Marie Doumic

Pierre Gabriel
email: gabriel@ann.jussieu.fr

Eigenelements of a General Aggregation-Fragmentation Model

Keywords: Aggregation-fragmentation equations, eigenproblem, size repartition, polymerization process, cell division, long-time asymptotic AMS Class. No. 35A05, 35B40, 45C05, 45K05, 82D60, 92D25

We consider a linear integro-differential equation which arises to describe both aggregationfragmentation processes and cell division. We prove the existence of a solution (λ, U, φ) to the related eigenproblem. Such eigenelements are useful to study the long time asymptotic behaviour of solutions as well as the steady states when the equation is coupled with an ODE. Our study concerns a non-constant transport term that can vanish at x = 0, since it seems to be relevant to describe some biological processes like proteins aggregation. Non lower-bounded transport terms bring difficulties to find a priori estimates. All the work of this paper is to solve this problem using weighted-norms.

Introduction

Competition between growth and fragmentation is a common phenomenon for a structured population. It arises for instance in a context of cell division (see, among many others, [START_REF] Adimy | Asymptotic behaviour of a singular transport equation modelling cell division[END_REF][START_REF] Basse | A mathematical model for analysis of the cell cycle in cell lines derived from human tumors[END_REF][START_REF] Brikci | Analysis of a molecular structured population model with possible polynomial growth for the cell division cycle[END_REF][START_REF] Brikci | An age-and-cyclin-structured cell population model for healthy and tumoral tissues[END_REF][START_REF] Doumic | Analysis of a population model structured by the cells molecular content[END_REF][START_REF] Farkas | Stability conditions for a nonlinear size-structured model[END_REF][START_REF] Gyllenberg | A Nonlinear Structured Population Model of Tumor Growth With Quiescence[END_REF][START_REF] Metz | The dynamics of physiologically structured populations[END_REF][START_REF] Perthame | Analysis of a cell system with finite divisions[END_REF]), polymerization (see [START_REF] Biben | Dynamics of Bio-Polymeric Brushes Growing from a Cellular Membrane: Tentative Modelling of the Actin Turnover within an Adhesion Unit; the Podosome[END_REF][START_REF] Destaing | Podosomes Display Actin Turnover and Dynamic Self-Organization in Osteoclasts Expressing Actin-Green Fluorescent Protein[END_REF]), telecommunication (see [START_REF] Baccelli | A mean-field model for multiple tcp connections through a buffer implementing red[END_REF]) or neurosciences (see [START_REF] Pakdaman | Dynamics of a structured neuron population[END_REF]). It is also a mechanism which rules the proliferation of prion's proteins (see [START_REF] Calvez | Size distribution dependence of prion aggregates infectivity[END_REF][START_REF] Greer | A mathematical analysis of the dynamics of prion proliferation[END_REF][START_REF] Laurençot | Well-posedness for a model of prion proliferation dynamics[END_REF]). These proteins are responsible of spongiform encephalopaties and appear in the form of aggregates in infected cells. Such polymers grow attaching non infectious monomers and converting them into infectious ones. On the other hand they increase their number by splitting.

To describe such phenomena, we write the following integro-differential equation,

               ∂ ∂t u(x, t) + ∂ ∂x τ (x)u(x, t) + β(x)u(x, t) = 2 ∞ x
β(y)κ(x, y) u(y, t) dy, x 0, u(x, 0) = u 0 (x), u(0, t) = 0.

(

) 1
The function u(x, t) represents the quantity of individuals (cells or polymers) of structured variable (size, protein content...) x at time t. These individuals grow (i.e., polymers aggregate monomers, or cells increase by nutrient uptake for instance) with the rate τ (x). Equation ( 1) also takes into account the fragmentation of a polymer (or the division of a cell) of size y into two smaller polymers of size x and y -x. This fragmentation occurs with a rate β(y) and produce an aggregate of size x with the rate κ(x, y). Equation ( 1) is a particular case of the more general one

∂ ∂t u(x, t) + ∂ ∂x τ (x)u(x, t) + [β(x) + µ(x)]u(x, t) = n ∞ x β(y)κ(x, y) u(y, t) dy, x x 0 , (2)
with the bound condition u(x 0 , t) = 0 (see [START_REF] Banasiak | On a coagulation and fragmentation equation with mass loss[END_REF][START_REF] Calvez | Size distribution dependence of prion aggregates infectivity[END_REF][START_REF] Laurençot | Well-posedness for a model of prion proliferation dynamics[END_REF]). Here, polymers are broken in an average of n > 1 smaller ones by the fragmentation process, there is a death term µ(x) ≥ 0 representing degradation, and a minimal size of polymers x 0 which can be positive. This more general model is biologically and mathematically relevant in the case of prion proliferation and is used in [START_REF] Calvez | Prion dynamic with size dependency -strain phenomena[END_REF][START_REF] Calvez | Size distribution dependence of prion aggregates infectivity[END_REF][START_REF] Greer | A mathematical analysis of the dynamics of prion proliferation[END_REF][START_REF] Laurençot | Well-posedness for a model of prion proliferation dynamics[END_REF] with a coupling to an ODE. Our results remain true for this generalization.

A fundamental tool to study the asymptotic behaviour of the population when t → ∞ is the existence of eigenelements (λ, U , φ) solution of the equation (

                           ∂ ∂x (τ (x)U (x)) + (β(x) + λ)U (x) = 2 ∞ x β (
For the first equation (equation on U ) we are looking for D ′ solutions defined as follows :

U ∈ L 1 (R + ) is a D ′ solution if ∀ϕ ∈ C ∞ c (R + ), - ∞ 0 τ (x)U (x)∂ x ϕ(x) dx+λ ∞ 0 U (x)ϕ(x) dx = ∞ 0 β(x)U (x) 2 ∞ 0 ϕ(y)κ(y, x) dy -ϕ(x) dx. (4)
Concerning the dual equation, we are looking for a solution φ ∈ W 1,∞ loc (0, ∞) such that the equality holds in L 1 loc (0, ∞), i.e. almost everywhere. When such elements exist, the asymptotic growth rate for a solution to [START_REF] Adimy | Asymptotic behaviour of a singular transport equation modelling cell division[END_REF] is given by the first eigenvalue λ and the asymptotic shape is given by the corresponding eigenfunction U . More precisely, it is proved for a constant fragmentation rate β that u(x, t)e -λt converges exponentially fast to ρU (x) where ρ = u 0 (y)dy (see [START_REF] Laurençot | Exponential decay for the growth-fragmentation/cell-division equation[END_REF][START_REF] Perthame | Exponential decay for the fragmentation or cell-division equation[END_REF]). For more general fragmentation rates, one can use the dual eigenfunction φ and the so-called "General Relative Entropy" method introduced in [START_REF] Michel | General entropy equations for structured population models and scattering[END_REF][START_REF] Perthame | Transport equations in biology[END_REF]. It provides similar results but without the exponential convergence, namely that ∞ 0 u(y, t)e -λt -u 0 , φ U (y) φ(y) dy -→ t→∞ 0 where u 0 , φ = u 0 (y)φ(y)dy (see [START_REF] Michel | General entropy equations for structured population models and scattering[END_REF][START_REF] Michel | General relative entropy inequality: an illustration on growth models[END_REF]).

The eigenvalue problem can also be used in nonlinear cases, such as prion proliferation equations, where there is a quadratic coupling of Equation ( 1) or (2) with a differential equation. In [START_REF] Calvez | Prion dynamic with size dependency -strain phenomena[END_REF][START_REF] Calvez | Size distribution dependence of prion aggregates infectivity[END_REF][START_REF] Engler | Analysis of a model for the dynamics of prions ii[END_REF][START_REF] Pruss | Analysis of a model for the dynamics of prion[END_REF] for instance, the stability of steady states is investigated. The use of entropy methods in the case of nonlinear problems remains however a challenging and widely open field (see [START_REF] Perthame | Nonlinear renewal equations[END_REF] for a recent review).

Existence and uniqueness of eigenelements has already been proved for general fragmentation kernels κ(x, y) and fragmentation rates β(x), but with very particular polymerization rates τ (x), namely constant (τ ≡ 1 in [START_REF] Perthame | Transport equations in biology[END_REF]), homogeneous (τ (x) = x µ in [START_REF] Michel | Existence of a solution to the cell division eigenproblem[END_REF]) or with a compact support (Supp τ = [0, x M ] in [START_REF] Doumic | Analysis of a population model structured by the cells molecular content[END_REF]).

The aim of this article is to consider more general τ as [START_REF] Calvez | Size distribution dependence of prion aggregates infectivity[END_REF][START_REF] Silveira | The most infectious prion protein particles[END_REF] suggest. Indeed, there is no biological justification to consider specific shapes of τ in the case when x represents a size (mass or volume) or some structuring variable and not the age of a cell (even in this last case it is not so clear that dx dt = 1, since biological clocks may exhibit time distorsions). For instance, for the prion proteins, the fact that the small aggregates are little infectious (see [START_REF] Lenuzza | Modélisation de la réplication des Prions: implication de la dépendance en taille des agrégats de PrP et de l'hétérogénéité des populations cellulaires[END_REF][START_REF] Silveira | The most infectious prion protein particles[END_REF]) leads us to include the case of rates vanishing at x = 0.

Considering fully general growth rates is thus indispensable to take into account biological or physical phenomena in their full diversity. The proof of [START_REF] Perthame | Transport equations in biology[END_REF] can be adapted for non constant rates but still positive and bounded (0 < m < τ (x) < M ). The paper [START_REF] Michel | Existence of a solution to the cell division eigenproblem[END_REF] gives results for τ (0) = 0, but for a very restricted class of shape for τ. The paper [START_REF] Doumic | Analysis of a population model structured by the cells molecular content[END_REF] gives results for τ with general shape in the case where there is also an age variable (integration in age then allows to recover Problem (1)), but requires a compact support and regular parameters. Here we consider polymerization rates that can vanish at x = 0, with general shape and few regularity for the all parameters (τ, β and κ).

From a mathematical viewpoint, relaxing as far as possible the assumptions on the rates τ, κ, β, as we have done in this article, also leads to a better understanding of the intrinsic mechanisms driving the competition between growth and fragmentation.

Theorem 1 (Existence and Uniqueness) Under assumptions (5)- [START_REF] Destaing | Podosomes Display Actin Turnover and Dynamic Self-Organization in Osteoclasts Expressing Actin-Green Fluorescent Protein[END_REF], there exists a unique solution (λ, U , φ) (in the sense we have defined before) to the eigenproblem (3) and we have λ > 0,

x α τ U ∈ L p (R + ), ∀α ≥ -γ, ∀p ∈ [1, ∞], x α τ U ∈ W 1,1 (R + ), ∀α ≥ 0 ∃k > 0 s.t. φ 1 + x k ∈ L ∞ (R + ), τ ∂ ∂x φ ∈ L ∞ loc (R + ).
The end of this paper is devoted to define precisely the assumptions and prove this theorem. It is organized as follows : in Section 2 we describe the assumptions and give some examples of interesting parameters. In Section 3 we prove Theorem 1 using a priori bounds on weighted norms and then we give some consequences and perspectives in Section 4. The proof of technical lemmas and theorem can be found in the Appendix.

Coefficients

Assumptions

For all y ≥ 0, κ(., y) is a nonnegative measure with a support included in [0, y]. We define κ on (R + ) 2 as follows : κ(x, y) = 0 for x > y. We assume that for all continuous function ψ, the application f ψ : y → ψ(x)κ(x, y) dx is Lebesgue measurable. The natural assumptions on κ (see [START_REF] Greer | A mathematical analysis of the dynamics of prion proliferation[END_REF] for the motivations) are that polymers can split only in two pieces which is taken into account by κ(x, y)dx = 1.

(

) 5
So κ(y, .) is a probability measure and f ψ ∈ L ∞ loc (R + ). The conservation of mass imposes

xκ(x, y)dx = y 2 , (6)
a property that is automatically satisfied for a symetric fragmentation (i.e. κ(x, y) = κ(y -x, y)) thanks to [START_REF] Brikci | Analysis of a molecular structured population model with possible polynomial growth for the cell division cycle[END_REF]. For the more general model [START_REF] Baccelli | A mean-field model for multiple tcp connections through a buffer implementing red[END_REF], assumption (6) becomes xκ(x, y)dx = y n to preserv the mass conservation. We also assume that the second moment of κ is less than the first one

x 2 y 2 κ(x, y)dx ≤ c < 1/2 (7)
(it becomes c < 1/n for model [START_REF] Baccelli | A mean-field model for multiple tcp connections through a buffer implementing red[END_REF]). We refer to the Examples for an explanation of the physical meaning.

For the polymerization and fragmentation rates τ and β, we introduce the set

P := f ≥ 0 : ∃µ, ν ≥ 0, lim sup x→∞ x -µ f (x) < ∞ and lim inf x→∞ x ν f (x) > 0 and the space L 1 0 := f, ∃a > 0, f ∈ L 1 (0, a) . We consider β ∈ L 1 loc (R + * ) ∩ P, ∃α 0 ≥ 0 s.t. τ ∈ L ∞ loc (R + , x α 0 dx) ∩ P (8) satisfying ∀K compact of (0, ∞), ∃m K > 0 s.t. τ (x) ≥ m K for a.e. x ∈ K (9)
(if τ is continuous, this assumption ( 9) is nothing but saying that for all x > 0, τ (x) > 0) and

∃b ≥ 0, Suppβ = [b, ∞). (10)
Assumption [START_REF] Calvez | Size distribution dependence of prion aggregates infectivity[END_REF] is necessary to prove uniqueness and existence for the adjoint problem.

To avoid shattering (zero-size polymers formation, see [START_REF] Banasiak | On a coagulation and fragmentation equation with mass loss[END_REF][START_REF] Laurençot | Well-posedness for a model of prion proliferation dynamics[END_REF]), we assume

∃ C > 0, γ ≥ 0 s.t. x 0 κ(z, y) dz ≤ min 1, C x y γ and x γ τ (x) ∈ L 1 0 (11)
which links implicitely τ to κ, and also

β τ ∈ L 1 0 . (12)
On the other hand, to avoid forming infinitely long polymers (gelation phenomenon, see [START_REF] Escobedo | Gelation and mass conservation in coagulation-fragmentation models[END_REF][START_REF] Escobedo | Gelation in coagulation and fragmentation models[END_REF]), we assume

lim x→+∞ xβ(x) τ (x) = +∞. ( 13
)
Remark 1 In case when [START_REF] Clairambault | Comparison of perron and floquet eigenvalues in age structured cell division cycle models[END_REF] is satisfied for γ > 0, then (7) is automatically fulfilled (see Lemma 3 in the Appendix).

Examples

First we give some examples of coefficients which satisfy or not our previous assumptions.

For the fragmentation kernel, we first check the assumptions ( 5) and [START_REF] Brikci | An age-and-cyclin-structured cell population model for healthy and tumoral tissues[END_REF]. They are satisfied for autosimilar measures, namely κ(x, y) = 1 y κ 0 ( x y ), with κ 0 a probability measure on [0, 1], symmetric in 1/2. Now we exhibit some κ 0 .

General mitosis : a cell of size x divides in a cell of size rx and one of size (1 -r)x (see [START_REF] Michel | Optimal proliferation rate in a cell division model[END_REF])

κ r 0 = 1 2 (δ r + δ 1-r ) for r ∈ [0, 1/2]. (14)
Assumption [START_REF] Clairambault | Comparison of perron and floquet eigenvalues in age structured cell division cycle models[END_REF] is satisfied for any γ > 0 in the cases when r ∈ (0, 1/2]. So [START_REF] Biben | Dynamics of Bio-Polymeric Brushes Growing from a Cellular Membrane: Tentative Modelling of the Actin Turnover within an Adhesion Unit; the Podosome[END_REF] is also fulfilled thanks to Remark 1. The particular value r = 1/2 leads to equal mitosis (κ(x, y) = δ x= y

2

). The case r = 0 corresponds to the renewal equation (κ(x, y) = 1 2 (δ x=0 + δ x=y )). In this case, we cannot strictly speak of mitosis because the size of the daughters are 0 and x. It appears when x is the age of a cell and not the size. This particular case is precisely the one that we want to avoid with assumption [START_REF] Biben | Dynamics of Bio-Polymeric Brushes Growing from a Cellular Membrane: Tentative Modelling of the Actin Turnover within an Adhesion Unit; the Podosome[END_REF] ; it can also be studied seperately with different tools (see [START_REF] Perthame | Nonlinear renewal equations[END_REF] for instance). For such a fragmentation kernel, assumption [START_REF] Clairambault | Comparison of perron and floquet eigenvalues in age structured cell division cycle models[END_REF] is satified only for γ = 0, and the moments z k κ 0 (z)dz are equal to 1/2 for all k > 0, so [START_REF] Biben | Dynamics of Bio-Polymeric Brushes Growing from a Cellular Membrane: Tentative Modelling of the Actin Turnover within an Adhesion Unit; the Podosome[END_REF] does not hold true. However, if we consider a convex combination of κ 0 0 with another kernel such as κ r 0 with r ∈ (0, 1/2], then [START_REF] Clairambault | Comparison of perron and floquet eigenvalues in age structured cell division cycle models[END_REF] remains false for any γ > 0 but (7) is fulfilled. Indeed we have for ρ ∈ (0, 1)

z 2 (ρκ 0 0 (z) + (1 -ρ)κ r 0 (z)) dz = ρ 2 + 1 -ρ 2 (r 2 + (1 -r) 2 ) = 1 2 (1 -2r(1 -r)(1 -ρ)) < 1 2 .
Homogeneous fragmentation :

κ α 0 (z) = α + 1 2 (z α + (1 -z) α ) for α > -1. (15)
It gives another class of fragmentation kernels, namely in L 1 (unlike the mitosis case). The parameter γ = 1 + α > 0 suits for [START_REF] Clairambault | Comparison of perron and floquet eigenvalues in age structured cell division cycle models[END_REF] and so [START_REF] Biben | Dynamics of Bio-Polymeric Brushes Growing from a Cellular Membrane: Tentative Modelling of the Actin Turnover within an Adhesion Unit; the Podosome[END_REF] is fulfilled. It shows that our assumptions allow fragmentation at the ends of the polymers (called depolymerization, see [START_REF] Lenuzza | Modélisation de la réplication des Prions: implication de la dépendance en taille des agrégats de PrP et de l'hétérogénéité des populations cellulaires[END_REF], when α is close to -1) once it is not the extreme case of renewal equation. Uniform repartition (κ(x, y) = 1 y 1l 0≤x≤y ) corresponds to α = 0 and is also included.

This last case of uniform repartition is useful because it provides us with explicit formulas for the eigenelements. For instance, we can consider the two following examples.

First example : τ (x) = τ 0 , β(x) = β 0 x. In this case, widely used by [START_REF] Greer | A mathematical analysis of the dynamics of prion proliferation[END_REF], the eigenelements exist and we have

λ = β 0 τ 0 , U (x) = 2 β 0 τ 0 X + X 2 2 e -X-X 2 2 , with X = β 0 τ 0 x, φ(x) = 1 2 (1 + X).
Second example : τ (x) = τ 0 x. For such β for which there exists eigenelements, we have

λ = τ 0 and φ(x) = x yU (y)
.

For instance when β(x) = β 0 x n with n ∈ N * , then the eigenelements exist and we can compute U and φ and we have the formulas in Table 1. In this table we can notice that U (0) > 0 but the boundary condition τ U (0) = 0 is fulfilled.

n = 1 λ = τ 0 U (x) = β 0 τ 0 e - β 0 τ 0 x φ(x) = β 0 τ 0 x n = 2 λ = τ 0 U (x) = 2β 0 πτ 0 e -1 2 β 0 τ 0 x 2 φ(x) = πβ 0 2τ 0 x n λ = τ 0 U (x) = β 0 nτ 0 1 n n Γ( 1 n ) e -1 n β 0 τ 0 x n φ(x) = β 0 nτ 0 1 n Γ( 1 n ) Γ( 2 n )
x Table 1: The example τ (x) = τ 0 x, β(x) = β 0 x n and uniform repartition κ(x, y) = 1 y 1l 0≤x≤y . The table gives the eigenelements solution to (3). Now we turn to non-existence cases. Let us consider constant fragmentation β(x) = β 0 with an affine polymerization τ (x) = τ 0 + τ 1 x, and any fragmentation kernel κ which satisfies to assumptions ( 5)-( 6). We notice that ( 13) is not satisfied and look at two instructive cases.

First case : τ 0 = 0. In this case assumption [START_REF] Clairambault | An inequality for the Perron and Floquet eigenvalues of monotone differential systems and age structured equations[END_REF] does not hold true. Assume that there exists U ∈ L 1 (R + ) solution of (3) with the estimates of Theorem 1. Integrating the equation on U we obtain that λ = β 0 , but multiplying the equation by x before integration we have that λ = τ 1 . We conclude that eigenelements cannot exist if τ 1 = β 0 . Moreover, if we take κ(x, y) = 1 y 1l 0≤x≤y , then a formal computation shows that any solution to the first equation of (3) belongs to the plan V ect{x -1 , x

- 2β 0 τ 1 }. So, even if β 0 = τ 1 , there does not exist an eigenvector in L 1 .
Second case : τ 0 > 0. In this case [START_REF] Clairambault | An inequality for the Perron and Floquet eigenvalues of monotone differential systems and age structured equations[END_REF] holds true but the same integrations than before lead to

xU (x) dx = τ 0 β 0 -τ 1 .
So there cannot exist any eigenvector U ∈ L 1 (x dx) for τ 1 ≥ β 0 .

Proof of the main theorem

The proof of Theorem 1 is divided as follows. We begin with a result concerning the positivity of the a priori existing eigenvectors (Lemma 1). We then define, in Section 3.2, a regularized and truncated problem for which we know that eigenelements exist (see the Appendix B for a proof using the Krein-Rutman theorem), and we choose it such that the related eigenvalue is positive (Lemma 2). In Section 3.3, we give a series of estimates that allow us to pass to the limit in the truncated problem and so prove the existence for the original eigenproblem (3). The positivity of the eigenvalue λ and the uniqueness of the eigenelements are proved in the two last subsections.

A preliminary lemma

Before proving Theorem 1, we give a preliminary lemma, useful to prove uniqueness of the eigenfunctions.

Lemma 1 (Positivity) Consider U and φ solutions to the eigenproblem (3).

We define m := inf

x,y

x : (x, y) ∈ Supp β(y)κ(x, y) . Then we have, under assumptions (5), ( 6), [START_REF] Calvez | Prion dynamic with size dependency -strain phenomena[END_REF] and ( 10)

Supp U = [m, ∞) and τ U (x) > 0 ∀x > m, φ(x) > 0 ∀x > 0. If additionaly 1 τ ∈ L 1 0 , then φ(0) > 0.
Remark 2 In case Supp κ = {(x, y)/x ≤ y}, then m = 0 and Lemma 1 and Theorem 1 can be proved without the connexity condition (10) on the support of β.

Proof. Let x 0 > 0, we define

F : x → τ (x)U (x)e x x 0
λ+β(s)
τ (s) ds . We have that

F ′ (x) = 2e x x 0 λ+β(s) τ (s) ds β(y)κ(x, y)U (y) dy ≥ 0. ( 16
)
So, as soon as τ U (x) once becomes positive, it remains positive for larger x.

We define a := inf{x : τ (x)U (x) > 0}. We first prove that a ≤ b 2 . For this we integrate the equation on [0, a] to obtain Assume by contradiction that m < a, integrating (3) multiplied by ϕ, we have for all

ϕ ∈ C ∞ c such that Supp ϕ ⊂ [0, a] ϕ(x)β(y)κ(x, y)U (y) dydx = 0. ( 17
)
By definition of m and using the fact that m < a, there exists (p, q) ∈ (m, a)× (b, ∞) such that (p, q) ∈ Supp β(y)κ(x, y). But we can choose ϕ positive such that ϕ(p)U (q) > 0 and this is a contradiction with [START_REF] Escobedo | Gelation and mass conservation in coagulation-fragmentation models[END_REF]. So we have m ≥ a.

To conclude we notice that on [0, m], U satisfies

∂ x (τ (x)U (x)) + λU (x) = 0.
So, thanks to the condition τ (0)U (0) = 0 and the assumption (9), we have U ≡ 0 on [0, m], so m = a and the first statement is proved.

For φ, we define G(x) := φ(x)e

-x
x 0 λ+β(s) τ (s) ds . We have that

G ′ (x) = -2e -x x 0 λ+β(s) τ (s) ds β(x) x 0 κ(y, x)φ(y) dy ≤ 0, (18)
so, as soon as φ vanishes, it remains null. Therefore φ is positive on an interval (0, x 1 ) with x 1 ∈ R * + ∪ {+∞}. Assuming that x 1 < +∞ and using that

x 1 > a = m because φ(x)U (x)dx = 1, we can find X ≥ x 1 such that X x 1 G ′ (x) dx = -2 X x 1 x 1 0 e x x 0 λ+β(s) τ (s) ds φ(y)β(x)κ(y, x) dy dx < 0.
This contradicts that φ(x) = 0 for x ≥ x 1 , and we have proved that φ(x) > 0 for x > 0. If 1 τ ∈ L 1 0 , we can take x 0 = 0 in the definition of G and so φ(0) > 0 or φ ≡ 0. The fact that φ is positive ends the proof of the lemma.

Truncated problem

The proof of the theorem is based on uniform estimates on the solution to a truncated equation. Let η, δ, R positive numbers and define

τ η (x) = η 0 ≤ x ≤ η τ (x) x ≥ η.
Then τ η is lower bounded on [0, R] thanks to [START_REF] Calvez | Prion dynamic with size dependency -strain phenomena[END_REF] and we denote by µ = µ(η, R) := inf [0,R] τ η . The existence of eigenelements (λ δ η , U δ η , φ δ η ) for the following truncated problem when δR < µ is standard (see Theorem 2 in the Appendix).

                           ∂ ∂x (τ η (x)U δ η (x)) + (β(x) + λ δ η )U δ η (x) = 2 R x β(y)κ(x, y)U δ η (y) dy, 0 < x < R, τ η U δ η (x = 0) = δ, U δ η (x) > 0, U δ η (x)dx = 1, -τ η (x) ∂ ∂x φ δ η (x) + (β(x) + λ δ η )φ δ η (x) -2β(x) x 0 κ(y, x)φ δ η (y) dy = δφ δ η (0), 0 < x < R, φ δ η (R) = 0, φ δ η (x) > 0, φ δ η (x)U δ η (x)dx = 1. ( 19
)
The proof of the theorem 1 requires λ δ η > 0. To enforce it, we take δR = µ 2 and we consider R large enough to satisfy the following lemma.

Lemma 2 Under assumptions (5), ( 8) and (13), there exists a R 0 > 0 such that for all R > R 0 , if we choose δ = µ 2R , then we have λ δ η > 0.

Proof. Assume by contradiction that R > 0 and λ δ η ≤ 0 with δ = µ 2R . Then, integrating between 0 and x > 0, we obtain τ (y) dy . But assumption [START_REF] Destaing | Podosomes Display Actin Turnover and Dynamic Self-Organization in Osteoclasts Expressing Actin-Green Fluorescent Protein[END_REF] ensures that for all n ≥ 0, there is a A > 0 such that

0 ≥ λ x 0 U (y) dy = δ -τ (x)U (x) -
β(x) τ (x) ≥ n x , ∀x ≥ A
and thus we have

τ (x)U (x) ≥ δ x A n , ∀x ≥ A.
Due to Assumption (8), we can choose n such that x -n τ (x) → 0 when x → +∞. Then there exists B > A such that x -n τ (x) ≤ µ 4A n for x ≥ B, and we have, for R > B,

1 = R 0 U (x) dx ≥ R B U (x) dx ≥ δ R B x n A n τ (x) dx ≥ 2 R (R -B)
what is a contradiction as soon as R > 2B; so Lemma 2 holds for R 0 = 2B. First estimate: λ δ η upper bound. Integrating equation ( 19) between 0 and R, we find

λ δ η ≤ δ + β(x)U δ η (x) dx,
then the idea is to prove a uniform estimate on βU δ η . For this we begin with bounding the higher moments x α βU δ η for α ≥ max (2, α 0 + 1) := m. Let α ≥ m, according to [START_REF] Biben | Dynamics of Bio-Polymeric Brushes Growing from a Cellular Membrane: Tentative Modelling of the Actin Turnover within an Adhesion Unit; the Podosome[END_REF] we have

x α y α κ(x, y) dx ≤ x 2 y 2 κ(x, y) dx ≤ c < 1 2
.

Multiplying the equation on U δ η by x α and then integrating on [0, R], we obtain for all

A ≥ η x α (1 -2c)β(x) U δ η (x) dx ≤ α x α-1 τ η (x)U δ η (x) dx = α x≤A x α-1 τ η (x)U δ η (x) dx + α x≥A x α-1 τ (x)U δ η (x) dx ≤ αA α-1-α 0 sup x∈(0,A) {x α 0 τ (x)} + ω A,α x α β(x)U δ η (x) dx,
where ω A,α is a positive number chosen to have ατ (x) ≤ ω A,α xβ(x), ∀x ≥ A. Thanks to [START_REF] Biben | Dynamics of Bio-Polymeric Brushes Growing from a Cellular Membrane: Tentative Modelling of the Actin Turnover within an Adhesion Unit; the Podosome[END_REF] and ( 13), we can choose A α large enough to have ω Aα,α < 1 -2c. Thus we find ∀α ≥ m, ∃A α : ∀η, δ > 0,

x α β(x)U δ η (x) dx ≤ αA α α-1-α 0 sup (0,A) {x α 0 τ (x)} 1 -2c -ω Aα,α := B α . ( 20
)
The next step is to prove the same estimates for 0 ≤ α < m and for this we first give a bound on τ η U δ η . We fix ρ ∈ (0, 1/2) and define x η > 0 as the unique point such that

xη 0 β(y)
τη (y) dy = ρ. It exists because β is nonnegative and locally integrable, and τ η is positive. Thanks to assumption [START_REF] Clairambault | An inequality for the Perron and Floquet eigenvalues of monotone differential systems and age structured equations[END_REF], we know that x η -→ η→0 x 0 where x 0 > 0 satisfies

x 0 0 β(y) τ (y) dy = ρ, so x η is bounded by 0 < x ≤ x η ≤ x.
Then, integrating (19) between 0 and x ≤ x η , we find

τ η (x)U δ η (x) ≤ δ + 2 x 0 β(y)U δ η (y)κ(z, y) dy dz ≤ δ + 2 β(y)U δ η (y) dy = δ + 2 xη 0 β(y)U δ η (y) dy + 2 ∞ xη β(y)U δ η (y) dy ≤ δ + 2 sup (0,xη ) {τ η U δ η } xη 0 β(y) τ η (y) dy + 2 x m η ∞ 0 y m β(y)U δ η (y) dy ≤ δ + 2ρ sup (0,xη ) {τ η U δ η } + 2 x m η B m .
Consequently, if we consider δ ≤ 1 for instance, we obtain sup x∈(0,x)

τ η (x)U δ η (x) ≤ 1 + 2B m /x m 1 -2ρ := C (21)
so τ η U δ η is uniformly bounded in a neighborhood of zero. Now we can prove a bound B α for x α βU δ η in the case 0 ≤ α < m. Thanks to the estimates ( 20) and ( 21) we have

x α β(x)U δ η (x) dx = x 0 x α β(x)U δ η (x) dx + R x x α β(x)U δ η (x) dx ≤ x α sup (0,x) {τ η U δ η } x 0 β(y) τ η (y) dy + x α-m R x x m β(x)U δ η (x) dx ≤ Cρx α + B m x α-m := B α . (22)
Combining ( 20) and ( 22) we obtain

∀α ≥ 0, ∃B α : ∀η, δ > 0, x α β(x)U δ η (x) dx ≤ B α , (23)
and finally we bound λ δ η

λ δ η = δ + βU δ η ≤ δ + B 0 . (24)
So the family {λ δ η } δ belong to a compact interval and we can extract a converging subsequence λ δ η -→ δ→0 λ η .

Second estimate : W 1,1 bound for x α τ η U δ η , α ≥ 0. We use the estimate [START_REF] Laurençot | Well-posedness for a model of prion proliferation dynamics[END_REF]. First we give a L ∞ bound for τ η U δ η by integrating (19) between 0 and x

τ η (x)U δ η (x) ≤ δ + 2 R 0 β(y)U δ η (y) dy ≤ δ + 2B 0 := D 0 . (25)
Then we bound x α τ η U δ η in L 1 for α > -1. Assumption (13) ensures that there exists X > 0 such that τ (x) ≤ xβ(x), ∀x ≥ X, so we have for R > X

x α τ η (x)U δ η (x) dx ≤ sup (0,X) {τ η U δ η } X 0 x α dx + R X x α+1 β(x)U δ η (x) dx ≤ sup (0,X) {τ η U δ η } X α+1 α + 1 + B α+1 := C α . Finally ∀α > -1, ∃C α : ∀η, δ > 0, x α τ η (x)U δ η (x) dx ≤ C α (26)
and we also have that x α U δ η is bounded in L 1 because τ ∈ P (see assumption ( 8)). A consequence of ( 23) and ( 26) is that x α τ η U δ η is bound in L ∞ for all α ≥ 0. We already have (25) and for α > 0, we multiply (19) by x α , integrate on [0, x] and obtain

x α τ η (x)U δ η (x) ≤ α R 0 y α-1 τ η (y)U δ η (y) dy + 2 R 0 y α β(y)U δ η (y) dy ≤ αC α + 2B α := D α , that give immediately ∀α ≥ 0, ∃D α : ∀η, δ > 0, sup x>0 x α τ η (x)U δ η (x) ≤ D α . (27)
To conclude we use the fact that neither the parameters nor U δ η are negative and we find by the chain rule, for α ≥ 0

∂ ∂x (x α τ η (x)U δ η (x)) dx ≤ α x α-1 τ η (x)U δ η (x) dx + x α ∂ x (τ η (x)U δ η (x)) dx ≤ α x α-1 τ η (x)U δ η (x) dx + λ δ η x α U δ η (x) dx + 3 x α β(x)U δ η (x) dx (28)
and all the terms in the right hand side are uniformly bounded thanks to the previous estimates.

Since we have proved that the family {x α τ η U δ η } δ is bounded in W 1,1 (R + ) for all α ≥ 0, then, because τ η is positive and belongs to P, we can extract from {U δ η } δ a subsequence which converges in L 1 (R + ) when δ → 0. Passing to the limit in equation ( 19) we find that

       ∂ ∂x (τ η (x)U η (x)) + (β(x) + λ η )U η (x) = 2 ∞ x β(y)κ(x, y)U η (y) dy, U η (0) = 0, U η (x) ≥ 0, U η = 1, (29)
with λ η ≥ 0.

3.4 Limit as η → 0 for U η and λ η

All the estimates ( 20)-( 28) remain true for δ = 0. So we still know that the family {x α τ η U η } η belongs to a compact set of L 1 , but not necessarily {U η } η because in the limit τ can vanish at zero. We need one more estimate to study the limit η → 0.

Third estimate: L ∞ bound for x α τ η U η , α ≥ -γ. We already know that x α τ η U η is bounded for α ≥ 0. So, to prove the bound, it only remains to prove that x -γ τ η U η is bounded in a neighborhood of zero. Let define f η : x → sup (0,x) τ η U η . If we integrate (29) between 0 and x ′ < x, we find

τ η (x ′ )U η (x ′ ) ≤ 2 x ′ 0 β(y)U η (y)κ(z, y) dy dz ≤ 2 x 0 β(y)U η (y)κ(z, y) dy dz
and so for all x f η (x) ≤ 2

x 0 β(y)U η (y)κ(z, y) dy dz.

We consider x η and x defined in the first estimate and, using ( 11) and ( 12), we have for all x < x η f η (x) ≤ 2

f η (y) y γ dy + 2C βU η L 1 x γ x γ η .
We set V η (x) = x -γ f η (x) and we obtain

(1 -2ρ)V η (x) ≤ K + 2C xη x β(y) τ η (y) V η (y) dy.
Hence, using Grönwall's lemma, we find that

V η (x) ≤ Ke 2Cρ 1-2ρ
1 -2ρ and consequently

x -γ τ η (x)U η (x) ≤ Ke 2Cρ 1-2ρ 1 -2ρ := C, ∀x ∈ [0, x]. ( 30
)
This last estimate allows us to bound U η by x γ τ which is in L 1 0 by the assumption [START_REF] Clairambault | Comparison of perron and floquet eigenvalues in age structured cell division cycle models[END_REF]. Thanks to the second estimate, we also have that x α U η is bounded in L 1 and so, thanks to the Dunford-Pettis theorem (see [START_REF] Brezis | Analyse fonctionnelle[END_REF] for instance), {U η } η belong to a L 1 -weak compact set. Thus we can extract a subsequence which converges L 1 -weak toward U . But for all ε > 0,

{x α U η } η is bounded in W 1,1 ([ε, ∞))
for all α ≥ 1 thanks to [START_REF] Michel | General entropy equations for structured population models and scattering[END_REF] and so the convergence is strong on [ε, ∞). Then we write

|U η -U | = ε 0 |U η -U | + ∞ ε |U η -U | ≤ 2 C ε 0 x γ τ (x) + ∞ ε |U η -U |.
The first term on the right hand side is small for ε small because x γ τ ∈ L 1 0 and then the second term is small for η small because of the strong convergence. Finally U η -→ η→0 U strongly in L 1 (R + ) and U solution of the eigenproblem (3).

Limit as δ, η → 0 for φ δ η

We prove uniform estimates on φ δ η which are enough to pass to the limit and prove the result.

Fourth estimate : uniform φ δ η -bound on [0, A]. Let A > 0, our first goal is to prove the existence of a constant C 0 (A) such that ∀η, δ, sup

(0,A) φ δ η ≤ C 0 (A).
We divide the equation on φ δ η by τ η and we integrate between x and x η with 0 < x < x η , where x η , bounded by x and x, is defined in the first estimate. Considering δ < µ (1-2ρ) x

(fulfilled for R > x 2(1-2ρ) since δ = µ 2R ), we find φ δ η (x) ≤ φ δ η (x η ) + 2 xη x β(y) τ η (y) y 0 κ(z, y)φ δ η (z) dz + x η δ µ φ δ η (0) ≤ φ δ η (x η ) + sup (0,xη) {φ δ η } 2 xη 0 β(y) τ η (y) y 0 κ(z, y) dz + x η δ µ
and we obtain sup

x∈(0,x) φ δ η (x) ≤ 1 1 -2ρ -δx/µ φ δ η (x η ).
Using the decay of φ δ η (x)e

φ δ η (x) ≤ C(A)φ δ η (x η ).
Noticing that φ δ η (x)U δ η (x)dx = 1, we conclude

1 ≥ xη 0 φ δ η (x)U δ η (x)dx ≥ φ δ η (x η )
φ δ η ≤ C 0 (A). ( 31
)
Fifth estimate : uniform φ δ η -bound on [A, ∞). Following an idea introduced in [START_REF] Perthame | Exponential decay for the fragmentation or cell-division equation[END_REF] we notice that the equation in [START_REF] Farkas | Stability conditions for a nonlinear size-structured model[END_REF] satisfied by φ δ η is a transport equation and therefore satisfies the maximum principle (see Lemma 4 in the Appendix). Therefore it remains to build a supersolution φ that is positive at

x = R, to conclude φ δ η (x) ≤ φ(x) on [0, R]
. This we cannot do on [0, R], but on a subinterval [A 0 , R] only. So we begin with an auxiliary function ϕ(x) = x k + θ with k and θ positive numbers to be determined. We have to check that on [A 0 , R]

-τ (x) ∂ ∂x ϕ(x) + (λ δ η + β(x))ϕ(x) ≥ 2β(x) κ(y, x)ϕ(y) dy + δφ δ η (0), i.e.
-kτ (x)x k-1 + (λ δ η + β(x))ϕ(x) ≥ 2θ + 2 κ(y, x)y k dy β(x) + δφ δ η (0).

For k ≥ 2, we know that κ(y, x) y k x k dy ≤ c < 1/2 so it is sufficient to prove that there exists A 0 > 0 such that we have

-kτ (x)x k-1 + (λ δ η + β(x))(x k + θ) ≥ (2θ + 2cx k )β(x) + δC 0 (1) (32)
for all x > A 0 , where C 0 is defined in [START_REF] Perthame | Transport equations in biology[END_REF]. For this, dividing (32) by x k-1 τ (x), we say that if we have

(1 -2c) xβ(x) τ (x) ≥ k + 2θβ(x) + δC 0 (1) x k-1 τ (x) , (33)
then [START_REF] Perthame | Exponential decay for the fragmentation or cell-division equation[END_REF] holds true. Thanks to assumptions ( 8) and ( 13) we know that there exists k > 0 such that for any θ > 0, there exists A 0 > 0 for which [START_REF] Perthame | Analysis of a cell system with finite divisions[END_REF] is true on [A 0 , +∞).

Then we conclude by choosing the supersolution φ(x)

= C 0 (A 0 ) θ ϕ(x) so that φ(x) ≥ φ δ η (x) on [0, A 0 ],
and on [A 0 , R], we have

   -τ (x) ∂ ∂x φ(x) + (λ δ η + β(x))φ(x) ≥ 2β(x) x 0 κ(y, x)φ(y) dy + δφ δ η (0), φ(R) > 0, (34)
which is a supersolution to the equation satisfied by φ δ η . Therefore φ δ η ≤ φ uniformly in η and δ and we get ∃k, θ, C s.t. ∀η, δ, φ δ η (x) ≤ (Cx k + θ).

Equation ( 19) and the fact that φ δ η is uniformly bounded in L ∞ loc (R + ) give immediately that ∂ x φ δ η is uniformly bounded in L ∞ loc (R + , τ (x)dx), so in L ∞ loc (0, ∞) thanks to [START_REF] Calvez | Prion dynamic with size dependency -strain phenomena[END_REF].

Then we can extract a subsequence of {φ δ η } which converges C 0 (0, ∞) toward φ. Now we check that φ satisfied the adjoint equation of (3). We consider the terms of (19) one after another.

First (λ δ η + β(x))φ δ η (x) converges to (λ + β(x))φ(x) in L ∞ loc . For ∂ x φ δ η ,
we have an L ∞ bound on each compact of (0, ∞). So it converges L ∞ - * weak toward ∂ x φ. It remains the last term which we write, for all x > 0,

x 0 κ(y, x)(φ δ η (y) -φ(y)) dy ≤ φ δ η -φ L ∞ (0,x) -→ η,δ→0 0.
The fact that φU = 1 comes from the convergence L ∞ -L 1 when written as

1 = φ δ η (x)U δ η (x) dx = φ δ η (x) 1 + x k (1 + x k )U δ η (x) dx -→ φ(x) 1 + x k (1 + x k )U (x) dx = φU .
At this stage we have found (λ, U , φ) ∈ R + × L 1 (R + ) × C(R + ) solution of (3). The estimates announced in Theorem 1 also follow from those uniform estimates. It remains to prove that λ > 0 and the uniqueness.

Proof of λ > 0

We prove a little bit more, namely that

λ ≥ 1 2 sup x≥0 {τ (x)U (x)}. (36)
We integrate the first equation of (3) between 0 and x and find

0 ≤ λ x 0 U (y) dy = -τ (x)U (x) - x 0 β(y)U (y) dy + 2 x 0 ∞ z β(y)κ(z, y)U (y) dy dz ≤ -τ (x)U (x) + 2 ∞ 0 ∞ z β(y)κ(z, y)U (y) dy dz = -τ (x)U (x) + 2 ∞ 0 β(y)U (y) dy = -τ (x)U (x) + 2λ,
Hence 2λ ≥ τ (x)U (x) and ( 36) is proved.

Uniqueness

We follow the idea of [START_REF] Michel | Existence of a solution to the cell division eigenproblem[END_REF]. Let (λ 1 , U 1 , φ 1 ) and (λ 2 , U 2 , φ 2 ) two solutions to the eigenproblem (3). First we have

λ 1 U 1 (x)φ 2 (x) dx = -∂ x (τ (x)U 1 (x)) -β(x)U 1 (x) + 2 ∞ x β(y)κ(x, y)U 1 (y) dy φ 2 (x) dx = τ (x)∂ x φ 2 (x) -β(x)φ 2 (x) + 2β(x) x 0 κ(y, x)φ 2 (y) dy U 1 (x) dx = λ 2 U 1 (x)φ 2 (x) dx and then λ 1 = λ 2 = λ because U 1 φ 2 > 0 thanks to Lemma 1.
For the eigenvectors we use the General Relative Entropy method introduced in [START_REF] Michel | General entropy equations for structured population models and scattering[END_REF][START_REF] Michel | General relative entropy inequality: an illustration on growth models[END_REF]. For C > 0, we test the equation on

U 1 against sgn U 1 U 2 -C φ 1 , 0 = ∂ x (τ (x)U 1 (x)) + (λ + β(x))U 1 (x) -2 ∞ x β(y)κ(x, y)U 1 (y) dy sgn U 1 U 2 (x) -C φ 1 (x) dx. Deriving U 1 U 2 (x) -C τ (x)U 2 (x)φ 1 (x) we find ∂ x (τ (x)U 1 (x))sgn U 1 U 2 (x) -C φ 1 (x) dx = ∂ x U 1 U 2 (x) -C τ (x)U 2 (x)φ 1 (x) dx + ∂ x (τ (x)U 2 (x)) U 1 U 2 (x)sgn U 1 U 2 (x) -C φ 1 (x) dx - U 1 U 2 (x) -C ∂ x (τ (x)U 2 (x)φ 1 (x)) dx
and then

∂ x (τ (x)U 1 (x))sgn U 1 U 2 (x) -C φ 1 (x) dx = 2 U 1 U 2 (x) -C x 0 β(x)κ(y, x)U 2 (x)φ 1 (y) dy - ∞ x β(y)κ(x, y)U 2 (y)φ 1 (x) dy dx +2 ∞ x β(y)κ(x, y)U 2 (y) dy U 1 U 2 (x)sgn U 1 U 2 (x) -C φ 1 (x) dx -(λ + β(x)) U 1 U 2 (x)sgn U 1 U 2 (x) -C U 2 (x)φ 1 (x) dx, ∂ x (τ (x)U 1 (x))sgn U 1 U 2 (x) -C φ 1 (x) dx = 2 β(y)κ(x, y) U 1 U 2 (y) -C - U 1 U 2 (x) -C U 2 (y)φ 1 (x) dxdy +2 ∞ x β(y)κ(x, y)U 2 (y) dy U 1 U 2 (x)sgn U 1 U 2 (x) -C φ 1 (x) dx -(λ + β(x)) U 1 U 2 (x)sgn U 1 U 2 (x) -C U 2 (x)φ 1 (x) dx. So 0 = 2 β(y)κ(x, y) U 1 U 2 (y) -C - U 1 U 2 (x) -C U 2 (y)φ 1 (x) dxdy +2 ∞ x β(y)κ(x, y)U 2 (y) dy U 1 U 2 (x)sgn U 1 U 2 (x) -C φ 1 (x) dx -2 ∞ x β(y)κ(x, y)U 1 (y) dy sgn U 1 U 2 (x) -C φ 1 (x) dx 0 = β(y)κ(x, y)U 2 (y) U 1 U 2 (y) -C 1 -sgn U 1 U 2 (x) -C sgn U 1 U 2 (y) -C φ 1 (x) dxdy. Hence 1-sgn U 1 U 2 (x)-C sgn U 1 U 2 (y)-C = 0 on the support of κ(x, y) for all C thus U 1 U 2 (x) = U 1
U 2 (y) on the support of κ(x, y) and

∂ x U 1 U 2 (x) = β(y)κ(x, y) U 1 U 2 (y) - U 1 U 2 (x) U 2 (y) U 2 (x) dy = 0 (37) so U 1 U 2 ≡ cst = 1.
We can prove in the same way that φ 1 = φ 2 even if we can have U ≡ 0 on [0, m] with m > 0. Indeed in this case we know that β ≡ 0 on [0, m] and so

φ i (x) = φ i (0)e x 0 λ τ (s) ds ∀x ∈ [0, m], i ∈ {1, 2}.

Conclusion, Perspectives

We have proved the existence and uniqueness of eigenelements for the aggregation-fragmentation equation ( 1) with assumptions on the parameters as large as possible, in order to render out the widest variety of biological or physical models. It gives access to the asymptotic behaviour of the solution by the use of the General Relative Entropy principle.

A following work is to study the dependency of the eigenvalue λ on parameters τ and β (see [START_REF] Michel | Optimal proliferation rate in a cell division model[END_REF]). For instance, our assumptions allow τ to vanish at zero, what is a necessary condition to ensure that λ tends to zero when the fragmentation tends to infinity. Such results give precious information on the qualitative behaviour of the solution.

Another possible extension of the present work is to prove existence of eigenelements in the case of time-periodic parameters, using the Floquet's theory, and then compare the new λ F with the timeindependent one λ (see [START_REF] Clairambault | Comparison of perron and floquet eigenvalues in age structured cell division cycle models[END_REF]). Such studies can help to choose a right strategy in order to optimize, for instance, the total mass xu(t, x)dx in the case of prion proliferation (see [START_REF] Calvez | Size distribution dependence of prion aggregates infectivity[END_REF]) or on the contrary minimize the total population u(t, x)dx in the case of cancer therapy (see [START_REF] Clairambault | Comparison of perron and floquet eigenvalues in age structured cell division cycle models[END_REF][START_REF] Clairambault | An inequality for the Perron and Floquet eigenvalues of monotone differential systems and age structured equations[END_REF]).

Finally, this eigenvalue problem could be used to recover some of the equation parameters like τ and β from the knowledge of the asymptotic profile of the solution, as introduced in [START_REF] Doumic | Numerical solution of an inverse problem in sizestructured population dynamics[END_REF][START_REF] Perthame | On the inverse problem for a size-structured population model[END_REF] for instance, we obtain

B Krein-Rutman

We prove existence of solution for the truncated equation [START_REF] Farkas | Stability conditions for a nonlinear size-structured model[END_REF]. In this part η and δ are fixed (with δR < µ), so we will omit these indices for τ, λ, U and φ but we keep in mind that τ (x) ≥ µ > 0.

We use the Krein-Rutman theorem which requires working in the space of continuous functions (see [START_REF] Perthame | Transport equations in biology[END_REF] for instance). First we define regularized parameters as follows :

τ ε = ρ ε * τ, β ε = ρ ε * β, and ∀y ≥ 0, κ ε (., y) = ρ ε * κ(., y),
where

ρ ε (x) = 1 ε ρ( x ε ) with ρ ∈ C ∞ c ((0, ∞))
, positive and such that ∞ 0 ρ = 1. Then we have the theorem Theorem 2 Under assumptions (5)-( 13) on the parameters and for all ε > 0, there is a unique solution

λ ε ∈ R and U ε , φ ε ∈ C 1 ([0, R]) to the regularized eigenproblem                              ∂ ∂x (τ ε (x)U ε (x)) + (β ε (x) + λ ε )U ε (x) = 2 R 0 β ε (y)κ ε (x, y)U ε (y) dy, 0 < x < R, τ ε U ε (x = 0) = δ R 0 U ε (y) dy, U ε (x) > 0, R 0 U ε (x)dx = 1, -τ ε (x) ∂ ∂x φ ε (x) + (β ε (x) + λ ε )φ ε (x) -2β ε (x) R 0 κ ε (y, x)φ ε (y) dy = τ ε (0)δφ ε (0), 0 < x < R, φ ε (R) = 0, φ ε (x) > 0, R 0 φ ε (x)U ε (x)dx = 1. ( 38
)
Proof. We follow the proof of [START_REF] Perthame | Transport equations in biology[END_REF]. We define linear operators on E := C 0 ([0, R]) to apply the Krein-Rutman theorem.

Direct equation. For ν > 0 we consider the following equation on E

         ∂ ∂x (n(x)) + ν + β ε (x) τ ε (x) n(x) -2 R 0 β ε (y) τ ε (y) κ ε (x, y)n(y) dy = f (x) τ ε (x) , 0 ≤ x ≤ R, n(x = 0) = δ R 0 n(y) τε(y) dy, (39)
and we prove that the linear operator A : f → n (solution of (39)) satisfies to the assumptions of the Krein-Rutman theorem.

First step: construction of A. Fix f ∈ E and for m ∈ E, we define n = T (m) ∈ E as the (explicit) solution to

         ∂ ∂x (n(x)) + ν + β ε (x) τ ε (x) n(x) = 2 R 0 β ε (y) τ ε (y) κ ε (x, y)m(y) dy + f (x) τ ε (x) , 0 ≤ x ≤ R, n(x = 0) = δ R 0 m(y) τε(y) dy,
We prove that T is a strict contraction. Therefore it has a unique fixed point thanks to the Banach-Picard theorem. This fixed point is a solution to (39).

In order to prove that T is a strict contraction, we consider m 1 and m 2 two functions in E, we compute for n

= n 1 -n 2 , m = m 1 -m 2 ,          ∂ ∂x (n(x)) + ν + β ε (x) τ ε (x) n(x) = 2 R 0 β ε (y) τ ε (y) κ ε (x, y)m(y) dy, 0 ≤ x ≤ R, n(x = 0) = δ R 0 m(y) τε(y) dy, therefore          ∂ ∂x |n(x)| + ν + β ε (x) τ ε (x) |n(x)| ≤ 2 R 0 β ε (y) τ ε (y) κ ε (x, y)|m(y)| dy, 0 ≤ x ≤ R, n(x = 0) ≤ δ R 0 |m(y)| τε(y) dy.
After integration, we obtain

|n(x)|e x 0 µ+βε τε ≤ δ R 0 |m(y)| τ ε (y) dy + x 0 e x ′ 0 ν+βε τε R 0 β ε (y) τ ε (y) κ ε (x ′ , y)|m(y)| dydx ′ and thus |n(x)| ≤ δ R 0 |m(y)| τ ε (y) dy + x 0 e -x x ′ ν+βε τε R 0 β ε (y) τ ε (y) κ ε (x ′ , y)|m(y)| dydx ′ ≤ m E 1 µ δR + x 0 e -x x ′ ν+βε τε R 0 β ε (y)κ ε (x ′ , y) dydx ′ ≤ m E 1 µ δR + R 0 β ε (y)κ ε (., y)dy L ∞ x 0 e - ν τε L ∞ (x-x ′ ) dx ′ ≤ m E 1 µ δR + ν -1 τ ε L ∞ R 0 β ε (y)κ ε (., y)dy L ∞ :=k .
Because δR < µ by assumption, we can choose ν large so that k < 1 and we obtain

n E ≤ k m E .
Thus T is a strict contraction and we have proved the existence of a solution to (39).

Second step: A is continuous. This relies on a general argument which in fact shows that the linear mapping A is Lipschitz continuous. Indeed, arguing as above

|n(x)|e x 0 ν+βε τε ≤ δ R 0 |n(y)| τ ε (y) dy + x 0 e x ′ 0 ν+βε τε R 0 β ε (y) τ ε (y) κ ε (x ′ , y)|n(y)| dydx ′ + x 0 e x ′ 0 ν+βε τε |f (x ′ )| τ ε (x ′ ) dx ′ ,
and thus

|n(x)| ≤ k n E + R 0 |f (x ′ )| τ ε (x ′ ) dx ′ ≤ k n E + R µ f E .
This indeed proves that

n E ≤ R µ(1 -k) f E .
Third step: A is strongly positive. For f ≥ 0, the operator T of the first step maps m ≥ 0 to n ≥ 0. Therefore the fixed point n is nonnegative. In other words n = A(f ) ≥ 0. If additionally f does not vanish, then n does not vanish either. Therefore n(0) = δ

where τε (x) = τ ε (R -x) and βε (x) = β ε (R -x). Then the same method than for the direct equation give the result, namely the existence of λ and φ solution to (40).

Finally we have proved existence of (λ U , U ε ) and (λ φ , φ ε ) solution to the direct and adjoint equations of (38). It remains to prove that λ U = λ φ but it is nothing but integrating the direct equation against the ajoint eigenvector, what gives

λ U U ε φ ε = λ φ U ε φ ε .
To have existence of solution for [START_REF] Farkas | Stability conditions for a nonlinear size-structured model[END_REF], it remains to do ε → 0. For this we can prove uniform bounds in L ∞ for U ε and φ ε because we are on the fixed compact [0, R]. Then we can extract subsequences which converge L ∞ * -weak toward U and φ, solutions to [START_REF] Farkas | Stability conditions for a nonlinear size-structured model[END_REF] because τ ε and β ε converge in L 1 toward τ and β. Concerning κ ε , we have that for all ϕ ∈ C ∞ c , ϕ(x)κ ε (x, y)dx -→ ϕ(x)κ(x, y)dx ∀y, and it is sufficient to pass to the limit in the equations.

C Maximum principle

Lemma 4 If there exists A 0 > 0 such that φ ≥ φ on [0, A 0 ] and φ a supersolution of (3) on [A 0 , R] with φ(R) ≥ φ(R), then φ ≥ φ on [0, R].

Proof. The proof is based on the same tools than to prove uniqueness (see above) or to establish GRE principles (see [START_REF] Michel | General entropy equations for structured population models and scattering[END_REF][START_REF] Michel | General relative entropy inequality: an illustration on growth models[END_REF] for instance). We know that φ ≥ φ on [0, A 0 ] and that φ is a supersolution to the equation satisfied by φ on [A 0 , R], i.e. there exists a function f ≥ δφ(0) such that Then, multiplying by 1l φ≥φ , we obtain (see [START_REF] Perthame | Transport equations in biology[END_REF] for a justification) and this can hold only if 1l φ≥φ ≡ 0 or φ(0) = 0. But we deal with the truncated problem with τ (x) ≥ η > 0, so 1 τ ∈ L 1 0 and φ(0) > 0 thanks to the lemma 1. Thus 1l φ≥φ ≡ 0 and the lemma 4 is proved.

  y)κ(x, y)U (y)dy, x 0,τ U (x = 0) = 0, U (x) ≥ 0, ∞ 0 U (x)dx = 1, -τ (x) ∂ ∂x (φ(x)) + (β(x) + λ)φ(x) = 2β(x) x)U (x)dx = 1.

  y)κ(x, y)U (y) dydx = 0, ∞ a β(y)U (y) a 0 κ(x, y) dxdy = 0. Thus for almost every y ≥ max(a, b), a 0 κ(x, y) dx = 0. As a consequence we have 1 = κ(x, y) dx = y a κ(x, y) dx ≤ 1 a xκ(x, y) dx = y 2a thanks to (5) and (6), and this is possible only if b ≥ 2a.

  )κ(z, y)U (y) dy dz = δ -τ (x)U (x) + x 0 β(y)U (y) dy + 2 , y) dz β(y)U (y) dy ≥ δ -τ (x)U (x) + x 0 β(y)U (y) dy. Consequently τ (x)U (x) ≥ δ + x 0 β(y) τ (y) τ (y)U (y) dy and, thanks to Grönwall's lemma, τ (x)U (x) ≥ δe x 0 β(y)

3. 3

 3 Limit as δ → 0 for U δ η and λ δ η Fix η and let δ → 0 (then R → ∞ since δR = µ 2 ).

x 0 β

 0 (y)U η (y)κ(z, y) dy dz = 2 β(y)U η (y)

  η (x) dx, so, as x η → x 0 and x 0 0 U (x)dx > 0 (thanks to Lemma 1 and because x 0 > b ≥ a), we have sup (0,A)

, 1 (

 1 in the case of symmetric division (τ = 1 and κ = δ x= y We use the first part of the proof to conclude. Taking η = min 1 4 4C) 1/γ

  and the lemma is proved for c = 1 2 -1 48 .

  dy > 0 and thusn(x) ≥ n(0) + e -′ ) τ ε (x ′ ) dx ′ > 0.Fourth step: A is compact. For f E ≤ 1, the third step proves that n is bounded in E and thus∂ ∂ x n = -ν + β ε τ ε n + β ε (y) τ ε (y) κ(x, y)n(y)dy + f τ ε is also bounded in E.Therefore by the Ascoli-Arzela theorem the family n is relatively compact in E.Adjoint equation. A function φ is a solution to the adjoint equation of (38) if and only if φ(x) := φ(R -x) ) + ( βε (x) + λ ε ) φ(x) -2 βε (x) R 0 κ ε (y, R -x) φε (y) dy = δ φε (R), 0 < x < R, φε (0) = 0,

  ) + (λ + β(x))φ(x) = 2β(x) x 0 κ(y, x)φ(y) dy + f (x), ∀x ∈ [A 0 , R].So we have for allx ∈ [A 0 , R] -τ (x) ∂ ∂x (φ(x)φ(x)) + (λ + β(x))(φ(x) -φ(x)) = 2β(x)x 0 κ(y, x)(φ(y) -φ(y)) dy -f (x).

2 R 0 ( 0 f 2 R 0 (

 20020 -τ (x) ∂ ∂x (φ -φ) + (x) + (λ + β(x))(φ -φ) + (x) ≤ 2β(x) x 0 κ(y, x)(φ -φ) + (y) dy -f (x)1l φ≥φ (x),and this inequality is satisfied on[0, R] since (φ -φ) + ≡ 0 on [0, A 0 ].If we test against U we have, using the fact that φ(R) = 0 < φ(R), R 0 (φ -φ) + (x) ∂ ∂x (τ (x)U (x)) dx + R 0 (λ + β(x))(φφ) + (x)U (x) dx ≤ φ -φ) + (y) R y β(x)κ(y, x)U (x) dxdy -R (x)1l φ≥φ (x)U (x) dx.But if we test the equation (3) satisfied by U against (φ -φ) + , we findR 0 (φ -φ) + (x) ∂ ∂x (τ (x)U (x)) dx + R 0 (λ + β(x))(φ -φ) + (x)U (x) dx = φ -φ) + (y)R y β(x)κ(y, x)U (x) dxdy, (x)U (x) dx ≤ 0

2

), by the use of inverse problems techniques. The method of [START_REF] Perthame | On the inverse problem for a size-structured population model[END_REF] has to be adapted to our general case, in order to model prion proliferation for instance, or yet to recover the aggregation rate τ ; this is another direction for future research.

Aknowledgment

The authors thank a lot Benoît Perthame for his precious help and his corrections. The two other assumptions ( 6) and ( 11) allow to control the mass of κ at the ends :

which gives the first assertion of the lemma. Now we can prove (7) :


Eugene Asarin

Nicolas Basset

Marie-Pierre Béal

Aldric Degorre

Dominique Perrin

Toward a Timed Theory of Channel Coding ⋆

The classical theory of constrained-channel coding deals with the following questions: given two languages representing a source and a channel, is it possible to encode source messages to channel messages, and how to realize encoding and decoding by simple algorithms, most often transducers. The answers to this kind of questions are based on the notion of entropy. In the current paper, the questions and the results of the classical theory are lifted to timed languages. Using the notion of entropy of timed languages introduced by Asarin and Degorre, the question of timed coding is stated and solved in several settings.

Introduction

This paper is the first attempt to lift the classical theory of constrained-channel coding to timed languages.

Let a language S represent all the possible messages that can be generated by a source, and C all the messages that can transit over a channel. Typical problems addressed by coding theory are:

-Is it possible to transmit any source generated message via the channel? -What would be the transmission speed? -How to encode the message before and to decode it after transmission?

The answers given by the theory of channel coding are as follows: to each language L is associated a non-negative real number h(L), called its entropy, which characterizes the quantity of information in bits per symbol. In order to transmit information in real-time (resp. with speed α) the entropy of the source should not exceed the one of the channel: h(S) ≤ h(C) (resp. αh(S) ≤ h(C)). For regular (or more precisely sofic) languages, whenever the information inequalities above are strict, the theory of channel coding provides a transmission protocol with a simple encoding and decoding (realized by a finite-state transducer). For the practically important case when S = Σ * and h(S) < h(C), the decoding can be made even simpler (sliding-window). A typical example is EFMPlus code [START_REF] Immink | EFMPlus: The coding format of the multimedia compact disc[END_REF] allowing writing any binary file (i.e. the source {0, 1} * , with entropy 1) onto a DVD (the channel C = (2, 10) -RLL admits all the words without factors 11, 101 and 0 11 , its entropy is 0.5418, see [START_REF] Blahut | Digital Transmission of Information[END_REF]) with almost optimal rate α = 1/2.

Classical theory of channel coding deals with discrete messages. It is, however, important to consider data words, i.e. discrete words augmented with data, e.g. real numbers. In this paper, we develop the theory of channel coding for the most studied class of data languages: timed languages. Several models of information transmission are possible for the latter:

the source is a timed language; the channel is a discrete language. In this case, lossless encoding is impossible, and we will consider encoding with some precision ε; -the source and the channel are timed languages, we are interested in exact (lossless) encoding; -the source and the channel are timed languages, some scaling of time data is allowed.

Our solution will be based on the notion of entropy of timed languages [START_REF] Asarin | Volume and entropy of regular timed languages: Discretization approach[END_REF]. For several models of transmission of timed data we will write information inequalities relating entropies of sources and channels with parameters of encodings (rate, precision, scaling, see below). Such an inequality is a necessary condition for existence of an encoding. On the other hand, whenever the information inequality holds (in its strict form) and the languages are regular (sofic), we give an explicit construction for simple timed encoding-decoding functions.

Related work. Constrained-channel coding theory for finite alphabets is a wellestablished domain; we refer the reader to monographs [START_REF] Lind | An Introduction to Symbolic Dynamics and Coding[END_REF][START_REF] Blahut | Digital Transmission of Information[END_REF][START_REF] Berstel | Codes and Automata[END_REF], handbook chapters [START_REF] Marcus | Constrained systems and coding for recording channels[END_REF][START_REF] Béal | Variable-length codes and finite automata[END_REF] and references therein. We started exploring information contents of timed languages in [START_REF] Asarin | Volume and entropy of regular timed languages: Analytic approach[END_REF][START_REF] Asarin | Volume and entropy of regular timed languages: Discretization approach[END_REF][START_REF] Asarin | Two Size Measures for Timed Languages[END_REF] where the notion of entropy was introduced and related to information measures such as Kolmogorov complexity and ε-entropy. Technically, we strongly build on discretization of timed languages, especially [START_REF] Asarin | Volume and entropy of regular timed languages: Discretization approach[END_REF][START_REF] Basset | Thin and thick timed regular languages[END_REF].

In a less formal way, a vision of timed languages as a special kind of languages with data [START_REF] Bouyer | An algebraic approach to data languages and timed languages[END_REF][START_REF] Bojanczyk | Data monoids[END_REF] was another source of inspiration.

Paper structure. In Sect. 2 we recall basic notions of the discrete theory of constrained-channel coding. In Sect. 3 we briefly recall some results and constructions on volume, entropy and discretization of timed languages. In Sect. 4 we state our main results on timed theory of constrained-channel coding. In Sect. 5 we discuss the rationale, perspectives and applications of this work.

Theory of channel coding for finite alphabet languages

In this section we give an elementary exposition3 of some basic notions and results from the theory of constrained-channel coding, see [START_REF] Marcus | Constrained systems and coding for recording channels[END_REF][START_REF] Lind | An Introduction to Symbolic Dynamics and Coding[END_REF][START_REF] Berstel | Codes and Automata[END_REF][START_REF] Béal | Variable-length codes and finite automata[END_REF] for more details.

Terminology

Let Σ be a finite alphabet. A factor of a word w ∈ Σ * is a contiguous subsequence of its letters. A language L is factorial whenever for each word w ∈ L every factor of it is in the language. A language L is extensible whenever for each word w ∈ L there exists a letter a ∈ Σ such that wa ∈ L. These two conditions are usual in the context of coding and can be justified in practice as follows. If we can encode (decode) some long word w (e.g. a movie file), then we want also to encode (decode) its contiguous fragments (e.g. a short scene in the middle of the movie). On the other hand, some extension of w should correspond to a longer movie.

A language L, which is regular, factorial and extensible, is called sofic. Sofic languages can be recognized by finite automata whose states are all initial and final (this ensures factoriality) and all have outgoing transitions (this ensures extensibility).

Given a language L, we denote by L n its sublanguage of words of length n. The entropy h(L) of a language over a finite alphabet is the asymptotic growth rate of the cardinality of L n , formally defined by (all the logarithms are base 2):

h(L) = lim n→∞ 1 n log |L n |.
The limit (finite or -∞) always exists if L is factorial. For a sofic language L recognized by a given automaton, its entropy h(L) can be effectively computed using linear algebra. In particular if L = Σ * for a k-letter alphabet Σ then h(L) = log k. Finally, the intuitive meaning of the entropy is the amount of information (in bits per symbol) in typical words of the language. Most of our coding functions have a special property defined below.

Definition 1 (almost injective

(wu) = φ(w ′ u ′ ) ⇒ w = w ′ .
Intuitively, if such a function is used to encode messages, then knowing the code of some message wu one can decode w, i.e. the whole message except its last d symbols. Thus the decoding is possible with delay d. This can be formalized as follows:

Definition 2 (almost inverse). For an almost injective function φ : Σ * → Γ * with delay d its d-almost inverse family of functions ψ n : Γ * → Σ n is characterized by the following property: for any w ∈ Σ n and v ∈ Γ * ,

w = ψ n (v) ⇔ ∃u ∈ Σ d : φ(wu) = v.
Lemma 1. If the domain of an almost injective function φ is extensible and ψ n is its almost inverse, then ψ n is a surjection to this domain (constrained to words of length n).

Coding: the basic case

Let A and A ′ be two alphabets (source and channel alphabets), S ⊂ A * and C ⊂ A ′ * factorial extensible languages and d ∈ IN. The aim is to encode any source message w ∈ S to a channel message φ(w) ∈ C. The latter message can be transmitted over the channel. The first condition means that the information is transmitted in real-time (with the transmission rate 1). The second one permits decoding.

A natural question is to find necessary and sufficient conditions on S and C for an (S, C)-encoding (with some delay) to exist. This question can be addressed by comparing the entropy of the languages S and C. Roughly, the channel language should contain at least as much information per symbol as the source language. Formally, we define the information inequality:

h(S) ≤ h(C). ( II1

⊓ ⊔

Thus, (II1) is necessary for existence of the coding. For sofic languages (if the inequality is strict) it is also sufficient. Moreover, the encoding can be realized by a sort of finite-state machine. We present this fundamental result in the following form which is essentially the finite-state coding theorem from [START_REF] Marcus | Constrained systems and coding for recording channels[END_REF], Theorem 10.3.7 in [START_REF] Lind | An Introduction to Symbolic Dynamics and Coding[END_REF].

Theorem 1. Let S and C be sofic languages. If the strict version of (II1) holds, then there exists an (S, C)-encoding realized by a finite-state transducer which is right-resolving on input and right-closing on output 4 .

The reader is now motivated to get through a couple of definitions.

Definition 4 (transducer).

A transducer is a tuple τ = (Q, A, A ′ , ∆, I, O) with a finite set Q of control states; finite input and output alphabets A and A ′ ; a set of transitions ∆ (each transition δ has a starting state ori(δ) and an ending state ter(δ)); input and output labeling functions The transducer is said to be right-resolving on input whenever for each state q ∈ Q every two different edges starting from q have different input labels. For such a transducer, the input automaton with a fixed initial state i, i.e. A i = (Q, A, ∆, I, i, Q) is deterministic, we denote by S i the language of this automaton.

I : ∆ → A and O : ∆ → A ′ .
The transducer is right-closing on output with delay d whenever every two paths π and π ′ of length d + 1 with the same output label and the same starting state q always have the same initial edge π 1 = π ′ 1 . A transducer τ satisfying both properties performs the encoding process in a natural way: an input word w of S is read from a state i along a path π w , and this path determines the output word O(π w ). The function φ i : S i → A ′ * defined as w → O(π w ) is length preserving and almost injective with delay d.

Example 1. Consider the source language S = {0, 1} * and the channel language C recognized by the sofic automaton on the left of Fig. 1. The language C is composed by all the words on {a, b, c} that do not contain any block bc. The entropy of the source is h(S) = 1, and the one of the channel is h(C) = 2 log (1 + √ 5)/2 ≈ 1.3885. The information inequality h(S) < h(C) holds and we can encode S in C using the transducer on the right of Fig. 1.

Other coding settings

Similarly to the previous section, other coding settings can be considered. For example, we can transmit information over a channel with some rate α = p q , when q letters of the channel message correspond to p letters of the source message (the previous section corresponds thus to the case α = 1). In this setting, the information inequality takes the form:

αh(S) ≤ h(C), ( II2
)
and it is a necessary and almost sufficient condition for the code to exist:

Proposition 2. Let S and C be factorial and extensible languages. If an (S, C)encoding with rate α exists, then (II2) necessarily holds.

Proposition 3. Let S and C be sofic languages. If a strict inequality (II2) holds, then an (S, C)-encoding of rate α exists. Moreover, it can be realized by a finitestate transducer of rate α.

We skip here a natural definition of such a transducer.

3 Preliminaries on timed languages

Timed alphabets, words, languages, and automata

We call k-M -alphabet a set A = [0, M ] × Σ where Σ is a k-letter alphabet and M a positive integer bound, so that every letter in A corresponds to a realvalued delay (seen as data) in [0, M ] and a discrete event in Σ. Timed words and languages are respectively words and languages over a k-M -alphabet. We define factor-closed and extensible timed language as in the untimed case. Timed automata as described below can be used to define timed languages, which are called regular. First we note G c,M the set of all c-dimensional Mbounded rectangular integer guards, i.e. Cartesian products of c real intervals

I i , i ∈ {1..c}, having integer bounds in {0..M }. A timed automaton is thus a tuple A = (Q, c, M, Σ, ∆, I, F ) where Q is a finite set of locations; c ∈ IN is the number of clocks; M ∈ IN is an upper bound on clock values; Σ is a finite alphabet of events; ∆ ⊆ Q × Q × Σ × G c,M × 2 {1..c} is a set of transitions; I : Q → G c,M maps each location to an initial constraint; F : Q → G c,M maps each location to a final constraint. A transition δ = ori(δ), ter(δ), L(δ), g(δ), r(δ) ∈ ∆ is such that ori(δ)
is its origin location, ter(δ) is its ending location, L(δ) is its label, g(δ) is the guard that clock values must satisfy for firing δ and r(δ) is the set of clocks reset by firing it.

A timed word (t 1 , a 1 ) . . . (t k , a k ) is in the language of a timed automaton if there exists a run (q i , x i ) i∈{0..k} with q i ∈ Q and x i ∈ [0, M ] c , such that x 0 ∈ I(q 0 ), x k ∈ F (q k ) and such that for all i, there exists

δ i ∈ ∆ satisfying ori(δ i ) = q i-1 , ter(δ i ) = q i , L(δ i ) = a i , x i-1 +t i ∈ g(δ i ) and x i = r(δ i )(x i-1 +t i ) (i.e. equal to x i-1 + t i
where coordinates in r(δ i ) are substituted by zeros).

A timed automaton is said to be right-resolving if outgoing transitions from the same location with the same label have pairwise incompatible guards. If we add the condition of having only one initial state we obtain the classical definition of determinism of [START_REF] Alur | A theory of timed automata[END_REF]. Languages recognized by right-resolving timed automata are also said right-resolving. Right-resolving factor-closed and extensible timed regular languages are called sofic.

An example of timed automaton is given in Fig. 2.

Volume and entropy

From now on A denotes a k-M -alphabet [0, M ] × Σ. Let L ⊆ A n be a timed language and n ∈ IN, we denote by L n the set of timed words of length n

p q r c, x ∈ [0, 3], x := 0 a, x ∈ [0, 3] d, x ∈ [0, 2], x := 0 b, x ∈ [0, 2]
Fig. 2. A timed automaton (all the states are initial and final).

in L. For each w = w 1 . . . w n ∈ Σ n we denote by L w the (possibly empty)

set of points t = (t 1 , . . . , t n ) ∈ R n such that (t 1 , w 1 ) . . . (t n , w n ) ∈ L n . Thus L n = w∈Σ n L w × {w} (by a slight abuse of notation).
The language L is said to be measurable whenever for all w ∈ Σ * , L w is Lebesgue measurable, i.e. its (hyper-)volume Vol(L w ) is well defined. The volume is just the n-dimensional generalization of interval length in R, area in R 2 , volume in R 3 ,. . . Examples of timed languages and their volume are given later, we also refer to the papers [START_REF] Asarin | Volume and entropy of regular timed languages: Discretization approach[END_REF][START_REF] Basset | Thin and thick timed regular languages[END_REF] for more detailed examples. For a timed regular language L, languages L w (w ∈ Σ * ) are just finite unions of convex polytopes (see for instance Fig. 4) and hence measurable [START_REF] Asarin | Volume and entropy of regular timed languages: Discretization approach[END_REF]. In the sequel, all timed languages considered are measurable. The volume of L n and the volumetric entropy of L are defined respectively as

Vol(L n ) = w∈Σ n Vol(L w ); H(L) = lim n→∞ 1 n log Vol(L n ),
and it can be shown that the limit exists for any factorial language.

For our running example on Fig. 2,

L ac = {(t 1 , t 2 ) | t 1 + t 2 ≤ 3} and L bd = {(t 1 , t 2 ) | t 1 + t 2 ≤ 2};
and their volumes are respectively 4.5 and 2; and thus L 2 (q), the sublanguage of L 2 of words accepted by runs starting from q, has volume 6.5. We have L 2n (q) = (L 2 (q)) n whose volume is 6.5 n , the entropy of the whole language H(L) is thus at least 0.5 log 6.5 (in fact it is exactly 0.5 log 6.5).

Discretization of languages and entropy

Here we adapt some definitions and results from [START_REF] Asarin | Volume and entropy of regular timed languages: Discretization approach[END_REF][START_REF] Basset | Thin and thick timed regular languages[END_REF]. For a k-M -alphabet A = [0, M ] × Σ, we denote by A ε its ε discretization:

A ε = {0, ε, 2ε, . . . , M } × Σ. We remark that A ε is a finite alphabet of size k( M ε +1
). An ε-discrete word is a timed word whose timed delays are multiples of ε. Given a timed language L ⊆ A * , its ε-discretization L ε is the discrete language on A ε composed by ε-discrete words in L:

L ε = L ∩ A ε * . Given an ε-discrete word w = (t 1 , a 1 ) . . . (t n , a n ), its ε-North-East-neighbor- hood is the timed set B N E ε (w) = {(u 1 , a 1 ) . . . (u n , a n ) | u i ∈ [t i , t i + ε], i = 1.
.n}. We extend this definition to associate timed languages with languages of εdiscrete words B N E ε (L ε ) = ∪ w∈Lε B N E ε (w). We will use discretization in a three-step reduction scheme:

1. discretize the timed languages S, C with a sampling rate ε to obtain S ε , C ε ; 2. use classical coding theorem 1 with S ε , C ε ; 3. go back to timed languages by taking ε-NE-neighborhood of S ε and C ε .

The following lemma is the main tool for this reduction scheme, it is a variant of results of [START_REF] Asarin | Volume and entropy of regular timed languages: Discretization approach[END_REF][START_REF] Basset | Thin and thick timed regular languages[END_REF]: Lemma 2. Let S be a timed sofic language. If H(S) > -∞ then for all positive small enough ε, one can compute ε-discrete sofic languages S - ε and S + ε that verify

B N E ε (S - ε ) ⊆ S ⊆ B N E ε (S + ε ), and
H(S) + log 1 ε = h(S - ε ) + o(1) = h(S + ε ) + o(1).

Timed coding

Similarly to classical results presented in Sect. 2, we will consider several settings for transmission of timed words over a channel. For every setting we will formulate an information inequality, and show that it is necessary and, with some additional hypotheses, sufficient for a coding to exist.

Timed source, discrete channel, approximate transmission

In practice, timed and data words are often transmitted via discrete (finite alphabet) channels. For example, a timed log of events in an operating system (a timed message) can be stored as a text file (ASCII message). The delays in the timed word cannot be stored with infinite precision, thus the coding is necessarily approximate. More precisely, the set of timed source messages w of a length n is infinite, while the set of discrete channel messages of the same length is finite. For this reason, the coding cannot be injective, and necessarily maps many timed words to a same discrete word. It is natural to require that all the timed words with the same code are ε-close to each other. This justifies Def. 6 below. We give first some notation. For two timed words of same length w = (t 1 , a 1 ) . . . (t n , a n ) and w = (t ′ 1 , a ′ 1 ) . . .

(t ′ n , a ′ n ), the distance dist(w, w ′ ) between w and w ′ is equal to +∞ if a 1 . . . a n = a ′ 1 . . . a ′ n , otherwise it is max 1≤i≤n |t i -t ′ i |.
Let A be a k-M alphabet, Σ ′ be a finite alphabet, S be a factorial extensible measurable timed language on A, C be a factorial extensible language on Σ ′ , and α, ε be positive reals and d be a non negative integer. Definition 6. Similarly to Def. 1 we say that a partial function φ : A * → Σ ′ * is almost approximately injective with precision ε and delay d if

∀n ∈ IN, w, w ′ ∈ A n ∀u, u ′ ∈ A d : φ(wu) = φ(w ′ u ′ ) ⇒ dist(w, w ′ ) < ε.
Its almost inverse is a multi-valued function family ψ n : Σ ′ * → A n characterized by the following property: for any w ∈ A n it holds that w ∈ ψ n (v) if and only if some u ∈ A d yields φ(wu) = v.

Lemma 3. Given an almost approximately injective function φ with precision

ε and delay d, let ψ n be its almost inverse family. Then for every v the diameter of ψ n (v) is at most ε. If the domain of φ is extensible, then the image of ψ n coincides with this domain (constrained to length n). The information inequality for this setting has the form:

α(H(S) + log(1/ε)) ≤ h(C), (II3)
which corresponds to information contents of S equal to H(S) + log(1/ε), see the formula for Kolmogorov complexity of timed words in [START_REF] Asarin | Volume and entropy of regular timed languages: Discretization approach[END_REF].

Proposition 4. For a factorial extensible measurable timed language S and a factorial extensible discrete language C the following holds. If an (S, C)-encoding of rate α, precision ε, and some delay d exists then necessarily (II3) must be satisfied.

Proof. Let φ be an (S, C)-encoding of rate α, precision ε and delay d, and ψ its almost inverse. By Lemma 3, for every n it holds that S n = ψ n (C ⌊(n+d)/α⌋ ). This leads to an inequality on volumes

Vol(S n ) ≤ v∈C ⌊(n+d)/α⌋ Vol(ψ n (v)).
Any ψ(v) has a diameter ≤ ε and thus is included in a cube of side ε and volume ε n . We have:

Vol(S n ) ≤ ε n |C ⌊(n+d)/α⌋ |. Thus α n log Vol(S n ) ≤ α n log ε n |C ⌊(n+d)/α⌋ | = α log ε + ⌊(n + d)/α⌋ n/α |C ⌊(n+d)/α⌋ | ⌊(n + d)/α⌋ .
Taking the limit as n tends to infinity we obtain αH(S) ≤ α log ε + h(C) and then (II3) holds.

⊓ ⊔

We strengthen a little bit (II3) to have a (partial) converse result for sofic timed languages. Proposition 5. For a sofic timed language S with H(S) > -∞, there exists a function R S such that lim x→0 R S (x) = 0 and the following holds. Whenever the entropy of a sofic discrete language C verifies the inequality α(H(S) + log(1/ε) + R S (ε)) < h(C), then there exists an (S, C)-encoding of rate α, precision ε and some delay d. Moreover it can be realized by a "real-time transducer" sketched below in the proof. Proof (Sketch). Let S be a sofic timed language. For ε > 0, let S + ε be its εdiscretized over-approximation given by Lemma 2. We define

R S (ε) = h(S + ε ) -H(S) -log(1/ε),
it satisfies the required condition: R S (ε) = o(1) (see Lemma 2). Let C be a sofic discrete language such that α(H(S)

+ log(1/ε) + R S (ε)) < h(C),
we prove that an (S, C)-encoding of rate α, precision ε and some delay d exists.

Lemma 2 gives us

S ⊆ B N E ε (S + ε ) and αh(S + ε ) = α(H(S) + log(1/ε) + R S (ε)) < h(C).
Thus by Prop. 3 an (S + ε , C)-encoding of rate α and some delay d exists and can be realized by a finite-state transducer τ ε of rate α and delay d. If we replace for each transition its input label (a, kε) by the label a and the guard t ∈ [kε, (k + 1)ε], we obtain a real-time transducer τ with input B N E ε (S + ε ) whose output is in C. The injectivity of τ ε ensures that τ realizes an approximately injective function with precision ε. ⊓ ⊔

Timed source, timed channel, exact transmission

Another natural setting is when a timed message is transmitted via a timed channel. In this case, the coding can be exact (injective). For the moment we consider length-preserving transmission (see Sect. 4.4 for faster and slower transmission). Let A, A ′ be a k-M and a k ′ -M ′ alphabet, S and C factorial extensible measurable timed languages on these alphabets, and d ∈ IN.

Let ℓ and σ be positive rationals. A function f : A ′n → A * is said to be ℓ-Lipshitz whenever for all x, y in its domain, dist(f (x), f (x ′ )) ≤ ℓ dist(x, x ′ ). We call a function σ-piecewise ℓ-Lipshitz if its restriction to each cube of the standard σ-grid on A ′n is ℓ-Lipshitz.

We can now state the definition of an (S, C)-encoding: The last condition rules out a possible cheating when all the time delays are divided by 1000 before transmission over the channel. We will come back to this issue in Sect. 4.3.

The information inequality in this setting takes a very simple form:

H(S) ≤ H(C). ( II4
)
The necessary condition for existence of a coding has a standard form (for technical reasons we require the channel to be sofic): Proposition 6. If for a factorial extensible measurable timed language S and a sofic timed language C an (S, C)-encoding of delay d exists, then (II4) holds.

Proof. We consider first the most interesting case when H(C) > -∞. + o(1). We choose ε dividing σ and small enough such that

H(C) + log 1 ε > lim n→∞ log K n,ε n -ζ. (1)
Thus we have S n ⊂ ψ n (C + n+d ), and, passing to volumes we get

Vol(S n ) ≤ Vol(ψ n (C + n+d )) ≤ K n+d,ε ε n , (2)
indeed, since ψ n is 1-Lipshitz on each ε-cube, ψ n -image of each such cube has a diameter ≤ ε and thus a volume ≤ ε n . Passing to logarithms, dividing by n and taking the limit as n → ∞ in (2) we get

H(S) ≤ lim n→∞ log K n+d,ε n + d + log ε,
and applying inequality (1) we obtain

H(S) ≤ H(C) + log 1 ε + ζ + log ε = H(C) + ζ,
which concludes the proof for the case when H(C) > -∞. The remaining case H(C) = -∞ is a simple corollary of the previous one.

⊓ ⊔

As usual, when both timed languages S and C are sofic and the information inequality (II4) strict, the converse holds.

Proposition 7. If for sofic timed languages S and C it holds that H(S) < H(C), then there exists an (S, C)-encoding (with some delay d). Moreover it can be realized by a "real-time transducer" described below in the proof.

Proof (Sketch). Let S and C be sofic timed languages whose entropies verify -∞ < H(S) < H(C). We prove that an (S, C)-encoding with some delay d exists. Let C - ε and S + ε be as in Lemma 2, i.e. such that

S ⊆ B N E ε (S + ε ); B N E ε (C - ε ) ⊆ C; H(S) + log 1 ε = h(S + ε ) + o(1); H(C) + log 1 ε = h(C - ε ) + o(1).
The discretization step ε can be chosen small enough such that h(S + ε ) < h(C - ε ). Thus by Theorem 1 a finite-state (S + ε , C - ε )-encoder τ ε exists. We replace each transition δ ǫ of τ ǫ with input label (a, kε) and output label (b, lε) by a transition δ with input label a, guards t ∈ [kε, (k + 1)ε], output We obtain what we call a realtime transducer τ . Its input language is B N E ε (S + ε ) and its output language is included in

p1 p0 q r (c, 0) (a, 0) (c, 1) (c, 0) (a, 1) (d, 0) (b, 0) p1 p0 p ′ 0 q q' r (c, 0) (a, 1) (a, 0) (c, 0) (c, 0) ( d, 0) (c, 1) (c, 0) (c, 1) (a, 0) (b, 0) (d, 0)
B N E ε (C - ε ) ⊆ C.
The encoding is performed as follows: a timed word (t 1 , a 1 ) . . . (t n , a n ) is in the input language if there is a path δ 1 . . . δ n such that I(δ i ) = a i and t i satisfies the guard of δ i :

t i ∈ [kε, (k + 1)ε]; the output timed word is in this case (t ′ 1 , b 1 ) . . . (t ′ n , b n ) with t ′ i = t i + c(δ i ), b i = O(δ i ). The collection of cubes B N E ε (w), w ∈ S + ε (resp. w ∈ C - ε ) forms a partition of timed languages B N E ε (S + ε ) (resp. B N E ε (C - ε ))
, they are cubes of the standard σ-grid with the step σ = ǫ. Transducer τ only translates cubes. Translations are 1-Lipshitz and thus the last condition of an (S, C)-encoding holds.

The remaining degenerate case when -∞ = H(S) < H(C) is an easy corollary of the non-degenerate one.

⊓ ⊔ ori(δ) ter(δ) I(δ) g(δ) O(δ) c(δ) q p 0 e [0, 1] a 0 q p 1 f [0, 1] a 1 q ′ p ′ 0 e [0, 1] a 0 q ′ r f [0, 1] b 1 p 0 q e [0, 1] c 0 p 0 q ′ f [0, 1] c 0 p ′ 0 q e [0, 1] c 1 p ′ 0 q ′ f [0, 1] c 1 p 1 q e [0, 1] c 0 p 1 q ′ f [0, 1] c 0 r q e [0, 1] d 0 r q ′ f [0, 1] d 0 Table 1.
The coding transducer

The following example illustrates the construction of the transducer. Let the source timed language be S = ([0, 1] × {e, f }) * and the channel timed language C be recognized by the automaton on Fig. 2. We have seen that the entropy H(C) is at least 0.5 log 6.5 and thus H(C) > H(S) = log 2. By Prop. 7 an (S, C)-encoding exists. To realize this encoding we take ε = 1. There are four cubes included in the language C 2 (q): (

[0, 1] × [0, 1] × {ac}, [0, 1] × [0, 2] × {ac}, [1, 0] × [0, 1] × {ac}, [0, 1] × [0, 1] × {bd}) while the cubes to encode (language S 2 ) are [0, 1]×{ee}, [0, 1]×{ef }, [0, 1]×{f e}, [0, 1]× {f f }.
The transducer will repeatedly map four "input cubes" to four "output cubes". We build an automaton for discrete words C - ε (as in Lemma 2, such words correspond to "output cubes") in Fig. 3, left. Then, as usual in coding, we first split the state p 0 and then the state q (each in two copies) to obtain an automaton with constant outdegree 2 (Fig. 3, right). This automaton accepts the same language C - ε , and can be transformed to the desired transducer just by adding input letters and increment/decrement to its transition. The transitions of the transducer are given in Table 1.

A variant: scaling allowed

In some situations, timed data can be scaled for coding, which leads to a new term in the information inequality. Let λ > 0 be a rational bound on scaling factor. We modify Def. 8 by replacing 1-Lipshitz by 1/λ-Lipshitz. The information inequality for this case becomes:

H(S) ≤ H(C) + log λ. (II5)
The problem of coding with scaling can be easily reduced to the one considered in the previous section. Indeed, for a timed language C let λC be the same language with all times multiplied by λ (the entropy of this language is H(λC) = H(C) + log λ). A function φ is an (S, C)-encoding with scaling λ if and only if the "λ-scaled" function λφ is an (S, λC)-encoding without scaling. Using this reduction, the results below are corollaries of Prop. 6-7.

A speedup and a slowdown lead to a collapse

For untimed channels, transmission with some rate α = 1leads to the factor α in information inequalities (II2), (II3). Unfortunately, for timed channels this does not work: any rate α = 1 leads to a collapse of the previous theory. For α > 1 no coding is possible, and for α < 1 it always exists. More precisely, the two following propositions hold.

Proposition 10. For factorial measurable timed languages S and C if H(S) > -∞ and α > 1, then no (S, C)-encoding with rate α exists.

Proof (Sketch). The proof follows the same lines as that of Prop. 6. Suppose φ is such an (S, C)-encoding, and ψ its almost inverse. By Lemma 1, for any natural n we have S n ⊂ ψ n (C ⌊(n+d)/α⌋ ). Each C n can be covered by C + n , a union of K n,ε cubes of size ε satisfying inequality [START_REF] Alur | A theory of timed automata[END_REF] We have S n ⊂ ψ n (C + ⌊(n+d)/α⌋ ), and, passing to volumes we get Vol(S n ) ≤ Vol(ψ n (C + ⌊(n+d)/α⌋ )) ≤ K ⌊(n+d)/α⌋,ε ε n . Passing to logarithms, dividing by n and taking the limit as n → ∞ we get

H(S) ≤ α -1 lim n→∞ log K ⌊(n+d)/α⌋,ε ⌊(n + d)/α⌋ + log ε,
and applying inequality (1) we obtain

αH(S) ≤ H(C) + log(1/ε) + ζ + α log ε = H(C) + ζ -(α -1) log(1/ε).
Choosing ε small enough makes the inequality wrong. This contradiction concludes the proof. ⊓ ⊔ Proposition 11. For sofic timed languages S and C (with H(C) > -∞) and any α < 1 there exists an (S, C)-encoding with rate α (and some delay d).

Moreover it can be realized by a kind of timed transducer.

The construction is non-trivial and uses spare time durations in the channel message to transmit discrete information.

Discussion and perspectives

In the previous section, we have established several results on timed channel coding following the standard scheme: a setting of information transmissioninformation inequality -coding existence theorem -synthesis of an encoder/decoder. We believe that this approach can be applied to various situations of data transmission (and compression). We also consider it as a justification of our previous research on entropy of timed languages [START_REF] Asarin | Volume and entropy of regular timed languages: Analytic approach[END_REF][START_REF] Asarin | Volume and entropy of regular timed languages: Discretization approach[END_REF][START_REF] Asarin | Two Size Measures for Timed Languages[END_REF]. In this concluding section, we explain some of our choices and immediate perspectives of this approach. The time is not preserved. In the central Def. 8 and Prop. 6,7, we consider codings of timed words that preserve the number of events, and not their duration. This choice is compatible to the general idea of dealing with data words (in our case, sequences of letters and real numbers), and less so with the standard timed paradigm. We use again the example of Fig. 2 to illustrate this feature. For n ∈ IN, the timed word w = [(0.5, e)(0.5, f )] n is encoded to the timed word w ′ = [(0.5, a)(1.5, c)] n , both have 2 events. However, the duration of w is (0.5 + 0.5)n = n, while the duration of w ′ is (0.5 + 1.5)n = 2n.

Other settings to explore. It would be still interesting to explore coding functions preserving durations. On the theoretical side, a more detailed analysis for transmission speeds different from 1, as in Sect. 4.4 would probably lead to information inequalities instead of a collapse. Many other settings of transmission of information could be practically relevant: approximated transmission of a timed source on a timed channel; coding of a discrete source on a timed channel; coding using transducers of a fixed precision; coding of other kinds of data languages. On the other hand, more physical models of timed and data channels would be interesting to study, one can think of a discrete channel with a fixed baud rate coupled with an analog channel with a bounded frequency bandwidth. Finally, some special codes, such as sliding-window, error-correcting etc., should be explored for timed and data languages.

p q b, x ∈ [0, 1] | a, x c, x ∈ [1, 2], x := 0 | a, x -1 p q a, x ∈ [0, 1], y ∈ [0, 1], x := 0 | b, x a, x ∈ [0, 1], y ∈ [0, 2], x := 0, y := 0 | c, 2x -y + 1 1 2 0 1 2 t2 t1 1
What is a timed transducer? In the classical theory of constrained channel coding, several kinds of transducers are used for encoding/decoding, such as those leading to a sliding-block window decoding. In this paper, we have realized our codes by using some very restricted ad hoc timed transducers. They behave like timed (in our case, real-time) automata on the input, and "print" letters and real numbers on the output; we believe that this is the correct approach. However, the right definition of a natural class of timed transducers adequate to coding remains an open question. As a preliminary definition, we suggest timed automata that output, on each transition, a letter and a real number (which is an affine combination of clock values). While reading a timed word, such a transducer would output another timed word. We illustrate this informal definition with the example of mutually inverse transducers τ 1 and τ 2 (encoder and decoder) on Fig. 4. Let us consider a run of the transducer τ 2 on the timed word (t 1 , b)(t 2 , c) . . . We start from q with x = 0, y = 0, after reading (t 1 , b) the value of x and y is t 1 , we fire the transition, the output is (t 1 , a), we pass in q where we read (t 2 , c), the value of x is t 2 and the value of y is t 1 + t 2 , we fire the transition, the output is (2t 2 -(t 1 + t 2 ) + 1, a) = (t 2 -t 1 + 1, a), we pass in p etc.

For τ 1 , the input language of timed words of length 2 starting from p is S 2 (p) = {(t 1 , b)(t 2 , c) | t 1 ∈ [0, 1], t 1 +t 2 ∈ [1, 2]}, while for the second transducer it is C 2 (p) = {(t 1 , a)(t 2 , a) | t 1 ∈ [0, 1], t 2 ∈ [0, 1]}. These two languages are depicted in Fig. 4, they have the same volume. It would be impossible to realize this kind of encoding using "rectangular" transducers as in Sect. 4.

How to improve code synthesis? The encoders in Sect. 4 are not completely satisfactory: they use non-integer guards even when the source and the target language are defined using integer timed automata. We believe that this issue can be avoided using a broader class of transducers as suggested just above.

What about timed symbolic dynamics? The classical theory of constrained-channel coding uses as a convenient terminology, and as a toolset, a branch of the theory of dynamical systems called symbolic dynamics. One of us, in [START_REF] Basset | Dynamique Symbolique et Langages Temporisés[END_REF], started a formulation of timed languages and automata in terms of symbolic dynamics. Relating it to timed channel coding remains a future work.

Applications. In practice, when transmitting (or storing) information containing discrete events and real-valued data, all the information is first converted to the digital form and next encoded for transmission or storage. Our paradigm combines both steps and, in principle, provides better bounds and codes. However, more research is needed to come up with useful practical codes.

Definition 3 .

 3 An (S, C)-encoding with delay d is a function φ : S → C (total on S but not necessarily onto C) such that it is length preserving: ∀w ∈ S, |φ(w)| = |w|, -it is almost injective with delay d.

Fig. 1 .

 1 Fig. 1. A sofic automaton of a channel C and a ({0, 1} * , C)-encoder

Definition 5 .

 5 An (S, C)-encoding with rate α ∈ Q + and delay d is a function φ : S → C (total on S and not necessarily onto C) such that it is of rate α, i.e. ∀w ∈ S, ⌈α|φ(w)|⌉ = |w|; -it is almost injective (with delay d).

Definition 7 .

 7 An (S, C)-encoding of a rational rate α, precision ε and delay d is a function φ : S → C (total on S) such that it is of rate α: i.e. ∀w ∈ S, ⌈α|φ(w)|⌉ = |w|; -it is almost approximately injective with precision ε and delay d.

Definition 8 .

 8 An (S, C)-encoding with delay d (and step σ) is a function φ : S → C such that it is length preserving: |φ(w)| = |w|, -it is almost injective (with delay d), -no time scaling: the almost inverse ψ n are σ-piecewise 1-Lipshitz.

Fig. 3 .

 3 Fig. 3. Left: A - ε : an automaton recognizing C - ε with ε = 1. Right: its split form.

Definition 9 .

 9 An (S, C)-encoding with delay d, scaling λ and step σ is a function φ : S → C such that it is length preserving: |φ(w)| = |w|, -it is almost injective (with delay d), -it has scaling at most λ: the almost inverse ψ n are σ-piecewise 1/λ-Lipshitz.

Proposition 8 .Proposition 9 .

 89 If for factorial extensible measurable timed language S and sofic timed C an (S, C)-encoding with scaling λ and delay d exists then (II5) holds. If for sofic timed languages S and C (with H(S) > -∞) the strict version of (II5) holds, then there exists an (S, C)-encoding with scaling λ (with some delay d). Moreover it can be realized by a "real-time transducer".

Definition 10 .

 10 An (S, C)-encoding with rational rate α, delay d and step σ is a function φ : S → C such that its rate is α, i.e. ∀w ∈ A * , ⌈α|φ(w)|⌉ = |w|; -it is almost injective (with delay d); -no time scaling: its almost inverse ψ is σ-piecewise 1-Lipschitz.

Fig. 4 .

 4 Top: transducers τ1 : S → C and τ2 : C → S. Bottom: languages S2 and C2.

  ) Let φ : S → C be an (S, C)-encoding with delay d. By Lemma 1 its almost inverse ψ maps C onto S. More precisely, for every n we have ψ n (C n+d ) = S n . Hence, the cardinalities should satisfy: |S n | ≤ |C n+d |. Finally we have

	Proposition 1. Let S and C be factorial and extensible languages. If an (S, C)-
	encoding exists then (II1) necessarily holds.
	Proof. lim n→∞	1 n	log |S n | ≤ lim n→∞	1 n	log |C n+d |
	and the expected inequality h(S) ≤ h(C).

  We will prove that for any ζ > 0 the inequality H(S) ≤ H(C) + ζ holds. Suppose φ is an (S, C)-encoding of delay d (and step σ), and ψ its almost inverse. By Lemma 1, for any natural n we have S n ⊂ ψ n (C n+d ).Since C is sofic, Lemma 2 applies, and for a fixed ǫ, each C n can be covered by C + n , a union of K n,ε cubes of size ε with H(C) + log 1 ε = lim n→∞

	log Kn,ε
	n

We avoid here the terminology of symbolic dynamics, standard in the area of coding.

Such a transducer is also called a finite-state (S, C)-encoder.

Acknowledgment. We thank the anonymous reviewers for valuable remarks that contributed to substantial improvement of this article.

⋆ The support of Agence Nationale de la Recherche under the project EQINOCS (ANR-11-BS02-004) is gratefully acknowledged.