C Chavaudret

S Marmi

Reducibility of quasiperiodic cocycles under a Brjuno-Rüssmann arithmetical condition

The arithmetics of the frequency and of the rotation number play a fundamental role in the study of reducibility of analytic quasi-periodic cocycles which are sufficiently close to a constant. In this paper we show how to generalize previous works by L.H.Eliasson which deal with the diophantine case so as to implement a Brjuno-Rüssmann arithmetical condition both on the frequency and on the rotation number. Our approach adapts the Pöschel-Rüssmann KAM method, which was previously used in the problem of linearization of vector fields, to the problem of reducing cocycles.

∀(θ, t) ∈ T d × R, X ′ (θ, t) = A(θ + tω)X(θ, t) [START_REF] Avila | A KAM scheme for SL(2, R) cocycles with liouvillean frequency[END_REF] where A is a continuous matrix-valued function on a torus T d and ω is a rationally independent vector of some space R d (the space of frequencies). Although the dynamics of such a system can be quite complicated, they are easily studied in case the system is reducible, i.e when there is a map Z, continuous on the double torus 2T d = R d /2Z d , taking its values in the group of invertible matrices and such that

∀θ ∈ 2T d , d dt Z(θ + tω) |t=0 = A(θ)Z(θ) -Z(θ)B
for some matrix B not depending on θ. Since smoothness is an issue, given a class of functions C, we will say that the cocycle is reducible in C if Z is in C. Here we will focus on the case in which A takes its values in sl(2, R), which is sufficient, for instance, for the study of the one-dimensional quasi-periodic Schrödinger equation. Moreover, we will consider solutions of (1) with A ∈ C ω r , the space of functions on T d having a holomorphic extension on {(z 1 , . . . , z d ) ∈ C d , ∀j | Im z j |< r}, whose "weighted norm" | • | r converges (see section 2.1).

The arithmetics of ω seem fundamental in the study of reducibility, as well as the arithmetics of the system's rotation number ρ (as it was defined in [START_REF] Johnson | The rotation number for almost periodic potentials[END_REF]; we recall the definition in Section 2.1). At least in the perturbative case, arithmetical conditions of diophantine type have long been used to obtain reducibility, which can be seen as the convergence of a certain sequence of analytic functions: a diophantine condition can be used to control small divisors and make sure that the sequence converges. This was achieved, in particular, by Eliasson in [4]:

Theorem 1.1 (Eliasson) Let r > 0, V ∈ C ω r (T d , R).
Suppose ω is a diophantine vector. There exists ǫ 0 depending only on r, ω such that if sup |Im θ|<r | V (θ) -V (0) |≤ ǫ 0 , then the cocycle which is solution of

d dt X(t, θ) = 0 V (θ + tω) -E 1 0 X(t, θ) (2)
is reducible for all E for which the rotation number is rational or diophantine with respect to ω.

In this article, we will give a reducibility result for analytic cocycles under a weaker arithmetical condition than the diophantine one. In order to obtain an analytic reducibility result, we will have to pick a frequency and a rotation number with good approximation properties, in the sense of Rüssmann ( [START_REF] Rüssmann | KAM iteration with nearly infinitely small steps in dynamical systems of polynomial character[END_REF]): ω will have to satisfy a strong irrationality condition controlled by an approximation function G, namely

∀m ∈ Z d \ {0}, | m, ω |≥ κ G(m)
for some positive κ (section 2.1), and ρ will have to satisfy a further arithmetical condition: its approximations by means of linear combinations of the frequencies are controlled by an approximation function g, i.e

∀m ∈ Z d \ {0}, | ρ -m, ω |≥ κ ′ g(m)
(we will say for short that ρ has g as approximation function with respect to ω with constant κ ′ ) with g, G satisfying some extra assumptions.

We will be particularly interested in the case of Brjuno frequency, i.e when

∞ 1 log G(t) t 2 dt < ∞ (3)
and of 1 2 -Brjuno rotation number (with respect to ω), i.e when

∞ 1 log g(t) t 3 2 dt < ∞ (4)
In dimension d = 2, Condition (3) coincides with the well-known Brjuno condition defined in terms of continued fraction expansion, which is closely related, as shown by Yoccoz, to the dynamical properties of the quadratic polynomial (see [START_REF] Yoccoz | Petits diviseurs en dimension 1, S.M.F[END_REF]). Classes of numbers defined by a condition analogous to (4) when d = 2, which is slightly stronger than Brjuno, were constructed in [START_REF] Marmi | The Brjuno functions and their regularity properties[END_REF]. Condition (3) was introduced by Rüssmann in KAM theory, making it possible to deal with a vector of frequencies. Brjuno-Rüssmann conditions are already known to be central in the study of the linearization of vector fields (see e.g. [START_REF] Giorgilli | Convergence radius in the Poincaré-Siegel problem[END_REF], [START_REF] Pöschel | KAM à la R[END_REF] and references therein).

The classical Brjuno condition on the frequency was also considered by Young in [START_REF] Young | Lyapunov exponents for some quasi-periodic cocycles[END_REF], who constructed examples of non-reducible discrete cocycles in this case. For discrete cocycles with one frequency, Zhou and Wang recently obtained in [START_REF] Wang | Reducibility results for quasiperiodic cocycles with liouvillean frequency[END_REF] a positive measure reducibility result for non-Brjuno frequencies for non-degenerate one-parameters families of cocycles. Other results have been obtained on quasiperiodic cocycles regardless of any arithmetic condition on the frequency, worth mentioning although they are not reducibility results in our sense. In [START_REF] Avila | A KAM scheme for SL(2, R) cocycles with liouvillean frequency[END_REF], it is shown that without any condition on the frequency, the Schrödinger cocycle (2) can be conjugated to a rotation-valued cocycle for a positive measure set of energies; in [START_REF] Wang | Reducibility results for quasiperiodic cocycles with liouvillean frequency[END_REF], Zhou and Wang showed that in the case d = 2, for any frequency, for a non-degenerate analytic one-parameter family which is close to a constant, the cocycle can be analytically diagonalized for a positive measure set of parameters.

Our main result states:

Theorem 1.2 Let ω be a Brjuno vector, A ∈ sl(2, R), r > 0, F ∈ C ω r (T d , sl(2, R)). Suppose ρ(A + F ) is a 1
2 -Brjuno number with respect to ω. There exists ǫ 0 depending only on ω, ρ(A + F ), r such that if | F | r ≤ ǫ 0 , then there exists r ′ ∈ (0, r) such that

A + F is reducible in C ω r ′ .
This dt: Brjuno-Rüssmann conditions are therefore not optimal in this problem as they might be in other dynamical problems.

In fact, our method gives this more explicit theorem: Theorem 1.3 Let κ > 0 and let G, g be positive increasing functions such that

• G(1) ≥ 1, g(1) ≥ 1, • +∞ 1 log G(t) + log g(t) t 2 dt < +∞, (5)
• the map t → g(t 2 ) G(t) is bounded.

Suppose ω has G as an approximation function with constant κ.

Let A ∈ sl(2, R), r > 0, F ∈ C ω r (T d , sl(2, R)). Let n 0 ∈ N.
There exist ǫ 0 depending only on g, κ, G, n 0 , r such that if

1. | F | r ≤ ǫ 0 ,
2. ρ(A + F ) has g as an approximation function with respect to ω with constant κ ′ > κ sup t≥n 0 g(t 2 )

G(t) ,
then there exists r ′ ∈ (0, r) such that

A + F is reducible in C ω r ′ (2T d , sl(2, R)).
A discussion on the dependence of ǫ 0 on g, G and the other parameters is given in subsection 3.4.

As an application, we consider the case when g and G look like exponentials (section 3.4):

Theorem 1.4 Let κ > 0, κ ′ > 0 and let G(t) = e t (log t) δ , g(t) = e t α , δ > 1, α < 1. Suppose ω has G as an approximation function with constant κ. Let A ∈ sl(2, R), r > 0, F ∈ C ω r (T d , sl(2, R)). There exist ǫ 0 depending only on α, κ, δ, κ ′ , r such that if 1. | F | r ≤ ǫ 0 ,
2. ρ(A + F ) has g as an approximation function with respect to ω with constant κ ′ , then there exists r ′ ∈ (0, r) such that

A + F is reducible in C ω r ′ (2T d , sl(2, R)).
Our aim is to adapt the Pöschel-Rüssmann method (see [START_REF] Rüssmann | KAM iteration with nearly infinitely small steps in dynamical systems of polynomial character[END_REF] and [START_REF] Pöschel | KAM à la R[END_REF]), which was used in the problem of linearization for vector fields, to the problem of reducing cocycles. It is a KAM-type method in which the speed of convergence is linear. First of all, we will build a setup in which a system A + F with A constant and F small is conjugated to another system which is arbitrarily close to a constant, in an analytic class which, however, cannot be well controlled: this follows the technique used in [START_REF] Eliasson | Almost reducibility of linear quasi-periodic systems[END_REF] and is obtained by iterating (as in subsection 3.1) infinitely many steps (described in section 2) in which one conjugates a system A n + F n to a system A n+1 + F n+1 where A n , A n+1 are constant and

| F n+1 | rn+1 ≤ C | F n | rn ,
with C < 1 being independent of n and r n being a decreasing sequence controlling how analytic a function is. Thus, if r n tends to a non zero limit, we have analytic reducibility. At each step, in order to proceed, the constant part has to be non resonant, and if it is resonant, then we will have to remove the resonances, as explained in subsection 2.2.

Our setup makes sure that r n tends to a non zero limit whenever there is only a small enough number of steps at which one has to remove resonances in the constant part. The Brjuno-Rüssmann condition on the frequency and on the rotation number of the cocycle is required exactly at this stage.

Acknowledgments The first author would like to thank the Centro Ennio de Giorgi for its hospitality and financial support during one year. Both authors would like to express their gratitude to Håkan Eliasson for useful discussions which allowed them to substantially improve the paper.

The basic step

In this section, we will prove the iterative step, which consists in conjugating a system to another one with a smaller non constant part, whether the constant part be resonant or not.

Definitions and notations

We will adopt the following conventions:

Definition: Let m = (m 1 , . . . , m d ) ∈ Z d . We denote by | m | its modulus: | m |= d j=1 | m j |. Definition: Let F ∈ C 0 (T d ) and r > 0; we say that F ∈ C ω r (T d ) if there exists an analytic continuation of F on a product of strips {(z 1 , . . . , z d ) ∈ C d , ∀j | Im z j |< r} and if the weighted norm | F | r = k∈Z d || F (k) || e 2π|k|r (6)
where || . || is the relevant norm for F (k) (for matrices, we use the operator norm), is finite.

Note that C ω r (T d ) is a Banach space.

Notation: For F ∈ C ω r (T d ), we denote its truncation by

F N (θ) = |m|≤N F (m)e 2iπ m,θ .
Remark: The weighted norms are particularly convenient since they satisfy, for any integer N,

| F -F N | r = k∈Z d ,|k|>N || F (k) || e 2π|k|r =| F | r -| F N | r . (7)
Moreover, they are related (although not equivalent) to the usual sup norms since it is easy to see that

sup |Im θ|<r || F (θ) ||≤| F | r .
For r ′ < r, we still have

| F | r ′ ≤ C(r -r ′ ) sup |Im θ|<r || F (θ) ||
where C(r -r ′ ) does not depend on F but depends on r -r ′ .

Definition:

If A ∈ C 0 (T d , sl(2, R)) and X is the solution of d dt X(t, θ) = A(θ+tω)X(t, θ); X(0, θ) = Id, the rotation number ρ(A) is the quantity ρ(A) = lim t→+∞ 1 t Arg(X(t, θ)φ -φ)
where φ ∈ R 2 ≃ C, θ ∈ T d and Arg stands for the variation of the complex argument (ρ(A) is independent of θ, φ).

Here and in what follows, we will fix ω ∈ R d rationally independent (i.e such that for all non zero m ∈ Z d , m, ω = 0): the vector ω will be the frequency of the cocycles we will consider. We will always assume that κ, κ ′ > 0 and that G, g are two positive continuous and strictly increasing functions such that G(1) ≥ 1, g(1) ≥ 1.

Definition: NR(κ, G) = {ω ∈ R d , ∀m ∈ Z d \ {0}, | m, ω |≥ κ G(|m|) }.
Remark: There exists a positive increasing and unbounded function

G ∈ C 0 (R * + ) with G(1) ≥ 1 and κ > 0 such that ω ∈ NR(κ, G). Indeed, one can take κ = min i | ω i | and G(N) = max |m|≤N κ | m,ω | . As noticed by H. Rüssmann ([10]), a condition NR(κ, G) with G such that ∞ 1 log G(t) t 2 dt < ∞ (8)
is fulfilled by all Bruno vectors (see [START_REF] Bruno | An analytic form of differential equations[END_REF]), i.e vectors satisfying:

k≥1 | log α 2 k -1 | 2 k < +∞ (9)
where

α k = min l≤k min j=1,...,d min |m|=l+1 | m, ω -ω j | . (10)
In [START_REF] Giorgilli | Convergence radius in the Poincaré-Siegel problem[END_REF] it is shown that condition ( 9) is equivalent to

k≥1 | log α k | k(k + 1) < +∞ (11)
and condition [START_REF] Marmi | The Brjuno functions and their regularity properties[END_REF] implies condition [START_REF] Yoccoz | Petits diviseurs en dimension 1, S.M.F[END_REF], so that ( 8), ( 9) and ( 11) are equivalent. Moreover, in dimension d = 2, these conditions are equivalent to the usual Brjuno condition on ω 2 ω 1 (see [START_REF] Yoccoz | Petits diviseurs en dimension 1, S.M.F[END_REF]). This suggests the following definition: Definition: The vector ω is a Brjuno vector if ω ∈ NR(κ, G) with G satisfying [START_REF] Marmi | The Brjuno functions and their regularity properties[END_REF].

We now have to introduce another type of arithmetic condition, related to the well-known "second Melnikov condition".

Definition: Let N ∈ N \ {0}; we set NR N ω (κ ′ , g) = α ∈ C, ∀m ∈ Z d \ {0}, 0 <| m |≤ N ⇒| α -iπ m, ω |≥ κ ′ g(| m |) (12)
and NR ω (κ ′ , g) = ∩ N ∈N NR N ω (κ ′ , g).

Remark: If g(t) = t τ for some τ > 1, this is a diophantine condition.

Definition: Let ν > 0. The number α is a ν-Brjuno number with respect to ω if α ∈ NR ω (κ ′ , g) with g satisfying ∞ 1 log g(t) t (1+ν) < +∞. (13)
This extended Brjuno condition was first considered in [START_REF] Marmi | The Brjuno functions and their regularity properties[END_REF].

Elimination of resonances

Now we shall prove the uniqueness of a resonance, i.e the situation of the spectrum being close to a number of the form m, ω , m ∈ Z d , when it exists.

Lemma 2.1 Let α ∈ C. Let N ∈ N \ {0}. There exists m ∈ Z d such that | m |≤ N and α -iπ m, ω ∈ NR N ω ( κ 4G(N ) , g); if m is non zero, then | α -iπ m, ω |< κ 4G(N)g(| m |) and α -iπ m, ω ∈ NR N ω ( κ G(N ) , g). Proof: Suppose α is not in NR N ω ( κ 4G(N ) , g), i.e there exists m ∈ Z d , 0 <| m |≤ N, such that | α -iπ m, ω |< κ 4G(N)g(| m |)
.

Then for all m ′ ∈ Z d with 0 <| m ′ |≤ N,

| α -iπ m + m ′ , ω |≥| π m ′ , ω | -| α -iπ m, ω |≥ κ G(| m ′ |) - κ 4G(N)g(| m |) (14) so | α -iπ m + m ′ , ω |≥ κ G(N)g(| m ′ |) (15)
and so α -iπ m, ω ∈ NR N ω ( κ G(N ) , g).

The following proposition explains how to eliminate resonances in the spectrum of a trace zero matrix.

Proposition 2.2 Let A ∈ sl(2, R) with eigenvalues ±α. Let N ∈ N. Suppose that α is not in NR N ω ( κ 4G(N ) , g). There exists Φ ∈ ∩ r ′ ≥0 C ω r ′ (2T d , SL(2, R)) and a numerical constant C ′ such that ∀r ′ ≥ 0, | Φ | r ′ ≤ C ′ e πN r ′ ; | Φ -1 | r ′ ≤ C ′ e πN r ′ (16)
and if à with eigenvalues ±α is such that

∂ ω Φ = AΦ -Φ Ã (17) then α ∈ NR N ω ( κ G(N ) , g). Moreover, | α |< κ 4G(N)
.

Proof: Lemma 2.1 gives a number m, 0 <| m |≤ N, such that letting

α = α -iπ m, ω then α ∈ NR N ω ( κ G(N ) , g).
Let P be such that P -1 AP is diagonal and || P ||= 1. We define Φ(θ) = P -1 e iπ m,θ 0 0 e -iπ m,θ P.

Relation (17) gives

à = P -1 α 0 0 -α P.
To obtain the estimate (16), we use an estimate shown for instance in [START_REF] Eliasson | Almost reducibility of linear quasi-periodic systems[END_REF], Lemma A:

|| P -1 ||≤ max 1, C. || A || 2 | α | 6
where C is a numerical constant, and since A is diagonalizable whenever its eigenvalues are non zero,

|| P -1 ||≤ max 1, C. | α | 2 | α | 6 ≤ C ′
where C ′ is a numerical constant, which gives (16).

Solution of the linearized homological equation

Our aim is to solve an equation of the form

∂ ω Z = (A + F )Z -Z(A ′ + F ′ )
where A and F are known, A ∈ sl(2, R) and F is analytic with values in sl(2, R). If A is non-resonant, we first solve

∂ ω X = [A, X] + aF N -a F (0)
where F N is some truncation of F and a is close enough to 1; then we define A ′ = A+a F (0) and F ′ by

∂ ω e X = (A + F )e X -e X (A ′ + F ′ )
and then we estimate

F ′ to get | F ′ | r ′ ≤ √ 1 -a | F | r .
If A is resonant, we conjugate A + F to a system à + F where à is non-resonant and we proceed in the same way as in the non-resonant case. So, from now on, to simplify the notations, we will assume that A, Ã, A ′ ∈ sl(2, R), that F, F , X, F ′ have values in sl(2, R) and Z, Φ have values in SL(2, R).

Proposition 2.3 Let N ∈ N and r, r ′ > 0. Let à with eigenvalues ±α ∈ NR N ω ( κ 4G(N ) , g). Let F ∈ C ω r (T d ). Then equation ∀θ ∈ T d , ∂ ω X(θ) = [ Ã, X(θ)] + F N (θ) -F (0); X(0) = 0 (18) has a unique solution X ∈ C ω r ′ (T d ) such that | X | r ′ ≤ 4 κ G(N)g(N) | F N | r ′ (19)
Proof: In Fourier series, equation ( 18) can be written:

∀m ∈ Z d ,0 <| m |≤ N ⇒ 2iπ m, ω X(m) = [ Ã, X(m)] + F (m); | m |∈ {0} ∪ [N + 1, +∞[⇒ 2iπ m, ω X(m) = [ Ã, X(m)]. ( 20
) So for | m |∈ {0} ∪ [N + 1, +∞[, X(m) = 0 is a solution (not necessarily unique).
For 0 <| m |≤ N, the solution is formally written as

X(m) = L -1 m F (m) (21)
where L m is the operator

L m : sl(2, R) → sl(2, R), M → 2iπ m, ω M -[ Ã, M]. Its spectrum is {2iπ m, ω -2 α, 2iπ m, ω + 2 α, 2iπ m, ω }. Since ω ∈ NR(κ, G) and α ∈ NR N ω ( κ 4G(N ) , g), L m is invertible and we have for all m ∈ Z d such that | m |∈ (0, N], || L -1 m ||≤ max{ G(| m |) κ , 4G(N)g(| m |) κ } = 4G(N)g(| m |) κ therefore for all m ∈ Z d such that 0 <| m |≤ N, || X(m) ||≤ 4G(N) g(| m |) κ || F (m) || (22) therefore | X | r ′ ≤ 4G(N) m∈Z d \{0},|m|≤N g(| m |) κ || F (m) || e 2π|m|r ′ ≤ 4 G(N)g(N) κ | F N | r ′ . (23)

Solution of the full homological equation without resonances

This section explains the basic step in case the constant part is non resonant, i.e when its eigenvalues are far from all m, ω , m ∈ Z d \ {0}.

Proposition 2.4 Let 0 < r ′ ≤ r, a ′ ∈ (0, 1], N ∈ N, F ∈ C ω r (T d ), Ã ∈ sl(2, R). If σ( Ã) = {±α}, α ∈ NR N ω ( κ 4G(N ) , g), then there exists X, F ′ ∈ C ω r ′ (T d )and A ′ ∈ sl(2, R) such that | A ′ -Ã |≤|| F (0) || (24) ∂ ω e X = ( Ã + F )e X -e X (A ′ + F ′ ) (25) | X | r ′ ≤ 4a ′ G(N)g(N) κ | F N | r ′ (26)
and

| F ′ | r ′ ≤ e | X| r ′ (1 -a ′ ) | F | r ′ +e | X| r ′ a ′ | F -F N | r ′ + e | X| r ′ | F | r ′ | X | r ′ (e | X| r ′ + a ′ + a ′ e | X| r ′ ). ( 27
)
Proof: Let X be a solution of

∀θ ∈ T d , ∂ ω X(θ) = [ Ã, X(θ)] + a ′ F N (θ) -a ′ F (0); X(0) = 0 (28)
as given by Proposition 2.3 (so it satisfies (26)). Let A ′ = Ã + a ′ F (0) so that (24) holds, and let F ′ be defined by

∂ ω e X = ( Ã + F )e X -e X (A ′ + F ′ ).
We have

F ′ =e -X ( F -a ′ F N ) + e -X F (e X -Id) + a ′ (e -X -Id) F (0) -e -X k≥2 1 k! k-1 l=0 Xl (a ′ F N -a ′ F (0)) Xk-1-l . ( 29
) Since | F -a ′ F N | r ′ ≤ a ′ | F -F N | r ′ +(1 -a ′ ) | F | r ′ . (30)
one easily obtains (27).

Remark: Denote ǫ =| F | r . Suppose 2G(N)g(N)ǫ ≤ κ(1 -a ′ ) 2 . (31)
Then (26

) implies | X | r ′ ≤ a ′ (1 -a ′ ) thus e | X| r ′ ≤ 2. By (27), if one assumes moreover that e -2πN (r-r ′ ) ≤ 1 -a ′ (32)
with r ′ > 0, then

| F -F N | r ′ ≤ (1 -a ′ ) | F -F N | r therefore | F ′ | r ′ ≤ 2(1 -a ′ )ǫ + 2a ′ (1 -a ′ )ǫ + 2ǫa ′ (1 -a ′ )(2 + 3a ′ ). (33)
Thus, if a ′ is close enough to 1 (i.e larger than 1 -1 14 2 ),

| F ′ | r ′ ≤ (1 -a ′ ) 1 2 ǫ. (34)

Solution of the full homological equation with resonances

This section presents the basic step when there are resonances in the constant part, i.e when its eigenvalues are too close to some m, ω , m ∈ Z d \ {0}.

Proposition 2.5 Let a ∈ (0, 1), c 0 > 0, C ′ as in Proposition 2.2, N ∈ N, r > 2 log(g(N )G(N )) πN , F ∈ C ω r (T d ), | F | r = ǫ, A ∈ sl(2, R). Suppose the eigenvalues ±α of A are not in NR N ω ( κ 4G(N ) , g). If 2G(N) 2 g(N) 2 ǫ ≤ (1 -a) 2 2 κ 2 (35)
and

eC ′ (G • g)(N + 1) -c 0 ≤ 1 -a (36)
then letting r ′ = r 2 -c 0 log(G•g)(N +1) 4πN
, there exists

F ′ ∈ C ω r ′ (T d ), A ′ ∈ sl(2, R) and Z ∈ C ω r ′ (2T d ) such that ∂ ω Z = (A + F )Z -Z(A ′ + F ′ ) (37)
and

| F ′ | r ′ ≤ (1 -a)ǫ. (38)
Proof: One first applies Proposition 2.2 on A. Let Φ, Ã be as in Proposition 2.2 so that σ

( Ã) = ±α, α ∈ NR N ω ( κ G(N ) , g) and | α |≤ κ 4G(N ) ; let F = ΦF Φ -1 .
Notice that, by construction of Φ, the map F remains continuous on T d . Apply Proposition 2.4 with a ′ = 1 and with

r ′ = r 2 -c 0 log(G•g)(N +1) 4πN to get X ∈ C ω r ′ (T d ), A ′ , F ′ ∈ C ω r ′ (T d
) such that (25) and (27) hold as well as

| X | r ′ ≤ 4 G(N)g(N) κ | F N | r ′ (39)
and let Z = Φe X ∈ C ω r ′ (2T d ) so that Z satisfies (37). Condition (35) implies that

(G • g)(N) | X | r ′ ≤ (1 -a) 2 | F N | r ′ ǫ so (27) with a ′ = 1 gives | F ′ | r ′ ≤ eC ′ | F -F N | r e -2πN (r-2r ′ ) + eC ′ | F N | r ′ e 2πN r ′ (1 -a) 2 (G • g)(N) (2e + 1) (40)
and by the choice of r ′ ,

| F ′ | r ′ ≤ eC ′ | F | r (G • g)(N + 1) -c 0 . (41)
This implies, by assumption (36), that

| F ′ | r ′ ≤ (1 -a) | F | r . ( 42
)
3 Iteration, reducibility and arithmetical conditions

Iteration

In this section we will first introduce the Brjuno-Rüssmann condition and then show how one can use it to control the convergence of the KAM iteration scheme.

Assumption 1 The functions g and G satisfy

∞ 1 log[g(t)G(t)] t 2 dt < ∞. (43)
In order to iterate the basic step, we will now fix the parameters as follows: let C ′ be as in Proposition 2.2. Furthermore, let r 0 > 0, n 0 ∈ N and choose

c 0 = r 0 4 n 0 +3 (sup t∈[1,n 0 ] log(G•g)(t+1) t + 1) Let a ∈ [1 -ā, 1) where ā = min( 1 14 2 , 1 (G•g)(2) 2 )
. Let ǫ 0 > 0 be small enough to assure that

∞ (G•g) -1   κ 2(1-a) n 0 -5 4 ǫ 1 2 0   log(G • g)(t) t 2 dt ≤ r 0 4 n 0 (44)
and

eC ′ ǫ c 0 4 0 ≤ (1 -a) 2 κ 2 . ( 45
)
For all n ∈ N, let ǫ n = (1 -a) n 2 ǫ 0 and let N n be the biggest integer such that

(G • g)(N n ) 2 ≤ (1 -a) 2 4ǫ n κ 2 (N n exists since ǫ n ≤ (1-a) 2 κ 2 4e(G•g)(1) 2 ).
The above choices of the sequences ǫ n and N n are made in such a way that the following holds:

∞ Nn 0 log(G • g)(t) t 2 dt ≤ r 0 4 n 0 +2 . ( 46
)
Remark: : The number ǫ 0 will then only depend on a, κ, g, G, n 0 and r 0 (the larger n 0 and the smaller r 0 are, the smaller ǫ 0 will be).

To simplify the notations, from now on the functions A n are understood to be in sl(2, R), while the F n have their values in sl(2, R) and Z ′ n , Z n have their values in SL(2, R).

Proposition 3.1 Let A ∈ sl(2, R) and F ∈ C ω r 0 (T d ). If | F | r 0 ≤ ǫ 0 , then there exist sequences (r n ) n∈N , r n > 0, Z n , ∈ C ω rn (2T d ), A n with spectrum ±α n , F n ∈ C ω rn (2T d )
, and m n ∈ Z d , such that 1. if all m n are zero when n ≥ n 0 , then r n has a positive limit;

2. if m n = 0 then | α n -π m n , ω |≤ κ 4G(Nn) ; 3. m n has modulus less than N n , 4. | F n | rn ≤ ǫ n ; 5. ∂ ω Z n = (A + F )Z n -Z n (A n + F n ); 6. | α n-1 -iπ m n-1 , ω -α n |≤ √ ǫ n-1 .
Remark: Proposition 3.1 implies that A + F is reducible in C ω r ′ for some r ′ > 0 if all m n are zero for n ≥ n 0 .

Proof: This proposition is shown by recurrence. Suppose these sequences are defined up to some n ∈ N and suppose that for all n ′ ≤ min(n -1, n 0 ), r n ′ +1 ≥ r n ′ 4 . We must distinguish two cases according to the possibility that the spectrum of A n is resonant or not.

First case: α

n ∈ NR Nn ω ( κ 4G(Nn) , g). Let r n+1 = r n -c 0 |log(1-a)| 2πNn , so that r n+1 ≥ rn 2 if n ≤ n 0 . One can apply Proposition 2.4 with r = r n , r ′ = r n+1 , N = N n , F = F n and à = A n and obtain Z ′ n = e Xn ∈ C ω r n+1 (T d ), F n+1 ∈ C ω r n+1 (T d ), A n+1 such that ∂ ω Z ′ n = (A n + F n )Z ′ n -Z ′ n (A n+1 + F n+1 ) (47)
and

| F n+1 | r n+1 ≤ (1 -a) 1 2 ǫ n = ǫ n+1 . (48)
One then takes Z n+1 = Z n Z ′ n .

Second case: α n / ∈ NR Nn ω ( κ 4G(Nn) , g). Assumption (35) is satisfied by definition of N n ; assumption (36) is also satisfied since, by maximality of N n ,

(G • g)(N n + 1) -c 0 ≤ 2ǫ n (1 -a) 2 κ 2 c 0 2 ≤ 2ǫ 0 (1 -a) 2 κ 2 c 0 2 (49)
which, together with (45), implies that

(G • g)(N n + 1) -c 0 ≤ 1 -a eC ′ . (50)
Therefore, one can apply Proposition 2.5 with r = r n , r

′ = r n+1 = rn 2 -c 0 log(G•g)(Nn+1) πNn , so that r n+1 ≥ rn 4 if n ≤ n 0 , and N = N n . It follows that there exists A n+1 ∈ sl(2, R), F n+1 ∈ C ω r n+1 (T d ) and Z ′ n ∈ C ω r n+1 (2T d ) such that ∂ ω Z ′ n = (A n + F n )Z ′ n -Z ′ n (A n+1 + F n+1 ) and | F n+1 | r n+1 ≤ (1 -a) | F n | rn ≤ ǫ n+1 One then takes Z n+1 = Z n Z ′ n .
To complete the proof, we now need to show that (r n ) n has a positive limit if all m n are zero for n ≥ N 0 . We have

lim n r n = r n 0 - ∞ k=n 0 (r k -r k+1 ) ≥ r 0 4 n 0 - k≥n 0 | log(1 -a) | 2πN k . (51)
Now, for all n,

N n = E (G • g) -1 (1 -a)κ 2 √ ǫ n thus lim n r n ≥ r 0 4 n 0 - | log(1 -a) | 2π ∞ n 0 (G • g) -1 κ 2(1 -a) n 4 -1 -1 dn. (52)
Through the change of variables X

= κ 2(1-a) n 4 -1 , lim n r n ≥ r 0 4 n 0 - ∞ (G•g)(Nn 0 ) 1 π(G • g) -1 (X)X dX. (53)
Letting now Y = (G • g) -1 (X), the integral becomes

∞ (G•g)(Nn 0 ) 1 π(G • g) -1 (X)X dX = ∞ Nn 0 1 πY (G • g)(Y ) d(G • g)(Y ) = -log(G • g)(N n 0 ) N n 0 + ∞ Nn 0 log(G • g)(Y ) Y 2 dY. (54)
Therefore (56) which is positive.

lim n r n ≥ r 0 4 n 0 + log(G • g)(N n 0 ) πN n 0 - 1 π ∞ Nn 0 log(G • g)(Y ) Y 2 dY (
ρ(A + F ) = ρ(A ∞ ) + π j≥0 m j , ω (59)
(see also [START_REF] Eliasson | Floquet solutions for the 1-dimensional quasi-periodic Schrödinger equation[END_REF]). Therefore

ǫ n ≤ (1 -a) 2 κ 2 4G(N n ) 2 g(N n ) 2 then κ ′ ≤ κ G(N n ) g( j≤n | m j |). (64)
Now, note that j≤n | m j |≤ N 2 n . This comes from the fact that, denoting by m j k the subsequence of non-zero m j 's, then for all k,

| m j k+1 , ω | <| m j k+1 , ω -α j k+1 | + | m j k , ω -α j k + α j k+1 | + | m j k , ω -α j k | < κ 4G(N j k+1 ) + 2 √ ǫ j k + κ 4G(N j k ) ≤ κ 4G(N j k+1 ) + κ 2G(N j k ) + κ 4G(N j k ) (65)
which together with the arithmetic condition on ω implies that N j k+1 > N j k ≥ k. Therefore

κ ′ ≤ κ G(N n ) g(N 2 n ) (66)
and since by assumption g(t 2 ) G(t) is bounded,

κ ′ ≤ κ sup t≥n g(t 2 ) G(t) . (67)
In other words, if

κ ′ > κ sup t≥n g(t 2 ) G(t) (68)
then m n = 0 for all n ≥ n 0 ; and so A + F is analytically reducible.

This proves Theorem 1.3. Here is an easy consequence of the main result:

Corollary 3.3 Let A ∈ sl(2, R), r > 0, F ∈ C ω r (T d ). Assume
1. the map t → g(t 2 ) G(t) tends to 0, 2. ρ(A + F ) ∈ NR ω (κ ′ , g) for some κ ′ > 0.

There exist ǫ 0 depending only on g, κ, G, ρ(A + F ), r such that if

| F | r ≤ ǫ 0 ,
then there exists r ′ ∈ (0, r) such that A + F is reducible in C ω r ′ (2T d ).

|√ ǫ j = 1 1- 4 √

 14 ρ(A + F ) -π j≤n m j , ω | =| ρ(A ∞ ) + π j≥n+1 m j , ω | ≤| α n+1 | + j≥n+1 | α j -π m j , ω -α j+1 | ≤| α n+1 | + j≥n+1 √ ǫ j (60) Suppose ρ(A + F ) satisfies ∀m ∈ Z d , | ρ(A + F ) -π m, ω |≥ κ ′ g(| m |) .In particular,| ρ(A + F ) -π j≤n m j , ω |≥ κ ′ g(| j≤n m j |)and soκ ′ g(| j≤n m j |) ≤ j≥n+1 √ ǫ j + | α n+1 | .(61)Let n > n 0 . Assume m n = 0. Then we have| α n+1 |≤| α n -π m n , ω | + | α n -π m n , ω -α n+1 |≤ κ 4G(N n ) + (1 -a) 1 ǫ n ≤ 2 √ ǫ nand since, by definition of N n ,

A link between the Brjuno sum and the allowed perturbation

It is easily seen that the condition on ǫ 0 can also be expressed more conveniently as the following sufficient condition:

Indeed, we have the following bound:

and the conclusion follows easily from the upper bound on t.

Reducibility theorem

We will now need one more assumption on the approximation functions G and g.

Assumption 2

The map t → g(t 2 ) G(t) is bounded.

Now we can prove the main result:

Under assumptions 1 and 2 on the approximation functions g and G, there exist ǫ 0 > 0 depending only on g, κ, G, n 0 , r such that if

n∈N as defined at the beginning of section 3.1. Let (r n ), (α n ), (m n ), (A n ), (F n ), (Z n ) be the sequences given by Proposition 3.1. The sequence (A n ) is bounded in sl(2, R) for the operator norm so taking a subsequence (A n k ), we find that A n k tends to some A ∞ ∈ gl(2, R). Now ρ(A ∞ ) is the limit of ρ(A n k ) (see [START_REF] Eliasson | Floquet solutions for the 1-dimensional quasi-periodic Schrödinger equation[END_REF], Lemma A.3) which implies that for all n,

Moreover,

Proof: Let n 0 be the smallest integer such that κ ′ > sup t≥n 0 g(t 2 ) G(t) . Take ǫ 0 as in Theorem 3.2 so that it really depends on g, κ, G, ρ(A + F ), r and apply Corollary 3.2.

Theorem ?? is a particular case of Corollary 3.3 since we can take g

, as we will see in the next section. Our main result will be made more convenient by the following corollary:

Assume ω is a Brjuno vector and ρ(A + F ) is a 1 2 -Brjuno number with respect to ω. There exists ǫ 0 depending only on

By assumption, there exists κ ′ > 0 and g positive increasing and continuous such that

dt < +∞ and ρ(A + F ) ∈ NR ω (κ ′ , g); there also exists κ > 0 and G ′ positive increasing and continuous such that

dt < +∞ and ω ∈ NR(κ, G ′ ). Now let for all t, G(t) = t max(G ′ (t), g(t 2 )). The function G is positive increasing and continuous and ω ∈ NR(κ, G). Since g(t 2 ) G(t) ≤ 1 t for all t, we can apply the previous corollary.

Possible choices of approximation functions

Here we give a few examples of approximation functions to which Theorem 3.2 can be applied.

Verification of Assumption 2

Here are a few examples where Assumption 2 holds, i.e g(t 2 ) G(t) is bounded:

In the example 1, and if µ ′ > µ 2 , then, as noted in section 3.4, the condition on ǫ 0 does not depend on n 0 and κ ′ might be arbitrarily small, which corresponds to Eliasson's full-measure reducibility result in [START_REF] Eliasson | Floquet solutions for the 1-dimensional quasi-periodic Schrödinger equation[END_REF].

Smallness conditions

We shall make conditions (44) and (45) more explicit for the particular cases that we mentioned before, namely, when

or if

Proof: Rewrite Condition (44) as

(71)

. Integrating by parts, this is

It is enough that

that is,

which is true if (70) is satisfied. If moreover

then ( 74) is satisfied as long as

Proof: By plugging e t α +t α ′ into (44) and recalling that g • G is increasing, one finds that Condition (44) holds if

so, in particular, (77) is a sufficient condition.

.

(79)

Proof: In this case, (44) can be rewritten

. Integrating by parts, we compute

which implies (89)


Thomas Gauthier
email: thomas.gauthier@math.univ-toulouse.fr

PERTURBATIONS OF FLEXIBLE LATT ÈS MAPS

Introduction

A rational map of degree D ≥ 2 is strictly postcritically finite if the orbit of each critical point intersects a repelling cycle. Among those, flexible Latts maps (the definition is given below) play a special role. The following result answers a question raised in [BE].

Theorem. Every flexible Lattès map can be approximated by strictly postcritically finite rational maps which are not Lattès maps.

Given D ≥ 2, denote by Rat D the space of rational maps of degree D. A rational map f ∈ Rat D has a Julia set J f which may be defined as the closure of the set of repelling cycles of f . The Julia set is the support of a measure µ f which may be defined as the unique invariant measure of (maximal) entropy log D.

The bifurcation locus in Rat D is the closure of the set of discontinuity of the map f → J f . Laura DeMarco [DeM] proved that the bifurcation locus is the support of a positive closed (1, 1)-current T bif :=dd c L, where L is the plurisubharmonic function which sends a rational map f to its Lyapunov exponent with respect to µ f .

Möbius transformations act by conjugacy on Rat D and the quotient space is an orbifold known as the moduli space M D of rational maps of degree D. Giovanni Bassanelli and François Berteloot [BB1] introduced a measure µ bif on this moduli space, which may be obtained by pushing forward T ∧(2D-2) bif . In [BE], the first author and Adam Epstein, using a transversality result in Rat D , proved that the conjugacy class of a strictly postcritically finite map which is not a flexible Lattès map is in the support of µ bif . Since the support of µ bif is closed and since the conjugacy class of a strictly postcritically finite rational maps which is not a Lattès maps is in this support, our result has the following consequence.

Corollary 1. The classes of flexible Lattès maps in M D lie in the support of the bifurcation measure µ bif .

In [G1], the second author proved that the support of the bifurcation measure has maximal Hausdorff dimension, i.e. has dimension 2(2D -2), and that it is homogeneous (the support of µ bif has maximal dimension in any neighborhood of its points). Corollary 1 thus yields the following result.

f . Then, dim H (supp(µ bif ) ∩ V ) = 2(2D -2).
The research of the first author was supported in part by the grant ANR-08-JCJC-0002 and the IUF.

Bassanelli and Berteloot [START_REF] Bassanelli | Lyapunov exponents, bifurcation currents and laminations in bifurcation loci[END_REF] proved that every point in the support of µ bif can be approximated by rational maps having 2D -2 distinct neutral cycles. Their argument can be adapted to prove that the support of µ bif can be approximated by hyperbolic maps having 2D -2 distinct attracting cycles (see [B] Section 5.2). By Corollary 1, we have the following result.

Corollary 3. Any flexible Lattès map f ∈ Rat D can be approximated by hyperbolic rational maps having 2D -2 distinct attracting cycles.

The approach for solving this problem was suggested by John Milnor. We wish to express our gratitude. We also wish to thank the Banff International Research Station for hosting the workshop "Frontiers in Complex Dynamics (Celebrating John Milnor's 80th birthday)" during which we developed our proof.

Flexible Lattès maps

Following Milnor [M], we define a flexible Lattès map of degree D ≥ 2 to be a rational map f : C → C for which there is a commutative diagram :

C/Λ L / / Θ C/Λ Θ C f / / C where • Λ ⊂ C is
a lattice of rank 2; • T :=C/Λ is the quotient torus;

• L : T ∋ τ → aτ + b ∈ T with a ∈ Z, a 2 = D, and 2b ∈ Λ/ 2Λ + (a -1)Λ ;

• Θ : T → C is a 2-to-1 holomorphic map ramifying at points in Λ/2.

In the rest of the article, the lattice Λ will be of the form Λ = Z ⊕ γZ, where γ is a complex number which is allowed to vary in the upper half-plane H of complex numbers with positive imaginary part. We shall use the following normalization for Θ γ : T → C :

Θ γ (0) = 0, Θ γ γ + 1 2 = ∞, Θ γ 1 2 = v(γ) and Θ γ γ 2 = w(γ)
where v : H → C -{0} and w : H → C -{0} are holomorphic functions (this may be achieved for example by requiring that v ≡ 1). Then, for all γ ∈ H, v(γ) = w(γ).

The derivative of the torus endomorphism L γ : T γ → T γ will be a fixed integer a which does not depend on γ. When a = ±2, the critical value set of the Lattès map

f γ : C → C is ∞, v(γ), w(γ) and if |a| ≥ 3, the critical value set is 0, ∞, v(γ), w(γ) . In all cases, the postcritical set of f γ is 0, ∞, v(γ), w(γ) .
More precisely, we will have to study three cases :

• Case 1 : a is even and L γ (τ ) = aτ . In that case, all the critical values are mapped to 0 which is a fixed point of f . • Case 2 : a is odd and L γ (τ ) = aτ . In that case, all the critical values are fixed with multiplier a 2 .

• Case 3 : a is odd and L γ (τ ) = aτ + (γ + 1)/2. In that case, the Lattès map permutes the critical value at 0 with that at infinity. It also permutes the critical value at v(γ) with the critical value at w(γ). The multiplier of each cycle is a 4 .

From now on, we assume that we are in one of those three cases, and we consider the analytic family of Lattès maps

H ∋ γ → f γ ∈ Rat D ,
where Rat D is the space of rational maps of degree D. We shall use the notation f , v, w, . . . in place of f γ , v(γ), w(γ), . . . when γ is assumed to be fixed and there is no confusion.

Estimates for Θ

Lemma 1. As τ → 0, we have the following expansion

Θ(1/2 + τ ) = v + λτ 2 + o(τ 2 ) and Θ(γ/2 + τ ) = w + µτ 2 + o(τ 2 ) with λ v = - µ w = 0.
Proof. Since Θ has simple critical points at 1/2 and γ/2, we have an expansion as in the statement with λ = 0 and µ = 0. Our work consists in proving the relation between λ/v and µ/w. Let q be the meromorphic quadratic differential on C defined by :

q:= dz 2 z(z -v)(z -w)
.

Since q has simple poles and since Θ is totally ramified above the polar set of q, the quadratic differential Θ * q is holomorphic on T , whence

Θ * q = κ • dτ 2 with κ ∈ C -{0}.
Let τ tend to 0 and set z = Θ(1/2 + τ ). Then, z -v ∼ λτ 2 and dz 2 ∼ 4λ 2 τ 2 dτ 2 . It follows that

dz 2 z(z -v)(z -w) ∼ 4λ 2 τ 2 dτ 2 v • (v -w) • λτ 2 = 4λdτ 2 v(v -w) .
This show that

κ = 4λ v(v -w)
.

Using the expansion of Θ at γ/2 yields similarly

κ = 4µ w(w -v)
.

The result now follows easily.

Hyperbolic sets

From now on, we let α, α ′ , β and β ′ be rational numbers. We assume that the denominators of α and β are coprime with a and 2. We let σ and τ be the points in the torus T defined by σ:=α + α ′ γ and τ :=β + β ′ γ.

Remark that, as σ and τ have rational coordinates, they are (pre)periodic under multiplication by a and that σ, τ , aσ and aτ are (pre)periodic under multiplication by a 2 . For each integer k ≥ 1, let τ k and σ k be the points in T defined by

σ k := 1 2 + σ a k and τ k := γ 2 + τ a k
and let x k and y k be the points in C defined by

x k :=Θ(σ k ) and y k :=Θ(τ k ).
We shall denote by X, Y , and Z the following subsets of C :

X:= k≥1 n≥0 f •n (x k ), Y := k≥1 n≥0 f •n (y k ) and Z:=X ∪ Y ∪ {0, ∞, v, w}.
Note that Z is an invariant set.

Lemma 2. There is a neighborhood G of f in Rat D and a dynamical holomorphic motion φ : G × Z → C such that for all g ∈ G and all z ∈ Z, we have g φ(g, z) = φ g, g(z) .

Proof. Case 1 : a is even. In that case, f •k maps x k to Θ(σ) and y k to Θ(τ ). Since σ and τ are (pre)periodic under the action of L, the points x k and y k are (pre)periodic.

Case 2 : a is odd and b = 0. In that case, f •k maps x k to Θ( 1 2 + σ) and y k to Θ( γ 2 + τ ). As L( 1 2 + σ) = 1 2 + aσ and L( γ 2 + τ ) = γ 2 + aτ and as σ and τ are (pre)periodic under multiplication by a, the points 1 2 +σ and γ 2 +τ are (pre)periodic under the action of L, which means that the points x k and y k are (pre)periodic.

Case 3 : a is odd and b = (γ + 1)/2. In that case, we use that

L •2 1 2 + z = 1 2 + a 2 z and L •2 γ 2 + z = γ 2 + a 2 z for any z ∈ T . If k is even, f •k maps x k to Θ( 1 2
+σ) and y k to Θ( γ 2 +τ ). Since σ and τ are (pre)periodic under multiplication by a 2 , 1 2 + σ and γ 2 + τ are (pre)periodic under iteration of L •2 and the points x k and y k are (pre

)periodic. Now, if k is odd, f •k maps x k to Θ( γ 2 + σ) and y k to Θ( 1 2 + τ ). Since L( γ 2 + σ) = 1 2 + aσ and L( 1 2 + τ ) = γ
2 + aτ and since aσ and aτ are (pre)periodic under multiplication by a 2 , 1 2 + aσ and γ 2 + aτ are (pre)periodic under the action of L •2 , which implies that the points x k and y k are (pre)periodic.

In all cases, the set Z is closed and each point in Z is (pre)periodic to a repelling cycle of f . Since the denominators of α and β are coprime with a and 2, the cycles capturing points in X and Y are disjoint from the postcritical set of f . In particular, Z does not contain critical points. As a consequence, Z is a hyperbolic invariant compact set (see [ST] Theorem 1.2 page 266) and the result follows (see [S] (1.2) section 2 for a sketch of the proof, or [G2] section 2.1 for a proof).

For all g ∈ G, the map φ g : z → φ(g, z) is a homeomorphism from Z to Z g :=φ g (Z) which conjugates f : Z → Z to g : Z g → Z g . Every point z ∈ Z g is (pre)periodic under iteration of g to a repelling cycle of g. For k ≥ 1, we shall use the notation x k (g):=φ g (x k ) and y k (g):=φ g (y k ). As k → +∞, the sequence x k tends to x ∞ :=v. We set x ∞ (g):=φ g (v). Similarly, as k → +∞, the sequence y k tends to y ∞ :=w. We set y ∞ (g):=φ g (w).

A family of perturbations

Let us now restrict to the family of rational maps t → f t which is defined for t ∈ C by f t :=(1 + t) • f.

If t = -1, the rational map f t has degree D and f 0 = f . The critical values of the rational map f t are contained in the set {0, ∞, (1 + t) • v, (1 + t) • w} and for t sufficiently close to 0, the point 0 is either a repelling fixed point or a repelling periodic point of period 2, and ∞ is either mapped to a repelling fixed point at 0, or a repelling fixed point, or a repelling periodic point of period 2. We will now study the relative motion of the critical values v(f t ):=(1 + t) • v and w(f t ):=(1 + t) • w with respect to the points x ∞ (f t ) and y ∞ (f t ). We will use the notations

v:= dv(f t ) dt t=0 ∈ T v C, ẇ:= dw(f t ) dt t=0 ∈ T w C, ẋ∞ := dx ∞ (f t ) dt t=0 ∈ T v C, and ẏ∞ := dy ∞ (f t ) dt t=0 ∈ T w C.
We shall also use the notation ḟ := df t dt t=0

which is a section of the pullback bundle

f ⋆ T C : for all z ∈ C, ḟ (z) ∈ T f (z) C. Lemma 3. We have ẋ∞ - v v = ẏ∞ - ẇ w = 0. Proof. By construction, v = v d dz and ẇ = w d dz .
In addition, for all z ∈ C,

ḟ (z) = f (z) d dz .
Case 1 : a is even. In that case, f maps x ∞ = v to 0 which is a fixed point. As t varies, f t still maps v to 0 and still fixes 0. Thus, x ∞ ≡ v and ẋ∞ = 0, whence

ẋ∞ -v = -v. Similarly ẏ∞ -ẇ = -ẇ.
Case 2 : a is odd and b = 0. In that case,

f fixes x ∞ = v. So, x ∞ (f t ) is the fixed point of f t which is close to v. Derivating f t x ∞ (f t ) = x ∞ (f t )
with respect to t, and evaluating at t = 0, we get :

ḟ (v) + D v f ( ẋ∞ ) = ẋ∞ . Since ḟ (v) = v d dz = v and D v f is multiplication by a 2 , we get ẋ∞ = v 1 -a 2 whence ẋ∞ -v = 1 1 -a 2 -1 v = a 2 1 -a 2 v. Similarly ẏ∞ -ẇ = a 2 1 -a 2 ẇ.
Case 3 : a is odd and b = (γ + 1)/2. In that case f permutes x ∞ = v and y ∞ = w.

Derivating f t x ∞ (f t ) = y ∞ (f t ) and f t y ∞ (f t ) = x ∞ (f t )
with respect to t, and evaluating at t = 0, we get :

(1) ḟ (v) + D v f ( ẋ∞ ) = ẏ∞ and ḟ (w) + D w f ( ẏ∞ ) = ẋ∞ .
On the one hand ḟ (v) = w d dz = ẇ and ḟ (w) = v d dz = v. On the other hand,

L 1 2 + τ = γ 2 + aτ and L γ 2 + τ = 1 2 + aτ.
Thus, f v+λτ 2 +o(τ 2 ) = w+µa 2 τ 2 +o(τ 2 ) and f w+µτ 2 +o(τ 2 ) = v+λa 2 τ 2 +o(τ 2 ) which implies λf ′ (v) = µa 2 and µf ′ (w) = λa 2 . By Lemma 1, it follows that

D v f ( ẋ∞ ) = µa 2 λ ẋ∞ = - a 2 w v ẋ∞ .
Similarly,

D w f ( ẏ∞ ) = - a 2 v w ẋ∞ . So, equation (1) yields ẇ - a 2 w v ẋ∞ = ẏ∞ and v - a 2 v w ẏ∞ = ẋ∞ which boils down to ẋ∞ = 1 -a 2 1 -a 4 v and ẏ∞ = 1 -a 2 1 -a 4 ẇ. As a consequence ẋ∞ -v = 1 -a 2 1 -a 4 -1 v = - a 2 1 + a 2 v. Similarly ẏ∞ -ẇ = - a 2 1 + a 2 ẇ.
In all cases, there is a constant c = 0 (c = -1 in the first case, c = a 2 /(1 -a 2 ) in the second case and c = -a 2 /(1 + a 2 ) in the third case) such that

ẋ∞ -v = c v = cv d dz and ẏ∞ -ẇ = c ẇ = cw d dz .
This shows that ẋ∞ -

v v = ẏ∞ - ẇ w = c d dz = 0.

Proof of the Theorem

Let γ 0 :=x 0 +iy 0 ∈ H be a point with rational real part x 0 and rational imaginary part y 0 which are coprime with a and 2. Set α:= -x 0 , α ′ :=1, β:=y 0 and β ′ :=0. We will now allow γ to vary in H. So, σ, τ , v, w, λ, µ, σ k , τ k , x k , y k , x ∞ , y ∞ are functions of γ. For example,

σ : H ∋ γ → α + α ′ γ ∈ H and τ : H ∋ γ → β + β ′ γ ∈ H.
Note that the Möbius transformation σ/τ sends γ 0 to i.

We still consider the family of flexible Lattès maps H ∋ γ → f γ ∈ Rat D and we introduce the family of perturbed maps

H × (C -{-1}) ∋ (γ, t) -→ f γ,t :=(1 + t) • f γ ∈ Rat D .
As in the previous section, we will consider the critical value functions

v : f γ,t → (1 + t) • v(γ) and w : f γ,t → (1 + t) • w(γ).
The notation v, ẇ, ẋ∞ , ẏ∞ refers to the derivative with respect to t evaluated at t = 0. In particular, those are functions of γ ∈ H. Lemma 4. There are a neighborhood Γ of γ 0 in H, an integer k 0 and for k ≥ k 0 , holomorphic functions s k : Γ → C and t k : Γ → C such that :

• the sequences (s k ) and (t k ) converge uniformly on Γ to 0,

• x k (f γ,s k (γ) ) = v(f γ,s k (γ) ), y k (f γ,t k (γ) ) = w(f γ,t k (γ)
) and • the sequence (s k /t k ) converges uniformly on Γ to -σ 2 /τ 2 .

Proof. First, note that as k → +∞, the holomorphic functions

x k : H → C and v : H → C satisfy x k -v ∼ λ • σ k - 1 2 2 = λσ 2 a 2k
. Second, let us consider the sequence of functions

X k :=(γ, u) → a 2k • x k (f γ,u/a 2k ) -v(f γ,u/a 2k ) . Note that as k → +∞, X k (γ, u) = a 2k • x k (γ) + u a 2k • ẋ∞ (γ) -v(γ) - u a 2k • v(γ) + o(1/a 2k ) = λ(γ)σ 2 (γ) + u • ẋ∞ (γ) -v(γ) + o(1).
So, as k → +∞, the sequence (X k ) converges uniformly on every compact subset of H × C to the function

X ∞ : (γ, u) → λσ 2 + u • ( ẋ∞ -v).
According to Lemma 3, ẋ∞ -v does not vanish on H. Thus, the function X ∞ vanishes along the graph of a holomorphic function u

∞ : H → C which satisfies u ∞ • ( ẋ∞ -v) = -λσ 2 .
Since (X k ) converges locally uniformly to X ∞ , given any neighborhood Γ of γ 0 compactly contained in H, there are, for k large enough, holomorphic functions

u k : Γ → C which satisfy (∀γ ∈ Γ) X k γ, u k (γ) = 0, the sequence u k converging uniformly on Γ to u ∞ . Now, set s k := u k a 2k : Γ → C.
Then the sequence (s k ) obviously converges uniformly to 0 on Γ and

x k (f γ,s k (γ) ) = v(f γ,s k (γ) ) and a 2k • s k • ( ẋ∞ -v) -→ k→+∞ -λσ 2 .
Similarly, there are, for k large enough, holomorphic functions t k : Γ → C which converge uniformly on Γ to 0 and satisfy

y k (f γ,t k (γ) ) = w(f γ,t k (γ) ) and a 2k • t k • ( ẏ∞ -ẇ) -→ k→+∞ -µτ 2 .
Those uniform convergences on Γ may be reformulated as

a 2k • s k σ 2 -→ k→+∞ - ẋ∞ - v λ and a 2k • t k τ 2 -→ k→+∞ - ẏ∞ - ẇ µ .
According to Lemma 1 and Lemma 3, ẋ∞ -v λ = -ẏ∞ -ẇ µ are non-vanishing functions. This shows that the sequence (s k /t k ) converges uniformly on Γ to -σ 2 /τ 2 . The previous construction yields, for k large enough, functions s k and t k which are defined and holomorphic near γ 0 . As k → +∞, the sequence of functions (1 -s k /t k ) converges to 1 + σ 2 /τ 2 which has a simple zero at γ 0 . Therefore, for k large enough, the function s k /t k takes the value 1 at a point γ k close to γ 0 :

s k (γ k ) = t k (γ k ) with γ k -→ k→+∞ γ 0 .
Let g k ∈ Rat D be the rational map

g k :=(1 + r k ) • f γ k with r k :=s k (γ k ) = t k (γ k ).
Note that r k tends to 0 and that g k converges to f γ0 as k tends to +∞. The critical values of g k are contained in the set 0, ∞, (1 + r k )v, (1 + r k )w (if a = 2, then 0 is not a critical value). By construction, 0 and ∞ are (pre)periodic to repelling cycles. In addition, (1 + r k )v coincides with x k (r k ) which is (pre)periodic to a repelling cycle and (1 + r k )w coincides with y k (r k ) which is also (pre)periodic to a repelling cycle. Thus, g k is strictly postcritically finite but is not a Lattès map.

We have just proved that f γ0 can be approximated by rational maps g k which are strictly postcritically finite but which are not Lattès maps. Since the set of allowable γ 0 is dense in the upper-half plane, this shows that any flexible Lattès map can be approximated by rational maps which are strictly postcritically finite but which are not Lattès maps.

Corollary 2 .

 2 Let f ∈ Rat D be a flexible Lattès map and let V ⊂ M D be an open neighborhood of the conjugacy class of


M A Sbai
email: a.sbai@brgm.fr

M Azaroual

A Mathematical Model for Nanoparticles Transport within Twophase Flow in Heterogeneous Porous media

A mathematical model for nanoparticles transport within two-phase flow in heterogeneous porous media -Transport of nanoparticles in porous media is a growing concern among scientific research and technological development, such as in the preservation of groundwater quality, pollution control and remediation, and enhancement of oil recovery. Indeed, recent advances in the later field suggested significant improvements in the recovered volumes by injecting hydrophobic nanoparticles which enhance or reverse the initial reservoir wettability favoring an increase in the relative permeability of the oil phase and the capillary pressure gap between phase pressures. A new mathematical model, based on previous work by Sbai and Azaroual [Adv. Water Resour. 34(1), 62-68, 2011] is developed which couples the incompressible two-phase fluid flow reservoir equations at the macroscopic level to equations of nanoparticles transport at a smaller, but still macroscopic, secondary scale. The latter accounts for pore scale processes of colloidal and hydrodynamic release from pore surfaces, deposition in pore surfaces and possibly clogging the pore throats. The mechanism of particles interphase transfer is accounted for based on the wettability of nanoparticles surfaces. Nanoparticles are divided among several classes based on their size and wettability properties. Each class is assumed to have a constant colloidal and hydrodynamic detachment rates taken as first-order kinetics, a constant surface deposition rate in pore walls of the same wettability according to filtration theory, and a constant mass capture rate at pore throats by blocking and bridging. For each nanoparticle class a mass convection-diffusion equation with a total rate of all related mechanisms is solved numerically with a finite volume method. Change of absolute permeability is correlated to porosity change resulting from mass distributions of flowing, surface, and pore-throat deposited nanoparticles. Relative permeability and capillary pressure functions are optionally allowed to be time-dependent as a function of the mass fraction of deposited nanoparticles of the same wettability. Many multi-dimensional applications of the presented model are given for performance evaluation. The newly developed model has many applications and opens interesting perspectives for environmental remediation, well injectivity design, re-engineered fluids injection, and CO 2 sequestration in deep geological formations. Finally, a nested multiscale finite element method is applied to this problem where three hierarchical scales are considered for the resolution of the global pressure, fluid saturation, and nanoparticles transport equations. An important speedup is achieved by this procedure allowing for realistic engineering grade applications to be tackled efficiently.

INTRODUCTION

Transport of nanoparticles (NP's) in porous media is a growing concern among scientific research and technological development. In earth sciences, several emerging applications have been reported in several fields, such as in the preservation of groundwater quality [START_REF] Cullen | Simulation of the subsurface mobility of carbon nanoparticles at the field scale[END_REF], pollution control and remediation, fines migration control [START_REF] Huang | Using nanoparticle technology to control formation fines migration[END_REF], and enhancement of oil recovery [START_REF] Ju | Enhanced oil recovery by flooding with hydrophilic nanoparticles[END_REF][START_REF] Fletcher | How EOR can be transformed by nanotechnology[END_REF]. Recent advances in the later field suggested significant improvements in the recovered volumes by injecting hydrophobic nanoparticles which enhance or reverse the initial reservoir wettability favoring an increase in the relative permeability of the oil phase and the capillary pressure drop between phase pressures [START_REF] Ju | Enhanced oil recovery by flooding with hydrophilic nanoparticles[END_REF][START_REF] Onyekonwu | Investigating the use of nanoparticles in enhancing oil recovery[END_REF]. A new mathematical model have been developed for the purpose of numerical interpretation and analysis of laboratory experiments of flooding petrographic cores with different nanofluids. The numerical model is elaborated for prediction of micro-and nano-particle mobilization, migration, and capture in porous media systems at the aquifers and reservoirs scale. This extended abstract highlights several key features of our numerical engine including physical and chemical processes taken into account. The developed framework involves solving a set of partial differential equations (PDE's) coupled to particle kinetics in particular. Resolution of this system is notoriously a computationally challenging task. We highlight the main numerical methods used for spatial discretization of Flows and mechanics in natural porous media from pore to field scale. Pore2Field 16-18 November 2011, IFP Energies nouvelles (France) these equations and describe our current efforts to improve computational efficiency and discretization on flexible and fully unstructured grids to be able to discretize the underlying equations on different subscales. Next, a couple of example applications are shortly described and commented. The first example involves the mobilization, migration, and capture in pore throats of endogenous micron size particles in the context of CO 2 injection in sandstones. The impact of precipitates from strong CO 2 -Water-Rock interactions on the injectivity while considering fine scale heterogeneity of the reservoir is analyzed. The second example describes a simulation of enhanced oil recovery in a heterogeneous reservoir while flooding with hydrophilic nanoparticles. It is shown that the numerical model is able to predict an improvement in the recovered volumes due to surface wettability effects, while noting a slight decrease in the absolute permeability. This suggests development of a modeling-optimization procedure for an optimal oil recovery while maintaining the well injectivity.

PHYSICAL PROCESSES AND CONCEPTUAL MODEL

Sbai and Azaroual (2011) describe in detail the conceptual model for particulate transport processes in two-phase flow systems. Their work has involved model verification on previously published experimental data.

The developed numerical engine was used equally to predict mobilization, migration, and capture of CO 2 -wet particles in homogeneous and heterogeneous permeability fields to assess the impact of fine scale geological heterogeneity on the injectivity during CO 2 storage. The mathematical model is based on a coupling between incompressible two-phase flow equations for reservoir simulation and a deep bed filtration model, which considers migration in each fluid phase and interphase particulate mass transfer. This set of governing equations is complemented by an optional advection-dispersion solute transport equation in the aqueous phase to calculate the spatial distribution of salinity, which could trigger the mobility of in-situ colloidal fines migrating in the aqueous phase. Pore scale particulate transport processes are described by a set of coupled convection-diffusion equations for each subset of NP's. The couplings occur only over quasilinear rate laws for basic processes of release from pores bodies, mobility in fluid phases, and capture in pores throats. Quantitative assessment of the permeability decline is strongly dependent on the accuracy of experimental data on permeability-porosity relationships as was discussed and shown previously [START_REF] Sbai | Numerical modelling of formation damage by two-phase particulate transport processes during CO 2 injection in deep heterogeneous porous media[END_REF]).

NUMERICAL METHODS

A fully coupled solution of the governing equations would be a daunting task. We choose instead a sequential formulation, which has the add-on advantage to select among other on-the-shelf reservoir codes (i.e. a threephase flow Black oil or compositional model). The incompressible two-phase flow model is based on a fully implicit solution of the global pressure equation using the standard TPFA first-order upwind finite-volume scheme. Coupling between the global pressure and phase saturation equations is performed with an IMPES or IMPSAT schemes favoring sharp fronts localization and computational efficiency, respectively. The intermediate algebraic nonlinear equations related to the saturation equation are solved with a Newton-Raphson adaptive procedure combined with a line-search technique. After recovery of the phase fluid velocities in an intermediate step, the convection-diffusion-reaction like equations for a set of involved particles are solved by operator splitting techniques. The chief advantage of such techniques being use of highly specialized solvers for each module. Particle kinetics are integrated with DASPK computer package. Particle rates are estimated in a postprocessing step. Finally, at the end of each time step porosity and permeability changes are evaluated based on the mass of particles being deposited and/or mobilized in pore surfaces and pore throats. We consider a hierarchical time stepping loop involving the pressure, phase saturation, and particles concentrations variables, respectively.

APPLICATIONS

Modeling formation damage by CO 2 injection in a heterogeneous subsurface aquifer

CO 2 injection for sequestration projects in sandstone formations involves mobilization, migration, and capture of endogenous and exogenous fine particles in porous media. Examples of endogenous particles are salt deposits, and precipitating secondary mineral phases due to strong geochemical reactivity with the primary minerals. Examples of endogenous particles are mobilized metal oxides from corroded metal pipes or those present in the gas stream as a result of the CO 2

Flows and mechanics in natural porous media from pore to field scale. Pore2Field 16-18 November 2011, IFP Energies nouvelles (France)

capture process together with CO 2 that cannot be completely avoided for economic reasons.

In this application we consider a two-dimensional permeability map extracted from the full permeability field used by the SPE-10 2 nd comparative solution project. This map corresponds to the 36 th layer in the Upper-Ness formation sequence where permeability is fluvial and could be characterized as a high-contrast heterogeneous porous medium. The selected layer is comprised of 60 x 200 (13,200) cells. The porosity is strongly correlated with the horizontal permeability and contains about 2.5% null values. To avoid division by zeros in these cells, we simply replace the null values by a threshold minimal value. The oil phase is extracted from four production wells at the domain corners with an equal flow rate and CO 2 is injected into the formation from an injection well in the center.

Figure 1 shows simulation results after 30 days of CO 2 injection. The injectivity decline is slightly higher due to an already well-established high-permeability channel and a high contrast with its surrounding. Here the particle capture mechanisms are exclusively controlled by the medium heterogeneity and not by the hydraulic interactions. Indeed the narrow channel volume intercepts most of the particles at the pore surfaces and pore throats. Additionally, we can see that injectivity loss distribution in the reservoir is not gradually decreasing radially from the injection well. It is not anymore restricted close to the injector and could be reached in other parts of the reservoir, as in the most permeable channel composed from fluvial sediments, or in fragmented sets of spots at the vicinity of some production wells.

Improving oil recovery in a permeable reservoir by injection of hydrophilic NP's

Ju and Fun (2009) combine an experimental and a onedimensional core-scale numerical modeling for flooding oil-wet rock with lipophobic and hydrophilic polysilicon nanaparticles (LHPN's). They noticed a change of the rock wettability, during the experiment, from oil-wet to

Flows and mechanics in natural porous media from pore to field scale. Pore2Field 16-18 November 2011, IFP Energies nouvelles (France) strongly water wet. The contact angle between the rock surface and the water phase decreased sharply from above to below 90°. Improvement in the recovered oil volume is due to two processes, (i) the change in the rock wettabillity, and (ii) the decrease of the interface tension when injecting a LHPN.

Here, we consider a two-dimensional domain having identical size and dimensions of the first example discussed above, but the permeability distribution is rather smooth and is extracted from the top layer of the SPE-10 2 nd comparative solution project's full grid. We consider the same well pattern except for the total flow rate, which equals 50 m 3 /h.

Figure 2 shows the water and oil cuts during the first 30 days of production from the upper right well. The numerical model clearly predicts an improved oil recovery when the reservoir is flooded with LHPN's, while noting a decrease in the mobility of the water phase and a relative increase in the oil mobility. This induces a higher residence time of the water-wet nanoparticles, and hence treatment with LHPN's shows a net decrease of the intrinsic permeability nearby the injection well. For a highly permeable reservoir, it is expected that such formation damage does not offset the benefit of reservoir treatment by LHPN's. However, for reservoirs with low initial permeability it is advised to perform such modelling coupled to an optimization procedure.

Figure 1 -

 1 Figure 1 -Simulation results after 1 month of CO2 injection in a five-spot well pattern showing spatial distribution of (A) the initial reservoir permeability (log10-scale) (B) CO2 phase saturation, mass concentrations of (C) suspended particles, (D) deposited particles in pore surfaces, and in (E) pore throats, and (F) permeability reduction factor.

Figure 2 -

 2 Figure 2 -Comparison between oil and water cuts obtained by injection of water with and without LHPN's. The curves indicate a net improvement in the net recovered oil volume.


Maria Papaiordanidou

Mitsuhiro Hayashibe

Alain Varray

David Guiraud

Charles Fattal

Charles Fattal Model

Model-based fatigue assessment

Keywords: muscle fatigue, contractile apparatus, Sigma-Point Kalman Filter, paraplegia, muscle model

Muscle fatigue is a complex phenomenon that limits the application of functional Electrical Stimulation (ES), used to activate skeletal muscle in order to perform functional movements. The purpose of the present study was to track the development of neuromuscular fatigue under intermittent FES applied to the triceps surae muscle of 5 subjects paralyzed by Spinal Cord Injury (SCI). Experimental results gave evidence of neuromuscular fatigue development attributed to muscle contractile properties impairment. Classical parameters representing muscle contractile properties (peak twitch, Pt and twitch contraction and relaxation parameters) significantly decreased at the end of the protocol. These experimental data were used to identify the parameters of a previously developed physiological mathematical model describing all possible contractive states occurring in a stimulated muscle. The sigma-point Kalman filter was used for the identification of the model's parameters and simulation results prove that the model was capable to track fatigue and under the present stimulation conditions even predict muscle contractile behavior. This work reinforces clinical research with a tool allowing clinicians to monitor the current state of the stimulated muscle for its optimal solicitation.

Introduction

Control of the electrically induced movement of the neuro-musculo-skeletal system, a highly complex, nonlinear and time-varying system, is a fundamental problem in FES research [START_REF] Riener | A physiologically based model of muscle activation verified by electrical stimulation Bioelectr[END_REF]. Development of mathematical models describing this complex system can assist in the design of efficient FES control strategies. Muscle fatigue, one of the major limitations in the clinical application of FES, further complicates the control task. The muscle contractile properties appear the most relevant parameter of fatigue development in subjects with SCI [START_REF] Klein | Fatigue properties of human thenar motor units paralyzed by chronic spinal cord injury[END_REF] and any attempt to track fatigue evolution should be focused on the study of this peripheral fatigue parameter. When ES is applied to paralyzed muscles the stimulation sequence is not predefined and on / off periods are not known in advance. Considering that, for optimal control of paralyzed muscles, stimulation parameters should change according to muscle's current state, our approach to the problem of muscle fatigue is to find muscle's current state with the least laps of time, independently of the stimulation pattern. For this, mathematical models that are based on physiological parameters are necessary. A previously developed physiological muscle model that accurately predicts muscle response to different stimulation patterns [START_REF] Maksoud | Mathematical muscle model for functional electrical stimulation control strategies Proc[END_REF] was used in the present study. The model's parameters were identified using experimental force data. The muscle mechanical response on a simple twitch was studied and its time-course under a preselected ES protocol was used to identify the model's parameters and to track fatigue. Fatigue studies classically use muscle contractile properties as index of peripheral fatigue development. This classical evaluation is pertinent for off-line analysis of muscle fatigue, but cannot be used for FES parameters modulation. The objective of the present study was to propose a method allowing the extraction of current muscle's state.

Material and Methods

Experimental design

Five motor complete SCI subjects volunteered to participate in the present study (ASIA A) which was approved by the ethical committee for persons' protection of Nîmes (2008-A00068-47/1). All subjects signed the consent form. Subjects came to the laboratory on two sessions separated by 2-3 days. On their first visit the intensities were set in order to obtain the maximal torque (Tmax) and the maximal mechanical response on a simple twitch (recruitment curves). Tmax was obtained through myostimulation of the triceps surae muscle (train at 30 Hz, 450 µs, 1 s on -10 s off), while the twitch was delivered via neural stimulation (single twitch of 500-µs-duration) of the tibial nerve at the popliteal fossa. After this first visit, subjects returned for a second session during which the aforementioned intensities were re-adjusted and the intermittent fatiguing ES protocol was performed.

Experimental protocol

The subjects' right triceps surae muscle was fatigued with 5 series of 5 trains (30 Hz, 450 µs, 2 s on -2 s off, at intensity evoking 50% Tmax). Three twitches at supramaximal intensity (+10% intensity eliciting maximal twitch mechanical response) were delivered before and after every 5train-series. Torque evoked by myo-and neural stimulation was continuously recorded.

Data analysis

From the muscle mechanical response the following contractile parameters were calculated: Peak Twitch (Pt), defined as the maximal value of torque evoked on a twitch, Contraction Time (CT), defined as the time elapsed from the beginning of the contraction until Pt and Half Relaxation Time (HRT), defined as the time necessary for torque to decrease to 50% of maximum. All parameters were statistically analyzed using a Friedman ANOVA test for repeated measures. Values after each stimulation series (post5, post10, post15, post20, post25) were compared to baseline values (pre). In the case of a significant effect of time over the variables, a post-hoc test was performed. Significance level was set at P < 0.05.

Presentation of the multi-scale physiological muscle model

The physiological muscle model used for identification and simulation of the muscle mechanical response is based on a multi-scale approach, from the sarcomere scale to the muscle scale [START_REF] Maksoud | Mathematical muscle model for functional electrical stimulation control strategies Proc[END_REF] which results in the following differential equations for stiffness, force development and contractile element elongation:

€ ˙ k c = -(u + ˙ ε c )kc + αkmΠc(t)Uc ˙ F c = -(u + ˙ ε c )Fc + αFmΠc(t)Uc + kcLc0˙ ε c € ˙ ε c = Fcu -αFmΠc(t)Uc ksLc0 + kcLc0 -S ˙ ε cFc
where the chemical input u is defined as follows:

€

u(t) = Πc(t)Uc + (1 -Πc(t))Ur € Πc(t) =

Identification method

The input controls of the mechanical model are the recruitment rate α and the chemical input u. Under the present experimental conditions, α is considered to be 1, since the neural stimulation permitting to evoke the muscle mechanical response was delivered at supramaximal intensity, ensuring synchronous recruitment of all muscle fibers. The chemical input u is defined by 2 parameters U c and U r . The switching time from relaxation state to contraction state can be obtained by the twitch stimulation timing. The electromechanical delay can directly be measured as the time laps between stimulation and torque production. k c , F c and ε c are time-varying states and only F c is accessible from experimentation. L c0 , k s , k m , F m , U c and U r are unknown parameters.

In the present experiment we identified the timevarying parameters under isometric conditions and were interested in their fatigue-induced changes. L c0 and k s are obtained from the literature [START_REF] Delp | Surgery simulation: a computer graphics system to analyze and design musculoskeletal reconstructions of the lower limb[END_REF]. Under isometric conditions k m is not a sensitive parameter, while U c is not significantly affected by fatigue. On the contrary, F m and U r (linked to Pt and HRT respectively), known to undergo the most prominent alterations under fatigue conditions [9], are the parameters to be identified by the muscle model in the present fatigue protocol. Finally, τ c and τ r are adjusted evaluating directly CT with which they are directly linked.

For the identification procedure of F m and U r the Sigma-Point Kalman Filter (SPKF) was used.

After identification, F m and U r values were put in the muscle model in order to simulate muscle mechanical response during the fatiguing ES protocol. The time window used both for identification and simulation assessment is determined by the beginning of the twitch response up to roughly the time when 50% of Pt is obtained to be consistent with the comparison with classical HRT measurements.

Results

Evolution of the contractile properties during the intermittent ES protocol

Pt significant decreased throughout the ES protocol (P<0.001). Pt is classically used as index of peripheral fatigue development and is closely associated with cross-bridge binding events. At post25 it wad decreased by 13%. Parameters of the mechanical response (CT and HRT) did not significantly change throughout the intermittent ES protocol.

Identification and correlations with classical parameters

during contraction phase τ c during transition phase else Identification of the muscle parameters was performed as explained in the previous section. F m values were highly correlated with Pt values during the ES protocol. In tandem with this correlation, U r had a high negative correlation with HRT. It shows that even if HRT evolves slightly, U r is able to track these changes. Moreover, the negative correlation was expected because HRT is a time (s) and U r a rate (Hz). These correlations were 0.98 and -0.95 for F m -Pt and for U r -HRT respectively.

Estimation results

Identification quality was evaluated by means of the normalized root mean square error (NRMSE).

Error values for all subjects at the different stages of the protocol ranged from 0.022 to 0.052. Figure 1 shows an example of model estimation for different fatigue states.

Discussion

The present study aimed at tracking the phenomenon of neuromuscular fatigue under ES. For this, experimental data collected on subjects with SCI were used to identify the two free parameters of the previously developed muscle model [START_REF] Maksoud | Mathematical muscle model for functional electrical stimulation control strategies Proc[END_REF]. We hypothesized that these two parameters would be the most sensible to the fatigue phenomenon. The triceps surae muscle was fatigued with an intermittent ES protocol and response to maximal neural stimulation was collected and analyzed for various mechanical parameters. When the model's free parameters (F m and U r ) were identified using the collected experimental data, it gave the possibility to track the evolution of muscle's mechanical response throughout the protocol. The evaluation of isometric twitches has been widely used in muscle fatigue studies and proved useful for identification of cellular mechanisms of muscle fatigue [START_REF] Fitts | Cellular mechanisms of muscle fatigue[END_REF]. Pt and HRT were used to identify the model's free parameters (F m and U r ). F m represents the maximal force evoked by the muscle in tetanic contraction and Pt is considered as the maximal force on a single neural twitch, assuming that all muscle fibers are recruited simultaneously. U r expresses the chemical kinetics during the relaxation phase of muscle contraction and HRT is the time needed for torque to reach 50% of its peak value on a simple twitch. The strong correlations between F m and Pt and U r and HRT further reinforce the relationship between the model's parameters and experimental data. To our knowledge this is the first time a muscle model was identified using torque obtained from experimental data whose physiological meaning is strongly implicated in the development of muscle fatigue.

Model estimation results of the force evoked by neural stimulation under fatigue conditions were rather satisfactory. We can thus accept that the model is able to track the fatigue developed by an intermittent ES protocol.

Conclusions

In the present study we managed to track the phenomenon of muscle fatigue using a physiologybased muscle model. These results can provide clinicians with a tool allowing them to monitor the muscle condition under ES application and to accordingly change stimulation patterns for optimal stress of the patients' musculo-skeletal system.

Figure 1 :

 1 Figure 1: Estimation results on a representative subject. The second half of the descending part of the torque curve was not used for identification. Red line represents estimation values while the blue line shows measured values.


Marion Davin
email: marion.davin@univmed.fr.

Karine Gente
email: karine.gente@univmed.fr

Carine Nourry
email: carine.nourry@univmed.fr

Social Optimum in an OLG Model with Paternalistic Altruism

Keywords: Optimal balanced growth path, social welfare function, paternalistic altruism JEL Classification Number : D64, D90, O41

There is no consensus yet on the correct way to write the social utility function in presence of paternalistic altruism. This note shows that the specification of the central planner objective is crucial for optimal capital intensity and optimal growth in a one and a two-sector models. In a one-sector model, optimal growth depends on preferences when paternalistic altruism enters the social utility function; otherwise it does only depend on the capital share as in the standard golden rule. In a twosector model, optimal growth depends on preferences and relative capital intensities when paternalistic altruism enters the social utility function; otherwise it does only depend on the capital share of the investment good sector. Moreover, both in a one and a two sector model, the optimal growth rate tends to be higher when warm-glow altruism enters the social utility function.

Introduction

How does the inclusion of paternalistic altruistic feelings in the social planner's objective affect the optimal growth rate? According to a strict definition, the social utility is the discounted sum of individual utility functions, namely a utilitarian social function. However, [START_REF] Harsanyi | A theory of prudential values and a rule utilitarian theory of morality[END_REF] recommends to exclude all external preferences from the social utility function, what we will call an Harsanyi social function. Thus there is no consensus yet on the correct way to write this social utility function. On one side, considering an overlapping generations one-sector model with consumption separable utility function, [START_REF] Cremer | Intergenerational Transfer of Human Capital and Optimal Education policy[END_REF] underline that optimal policy depends on the specification of the social utility function.

Nevertheless, they do not clearly examine the implications on the optimal balanced growth path. On the other side, de la Croix and Monfort (2000) do not include the "joy of giving" term in the welfare function. In this paper, we show, in both a one and a two-sector model, that the way to write the central planner objective is crucial for the optimal growth path. In this purpose, we will consider the example of a paternalistic altruism where agents are concerned by the level of human capital of their children. We 1 also assume that human capital is a simple function of parents investment in their child's education.

Our contribution is double. First, excluding paternalistic altruism (Harsanyi social function), we show that the optimal growth rate is determined only by the technological parameters, whereas if the central planner maximizes the discounted sum of individual utilities (utilitarian social function), the optimal growth rate depends also on the preference parameters. Second, in a two-sector model, we obtain that optimal growth in the Harsanyi case only depends on the investment good sector technological parameters. Whereas in the utilitarian case, relative capital intensities do matter.

These results come from the fact that as long as education is ignored in the social welfare function, the only way to increase welfare is to maximize consumption. Conversely, when child's education provides direct welfare to parents, there is an arbitrage in the social utility function between consumption and education. This is the reason why preference parameters enter the optimal growth rate with the utilitarian social function. In the two-sector case, welfare depends only on the consumption good with the Harsanyi social function whereas it depends also on the investment good through education in the utilitarian social function. Thus, in the Harsanyi social function, the optimal capital intensity is given by a standard modified golden rule where matters the physical capital share of the investment good sector only; whereas in the utilitarian social function, the central planner objective is to produce efficiently over the two sectors since both types of goods (investment good through education and consumption good) matter for welfare.

Moreover, the optimal growth rate tends to be higher with a utilitarian social welfare function than Harsanyi social welfare function.

Social Optimum in a one-sector framework

Consider a perfectly competitive economy in which the final output is produced using physical capital K and human capital H. The production function of a representative firm is an homogeneous function of degree one: F (K, H). We assume for simplicity a complete depreciation of the capital stock within one period. Denoting, for any H = 0, k = K/H the physical to human capital ratio, we define the production function in intensive form as f (k).

Assumption 1 f (k) is defined over R + , C r over R ++ for r large enough, increasing (f (k) > 0) and concave (f (k) < 0). Moreover, lim k→0 f (k) = +∞ and lim k→+∞ f (k) = 0.
We can also compute the share of capital in total income:

s(k) = kf (k) f (k) ∈ (0, 1) (1)
As in [START_REF] Michel | Economic Integration and Growth under Intergenerational Financing of Human-Capital formation[END_REF], we consider a three-period overlapping generations model. In their first period of life, individuals are reared by their parents. In the second period they work and receive a wage, consume, save and rear their own children. In their last period of life, they are retired and consume their saving returns. Following Glomm and Ravikumar (1992), we consider a paternalistic altruism according to which parents value the quality of education received by their children. Thus, the preferences of a paternalistic agent born in t are represented by:

U t = u(c t , d t+1 ) + v(h t+1 ) (2)
where c t and d t+1 represent adult and old aggregate consumption, and h t+1 child's human capital.

Assumption 2. i) u(c, d) is C 2 , increasing with respect to each argument (u c (c, d) > 0, u d (c, d) > 0)
, concave and homogeneous of degree a with a ∈ (0, 1). Moreover, for all d > 0, lim c→0 u c (c, d) = +∞, and for all c > 0,

lim d→0 u d (c, d) = +∞. ii) v(h) is C 2 , increasing (v (h) > 0), convex (v (h) > 0) and satisfies lim h→0 v (h) = +∞.
Parents devotes e t to his child's education, so human capital in t + 1 is given by:

h t+1 = G(e t ) (3)
Assumption 3. The human capital production function G(e) is C 2 , strictly increasing with e.

Grandparents' expenditures in education generate a positive intergenerational external effect in human capital accumulation. Indeed, parents decide for their child's education but do not consider the impact of this decision on their grand child's education. We assume that population is constant over time and is normalized to 1, i.e N t = N = 1. Moreover, clearing condition on the labor market gives

H t = N h t = h t .
The social planner maximizes the discounted sum of the life-cycle utilities of all current and future generations under the resource constraint of the economy and the human capital accumulation equation.

max ct,dt,Kt+1,Ht+1 ∞ t=-1 δ t (u(c t , d t+1 ) + v(H t+1 )) (4)
subject to: ∀t 0

F (K t , H t ) = c t + d t + K t+1 + e t H t+1 = G(e t )
K 0 , H 0 and c -1 given [START_REF] Harsanyi | A theory of prudential values and a rule utilitarian theory of morality[END_REF] with δ ∈ (0, 1), and taking alternatively the extreme values 0 (Harsanyi social function) and 1 (utilitarian social function).

The Lagrange function is:

L = δ -1 [u(c -1 , d 0 ) + v(H 0 )] + ∞ t=0 δ t u(c t , d t+1 ) + v(H t+1 ) + q t F (K t , H t ) -c t -d t -G -1 (H t+1 ) -K t+1 (6)
where q t the Lagrange multiplier associated with the constraint. First order conditions for all t 0 are:

u c (c t , d t+1 ) = q t (7) u d (c t-1 , d t ) = δq t (8) δ t+1 F K (K t+1 , H t+1 )q t+1 = δ t q t (9) δ t v (H t+1 ) + δ t+1 q t+1 F H (K t+1 , H t+1 ) = δ t q t G -1 (H t+1 ) (10)
For initial conditions K 0 and H 0 and for all t 0, optimal solutions satisfy (7), ( 9) and (10).

From ( 8) and (9) we can rewrite the condition that gives optimal physical capital accumulation:

f (k t+1 ) = u c (c t , d t+1 ) u d (c t , d t+1 ) = q t δq t+1 (11)
From ( 7), ( 9) and (10), we obtain the following expression that determines optimal human capital accumulation:

M RS e/c = G -1 (H t+1 ) -k t+1 1 s(k t+1 ) -1 (12)
with M RS e/c ≡ v (Ht+1) uc(ct,dt+1) the marginal rate of substitution between education and first period consumption.

Definition 1 Along the balanced growth path per-capita variables grow at a constant rate g.

As marginal utility of consumption is homogeneous of degree (a -1), from (11), we have δf (k) = (1 + g) 1-a , where k the physical to human capital ratio along the balanced growth path, is determined by equation ( 12). Proposition 1 summarizes the different cases.

f (k t+1 ) = (f (k t+1 ) -k t+1 f (k t+1 )) dH t+1 de t (13)
The optimal return on investment in human capital (through education) is equal to the return on physical capital since the central planner does not differentiate between physical and human capital accumulation.

The welfare increases only with consumption. Thus optimal k corresponds to the standard Modified Golden Rule.

Conversely, with utilitarian social function ( = 1), from equation (12) we get:

f (k t+1 ) > (f (k t+1 ) -k t+1 f (k t+1 )) dH t+1 de t (14)
The optimal return on investment in human capital (through education) is lower than the one on physical capital because human capital accumulation provides direct welfare. There is a trade-off between consumption and education. Then, we depart from the Modified Golden Rule through the M RS term.

Social Optimum in a two-sector framework

We introduce a two-sector setting à la [START_REF] Galor | A Two-Sector Overlapping-Generations Model: a Global Characterization of the Dynamical System[END_REF]. There are a consumption good, produced by sector 0, and an investment good, produced by sector 1. Denoting respectively by K i and H i , the i-sector physical and human capital. Production functions are

Y 0 t = F 0 (K 0 t , H 0 t ) and Y 1 t = F 1 (K 1 t , H 1 t ), with F i (K i t , H i t
) homogeneous of degree one. Production factors are mobile between sectors and fully employed: K = K 0 + K 1 and H = H 0 + H 1 . Let k i = K i /H i be the physical to human capital ratio in sector i, then H 0 k 0 + H 1 k 1 = k. We define the production functions in intensive forms as f 0 (k 0 ) and

f 1 (k 1 ).
Assumption 4 f 0 (k 0 ) and f 1 (k 1 ) are defined over R + , C r over R ++ for r large enough, increasing

(f i (k i ) > 0 ) and concave (f i (k i ) < 0) . Moreover, lim k i →0 f i (k i ) = +∞ and lim k i →+∞ f i (k i ) = 0.
We can also compute the share of capital in total production for each sector:

s i (k i ) = k i f i (k i ) f i (k i ) ∈ (0, 1) (15)
The social planner objective becomes:

max ct,dt,Kt+1,Ht+1 ∞ t=-1 δ t (u(c t , d t+1 ) + v(H t+1 )) (16)
subject to: ∀t 0

F 0 (K 0 t , H 0 t ) = c t + d t F 1 (K 1 t , H 1 t ) = K 0 t+1 + K 1 t+1 + e t H 0 t+1 + H 1 t+1 = G(e t ) K 0 , H 0 and c -1 given (17)
And the Lagrange function:

L = δ -1 (u(c -1 , d 0 ) + v(H 0 )) + ∞ t=0 δ t u(c t , d t+1 ) + v(H t+1 ) + q 0 t (F 0 (K 0 t , H 0 t ) -c t -d t ) + ∞ t=0 δ t q 1 t F 1 (K 1 t , H 1 t ) -K 0 t+1 -K 1 t+1 -G -1 (H 0 t+1 + H 1 t+1 ) (18)
with q 0 t and q 1 t the Lagrange multipliers. First-order conditions for all t 0 are:

u c (c t , d t+1 ) = q 0 t (19) u d (c t-1 , d t ) = δq 0 t (20) δ t+1 F 0 K (K 0 t+1 , H 0 t+1 )q 0 t+1 = δ t q 1 t (21) δ t+1 F 1 K (K 1 t+1 , H 1 t+1 )q 1 t+1 = δ t q 1 t ( 22
)
δ t v (h t+1 ) + δ t+1 q 0 t+1 F 0 H (K 0 t+1 , H 0 t+1 ) = δ t q 1 t G -1 (H 0 t+1 + H 1 t+1 ) ( 23
)
δ t v (h t+1 ) + δ t+1 q 1 t+1 F 1 H (K 1 t+1 , H 1 t+1 ) = δ t q 1 t G -1 (H 0 t+1 + H 1 t+1 ) (24)
F 0 K (K 0 0 , H 0 0 )q 0 0 = q 1 0 F 1 K (K 1 0 , H 1 0 ) (25)
F 0 H (K 0 0 , H 0 0 )q 0 0 = q 1 0 F 1 H (K 1 0 , H 1 0 ) (26)
K 0 = K 0 0 + K 1 0 ( 27
)
H 0 = H 0 0 + H 1 0 (28)
We obtain the following Proposition:

Proposition 2 In a two-sector setting, under assumptions 2, 3 and 4, optimal capital intensities for each sector depend on the definition of the social welfare function.

i) If = 0 (Harsanyi social function) in each sector i, optimal sectoral physical to human capital ratio k i is determined by both technological parameters of this sector i and the education technology, as M RS e/c = 0.

ii) If = 1 (utilitarian social function) optimal physical to human capital ratios, k 1 and k 2 , are determined by both sectors technological parameters and preference parameters as M RS e/c > 0.

Proof:

From equations ( 19), ( 21), ( 22) and ( 23), we obtain the following expression that determines optimal human capital in the consumption sector:

v (H t+1 ) f 0 (k 0 t+1 )δu c (c t+1 , d t+2 ) = G -1 (H t+1 )-k 0 t+1 1 s 0 (k 0 t+1 ) -1 = G -1 (H t+1 )-k 1 t+1 1 s 1 (k 1 t+1 ) -1 (29)
Thus when = 0, we obtain i) from:

k 0 t+1 1 s 0 (k 0 t+1 ) -1 = k 1 t+1 1 s 1 (k 1 t+1 ) -1 = G -1 (H t+1 ) (30)
From ( 21) and ( 22) we obtain the following expression that determines optimal physical capital accumulation:

δf 0 (k 0 t+1 ) = q 1 t q 0 t+1 δf 1 (k 1 t+1 ) = q 1 t q 1 t+1 (31)
Using equation ( 19) and (31), we can write:

u c (c t+1 , d t+2 ) u c (c t , d t+1 ) = f 0 (k 0 t ) f 0 (k 0 t+1 ) 1 δf 1 (k 1 t+1 ) (32)
Finally, using equations (30) and (32), we obtain ii) from:

M RS e/c = f 0 (k 0 t ) f 1 (k 1 t+1 ) G -1 (H t+1 ) -k 0 t+1 1 s 0 (k 0 t+1 ) -1 = f 0 (k 0 t ) f 1 (k 1 t+1 ) G -1 (H t+1 ) -k 1 t+1 1 s 1 (k 1 t+1 ) -1 (33)
From equation (32), the growth rate along the balanced path is given by 1 + g = δf 1 (k 1 ) 1 1-a . This optimal growth rate only depends on the marginal productivity of capital in the sector producing the investment good. From Proposition 2, optimal capital intensity determinants vary with the specification of the social utility function, and so does the optimal growth rate.

Considering an Harsanyi social function ( = 0), optimal k does not depend on agents preferences (impatience, altruism) nor on sector 0 technology. In this two-sector framework, producing the investment good (good 1) is the only way to optimally allocate wealth between periods and generations. Thus the fact that sector 0 technology does not matter for the two-sector Modified Golden Rule is rather intuitive.

Conversely, with a utilitarian social function, both education (good 1) and consumption good (good 0) are now welfare improving. Thus the central planner has to produce efficiently over the two sectors and optimal k is directly linked to preferences and relative capital intensities.

As regards the growth effect of including paternalistic altruism in the social welfare function, we can deduce the following proposition: Proposition 3 When G(.) is linear, under assumptions 2, 3, and 1 or 4, along the balanced growth path, the optimal growth rate is higher in the case with utilitarian social function than in the case with Harsanyi social function.

Proof:

Along the balanced growth path, equation (12) can be rewritten:

k 1 s(k) -1 = G -1 (H t+1 ) -M RS e/c (34)
The left hand side of this equality is an increasing function of k. As G -1 (H t+1 ) is independent of k, and

M RS e/c ≥ 0, then k is decreasing with M RS e/c . The growth rate is increasing with M RS e/c .

In the two-sector framework, equation (33) becomes:

k 1 1 s 1 (k 1 ) -1 = G -1 (H t+1 ) - f 1 (k 1 ) f 0 (k 0 ) M RS e/c (35)
As f 1 (k 1 ) f 0 (k 0 ) M RS e/c ≥ 0, k 1 is decreasing with M RS e/c and hence the growth rate is increasing with M RS e/c .

Proposition 1

 1 Under assumptions 1, 2 and 3, the balanced growth path depends on the definition of the social welfare function. If = 0 (Harsanyi social function) optimal physical to human capital ratio, k, and optimal growth rate g are directly determined both by technological parameters and the education technology, as M RS e/c = 0. If = 1 (utilitarian social function) optimal physical to human capital ratio, k, and optimal growth rate g are determined by both technological parameters and preference parameters as M RS e/c > 0. Proposition 1 emphasizes the importance of paternalistic altruism to design balanced growth path. With Harsanyi social function ( = 0), equation (12) becomes: