extending to a C 0 -group. Notice that it would be possible to investigate the spectral properties of (U 3 (t)) t∈R within the general framework developed in [13, Chapter 6] (see also [START_REF] Yu | Weighted translation operators and linear extensions of dynamical systems[END_REF]). The approach of [START_REF] Chicone | Evolution semigroups in dynamical systems and differential equations[END_REF] uses sophisticated tools from dynamical systems theory while our approach is completely different and relies on general results concerning the spectral properties of C 0 -groups in L p -spaces given in the Appendix.

1.1. Preliminaries and motivation. If τ (x) is finite, then the function X : s ∈ I x -→ Φ(x, s) is bounded since F is Lipschitz continuous. Moreover, still by virtue of the Lipschitz continuity of F , the only case when τ ± (x) is finite is when Φ(x, ±s) reaches the boundary ∂Ω so that Φ(x, ±τ ± (x)) ∈ ∂Ω. We finally mention that it is not difficult to prove that the mappings τ ± : Ω → R + are lower semi-continuous and therefore measurable [7, p. 301]. Note that, since the field F is not assumed to be divergence-free, then the transformation induced by the flow Φ is not measure-preserving. Precisely, one has the following [START_REF] Lasota | Chaos, Fractals, and Noise. Stochastic Aspects of Dynamics[END_REF][START_REF] Arlotti | On transport equations driven by a non-divergence free force field[END_REF]: Proposition 1.2. Let ̺ t denote the image of the N -dimensional Lebesgue measure m through the transformation T t : x → Φ(x, -t), (t ∈ I x ). Then, ̺ t is absolutely continuous with respect to m, and its Radon-Nikodym derivative d̺t dm with respect to m is given by d̺ t dm (x) = exp t 0 div(F )(Φ(x, s))ds for ma.e. x ∈ Ω, t ∈ I x .

The semiflow Φ enjoys the following elementary properties [START_REF] Banasiak | Perturbations of positive semigroups with applications[END_REF] Proposition 1.3. Let x ∈ Ω and t ∈ R be fixed. Then, (i) Φ(x, 0) = x.

(ii) Φ(Φ(x, s 1 ), s 2 ) = Φ(x, s 1 + s 2 ), ∀s

1 ∈ I x , s 2 ∈ (-s 1 -τ -(x), τ + (x) -s 1 ). (iii) |Φ(x 1 , t) -Φ(x 2 , t)| exp(κ|t|)|x 1 -x 2 | for any x 1 , x 2 ∈ Ω, t ∈ I x 1 ∩ I x 2 .
In all the paper, we fix 1 p < ∞ and set X = L p (Ω, dm).

The following classical result (see, e.g., [START_REF] Banasiak | Perturbations of positive semigroups with applications[END_REF][START_REF] Arlotti | On transport equations driven by a non-divergence free force field[END_REF][START_REF] Bardos | Problèmes aux limites pour les équations aux dérivées partielles du premier ordre à coefficients réels; théorèmes d'approximation; application à l'équation de transport[END_REF]) asserts that the family (U(t)) t 0 given by (1.1) is a strongly continuous semigroup of bounded operators in X.

Theorem 1.4. Let

U(t)f (x) = exp - t 0 ν(Φ(x, -s))ds f (Φ(x, -t))χ {t<τ -(x)} (x), x ∈ Ω, f ∈ X,
where χ A denotes the characteristic function of a set A. The family (U(t)) t 0 is a positive C 0semigroup of bounded operators in X. We shall denote by (T , D(T )) its generator.

In the present paper, we do not need to explicit further the generator (T , D(T )) of (U(t)) t since our spectral analysis does not depend on its description. Note however that, if Ω is a sufficiently smooth open subset of R N , then the generator (T , D(T )) is explicitly described in [START_REF] Arlotti | On transport equations driven by a non-divergence free force field[END_REF] for h = 0. In some sense, which we do not explicit here (see for instance [START_REF] Arlotti | On transport equations driven by a non-divergence free force field[END_REF][START_REF] Arlotti | Semigroups for general transport equations with abstract boundary conditions[END_REF]), the semigroup (U(t)) t 0 governs the following advection equation:

∂ t f (x, t) + ∇ x • (F (x)f (x, t)) + h(x)f (x, t) = 0 (x ∈ Ω, t > 0), (1.4a)
supplemented by the boundary condition

f |Γ -(y, t) = 0, (y ∈ Γ -, t > 0), (1.4b)
and the initial condition

f (x, 0) = f 0 (x), (x ∈ Ω), (1.4c)
where Γ -is the incoming part of the boundary of Ω (we refer the reader to [START_REF] Bardos | Problèmes aux limites pour les équations aux dérivées partielles du premier ordre à coefficients réels; théorèmes d'approximation; application à l'équation de transport[END_REF][START_REF] Banasiak | Perturbations of positive semigroups with applications[END_REF] for details on the matter), i.e. f (x, t) = [U(t)f 0 ](x) at least for regular initial data f 0 . A typical example of such an absorption equation is the so-called Vlasov equation for which: i) The phase space Ω is given by the cylindrical domain Ω = D × V ⊂ R 6 where D is a smooth open subset of R 3 , referred to as the position space, and V ⊂ R 3 is referred to as the velocity space. ii) For any x = (x, v) ∈ D × V ,

F (x) = (v, F(x, v)) ∈ R 6 (1.5)
where F = (F 1 , F 2 , F 3 ) is a time independent globally Lipschitz field (the force field) over D × V .

With the above choice, Eq. (1.4a) reads:

∂ t f (x, v, t) + v • ∇ x f (x, v, t) + F(x, v) • ∇ v f (x, v, t) + ν(x, v)f (x, v, t) = 0, (1.6)
where ν(x, v) = h(x, v) + div v (F)(x, v), supplemented with suitable initial and boundary conditions (see (1.4b)). More general problems can be handled with. For instance, one can treat with our formalism the collisionless version of the linear Boltzmann arising in semiconductor theory:

∂ t f (r, k, t) + 1 ℏ ∇ k ǫ(k) • ∇ k f (r, k, t) + e ℏ E • ∇ k f (r, k, t) + ν(r, k)f (r, k, t) = 0
where the unknown f (r, k, t) is the density of electrons having the position r ∈ R 3 , the wavevector k ∈ R 3 at time t > 0. The parameters e and 2πℏ are the positive electron charge and the Planck constant respectively while ǫ(k) represents the electron energy and E = E(r, k) is an external electric field.

The abstract equation (1.4) allows also to consider collisionless kinetic equation for relativistic models for which

v = v(p) = p m 1 + p 2 /c 2 m 2 ,
where m stands for the mass of particles and c being the velocity of light. For all this kind of models, the solution f to the collisionless kinetic equation is given, under suitable conditions on the data, by f (x, t) = U(t)f 0 (x) where f 0 corresponds to the initial state of the system and U(t) is a weighted shift semigroup of the shape (1.1). We provide in the rest of the paper a large number of examples arising in kinetic theory for which our abstract results apply. Spectral properties of systems of scalar advective equations on the torus coupled by a pseudo-differential operator of order zero (motivated by fluid mechanics ) are dealt with by Shvydkoy [START_REF] Shvydkoy | The essential spectrum of advective equations[END_REF] (see also [START_REF] Shvydkoy | Essential spectrum of the linearized 2D Euler equation and Lyapunov-Oseledets exponents[END_REF][START_REF] Shvydkoy | Cocycles and Mañe sequences with an application to ideal fluids[END_REF][START_REF] Shvydkoy | On the spectrum of the linearized 3D Euler equation[END_REF][START_REF] Chicone | The Annular Hull Theorems for the kinematic dynamo operator for ideally conducting fluid[END_REF]). We note certain similarities between some of those results and our results concerning the group (U 3 (t)) t∈R .

Main results and methodology.

To describe the spectral features of the semigroup (U(t)) t 0 and its generator T , we generalize the approach initiated in a recent work of the second author [START_REF] Mokhtar-Kharroubi | Spectral properties of a class of positive semigroups on Banach lattices and streaming operators[END_REF]. The analysis of [START_REF] Mokhtar-Kharroubi | Spectral properties of a class of positive semigroups on Banach lattices and streaming operators[END_REF] is restricted to the neutron transport equation, corresponding to the choice of F = 0 in the above equation (1.6). We generalize [START_REF] Mokhtar-Kharroubi | Spectral properties of a class of positive semigroups on Banach lattices and streaming operators[END_REF] to general vector field F . Precisely, thanks to a suitable decomposition of the phase space Ω according to the finiteness of τ -(•) and τ + (•), we show that a general weighted shift semigroup (U(t)) t 0 admits a canonical decomposition into three semigroups (U i (t)) t 0 , (i = 1, 2, 3) with independent dynamics. The third semigroup (U 3 (t)) t 0 actually extends to a C 0 -group, which corresponds to the global in time flow Φ(t, •) already investigated in [START_REF] Yu | Weighted translation operators and linear extensions of dynamical systems[END_REF]. Concerning the spectral properties of the semigroups (U 1 (t)) t 0 and (U 2 (t)) t 0 (which correspond to trajectories (Φ(x, t)) t such that either τ + (x) or τ -(x) is finite), they both enjoy the same spectral structure:

S(U i (t)) = {µ ; |µ| exp(-γ i t)}, S(T i ) = {λ ; Reλ -γ i }, i = 1, 2 (1.7)
where γ i (i = 1, 2) are positive constants, depending on h and F (see Theorems 2.4 & 2.6). The above description relies on several abstract results on positive semigroups on Banach lattices and, in particular, on the following property, proved in [START_REF] Mokhtar-Kharroubi | Spectral properties of a class of positive semigroups on Banach lattices and streaming operators[END_REF]:

Proposition 1.5. Let (T (t)) t 0 be a C 0 -semigroup of positive operators on a complex Banach lattice X. Let Y denote the subspace of local quasinilpotence of (T (t)) t 0 :

Y = u ∈ X ; lim t→∞ exp 1 t log T (t)(|u|) X = 0 . If Y is dense in X then [0, exp(ω 0 (T )t)] ⊂ S ap (T (t)) for any t 0 while (-∞, s(A)] ⊂ S ap (A), where ω 0 (T ) ∈ [-∞, ∞)
is the type of (T (t)) t 0 and s(A) denotes the spectral bound of the generator A of (T (t)) t 0 .

Such a result allows to describe very precisely the real spectrum of U 1 (t) and U 2 (t). Then, to deal with the non-real spectrum, we prove the invariance by rotation of the spectrum of U 1 (t) and U 2 (t) and the invariance by translation along the imaginary axis of their generators. This is done in the spirit of [START_REF] Voigt | Positivity in time dependent linear transport theory[END_REF], see Proposition 2.2.

The description (1.7) shows in particular that, whenever the flow Φ (and the geometry Ω) do not allow trajectories that are global in both positive and negative times (i.e. either τ + (x) or τ -(x) is finite), then the spectrum of the generator T is a left-half plane and the Spectral Mapping Theorem S(U(t)) \ {0} = exp (S(T )t) , t 0, (1.8) holds. Such a result seems to be new. Actually, we show that only the existence of periodic orbits and/or stationary points could prevent the Spectral Mapping Theorem (1.8) to hold. Indeed, when dealing with the C 0 -group (U 3 (t)) t∈R , we show that, here again, this group admits a canonical decomposition into three groups with independent dynamics (U rest (t)) t∈R , (U per (t)) t∈R and (U ∞ (t)) t∈R corresponding respectively to stationary orbits, periodic orbits and infinite orbits which are neither stationary nor periodic. The spectral analysis of (U rest (t)) t∈R is really easy to derive since U rest (t) acts as a multiplication group. On the other hand, the group (U ∞ (t)) t∈R falls within the general theory of Mather groups associated to aperiodic flow investigated in [START_REF] Chicone | Evolution semigroups in dynamical systems and differential equations[END_REF] and [START_REF] Yu | Weighted translation operators and linear extensions of dynamical systems[END_REF]. Concerning the delicate case of periodic trajectories, we deal with a description of the spectrum of the generator T per and prove (thanks to a general result on positive C 0 -groups on L p -spaces) that (U per (t)) t∈R fulfils the so-called Annular Hull Theorem

T • S(U per (t)) = T • exp t S(T per ) ∩ R , ∀t ∈ R.
Notice that, in full generality, U per (t) does not satisfy the Spectral Mapping Theorem (1.8) (see Example 4.1). The organization of the paper is as follows: in Section 2, we establish the aforementioned decomposition of (U(t)) t 0 into three semigroups (U i (t)) t 0 , (i = 1, 2, 3) with independent dynamics. Moreover, we provide a complete description of the spectrum of (U 1 (t) t 0 , (U 2 (t)) t 0 and their generators and illustrate our results by several examples from kinetic theory. The properties of the group (U 3 (t)) t 0 are investigated in Section 3 where we deal only with stationary or aperiodic flow. In Section 4, we investigate the more delicate case of a periodic flow. Finally, in the Appendix, we state some known and new abstract results on the spectral properties of positive C 0 -semigroups in L p -spaces, 1 p < ∞.

SPECTRAL PROPERTIES OF THE STREAMING SEMIGROUP (U(t)) t 0

Let us define the following subsets of Ω:

Ω 1 = {x ∈ Ω ; τ + (x) < ∞} , Ω 2 = {x ∈ Ω ; τ + (x) = ∞ and τ -(x) < ∞}
and

Ω 3 = {x ∈ Ω ; τ + (x) = τ -(x) = ∞} .
Moreover, define

X i = {f ∈ X ; f (x) = 0 m -a.e. x ∈ Ω \ Ω i } , i = 1, 2, 3.
In the sequel, we shall identify

X i with L p (Ω i , dm), i = 1, 2, 3. Since (Ω i ) i=1,...,3 is a partition of Ω, it is clear that X = X 1 ⊕ X 2 ⊕ X 3 .
Of course, if m(Ω i ) = 0, the space X i reduces to {0} and drops out in the direct sum (i = 1, 2, 3). Following [START_REF] Mokhtar-Kharroubi | Spectral properties of a class of positive semigroups on Banach lattices and streaming operators[END_REF], we can state the following:

Theorem 2.1. For any i = 1, 2, 3, X i is invariant under (U(t)) t 0 . Define U i (t) = U(t) |X i for any t 0, i = 1, 2, 3. Then, (U i (t)) t 0 is a positive C 0 -semigroup of X i (i = 1, 2, 3) and

S(U(t)) = S(U 1 (t)) ∪ S(U 2 (t)) ∪ S(U 3 (t)) (t 0) (2.1)
where S(U i (t)) stands for the spectrum of U i (t) in the space X i (i = 1, 2, 3). Moreover, (U 3 (t)) t 0 extends to a C 0 -group in X 3 .

Proof. Let f ∈ X and recall that

U(t)f (x) = exp - t 0 ν(Φ(x, -s))ds f (Φ(x, -t))χ {t<τ -(x)} (x) (t 0). Consequently, if U(t)f (x) = 0, then t < τ -(x)
U(t)f (x) = 0 m-a.e. on Ω \ Ω 1 for any f ∈ X 1 . This shows that X 1 is invariant under (U(t)) t 0 .
In the same way, still using Proposition 1.3, one observes that

τ -(Φ(x, -t)) = τ -(x) -t ∀t < τ -(x) so that τ -(x) < ∞ if and only if τ -(Φ(x, -t)) for any t < τ -(x).
As before, this leads to the invariance of both X 2 and X 3 under (U(t)) t 0 . Thus, the triplet (X 1 , X 2 , X 3 ) reduces (U(t)) t 0 and (2.1) follows. Finally, defining U 3 (-t)f (x) for any t 0, x ∈ Ω 3 and f ∈ X 3 as

U 3 (-t)f (x) = exp 0 -t ν(Φ(x, -s))ds f (Φ(x, t))
one obtains easily the last assertion.

From the above Theorem, to describe the spectra of (T (t)) t 0 and T , one can deal separately with the spectral properties of the various semigroups (U i (t)) t 0 and their generator T i on X i , i = 1, 2, 3. We will show in the rest of this paper that our analysis of (U 1 (t)) t 0 and (U 2 (t)) t 0 differs very much from that of the group (U 3 (t)) t∈R . In particular, to show the rotational invariance of the spectra of the formers, we will make use of the following result [START_REF] Voigt | Positivity in time dependent linear transport theory[END_REF]:

Proposition 2.2. Let Ω be a measurable subset of Ω such that X = f ∈ X ; f (x) = 0 m -a.e. x ∈ Ω \ Ω
is invariant under (U(t)) t 0 and let ( U (t)) t 0 and T be the restrictions to X of (U(t)) t 0 and T respectively. Assume that there exists a measurable mapping α(•) :

Ω → R such that (a) |α(x)| is finite for almost any x ∈ Ω; (b) for almost any x ∈ Ω such that t < τ -(x), α(Φ(x, -t)) = α(x) + t. (2.2)
Then, for any η ∈ R, the mapping

M η : f ∈ X -→ [M η f ](x) = exp(-iηα(x))f (x) ∈ X is an isometric isomorphism of X such that M -1 η T M η = T + iηId and M -1 η U(t)M η = e iηt U(t) (t 0). (2.3)
Consequently, S( T ) = S( T ) + iR and S( U(t)) = S( U (t)) • T for any (t 0) where T is the unit circle of the complex plane.

2.1. Spectral properties of (U 1 (t)) t 0 . To investigate the spectral properties of the restriction of the C 0 -semigroup (U(t)) t 0 to X 1 we employ a strategy inspired by [START_REF] Mokhtar-Kharroubi | Spectral properties of a class of positive semigroups on Banach lattices and streaming operators[END_REF] based upon Propositions 1.5 and 2.2. First, one gets the following invariance result as a straightforward application of Prop.

2.2:

Theorem 2.3. The spectra of (U 1 (t)) t 0 and T 1 in X 1 enjoy the following invariance properties:

S(T 1 ) = S(T 1 ) + iR and S(U 1 (t)) = S(U 1 (t)) • T (t 0)
where T is the unit circle of the complex plane.

Proof. The proof consists in applying the above Proposition 2.2 to the subset Ω 1 and the subspace X 1 using the function

α(x) = -τ + (x), x ∈ Ω 1 .
One checks immediately that α(•) fulfills (2.2). Note that α(•) is finite over Ω 1 .

According to the above Theorem, to describe the spectral features of (U 1 (t)) t 0 , it is sufficient to describe its real spectrum. We shall denote

Σ p (x) = h(x) + 1 p ⋆ div(F )(x), x ∈ Ω,
where p ⋆ is the conjugate of the exponent 1 p < ∞ we fixed at the beginning,

1 p ⋆ + 1 p = 1. Theorem 2.4. Assume m(Ω 1 ) = 0. One has S(U 1 (t)) = S ap (U 1 (t)) = {ξ ∈ C ; |ξ| exp(-γ 1 t)} (t 0) and S(T 1 ) = S ap (T 1 ) = {λ ; Reλ -γ 1 }
where

γ 1 = lim t→∞ inf t -1 t 0 Σ p (Φ(x, -s))ds ; x ∈ Ω 1 , t < τ -(x) .
Proof. Let us denote by ω 0 (U 1 ) the type of (U 1 (t)) t 0 and let us fix t 0 and f ∈ X 1 . One has

U 1 (t)f p X 1 = {τ + (x)<∞} exp -p t 0 ν(Φ(x, -s))ds f (Φ(x, -t)) p χ {t<τ -(x)} (x)dm(x).
Recalling that the finiteness of τ + (x) is equivalent to that of τ + (Φ(x, -t)) for any t < τ -(x), the change of variable x → y = Φ(x, -t) maps Ω 1 onto itself. Moreover, according to Proposition 1.2, the Jacobian of the transformation is given by

G(t, y) = exp t 0 div(F )(Φ(y, s))ds . Now, using that Φ(y, s) = Θ(x, t, s) = Φ(x, s -t) ∈ Ω as long as 0 < t -s < τ -(x), one sees that χ {τ + (x)<∞}∩{t<τ -(x)} (x) = χ {τ + (y)<∞}∩{τ + (y)>t} (y).
Therefore,

U 1 (t)f p X 1 = {τ + (y)<∞} exp -p t 0 ν(Φ(y, r))dr |f (y)| p χ {t<τ + (y)} (y)G(t, y)dm(y) = {τ + (y)<∞} exp -p t 0 Σ p (Φ(y, r))dr |f (y)| p χ {t<τ + (y)} (y)dm(y).
(2.4)

Consequently,

U 1 (t) B(X 1 ) = sup exp - t 0 Σ p (Φ(y, r))dr ; τ + (y) < ∞ , t < τ + (y)
or, equivalently,

log U 1 (t) B(X 1 ) = -inf t 0 Σ p (Φ(y, r))dr ; τ + (y) < ∞ , t < τ + (y) .
Performing now the converse change of variable y → x = Φ(y, t) one sees as previously that

log U 1 (t) B(X 1 ) = -inf t 0 Σ p (Φ(x, -s))ds ; τ + (x) < ∞ , t < τ -(x) .
Finally, recalling that ω 0 (U 1 ) = lim t→∞ t -1 log U 1 (t) B(X 1 ) , one deduces that ω 0 (U 1 ) = -γ 1 . Moreover, from the positiveness of (U 1 (t)) t 0 and [START_REF] Weis | The stability of positive semigroups in L p -spaces[END_REF], one gets s(T 1 ) = -γ 1 where s(T 1 ) is the spectral bound of T 1 . Let us now show that the set

Y 1 of local quasi-nilpotence of (U 1 (t)) t 0 is dense in X 1 . Indeed, set Ω m 1 = {x ∈ Ω ; τ + (x) m}, m ∈ N. One has Ω 1 = ∪ m 1 Ω m 1 . Set Y 1 = ∪ m 1 X m 1 where X m 1 = {f ∈ X 1 ; f (x) = 0 a.e. x ∈ Ω 1 \ Ω m 1 } . One checks easily that Y 1 is a dense sublattice of X 1 . Now, let f ∈ Y 1 . There exists some integer m 1 such that f (x) = 0 for almost every x ∈ Ω 1 with τ + (x) > m. From (2.4), one gets U 1 (t)(|f |) X 1 = 0 ∀t > m which clearly shows that Y 1 is a subset of Y 1 and that Y 1 is dense in X 1 . From Proposition 1.5, one gets then that (-∞, -γ 1 ] ⊂ S ap (T 1 ) and [0, exp(-γ 1 t)] ⊂ S ap (U 1 (t)) (t 0).
Now, the result follows from the invariance of S(T 1 ) and S(U 1 (t)) under vertical translations along the imaginary axis and under rotations respectively (Theorem 2.3).

2.2. Spectral properties of (U 2 (t)) t 0 . Let us investigate now the spectral properties of the restriction of (U(t)) t 0 to the subspace X 2 . As in the previous case, the main ingredient of our proof is the following special version of Proposition 2.2.

Theorem 2.5. The spectra of (U 2 (t)) t 0 and T 2 in X 2 enjoy the following invariance properties:

S(T 2 ) = S(T 2 ) + iR and S(U 2 (t)) = S(U 2 (t)) • T (t 0)
where T is the unit circle of the complex plane.

Proof. The proof consists in applying the above Proposition 2.2 to (U 2 (t)) t 0 by choosing

α(x) = τ -(x), x ∈ Ω 2 .
One checks immediately that α(•) fulfills (2.2) and that α(

•) is finite Ω 2 .
This leads to the following result whose proof is a technical generalization of [START_REF] Mokhtar-Kharroubi | Spectral properties of a class of positive semigroups on Banach lattices and streaming operators[END_REF] that we repeat here for the self-consistency of the paper.

Theorem 2.6. Assume m(Ω 2 ) = 0. One has S(U 2 (t)) = {ξ ∈ C ; |ξ| exp(-γ 2 t)} and S(T 2 ) = {λ ; Reλ -γ 2 }
where

γ 2 = lim t→∞ inf t -1 t 0 Σ p (Φ(x, s))ds ; x ∈ Ω 2 , t < τ -(x) .
Proof. The proof relies on duality arguments. We only prove it in the L 1 case, the case 1 < p < ∞ being simpler (see [START_REF] Mokhtar-Kharroubi | Spectral properties of a class of positive semigroups on Banach lattices and streaming operators[END_REF]). Since the dual semigroup (U ⋆ 2 (t)) t 0 of (U 2 (t)) t 0 is not strongly continuous on L ∞ (Ω 2 ), we have to introduce the space of strong continuity

X ⊙ 2 = g ∈ L ∞ (Ω 2 ) ; lim t→0 + sup x∈Ω 2 |U ⋆ 2 (t)g(x) -g(x)| = 0 .
The general theory of C 0 -semigroups [16, Chapter 2, Section 2.6] ensures that

X ⊙ 2 is a closed subset of L ∞ (Ω 2 ) invariant under (U ⋆ 2 (t)) τ 0 . For any t 0, define U ⊙ 2 (t) as the restriction of U ⋆ 2 (t) to X ⊙ 2 , i.e. (U ⊙ 2 (t)) t 0 is the sun-dual semigroup of (U 2 (t)) t 0 .. Then, the semigroup (U ⊙ 2 (t)) t 0 is a C 0 -semigroup of X ⊙ 2 whose generator is denoted by T ⊙ 2 with [16, Chapter 2, Section 2.6], D(T ⊙ 2 ) = g ∈ D(T ⋆ 2 ) ; T ⋆ 2 g ∈ X ⊙ 2 and T ⊙ 2 g = T ⋆ 2 g, ∀g ∈ D(T ⊙ 2 ), where T ⋆ 2 denotes the dual operator of T 2 whose domain D(T ⋆ 2 ) is dense in X ⊙ 2 . Moreover [16, Chapter 4, Proposition 2.18], S(T 2 ) = S(T ⋆ 2 ) = S(T ⊙ 2 ) and S(U 2 (t)) = S(U ⋆ 2 (t)) = S(U ⊙ 2 (t)) where the spectra of T ⋆ 2 and U ⋆ 2 (t) are intended in the space L ∞ (Ω 2 ) while that of T ⊙ 2 and U ⊙ 2 (t) stand for spectra in X ⊙ 2 . Define X ⊙ 2 = g ∈ X ⊙ 2 ; lim r→∞ sup τ -(x) r |g(x)| = 0 and let us show that X ⊙ 2 is invariant under (U ⊙ 2 (t)) t 0 . Indeed, let g ∈ X ⊙ 2 and f = (λ -T ⊙ 2
) -1 g where λ > 0 is large enough. Using again Proposition 1.2, one can check without difficulty that the dual of U 2 (t) is given by

U ⋆ 2 (t)g(x) = exp - t 0 h(Φ(x, s))ds g(Φ(x, t))
for any t 0, x ∈ Ω 2 and g ∈ L ∞ (Ω 2 ) where we recall that τ + (x) = ∞ for any x ∈ Ω 2 . Consequently,

f (x) = (λ -T ⊙ 2 ) -1 g(x) = ∞ 0 exp -λt - t 0 h(Φ(x, s))ds g(Φ(x, t))dt.
Recalling that τ -(Φ(x, t)) = τ -(x) + t and setting h = ess inf y∈Ω h(y), one has

sup τ -(x) r |f (x)| ∞ 0 exp(-(λ + h)t)dt sup τ -(y) r |g(y)|, ∀r > 0 which proves that f ∈ X ⊙ 2 and implies that X ⊙ 2 is invariant under the action of (U ⊙ 2 (t)) t 0 . Denote ( U ⊙ 2 (t)) t 0 the restriction of (U ⊙ 2 (t)) t 0 to X ⊙ 2 . Define now Y 2 as the set of g ∈ X ⊙ 2
for which there is some r

0 such that g(x) = 0 if τ -(x) r. One claims that Y 2 is dense in X ⊙ 2 . Indeed, for any integer m 1, let γ m (•) be a continuous function from [0, ∞) to [0, 1]
such that γ m (s) = 1 for any 0 s m and γ m (s) = 0 for any s 2m. For any g ∈ X ⊙ 2 , one can define g m (x) = γ m (τ -(x)) g(x) and prove without major difficulties that (g m ) m ⊂ Y 2 and g m -g L ∞ (Ω 2 ) → 0 as m → ∞. This proves our claim. Now, the set Y 2 of local quasinilpotence of ( U ⊙ 2 (t)) t 0 obviously contains Y 2 and is therefore dense in X ⊙ 2 . According to Proposition 1.5, one gets then

[0, exp(ω 0 ( U ⊙ 2 ) t)] ⊂ S ap ( U ⊙ 2 (t)) ∩ R and (-∞, ω 0 ( U ⊙ 2 )] ⊂ S ap ( T ⊙ 2 ) ∩ R
where T ⊙ 2 is the restriction of T ⊙ 2 to X ⊙ 2 and where we used the identity between the type ω 0 ( U ⊙ 2 ) of ( U ⊙ 2 (t)) t 0 and the spectral bound of T ⊙ 2 . Since X ⊙ 2 ⊂ X ⊙ 2 , one obtains the following inclusion

[0, exp(ω 0 ( U ⊙ 2 ) t)] ⊂ S ap (U ⊙ 2 (t)) ∩ R and (-∞, ω 0 (U ⊙ 2 )] ⊂ S ap (T ⊙ 2 ) ∩ R and Theorem 2.5 implies that S(U 2 (t)) = S(U ⊙ 2 (t)) = S ap (U ⊙ 2 (t)) = {ξ ∈ C ; |ξ| exp(ω 0 ( U ⊙ 2 ) t)} and S(T 2 ) = S(T ⊙ 2 ) = S ap (T ⊙ 2 ) = {λ ; Reλ ω 0 ( U ⊙ 2 )}. To conclude the proof, it remains only to show that ω 0 ( U ⊙ 2 ) = -γ 2 .
To do so, one notes easily that

U ⊙ 2 (t) B( X ⊙ 2 ) ω(t) := sup exp - t 0 h(Φ(x, s))ds ; x ∈ Ω 2 , t < τ -(x) .
Moreover, with the above notations, let ψ m (x) = γ m (τ -(x)), m 1. One has ψ m ∈ X ⊙ 2 and ψ m = 1 on the set {x ∈ Ω 2 ; τ -(x) m}. Therefore,

U ⊙ 2 (t) B( X ⊙ 2 ) U ⊙ 2 (t)ψ m X ⊙ 2 sup exp - t 0 h(Φ(x, s))ds ; x ∈ Ω 2 , t < τ -(x) m . Letting m → ∞, one gets U ⊙ 2 (t) B( X ⊙ 2 )
= ω(t). One obtains immediately that ω 0 (

U ⊙ 2 ) = -γ 2 .
The above results give a complete picture of the spectra of (U(t)) t 0 and T when m(Ω 3 ) = 0: Theorem 2.7. Assume that m(Ω 3 ) = 0. Then,

S(U(t)) = {ξ ∈ C ; |ξ| exp(-γt)}
and S(T ) = {λ ; Reλ -γ} where γ = min{γ 1 , γ 2 }, γ 1 , γ 2 being given by Theorems 2.4 and 2.6. In particular, (U(t)) t 0 satisfies a Spectral Mapping Theorem S(U(t)) \ {0} = exp (tS(T )) for any t 0.

We shall see in the following section that the picture is drastically different when m(Ω 3 ) = 0. First, we shall illustrate the above results by several examples.

Examples.

The above result is of particular relevance for applications when Ω is bounded in some directions. 6 where the force field F is given by the Lorentz force:

= (x, v) ∈ D × R 3 , let F (x) = (v, F(v)) ∈ R
F = F(v) = q(E + v × B)
where E ∈ R 3 stands for some given electric field, B ∈ R 3 denotes a given magnetic field and q is the electric charge of the particle. One assumes in this example that E and B are two constant fields such that E, B = 0 and that D is bounded in the B-direction, i.e.

sup{| x, B |, x ∈ D} < ∞,
then m(Ω 3 ) = 0. Indeed, one sees easily that the solution (x(t), v(t)) to the characteristic system

ẋ(t) = v(t), v(t) = q(E + v(t) × B), t ∈ R (2.5) with initial condition x(0) = x, v(0) = v, is such that x(t), B = q 2 E, B t 2 + v, B t + x, B .
Since the right-hand side is unbounded as t → ±∞ for any (x, v) ∈ Ω, necessarily x(t) escapes D in finite time and Ω 3 = ∅.

Remark 2.9. Note that, in general, the solution (x(t), v(t)) to the characteristic system (2.5) describes an helix with axis directed along B and radius proportional to 1/|q||B| (Larmor radius).

Example 2.10. Consider now the following one dimensional relativistic transport equation

∂ t f (x, p, t) + v(p)∂ x f (x, v, t) -pv(p) + (1 + p 2 ) -1/2 ) φ ′ (x)∂ p f (x, p, t) + ν(x, p)f (x, p, t) = 0, (x, v) ∈ (0, 1) × R (2.6)
where the potential φ is a given smooth function, say φ ∈ W 2,∞ (0, 1), and the relativistic velocity v(p) corresponding to the impulsion p ∈ R is given by v(p) = p √ 1+p 2 . The study of the above equation (2.6) is related to the relativistic Nordström-Vlasov systems for plasma (see [START_REF] Bostan | Mild solutions for the one dimensional Nordström-Vlasov system[END_REF][START_REF] Bostan | Stationary solutions for the one dimensional Nordström-Vlasov system[END_REF] and the references therein). We can then define a smooth vector field over Ω = (0, 1) × R by

F (x, p) = v(p), -pv(p) + (1 + p 2 ) -1/2 ) φ ′ (x) = p 1 + p 2 , -1 + p 2 φ ′ (x)
It can be proved [12, Corollary 2] that, whenever φ is convex, the exit time associated to the above field is finite for all characteristic curves with non zero impulsion, i.e., with the notations of the above section,

τ ± (x, p) < ∞, ∀(x, p) ∈ (0, 1) × (R \ {0}).
Therefore, Theorem 2.7 applies to such a problem.

SPECTRAL PROPERTIES OF (U 3 (t)) t∈R

We are dealing in this section with the spectrum of the C 0 -group (U 3 (t)) t∈R and its generator T 3 . We assume therefore for this section that m(Ω 3 ) = 0. For any x ∈ Ω 3 , the mapping s → Φ(x, s) is defined over R. For simplicity, for any t ∈ R, we will denote by ϕ t the mapping

ϕ t : x ∈ Ω 3 → ϕ t (x) = Φ(x, -t).
With this notation, the group (U 3 (t)) t∈R is given by

U 3 (t)f (x) = exp - t 0 ν(ϕ s (x))ds f (ϕ t (x)), ∀x ∈ Ω 3 , t ∈ R.
Notice that, arguing as above, it is possible to compute explicitly the type of both the semigroups (U 3 (t)) t 0 and (U 3 (-t)) t 0 (see [START_REF] Zhang | The spectrum of the linear Boltzmann operator with an external field[END_REF]Theorem 1]). As we shall see further on, the arguments of the previous section do not apply in general to the group (U 3 (t)) t∈R . However, for very peculiar cases, it is possible to proceed in the same way. More precisely, we recall the following definition of section, taken from [START_REF] Bathia | Stability theory of Dynamical Systems[END_REF].

Definition 3.1.

A set S ⊂ Ω 3 is said to be a section of Ω 3 (associated to the flow ϕ) if, for any x ∈ Ω 3 , there exists a unique real number θ(x) such that ϕ θ(x) (x) ∈ S.

One can state the following: Theorem 3.2. If there exists a measurable section S associated to the flow ϕ, then

S(T 3 ) = S(T 3 ) + iR and S(U 3 (t)) = S(U 3 (t)) • T, (t ∈ R).
As a consequence, S(U 3 (t)) = exp (tS(T 3 )) , (t ∈ R).

Proof. As in the previous section, the proof consists in exhibiting a function α(•) satisfying (2.2). Precisely, set α(x) = -θ(x),

x ∈ Ω 3 where θ(•) is provided by Definition 3.1. Since the section S is measurable, it is easy to see that α(•) is measurable and is finite almost everywhere. Moreover, from the definition of θ(x) as the unique 'time' at which a trajectory starting from x meets S, one has

α(ϕ t (x)) = t + α(x)
and we conclude the proof thanks to Proposition 2.2. Remark 3.3. Notice that there exists a section S associated to the flow (ϕ t ) t∈R if and only if (ϕ t ) t∈R does not admit periodic orbits nor rest points [9, p. 48]. However, it is a difficult task to provide sufficient conditions on the flow (ϕ t ) t∈R ensuring the section S to be measurable. Let us however mention that, whenever the flow (ϕ t ) t∈R is dispersive over Ω 3 in the sense of [9, Chapter IV] then S is measurable (and the function θ(•) provided by Definition 3.1 is continuous).

Example 3.4. Adopting notations from neutron transport theory, assume Ω = R N × V where V is a closed subset of R N . Then, for any x = (x, v) ∈ Ω, with x ∈ R N and v ∈ V , define F (x) = (v, 0). Then, the mapping α(x) = (x • v)/|v| 2 for any v = 0, fulfills (2.2) and the spectrum of (U(t)) t∈R and T are invariant by rotations and by vertical translations along the imaginary axis respectively (see [START_REF] Mokhtar-Kharroubi | Spectral properties of a class of positive semigroups on Banach lattices and streaming operators[END_REF][START_REF] Voigt | Positivity in time dependent linear transport theory[END_REF] for more details).

Apart from the very special result above, one has to provide an alternative approach to deal with the spectral properties of (U 3 (t)) t∈R . The aim of this section is to get a more precise picture of the spectrum of both (U 3 (t)) t∈R and T 3 . Recall now that, for a point x ∈ Ω 3 , since F is globally Lipschitz, three situations may occur:

(1) x is a rest point of the flow (ϕ t ) t , i.e. F (x) = 0;

(2) x belongs to a periodic orbit of (ϕ t ) t , i.e. there is some t 0 > 0 such that ϕ t 0 (x) = x.

(3) x belongs to an infinite but non closed orbit. This leads to the following splitting of Ω 3 :

Ω 3 = Ω rest ∪ Ω per ∪ Ω ∞
where

Ω rest = {x ∈ Ω 3 ; ϕ t (x) = x ∀t ∈ R}, Ω per = {x ∈ Ω 3 : ∃t > 0, ϕ t (x) = x} and Ω ∞ = Ω 3 \ (Ω rest ∪ Ω per ).
Notice that Ω rest is clearly a closed subset of Ω while Ω per is measurable (this follows from the fact that the sets Ω per,n defined by (4.2) are closed for any n ∈ N according to [3, p. 314]). It is not difficult to see that these sets are all invariant under the flow (ϕ t ) t∈R and consequently, under the action of U 3 (t) for any t ∈ R. Therefore, defining X rest , X per and X ∞ as the set of functions in X 3 which are null almost everywhere outside of the sets Ω rest , Ω per and Ω ∞ respectively, one can define the following restrictions of U 3 (t):

U rest (t) = U 3 (t) |Xrest , U per (t) = U 3 (t) |Xper and U ∞ (t) = U 3 (t) |X∞ .
Clearly, (U rest (t)) t∈R , (U rest (t)) t∈R and (U ∞ (t)) t∈R are positive C 0 -groups of X rest , X rest and X ∞ respectively, whose generators will be denoted respectively by T rest , T per and T ∞ . Arguing as in Theorem 2.1, the spectra of U 3 (t) and T 3 are given respectively by

S(U 3 (t)) = S(U rest (t)) ∪ S(U per (t)) ∪ S(U ∞ (t)), S(T 3 ) = S(T rest ) ∪ S(T per ) ∪ S(T ∞ ). (3.1)
Because of the possible existence of periodic orbits, it is not clear a priori that a result analogous to Theorems 2.3 and 2.5 can be proved in this case (even if ν(•) = 0). Indeed, if m(Ω per ) = 0, then there is no mapping α(•) satisfying (2.2) on Ω 3 . Indeed, if such a mapping were exist, in particular, for any x ∈ Ω per , α(ϕ t (x)) = α(x) + t, ∀t 0.

However, there exists a period t 0 = 0 such that ϕ t 0 (x) = x so that t 0 = α(ϕ t 0 (x)) -α(x) = 0 which is a contradiction. Of course, the impossibility to construct a function α(•) with the above properties does not imply that the spectrum of T 3 is not invariant by vertical translations along the imaginary axis.

Example 3.5. Let us consider the planar field F (x) = (-y, x) for any x = (x, y) ∈ Ω = R 2 . In such a case, the characteristic curves are circular, namely Φ(x, s) = (x cos s -y sin s, x sin s + y cos s), x = (x, y), s ∈ R.

In particular, for any x = (x, y) ∈ Ω one has τ ± (x) = ∞ and the mapping:

t ∈ R -→ ϕ t (x) = (x cos t + y sin t, y cos t -x sin t)
is 2π-periodic. This means that Ω 1 and Ω 2 are both empty sets. Consider for a while ν(•) = 0.

The semigroup (U(t)) t 0 extends to a group and one has

U(t)f (x) = f (ϕ t (x)), ∀t ∈ R, f ∈ X.
Clearly, (U(t)) t∈R is a positive and periodic C 0 -group of X with period 2π. As a consequence [START_REF] Eds | One-parameter semigroups of positive operators[END_REF], S(T ) = iZ. In particular, S(T ) = S(T ) + iR.

The previous example shows that one cannot hope to deduce the spectral properties of T 3 or (U 3 (t)) t∈R from their respective real spectrum as it was the case for the restrictions of (U(t)) t 0 to X 1 and X 2 .

Stationary flow.

We first deal with the spectral structure of the part (U rest (t)) t∈R of the C 0group (U(t)) t∈R acting on the set of rest points Ω rest . Since ϕ t (x) = x for any x ∈ Ω rest and any t ∈ R, it is clear that U rest (t) is given by:

U rest (t)f (x) = exp(-tν(x))f (x), x ∈ Ω rest , f ∈ X rest .
Hence, (U rest (t)) t∈R is a positive C 0 -group of multiplication. This leads naturally to the following description of its generator:

T rest f = -ν(•)f, f ∈ D(T rest )
which can easily be deduced from the definition of (U rest (t)) t∈R . Since the spectrum of T rest is real, one can deduce from the spectral mapping theorem for the real spectrum (see Theorem A. 3) the following:

Theorem 3.6. Assume that m(Ω rest ) = 0. The spectra of (U rest (t)) t∈R and its generator T rest satisfy the following spectral mapping theorem:

S(U rest (t)) = exp(tS(T rest )), ∀t ∈ R
where the spectrum of the generator is given by

S(T rest ) = R ess (-ν(•)) ⊂ R
where R ess (-ν(•)) denotes the essential range of -ν(•).

Remark 3.7. Note that the spectrum of T rest is not necessarily connected.

Aperiodic flow.

The study of the spectral properties of the C 0 -group associated to an aperiodic flow has been investigated in a framework of continuous functions in [START_REF] Arendt | The spectral mapping theorem for one-parameter groups of positive operators on C0(X)[END_REF] and in the more general framework of Mather semigroups in [START_REF] Chicone | Evolution semigroups in dynamical systems and differential equations[END_REF] under the supplementary assumption:

Assumption 3.8. The mapping x ∈ Ω ∞ -→ t 0 h(ϕ s (x))ds is continuous for any t ∈ R.
Namely, one can prove the following which is a consequence of [13, Theorem 6.37, p.188]: Theorem 3.9. Assume that m(Ω ∞ ) = 0 and Assumption 3.8 holds true. Then,

S(T ∞ ) = S(T ∞ ) + i R and S(U ∞ (t)) = S(U ∞ (t)) • Ì, (t ∈ R). As a consequence, S(U ∞ (t)) = exp (tS(T ∞ )) , (t ∈ R). (3.2)
Proof. Using the terminology of [START_REF] Chicone | Evolution semigroups in dynamical systems and differential equations[END_REF], the group (U ∞ (t)) t∈R is an evolution Mather group induced by a cocycle (Ψ t ) t over a flow (ψ t ) t . Indeed,

U ∞ (t)f (x) = J t (x)Ψ t (ψ -t (x)) f (ψ -t (x)), t ∈ R, x ∈ Ω ∞ , f ∈ X ∞
where the flow (ψ t ) t is given by ψ

-t = ϕ t while    Ψ : Ω ∞ × R → R (x, t) → Ψ t (x) = exp - t 0 h(ϕ s (x))ds and J t (x) = exp - t 0 div(F )(ϕ -s (x)
)ds is the Radon-Nikodym derivative of the dm • ψ t with respect to m (see Proposition 1.2). The fact that (Ψ t ) t is a cocycle of Ω ∞ over the flow (ψ t ) t is a consequence of the following straightforward identities

Ψ t+s (x) = Ψ t (ψ -s (x))Ψ s (x) and Ψ 0 (x) = 1 ∀x ∈ Ω ∞ , t, s ∈ R.
Therefore, one sees that [13, Theorem 6.37] applies since Assumption 3.8 implies that the mapping

(x, t) ∈ Ω ∞ × R → Ψ t (x) ∈ R is continuous.
Remark 3.10. Notice that a more precise picture of the spectrum of S(T ∞ ) is still missing. We also point out that that the rotational invariance of the spectrum S(U ∞ (t)) for any t ∈ R has been proved in [START_REF] Kitover | Spectrum of operators in ideal spaces[END_REF] in a direct way.

Remark 3.11. Assumption 3.8 is needed here in order to apply directly the results of [START_REF] Chicone | Evolution semigroups in dynamical systems and differential equations[END_REF]Chapter 6]. One may wonder if it is possible to get rid of such an assumption.

Remark 3.12. From the results of the previous sections, one can deduces that, when the flow ϕ(•) is aperiodic then the semigroup (U(t)) t 0 satisfies the spectral mapping theorem. Precisely, if m(Ω per ) = 0 and Assumption 3.8 holds, then

S(U(t)) \ {0} = exp(tS(T )), ∀t 0. (3.3)
If moreover, m(Ω rest ) = 0, then S(T ) = S(T ) + iR.

Note that practical criteria ensuring Ω per to be empty are well-known. For a C 1 -planar field F = (F 1 , F 2 ) (i.e. when N = 2), one can mention the so-called Dulac's criterion [START_REF] Amann | Ordinary Differential Equations. An introduction to nonlinear analysis[END_REF]Proposition 24.14] which states that, if Ω is simply connected, and if there exists ̺ ∈ C 1 (Ω, R) such that div(F ̺) is not identically zero and does not change sign in Ω, then there are no periodic orbits lying entirely in Ω. Generalizations to higher dimension (N 3) can also be provided (see e.g., [START_REF] Li | On Bendixon's criterion[END_REF][START_REF] Beretta | Negative criteria for the existence of periodic solutions in a class of delaydifferential equations[END_REF]). For planar C 1 field, one has also the following useful criterion:

Corollary 3.13. Let I be an open interval of R and let F ∈ C 1 (I × R ; R). Consider the planar field F (x) = (v, F (x, v)) for any x = (x, v) ∈ Ω := I × R. If F (x, 0) = 0, ∀x ∈ I,
then Ω rest = Ω per = ∅ and the spectral mapping theorem (3.3) holds for any h such that Assumption 3.8 is met.

Proof. Under the above assumption, F does not possess any equilibrium point in Ω, i.e. Ω rest = ∅. According to [1, Corollary 24.22, p. 346] there is no periodic orbit whose interior lies completely in Ω = I × R. Since I is an interval of R, Ω per = ∅ and we get the conclusion from Remark 3.12.

Finally, it is also known that planar gradient flows do not exhibit any periodic orbit [1, p. 241] leading to the following useful corollary Corollary 3.14. Assume that there exists V : R N → R such that F (x) = -∇V (x), ∀x ∈ R N . Then S(U (t)) \ {0} = exp(tS(T )), for any t 0 as soon as Assumption 3.8 is met.

We finally illustrate the above results by examples taken from various kinetic equations.

Examples. Let us begin with the classical Vlasov equation with a quadratic potential

Example 3.15. One considers, as in (1.5), a cylindrical domain

Ω = D × V ⊂ R 6 where D is a smooth open subset of R 3 . For any x = (x, v) ∈ D × V , let F (x) = (v, F(x, v)) ∈ R 6 (3.4)
where F = (F 1 , F 2 , F 3 ) is a time independent globally Lipschitz field (the force field) over D×V given by

F(x, v) = -x -∇ v U(v), ∀(x, v) ∈ Ω where U(v) is a Lipschitz space homogeneous potential U : V → R. Then, F is a gradient flow associated to the potential V (x, v) = v, x + U(v).
Therefore, the associated transport operator and semigroups are satisfying Corollary 3.14.

Example 3.16. Of course, the classical neutron transport equation for which

Ω = D × V ⊂ R 6 and F (x, v) = (v, 0), ∀x = (x, v) ∈ Ω is such that m(Ω per ) = 0.
More surprisingly, the above results also apply to kinetic equations of second order via a suitable use of Fourier transform [START_REF] Desvillettes | About the use of the Fourier transform for the Boltzmann equation[END_REF] : Example 3.17. Consider as in [START_REF] Desvillettes | About the use of the Fourier transform for the Boltzmann equation[END_REF] the Vlasov-Fokker-Planck equation with quadratic confining potential:

∂ t f + v • ∇ x f -x • ∇ v f = ∇ v • (∇ v f + vf ) (3.5) where f = f (x, v, t) ∈ L 2 (R N × R N )
for any t 0. The above equation is unitarily equivalent to the following first order equation in ξ and η:

∂ t f + η • ∇ ξ f + (η -ξ) • ∇ η f + |η| 2 f = 0 (3.6) where f (ξ, η, t) = R N ×R N exp(-i(x • ξ + v • η))f (x, v, t)dvdx
denotes the L 2 Fourier transform (in x and v) of f . The above equation (3.6) falls within the theory we developed in the previous sections. Precisely, let

F (x) = F (ξ, η) = (η, η -ξ), ∀x = (ξ, η) ∈ Ω = R N × R N .
Notice that divF (x) = N for any x ∈ Ω and, according to

(3.6), h(x) = h(ξ, η) = |η| 2 -N . The characteristic system ξ(t) = η(t), η(t) = (η(t) -ξ(t)), t ∈ R with initial condition ξ(0) = ξ 0 , η(0) = η 0 is explicitly solvable [15] with ξ(t) = 2 √ 3 exp(t/2) √ 3 2 cos √ 3 2 t - 1 2 sin √ 3 2 t ξ 0 + sin √ 3 2 t η 0 η(t) = 2 √ 3 exp(t/2) √ 3 2 cos √ 3 2 t + 1 2 sin √ 3 2 t η 0 -sin √ 3 2 t ξ 0 .
The flow ϕ t : (ξ 0 , η 0 ) → (ξ(t), η(t)) does not possess any periodic orbit while (0, 0) is the unique rest point. Therefore, according to the above Theorem 3.12, the C 0 -group governing (3.6) satisfies the Spectral Mapping Theorem 3.3. Turning back to the original variables, the Vlasov-Fokker-Planck (U V F P (t)) t∈R governing Eq. (3.5) also satisfies the Spectral Mapping Theorem 3.3:

S(U V F P (t)) = exp(tS(L V F P )), ∀t ∈ R,
where L V F P is the Vlasov-Fokker-Planck operator:

L V F P f (x, v) = -v • ∇ x f + x • ∇ v f + ∇ v • (∇ v f + vf ) , with its maximal domain in L 2 (R N × R N ).

PERIODIC FLOW

We now deal with the study of the periodic part of the group U(t), by studying the spectral properties of (U per (t)) t∈R on the space X per . In contrast to what happens for aperiodic flow, for periodic flow the spectral mapping theorem

S(U per (t)) = exp (tS(T per )) , ∀t ∈ R (4.1)
does not hold. Indeed, let us consider the following example Example 4.1. We turn back to the rotation group in R 2 introduced in Example 3.5. Recall that, for such an example Ω = R 2 , F (x) = (-y, x) for any x = (x, y) and the associated C 0 -group in L p (R 2 , dm) is given by U(t)f (x) = f (ϕ t (x)), t ∈ R where the flow ϕ t is given by the rotation of angle t ∈ R :

ϕ t (x) = cos t sin t -sin t cos t x y , x = (x, y) ∈ R 2 .
According to [START_REF] Eds | One-parameter semigroups of positive operators[END_REF], the spectrum of the generator T of (U(t)) t∈R is given by S(T ) = iZ. Moreover, for any t ∈ R, µ n (t) = exp(int) is an eigenvalue of U(t) for any n ∈ Z. If t/2π is irrational, the eigenvalues {µ n (t), n ∈ Z} describe a dense subset of T and the closedness of the spectrum of U(t) implies that S(U(t)) = T while exp(tS(T )) = exp(itZ) = T for such a t. This shows that, in general, the Spectral Mapping Theorem 4.1 fails for periodic flows.

For any x ∈ Ω per , one can define the prime period of x as

p(x) = inf{t > 0, ϕ t (x) = x}.
The main properties of the prime period are listed in the following Proposition. We refer the reader to [START_REF] Amann | Ordinary Differential Equations. An introduction to nonlinear analysis[END_REF][START_REF] Arendt | The spectral mapping theorem for one-parameter groups of positive operators on C0(X)[END_REF] for the first assertions while the last one is referred to as Yorke's Theorem [START_REF] Yorke | Periods of periodic solutions and the Lipschitz constant[END_REF]. where κ > 0 is the Lispchitz constant of the field F .

Notice that, a priori, the prime period p is an unbounded function. However, we shall prove that the description of S(T per ) relies actually on the behavior of T per on functions supported on sets where p is bounded. Precisely, for any n > 0, define Ω per,n as :

Ω per,n := {x ∈ Ω per ; p(x) n} . ( 4
(λ -T per,n ) -1 B(Xper,n) < ∞. (4.3)
Proof. Assume first that λ ∈ ̺(T per ) and let n ∈ N. Given g ∈ X per,n ⊂ X per , there is a unique f ∈ X per such that (λ -T per )f = g. It is clear that, f ∈ X per,n which proves that λ ∈ ̺(T per,n ). Moreover, f Xper,n (λ -T per ) -1 B(Xper) g Xper,n , which proves that

sup n∈N (λ -T per,n ) -1 B(Xper,n) (λ -T per ) -1 B(Xper)
.

Conversely, assume Eq. (4.3) to holds. Let g ∈ X per . For any n ∈ N, let g n = gχ Ωper,n . It is clear that g n ∈ X per,n . Since λ ∈ ̺(T per,n ), there is a unique f n ∈ X per,n such that (λ-T per,n )f n = g n . Now, one notes that, for any x ∈ Ω per , lim n→∞ χ Ωper,n (x) = 1. Consequently, g n converges to g in X per . Let us prove that (f n ) n also converge in X per . Given n 1 n 2 , one has

f n 1 -f n 2 ∈ X per,n 2 and f n 1 -f n 2 Xper = f n 1 -f n 2 Xper,n 2 sup n∈N (λ -T per,n ) -1 B(Xper,n) g n 1 -g n 2 Xper,n 2 sup n∈N (λ -T per,n ) -1 B(Xper,n) g n 1 -g n 2 Xper , n 1 n 2 .
Therefore, (f n ) n is a Cauchy sequence in X per . Let f denote its limit. Since f n → f and g n → g, one has T per,n f n converges to λf -g. Since T per is a closed operator, one gets f ∈ D(T per ) with (λ -T per )f = g. Now, let h ∈ D(T per ) be another solution to the spectral problem (λ -T per )h = g. Then, since λ ∈ ̺(T per,n ) for any n, one sees that hχ Ωper,n = f n for any n ∈ N. Using again the fact that χ Ωper,n (x) → 1 for any x ∈ Ω per , one sees that h(x) = f (x) for any x ∈ Ω per . This proves that λ ∈ ̺(T per ).

We describe now more precisely the spectrum S(T per,n ). For any λ ∈ C, define

ϑ(x) = - 1 p(x) p(x) 0 ν(ϕ s (x))ds and M λ (x) = exp {-p(x)(λ -ϑ(x))} ,
x ∈ Ω per .

Note that, for any t ∈ R and any x ∈ Ω per , p(ϕ t (x)) = p(x) so that ϑ(ϕ t (x)) = ϑ(x), and M λ (ϕ t (x)) = M λ (x) ∀x ∈ Ω per .

One has, Proof. Let n ∈ N be fixed. First, for any λ ∈ C, let

J λ : f ∈ X per,n → J λ f (x) = p(x) 0 exp(-λt)U per,n (t)f (x)dt, x ∈ Ω per,n .
The proof of the Proposition is based on the fact that J λ f ∈ D(T per,n ) for any f ∈ X per,n with

(λ -T per,n )J λ f (x) = (1 -M λ (x)) f (x), x ∈ Ω per,n . (4.4)
Indeed, given f ∈ X, since p(x) n for any x ∈ Ω per,n , one sees that J λ f ∈ X per,n with

J λ f Xper,n n 0 exp(-Reλt) U per,n (
(t)J λ f (x)] t=0 = λJ λ f (x)+exp (-λp(x)) U per,n (p(x))f (x)-f (x), x ∈ Ω per,n . Now, since [U per,n (p(x))f ](x) = exp - p(x) 0 ν(ϕ s (x))ds f (x) one has d dt [U per,n (t)J λ f (x)] t=0 = λJ λ f (x) + (M λ (x) -1) f (x), ∀x ∈ Ω per,n .
This proves that J λ f ∈ D(T per,n ) and Eq. (4.4) holds. Now, let us prove that λ ∈ S(T per,n ) implies 1 ∈ R ess (M λ ). To do so, assume 1 / ∈ R ess (M λ ), and define the operator

R λ f (x) = (1 -M λ (x)) -1 J λ f (x).
Since 1 / ∈ R ess (M λ ), one sees that R λ is a bounded operator in X per,n . Moreover, using the fact that p(•) and M λ (•) are invariant under the flow (ϕ t ) t , one easily sees that

R λ U per,n (t) = U per,n (t)R λ , t ∈ R.
Classically, this implies that R λ D(T per,n ) ⊂ D(T per,n ) and

R λ T per,n f = T per,n R λ f, ∀f ∈ D(T per,n ).
Using again the fact that M λ (•) is invariant under the flow (ϕ t ) t , one sees that T per,n R λ f = (1 -M λ (•)) -1 T per,n J λ f and Eq. (4.4) asserts that

R λ (λ -T per,n )f = (λ -T per,n )R λ f, ∀f ∈ D(T per,n ).
This proves that λ ∈ ̺(T per,n ) with (λ -T per,n ) -1 = R λ .

Remark 4.5. We conjecture that the inclusion in the above Theorem is an equality. More precisely, if there is some T 0 such that p(x) T for any x ∈ Ω, then, we conjecture that

λ ∈ S(T per ) ⇐⇒ 1 ∈ R ess (M λ ). ( 4

.5)

The following reasoning comforts us in our belief. If 1 ∈ R ess (M λ ), then, for any fixed ǫ > 0, the set

Ω ǫ = {x ∈ Ω per,n ; |1 -M λ (x)| < ǫ} is of non zero measure. Let f ǫ ∈ X per be such that f ǫ = 1 and Suppf ǫ ⊂ Ω ǫ . Let g ǫ = J λ f ǫ . Then, Eq. (4.4) implies that (λ -T per )g ǫ (x) = (1 -M λ (x))f ǫ (x),
so that (λ -T per )g ǫ ǫ. To prove that λ ∈ S(T per ), it would suffice to prove that g ǫ C > 0 for some constant C > 0 that does not depend on ǫ. We did not succeed in proving this point. Notice however that the identity (4.5) holds true in space of continuous functions [START_REF] Arendt | The spectral mapping theorem for one-parameter groups of positive operators on C0(X)[END_REF].

The following Proposition provides a complete picture of the set of λ for which 1 ∈ R ess (M λ ). Its proof is inspired by similar calculations already used in the study of 2D neutron transport equations [START_REF] Lods | On the spectrum of absorption operators with Maxwell boundary conditions arising in population dynamics and transport theory[END_REF]: Proposition 4.6. Assume that m(Ω per ) = 0 and there exists some T > 0 such that p(x) T for any x ∈ Ω per . Then,

1 ∈ R ess (M λ ) if and only if λ ∈ k∈Z R ess (F k )
where, for any k ∈ Z, F k : Ω per → C is a measurable mapping given by

F k (x) = ϑ(x) + i 2kπ p(x) , x ∈ Ω per .
Proof. Let us pick λ / ∈ k∈Z R ess (F k ). Then, for any k ∈ Z, there exists

β k > 0 such that |λ -F k (x)| β k a. e. x ∈ Ω per , i.e. | -p(x)ϑ(x) -2ikπ -λ p(x)| p(x) β k 2πβ k /κ a. e. x ∈ Ω per
where we made use of Yorke's Theorem for the last estimate. This means that, for any integer n 0, there exists c n > 0 such that Now, since

-p(x)ϑ(x) -λ p(x) 2π n ± i c n a. e. x ∈ Ω per , n 1, (4.6
e u -1 = ue u 2 ∞ n=1 1 + u 2 4π 2 n 2 , u ∈ C, and
∞ n=N 1 + u 2 4π 2 n 2 → 1 (N → ∞)
uniformly on any compact subset of C, for any M > 0, there exists N > 0 such that It remain to investigate the spectral properties of the group (U per (t)) t∈R . As we already saw it, the Spectral Mapping Theorem fails to be true in general. However, one can deduce from Proposition A. 2 the following version of the Annular Hull Theorem: Theorem 4.7. Assume m(Ω per ) = 0. Then,

|e u -1| 1 2 |u| e u 2 N n=1 i - u 2π n i + u 2π n , |u| < M.
T • S U per (t) = T • exp tS T per ∩ R , ∀t ∈ R. (4.8)
In particular, the semigroup (U(t)) t 0 fulfils the Annular Hull Theorem:

T • S(U(t)) \ {0} = T • exp tS(T ) , ∀t 0.
Remark 4.8. The above Annular Hull Theorem is known to be true for general weighted shift semigroups [START_REF] Yu | Weighted translation operators and linear extensions of dynamical systems[END_REF][START_REF] Chicone | Evolution semigroups in dynamical systems and differential equations[END_REF]. Notice that the proof of [START_REF] Yu | Weighted translation operators and linear extensions of dynamical systems[END_REF][START_REF] Chicone | Evolution semigroups in dynamical systems and differential equations[END_REF] rely on completely different arguments and involve very sophisticated tools. On the contrary, our result is a very easy consequence of a more general result on positive groups on L p spaces. Of course, the simplification relies on the fact that we are dealing here with C 0 -groups rather than semigroups.

In the L 1 case, one can strengthen this result thanks to the Weak Spectral Mapping Theorem for positive C 0 -group by W. Arendt and G. Greiner [START_REF] Arendt | The spectral mapping theorem for one-parameter groups of positive operators on C0(X)[END_REF]Corollary 1.4]. Precisely, Theorem 4.9. Assume p = 1, i.e. X per = L 1 (Ω per , dm). For any t ∈ R, the following weak spectral mapping theorem holds: S(U per (t)) = exp (tS(T per )) and, under the assumption 3.8, S(U(t)) = exp (tS(T )), ∀t 0.

APPENDIX: SPECTRAL PROPERTIES OF GENERAL POSITIVE SEMIGROUPS

Let X be a complex Banach lattice with positive cone X + and let (T (t)) t∈R be a positive C 0group in B(X) with generator A. Recall that the positivity of the group (T (t)) t∈R means that X + is invariant under T (t) for any t ∈ R. We establish in this Appendix several abstract results on (positive) C 0 -groups we used in the paper. The key point is the following spectral decomposition result for strongly continuous groups of positive operators due to Arendt [2, Theorem 4.2] and Greiner [START_REF] Greiner | A spectral decomposition of strongly continuous groups of positive operators[END_REF].

Theorem A. 1. Let (T (t)) t∈R be a C 0 -group of positive operators with generator A on some Banach lattice X. Let µ ∈ ̺(A) ∩ R. Then, X is the direct sum of the orthogonal projection bands:

I µ = {x ∈ X ; R(µ, A)|x| 0}
and J µ = {x ∈ X ; R(µ, A)|x| 0}.

Moreover, I µ and J µ are invariant under T (t) (t ∈ R) and S(A |Iµ ) = {λ ∈ S(A) ; Reλ < µ}, S(A |Jµ ) = {λ ∈ S(A) ; Reλ > µ} where A |Iµ (respectively A |Jµ ) denotes the generator of (T (t) |Iµ ) t∈R (resp. of (T (t) |Jµ ) t∈R ).

Using this result, G. Greiner [START_REF] Greiner | A spectral decomposition of strongly continuous groups of positive operators[END_REF] has been able to prove a Spectral Mapping Theorem for the real spectrum (Theorem A. 3 hereafter). Borrowing the ideas of Greiner, it is possible to prove the following more general result. Actually, we did not find this result in the literature and give here a simple proof of it. Proposition A. 2. Let (Σ, ̟) be a S-measured space and let (T (t)) t∈R be a positive C 0 -group in L p (Σ, d̟) (1 p < ∞), with generator A. Then, {|µ| ; µ ∈ S(T (t))} = exp {t (S(A) ∩ R)} for any t 0.

In particular, the following Annular Hull Theorem holds true: The Spectral Mapping Theorem for the real spectrum of Greiner [START_REF] Greiner | A spectral decomposition of strongly continuous groups of positive operators[END_REF] (see also [START_REF] Arendt | Kato's equality and spectral decomposition for positive C0-groups[END_REF] and [START_REF] Eds | One-parameter semigroups of positive operators[END_REF]Corollary 4.10]) is now a direct consequence of the above Proposition. Another consequence of Proposition A. 2 is the following spectral mapping theorem which applies to generator whose approximate spectrum is invariant by vertical translations: Theorem A. 4. Let (Σ, ̟) be a S-measured space and let (T (t)) t∈R be a positive C 0 -group in L p (Σ, d̟) (1 p < ∞), with generator A. If S ap (A) = S ap (A) + iR then

S(T (t)) = exp (tS(A))

for any t ∈ R.

Proof. It clearly suffices to prove the "⊂" inclusion. We first note that S(A) = S(A) + iR. Indeed, let λ ∈ S(A), assume that λ + iR S(A). Then, there is α ∈ R such that λ + iα lies in the boundary of S(A). In particular, λ + iα ∈ S ap (A), and by assumption, λ + iR ⊂ S(A) which is a contradiction. Therefore, S(A) = S(A) + iR. Now, let z / ∈ exp (tS(A)). Then, there is λ ∈ ̺(A) such that z = exp (λt), λ = α + iβ. Since the spectrum of A is invariant by vertical translations, α ∈ ̺(A). Hence exp (αt) / ∈ exp {t (S(A) ∩ R)}. According to Proposition A.2, and since |z| = exp (αt), this means that z ∈ ̺(T (t)).

Remark A. 5. As a consequence of the above result, one sees that, if (T (t)) t∈R is a positive groups of operator with generator A in some L p -space (1 p < ∞) and if S(A) = S(A) + iR, then S(T (t)) = S(T (t)) • T (t ∈ R) where T is the unit circle of C.

Example 2 . 8 .

 28 Let us consider the case of the classical Vlasov equation described in (1.6) with a Lorentz force. Namely, let Ω = D × R 3 where D is a smooth open subset of R 3 . For any x

Theorem 4 . 4 .

 44 For any n ∈ N, S(T per,n ) ⊂ {λ ∈ C ; 1 ∈ R ess (M λ )} where R ess (M λ ) denotes the essential range of M λ : Ω per,n → C.

  ) and |p(x)ϑ(x) -λ p(x)| c 0 a. e. x ∈ Ω per . (4.7)

Now, since ν is 2 N n=1 c 2 n

 22 bounded and p(x) T , there exists M > 0 (large enough), such that |p(x)ϑ(x) -λ p(x)| M a. e. x ∈ Ω per , one gets form (4.6) and (4.7) that | exp{p(x)ϑ(x) -λ p(x)} -1| C a. e. x ∈ Ω per , where C = ess inf x∈Ωper exp p(x) 2 (ϑ(x) -λ) . Hence, ess inf x∈Ωper |M λ (x) -1| > 0 which proves the first inclusion. To prove the second inclusion, it is enough to notice, from the continuity of the exponential function that, if there is some constant C > 0 such that |exp(p(x)ϑ(x) -λp(x)) -1| C > 0, a. e. x ∈ Ω per , then, for any k ∈ Z, there exists c k > 0 such that |p(x)ϑ(x) -λp(x) -2ikπ| c k for a. e. x ∈ Ω per . Now, since p(x) T for any x ∈ Ω per , one sees that |λ -F k (x)| c k /T for a. e. x ∈ Ω per .

T

  • S(T (t)) = T • exp t S(A) ∩ R = T • exp tS(A) , ∀t ∈ R. (4.9) Proof. It is clear that exp {t (S(A) ∩ R)} ⊂ {|µ| ; µ ∈ S(T (t))}. Let us show the converse inclusion. Let z ∈ R + \ exp {t (S(A) ∩ R)}. Then, z = exp (αt), with α ∈ ̺(A) ∩ R. Then, according to Theorem A. 1, there exist two projection bands I α and J α such that L p (Ω, d̟) = I α ⊕ J α and S(A |Iα ) = {z ∈ S(A) ; Rez < α}, while S(A |Jα ) = {z ∈ S(A) ; Rez > α}.Consequently, s(A |Iα ) < α and s(-A |Jα ) < -α where s(•) denotes the spectral bound. Since, in L p -spaces (1 p < ∞), the type of any positive C 0 -semigroup coincide with the spectral bound of its generator[START_REF] Weis | The stability of positive semigroups in L p -spaces[END_REF], one getsr(T (t) |Iα ) < exp (αt), r(T (-t) |Jα ) < exp (-αt).Hence, using that S(T (t) |Jα ) = {z ∈ C ; 1/z ∈ S(T (-t) |Jα )}, S(T (t) |Iα ) ⊂ {z ∈ C ; |z| < exp (αt)} whereas S(T (t) |Jα ) ⊂ {z ∈ C ; |z| > exp (αt)}.Now, since S(T (t)) = S(T (t) |Iα ) ∪ S(T (t) |Jα ), any µ ∈ C with |µ| = exp (αt) is such that µ / ∈ S(T (t)) which achieves the proof. The proof of the Annular Hull Theorem is then obvious since any µ ∈ S(T (t)) writes µ = |µ| exp(iθ) for some θ ∈ R while |µ| = exp(αt) for some α ∈ S(A) ∩ R.

Theorem A. 3 .

 3 Let (Σ, ̟) be a S-measured space and let (T (t)) t∈R be a positive C 0 -group in L p (Σ, d̟) (1 p < ∞), with generator A. Then,S(T (t)) ∩ R + = exp {t (S(A) ∩ R)}for any t 0.Proof. It is clear that S(T (t)) ∩ R + ⊇ exp {t (S(A) ∩ R)} for any t ∈ R. Now, since S(T (t)) ∩ R + ⊂ {|µ| ; µ ∈ S(T (t))}, the converse inclusion follows immediately from Proposition A.2.

  .2) Proposition 4.2 asserts that Ω per,n = ∅ for any 0 < n < 2π/κ and Ω per,n is invariant under the action of the flow (ϕ t ) t for any n > 0. As above, one can define X per,n asX per,n = {f ∈ X per ; f (x) = 0 ma.e. x ∈ Ω per \ Ω per,n } , ∀n 2π/κ,and let (U per,n (t)) t∈R be the restriction of (U per (t)) t∈R on X per,n . We denote by T per,n its generator. One has the following abstract result: Let λ ∈ C be given. Then, λ ∈ ̺(T per ) if and only if

	Theorem 4.3. λ ∈	n∈N	̺(T per,n ),	and	sup n∈N

  t)f Xper,n dt. Now, given t ∈ R and f ∈ X per,n , since p(•) is invariant under the action of the flow (ϕ t ) t , one sees easily that

			p(x)
		U per,n (t)J λ f (x) =	0	exp(-λs)U per,n (t + s)f (x)ds
					t+p(x)
		= exp(λt)	t	exp(-λs)U per,n (s)f (x)ds,	∀x ∈ Ω per,n ,
	and,
	d dt	[U per,n


domain: sdv sde

text:
Leaves 40 (1%) 9 (0.2%) Roots 30 (0.7%) 10 (0.2%) (total number of analyzed cDNA elements: 4062)

Water transfert

Aquaporin TIP (L)

Differentially expressed transcripts Introduction Introduction

Drought greatly affects the growth and development of plants. In order to understand the molecular basis of the response of poplar trees to water-deficit and identify candidate genes, we carried out an integrated approach combining functional genomics and ecophysiological approaches on Populus trichocarpa x deltoides cv Beaupré .

Material and methods

Material and methods

The response to drought was assessed using experimental conditions that closely approximate stress development in the field. Drought level was monitored through transpiration rate. 3 stress levels (50, 25 and 10% of control transpiration) were combined to 4 durations (t1, t2, t3 and t4 corresponding to 3, 7, 10 and 20 days, respectively). Physiological status was characterised at each time point on the harvested plants. Growing leaves and roots were collected for transcriptome analysis (2-3 replicates). cDNA arrays contained 4608 ESTs from adventitious roots and leaves of Populus trichocarpa x deltoides cv Beaupré (Kohler, 2003). Signal intensities lower than 2x mean of (background + standard deviation) were eliminated and central normalization was performed on the remaining data.

A Bayesian statistical method (CyberT), based on the t-test, was used to test for statistically significant differences in gene expression.

  PhotosynthesisPSII 5kd protein precursor (L) RUBISCO small subunit (L)Stress/Defense-related Metallothionein PtdMT1a, PtdMT1b (R)

DROUGHT DROUGHT

Summary Summary

3 levels of analysis : physiology, transcriptome and proteome (cf. Meddour poster)


Project ANALYSIS OF AGRO-ENVIRONMENTAL SCHEMES

The research is

The agricultural sector has experienced this evolution with the development of contractual approaches based on the free choice of farmers to undertake environmental-friendly measures. Even if national schemes have developed . given the share of expenditure devoted to this sector in Europe, environmental contracting in agriculture has especially been administered at the European level. Severa! European regulations have instituted environmental contracting (Regulation EU 797/87 (article 19), Regulation 2078/92, Regulation 1257/99).

Given this context. two questions arise:

1) What organisational and institutional factors help increase economic efficiency and env ironmental effectiveness of agro-environmental schemes? 2) How do they compare to other environmental policy tools?

Some.fàcts

Previous studies report t\\'o important facts to be taken into account in analysing agro-environmental schcmes: the high level of transaction costs and the high organisa1ional and institutional diversity of existing agrn-em ironmental schemes a cross Europe and across the ,, 0rld. High transaction costs: Severa! studies repo1t high level of transaction costs in setting and running agroenvironmental schemes. They encompass costs born by public authorities (administrative transaction costs) and by farmers (private transaction costs).

UMR SMART Documentation

As fo r private transaction costs, Falconner (2000) reports: « Private transaction costs might amount to around 5% of the compensation payments made to farmers for lost agricultural income in average. through with a potentially wide range that has not been investigated ». As for administrative costs. Falconner et al. (2001) have studied their evolution for the English scheme (Environmental Sensitive Areas). Figure I reports the evolution of the level of administrative transaction costs as compared to the level of payments received by farmers. We may give two comments. First. these costs are relatively high . Second, they seem to decrease across years. Two factors influence the level of these costs: learning effects and the increasing number of participants in the scheme which enables to distribute fixed costs across more farmers. --------------------------------

i 30 1------------,--------1 25 1 ----------1 20 -1-----------t ..... 15 +-----1 ---....__ 10 ,_- 5 ,- - - - --- ,_.
,__ High organ isational and institutional diversity of existing agro-environmental schemes: First, the 9 case studies across Europe in the IT AES project reveal a great diversity in the application of European regulations. Although European regulations constrain regions or countries on some variables. case studies show diversity in the design, contracting, monitoring and enforcement phases (involved actors, level of decision). in the monitoring and enforcement phases. in the freedom of choices of measures left to farmers . Second, a preliminary review of the literature shows how diverse are policies aimed at providing public goods on private land. Among these are trusts, auctions. ecolabeling . . .. ln view of the high level of transaction costs and the high organisational and institutional diversity of agro-environmental schemes. transaction costs economics appears to be a good framework to elabo rate hypotheses and test them. This kind of analysis should lead to propositions to enhance exist ing policies.

Research steps

Two steps may be identified :

Step 1: Two theoretical questions How to define the environmental transaction in the framework of transaction costs economics? What criteria enable to discrim inate among contractual arrangements?

We began working on this step. Indeed. environmental economics bas ernployed the seminal contribution of Ronald 11. [START_REF] Coase | The Problem of Social Cost[END_REF] operationalization à la Williamson. Further. we sho,, that extending transaction costs economics to environrnental related transactions constitutes a promising challenge. especially in order to provide guidance about the choice of regulatory instruments. Bougherara et al. (20056) apply the analytical framework of discrete structural alternatives to the choice of regulatory instruments. Environmentalrelated transactions, which differ in their attributes. are aligned with categories of policy instruments . . which differ in their cost and competence. so as to effect a discriminating -mainly transaction costseconomizing -result.

Step 2: An applied research question How does transaction cost economics help to analyse the econom1c efficiency and environmental effectiveness of agro-environmental schemes?

We focused our research project presentation on the second step. The tirst resulls on the analysis of agroenvironmental schemes is presented.

Tnmsactùm charncteristics in agro-e11vir<mme11ta/ ,\•chemes (examples from case studies)

Given that transaction costs depend on transaction attributes, transactions have to be primarily characterised before analysing the role of organisations and institutions in agro-environmental schemes environmental effectiveness (Williamson, 1985). According to the transaction cost econornics perspective. the transaction is the unitary element of analysis. First of ail parties and the nature of the good wh ich is exchanged are considered before turning to transaction attributes.

Who are the parties and what is exchanged?

Case studies show that most agro-environmental schemes involve two parties: a farmer applying for a contract offered by a governmental organisation operating at one territorial level or anothtr. Sorne contracts are more complcx and look like a kind of collective agreement with several parties contracting more or less directly.

Note that the two invoh ed pai1ies contract for the exchange of environmental goods. Indeed, the European viewpoint based on the provider-gets principle reckons farmers a property right on land use. They bcnefit a compensatory payment for using land so as to provide non-commodity output [START_REF] Hagedorn | Agri-environmental policy: A UK Perspective[END_REF]. According to case studies. there is evidence that ail agro-environmental schemes involve compensatory payments for undertaking environmental friendly practices. Whereas most agroenvironmental schemes are contracted for fi, e years, several regions consider organic fanning and conversion to grassland \.\Orth compensating only for a more limited period of lime.

Measures aiming at providing environmental goods may be classified into three types:

-Maintenance measures encompass the promotion of existing practices that are not harmful for the environment. They cou Id also be named • status quo' measures.

-Externality reduction measures include measures aiming at improving the environment through a reduction of adverse effects or farming.

-Public good provision measures addressed the promotion of positive externalities of agricultural An alternative classification focusing on the technology is helpful to understand what is required to farmers entering agro-environmental scheme. Indeed. the concept of jointness is instrumental since it matters whether an environmental good is a complement or not of commodities.

A lthough agro-environmental schemes target a shift from existing practices there is evidence that farmers may get money for already operating practices. When farmers have the power to influence the design of agro-environmental schemes, they may twist scheme requirements towards what they already do, so that compliance costs are nil. Agro-environmental schemes' payments may compensate farmers for practices they would have undertaken without adopting agro-environmental schemes. The problem remains to distinguish among environmental practices those that would have been undertaken should the agroenvironmental scheme not have been signed.

Asset specificity

Asset spec ificity refers to the relative lack of transferability of assets intended for use in a given transaction to other uses. Highly specific assets lead to sunk costs, which have relatively little value beyond their use in the context of a speci fic transaction. The higher the asset specificity is the greater the interdependence of parties is. If asset specificity is low, firms can easily change their partnership and modify their output mix in order to comply with market opportunities. The demanding partner will choose to obtain inputs in the market place. However, if asset specificity is high, potential suppliers may find it too risky to make the necessary investments without the guarantee of adequate outlets.

Is there such specificity in the agro-environmental contracts described in case studies? Yes, although it ranges on a continuum from weak to high spccificity. At least three types of asset specificity must be considered [START_REF] Hagedorn | lnstitutional Arrangements for Environmental Co-operatives: a Conceptual Framework[END_REF]: site. physical and human asset specificity.

-Site asset specifïcity: the asset looses value if invested in another location. Measures for protecting a specific ecosystem are not effective if undertaken elsewhere since ecosystems of spccific interest are attached to a given area. Most agro-environmental schemes display such specificity especially when eligible areas rules are set.

-Physical asset spec ificity: One party invests in specific equipments that loose value if used elsewhere. For example, hedgerow maintenance requires specific machines.

-Human asset specificity: refers to the accumulation of specific knO\vledge to a trading partner or to a transaction, which looses value for another transaction. For example, farmers may invest in knowledge about specific measures to enhance the environment or about the way to monitor the contracts they sign (paperwork). Conversion to organic farming also required a precise training and learning. Otherwise. there are several cases involving specific production methods.

The agent\\ ho invests in a highly specific asset requires the other party that the contract will last. Severa! factors may affect asset specificity. Under joint production, the provision of environmental good depends on the provision of market commodities. Therefore. efforts to deliver environmental goods are still more or less valued through the market. Joint production is likely to lower asse! specificity. When alternative channels such as green tourism exist to monetise environmental friendl y effo rt. farmers are fewer dependants on agro-environmental schemes. Asset specificity may then also be lowered . One must however be careful with respect to agro-tourism as a means to value farmers" efforts since tourists generally en_joy more the existing environmental and rural assets than the e ffort s of farmers.

As de, e loped in the next sub-section. env iro nmental outcomes arc strongly linked to knowledge about the agro-em ironmental technolog~. lt fol lO\\S that measures in agro-el1\ ironmental schemes should refled this nced to invest in specilic human capital. 1-lowever. the anal) sis or case-study reports \\eakl) revcal orgarnscd education and training of fàrmers before undertaking agro-environmental schemes.

  Structures et Marchés Agricoles. Ressources et Territoires INRA-Agrocampus Ouest 4 allée Adolphe Bobierre-CS 61103 35011 Rennes Cedex DOCUMENTATION ÉCONOMIE RURAL E RENNES I IIIIII IIIII IIIII IIIII IIIII IIIII IIII IIII * 0 0 9 9 5 1 * Boughernrn IJ . 2005. Ncw-I11st1tut1011al l:conom ,cs Anali•sis ,if Agio-Fm 11011mental Schcmcs [ uropcan Sthool for Nco-l nstitutional ~:rnnomit s. Ca,gè,c. Corse. 2-6 mai 2005

Figure 1 :

 1 Figure 1: Evolution of administrati0n costs and payments to fanners in the Envirnnmental Sensit ive Areas schemc in the UK

  practices. The enhancement of' landscapes in the ESA and CS programmes in Cngland are t> pica!

examples ()f lww agro-cll\ ironmental schemes aim al providing hiodi, crsity. amenities and bcautiful landscapes. 13nughera,a Il. 21105 New-l11s11tutumal Ec"1101111cs Anal) s,s of Agnl-1'11\'ITOllmrntal Sclwmcs E11.-opra11 Srhool for Nro-lnstitutional Ernnomirs. Ca,gc,c Corse. 2-6 ma, 2005

Researclt programme part of the 1T AES programme which involves 9 European research teams. IT AES is the acronym of "Integrated Tools to design and implement Agro-Environmental Schemes"_ it is a Specific Targeted Research Project of the six th Framework Programme launched by the European Commission in 2003. institut ional process and the environmental outcome. governance mechanisms.

Uncertainty

Uncertainty is pervasive in agro-environmental contracts. lt potentially concerns parties' behaviour and the environmental context of the transaction.

(i) Each party in the contract benefits from a power to influence the output of the contract. the environmental effectiveness.

On the one hand, there is an informational asymmetry between farmers and the State about measures effectively undertaken and the value of the land they contract on. Farmers may benefit from an informational rent since their efforts are costly to monitor and information costly to collect for the State. For example, non-point pollution prevents the State from determining which farrner has not lived up to his contract.

On the other hand, the State also benefits from an • institutional rent". lndeed, the State has the power to influence the ru les of the game. Farrners then face a risk of contract breaching from the State after having invested according to the former contract. This happened to some French farmers. ln 2002. CADs replace CTEs as a new tool for agro-environmental contracts. Only the CTEs that were already in the pipeline were signed at that tirne, while the ones that were about to be submitted ,vere cancelled, although farmers had already spend some money on it. 1 n front of some uncertainty on the stability of ru les for applying for agro-environmental schemes. farmers may lack motivations to commit.

(ii) Uncertainty arises also because unequivocal causality between farming practices and environmental impact is infrequent.

Natural processes suffer from threshold effects for example. Farmers may live up to their commitment but may not be numerous enough to reach a threshold enabling the environmental improvement to happen.

Uncertainty also cornes from natural processes like the weather. The Dutch case study reveals that breaching of contracts happened sometimes because bad weather conditions provided incentives for cutting the grass earlier.

Other sources of uncertainty cover the delay between the application of a measure and its effect, the difficulty of assessing the specific effect of agro-environmental schemes independently of other measures, the uncertainty about the future va lue of environmental benefits (option value).

Frequency of transactions and duration of agreement

Frequency of transactions between two partners is important since it enables costs incurred in establishing the con tract to be distributed over man y transactions. The higher the frequency of transactions. the easier it is to invest in specitïc assets. For example, investments in machines for hedgerows maintenance may be recovered if transactions are repeated over a certain period of time. Thus. another parameter to be considered is the duration of the con tract.

Case studies show that agro-environmental schemes are usually fïve-year contracts without re-negotiation of the contract. Sorne contracts however are long-term like the long-term set aside which lasts 20 years. As noted in the French case studies. the uptake for this kind of contract remained low since it was impossible for municipalities (main actors of this scheme) to set up a 20-year budget.

H,m ever, it is often stated that five-year con tracts are too short to observe any environmental improvement. Longer agreemtnts ma> not onl) frl\ our specific i11\'est ments but also reduce the frequency of contract re-negotiation \\ hich generall_\ incurs high costs. For instancé'. in England. the ESA scheme (Erwironmental Sensiti, e /\rcas) is a 10-) ear agreement wi1h an option to withdrav. after fi, e years. lt certain ly he lps sa, e re-negot iat ion transaction cnsts. Output measurability This attribute of transactions appeared through the development of the former attributes we characterised.

Il is however relevant to stress the measurability output issue for agro-environmental schemes. Output measurability critically increases the complexity of implementing agro-environmental schemes.

Environmental conservation or improvement is usually a hard and overall costly task because of nonpoint source pollution or because environmental impacts of practices appear after a more or less impo1tant delay. Only in few cases. output measurement is straightforward. The example of Netherlands where counting birds is a joint product of birds vvatching is seldom met. But connecting supply and demand reveals also a challenge. That is why case studies show that agro-environmental schemes usually rely on proxies like specific farming practices that are expected to lead to environmental good conservation or provision. Thus. although contracts ought to be objective oriented, they are usually practice oriented.

Although it spares from measuring the level of environmental output, costs of measuring whether the farmer undertook the required practices remain high. Practice-oriented contracts at least enable to reduce uncertainty on the farmer side.

One must be careful when considering the existence of performance indicators. Sometimes. a proxy is used to define objectives but is closer to practice requirements than performance requirement. In England, the performance indicator for landscape enhancement is a threshold of agreements with a conservation plan which gives little information if not on actual improvements. lt is especially difficult to set performance objectives for measures that are to some extent subjective. like aesthetic value of a landscape. The measurability of proxies is dependent on the way measures are defined. For instance, payments for stone walls maintenance are rather easy to enforce since measurement costs are low.

As a conclusion to the analysis on transaction attributes, we may say that agro-environmental schemes in Europe cover a wide diversity. As suggested above, the higher asset specificity, the higher uncertainty. the lower the frequency of transactions, and the higher output measurement costs then the higher transaction costs. Severa! suggestions follow from our discussion to enhance economic efficiency and environmental effectiveness: longer con tracts, investments in human capital and a form of coordination of farrners in undertaking measures. lndeed. the costs of transacting wil I more or less affect contract efficiency depending on the way transactions are governed. The several actors in agro-environmental schemes are expected to adopt governance structures minimising transaction costs. Severa! organisational and institutional parameters may affect the level of these costs and what is also important, the distribution of transaction costs among parties in the transaction.