C Chavaudret

S Marmi

Analytic linearization of a generalization of the semi-standard map: radius of convergence and Brjuno sum

Analytic linearization of a generalization of the semi-standard map: radius of convergence and Brjuno sum

Claire Chavaudret, Stefano

Introduction

Consider the following discrete dynamical system:

x n+1 = x n + y n + A(x n ) y n+1 = y n + A(x n ) ( 1
)
where A is a trigonometric polynomial with positive Fourier modes: A(x) = N K=1 a K e iKx , N ∈ N, a K ∈ C. This can be viewed as a generalization of the semi-standard map, which is given by A(x) = e ix . The semi-standard map was introduced as a model which is similar to, but easier to study than, the standard map given by A(x) = sin x, where the coexistence of both positive and negative Fourier modes makes the linearization problem more difficult to solve. The standard map, or Taylor-Chirikov map, is the Poincaré section of a kicked rotator and can be used as an approximating model in several areas of physics. It is an example of a simple low-dimensional system displaying chaos and numerical studies (see for instance [START_REF] Greene | Hamiltonian maps in the complex plane[END_REF]) have investigated the boundary between the KAM regime and chaos for this system. However few analytical results are known on the standard map.

With the change of variables z n = e ixn , λ n = e iyn , system (1) is conjugated to the system

z n+1 = λ n z n N K=1 e ia K z K n λ n+1 = λ n N K=1 e ia K z K n (2)
for which the set {0} × S 1 is invariant. Let F (λ, z) = (λz N K=1 e ia K z K , λ N K=1 e ia K z K ) so the system (2) can be written (z n+1 , λ n+1 ) = F (z n , λ n ). One looks for a linearization of the system, that is to say, a map H(z, λ) = (h(z, λ), h 2 (z, λ)), close to the identity, such that F • H = H • R where R(z, λ) = (λz, λ). If there is such a linearization and if it is analytic, then the invariant curves of the rotation R are smoothly preserved. Then one has a family of invariant closed curves in C 2 , corresponding to the numbers λ with modulus 1 and z in a neighbourhood of 0 where the linearization is analytic. Now if λ is rational, there is no way of finding a linearization which is analytic in z. Thus we will have to assign a value to the parameter λ, with |λ| = 1 and arg(λ) ∈ R \ Q, to construct a linearization which is close to the identity and analytic; its radius of convergence will depend on the arithmetical properties of λ.

Davie [START_REF] Davie | The critical function for the semistandard map[END_REF] and Marmi [START_REF] Marmi | Critical functions for complex analytic maps[END_REF] proved that concerning the semi-standard map, the radius of convergence ρ(α) of the linearization is bounded as follows:

exp(-2B(α) -C) ≤ ρ ≤ exp(-2B(α) + C )
where C > 0, C > 0 do not depend on the complex argument α of λ and where B(α) is the Brjuno sum of α. In particular, if B(α) diverges, then there is no analytic linearization around 0. This can be reformulated as stating that the error function α → B(α) + ln ρ(α) is bounded.

After the numerical evidence in [START_REF] Marmi | On the standard map critical function[END_REF], a similar result about the standard map, in the perturbative case, was proved in [START_REF] Berretti | Bryuno Function and the Standard Map[END_REF] and [START_REF] Berretti | Periodic and quasi-periodic orbits for the Standard Map[END_REF]. However, concerning the semi-standard map and in the present paper, the strong assumption of Fourier modes being only positive makes it possible to remove the perturbative assumption.

The Brjuno sum was first introduced in [START_REF] Brjuno | Analytic forms of differential equations[END_REF] to give a sufficient condition to the convergence of the linearization for analytic vector fields around a fixed point. [START_REF] Yoccoz | Petits diviseurs en dimension 1[END_REF] that the Brjuno condition (i.e the convergence of the Brjuno function, or equivalently of the Brjuno sum) is necessary and sufficient to the analytic linearization of the quadratic polynomial and of germs of diffeomorphisms of (C, 0). This resulted in the study of the error function Φ + ln r, where Φ is the Brjuno function and r is the radius of convergence of the linearization for the quadratic polynomial. It was conjectured in [START_REF] Marmi | The Brjuno functions and their regularity properties[END_REF] that this function is 1/2-Hölder and Buff and Chéritat showed first that it is bounded in [START_REF] Buff | Upper bound for the size of quadratic Siegel disks[END_REF], then that it is continuous in [START_REF] Buff | The Brjuno function continuously estimates the size of quadratic Siegel disks[END_REF]. Cheraghi-Chéritat then proved that a restriction of this error function is 1/2-Hölder.

Yoccoz proved in

Brjuno's lower bound on the convergence radius for linearization of analytic vector fields was improved in [START_REF] Giorgilli | Improved estimates for the convergence radius in the Poincaré-Siegel problem[END_REF]. In [START_REF] Stolovitch | Singular complete integrability[END_REF], Stolovitch replaced the arithmetical condition on the spectrum of the linear part by a condition of algebraic nature. The Brjuno sum was also proved to play a role in other analytic linearization problems, as for instance linearization of vector fields around an invariant torus (see [START_REF] Aurouet | Normalisation de champs de vecteurs holomorphes et équations différentielles implicites[END_REF] and [START_REF] Chavaudret | Normal form of holomorphic vector fields with an invariant torus under Brjuno's A condition[END_REF]), or reducibility of quasiperiodic cocycles ( [START_REF] Chavaudret | Reducibility of quasiperiodic cocycles under a Brjuno-Rüssmann arithmetical condition[END_REF]).

The optimality of the Brjuno condition was also studied in other linearization problems. Carletti-Marmi proved in [START_REF] Carletti | Linearization of analytic and non analytic germs of diffeomorphisms of (C, 0)[END_REF] that the Brjuno condition is also necessary to linearize analytically a germ of diffeomorphism around a fixed point, and generalize it to Gevrey classes. However the continuity of the analogue of the error function for linearization problems more general than the quadratic polynomial remains open up to now.

Our main result is stated in the following two theorems:

Theorem 1 Let ρ be the radius of convergence of the linearization of the system (2). Let d be the greatest common divisor of the indices of the Fourier modes of A. There exists C ≥ 0, which is a non decreasing function of the greatest coefficient of A and which does not depend on α, such that

ρ ≥ exp(- 2 d B(dα) -C) (3)
Theorem 2 Assume that the coefficients of A satisfy the following assumption: there exists θ ∈ R such that for all k = 1, . . . , N with a k = 0, the complex argument of the number i k-1 a k is kθ.

Let ρ be the radius of convergence of the linearization of the system (2). Let d be the greatest common divisor of the indices of the Fourier modes of A. There exists C ≥ 0, which is a non decreasing function of the smallest coefficient of A and does not depend on α such that

ρ ≤ exp(- 2 d B(dα) + C ) (4)
In particular, if B(dα) diverges, then there is no analytic linearization.

Remark: Let κ 0 be the smallest integer such that a κ 0 = 0. The assumption in Theorem 2 says that the argument of every non zero a k is the following function of a κ 0 :

Arg(a k ) = k κ 0 Arg(a κ 0 ) + π 2 (1 - k κ 0 )
This assumption is in the spirit of Cremer's couterexample of non-linearizable germs, except that the arguments are defined from the beginning instead of recursively.

Remark: It is also worth mentioning that dα is a Brjuno number if and only if α is a Brjuno number (see Appendix, section 8.2 for a proof).

The main result is obtained by a direct analysis of the coefficients of the formal linearization (which always exists, and is unique if one requires it to be formally close to the identity); as in [START_REF] Marmi | Critical functions for complex analytic maps[END_REF] and [START_REF] Davie | The critical function for the semistandard map[END_REF], it appears that there is a link between those coefficients and the Brjuno sum of dα. To get an upper bound of the radius of convergence, that is to say, a lower bound on the coefficients of the linearization, one uses a strong assumption on the complex arguments of the coefficients of A, in order to be able to bound the sum from below by just one of its terms. However the lower bound on the radius of convergence does not use this assumption.

Notations

Let x ∈ R, one denotes by ||x|| Z the distance between x and the closest integer: ||x|| Z = min p∈Z |x -p|.

Considering the trigonometric polynomial A(x) defined at the beginning, denote by κ 0 < κ 1 < • • • < κ N all indices of Fourier modes of A: thus for all 0 ≤ i ≤ N , a κ i = 0, and if ∀i = 0, . . . , N , K = κ i , then a K = 0. Also denote by κ min (resp. κ max ) the number κ i minimizing (resp. maximizing) {|a κ i |, i = 0, . . . , N }.

Denote by M ⊂ N the additive semi-group generated by {κ 0 , . . . , κ N }:

M = {p 0 κ 0 + • • • + p N κ N > 0, p 0 ∈ N, . . . , p N ∈ N} (5)

Analysis of the linearization

With a reasoning similar to the one in [START_REF] Berretti | Limit at resonances of linearizations of some complex analytic dynamical systems[END_REF] (which is reproduced and adapted to the present model in the appendix), one proves that if H linearizes the system and is close to the identity, that is, its Taylor series starts with (z, λ), then the first component of the linearization is a function h(z, λ) = ize Φ λ (z) satisfying (mod 2iπ):

N K=1 ia K (iz) K e KΦ λ (z) = Φ λ (λ -1 z) + Φ λ (λz) -2Φ λ (z) (6)
Let λ ∈ C with modulus 1 and α = Arg(λ) 2π . Denoting Φ λ (z) = l≥1 φ l z l and

d l,λ = λ l 2 -λ -l 2 = 2i sin(πlα) (7)
one gets the following:

K∈{κ 0 ,...,κ N } i K+1 a K z K 1 + p≥1 1 p! ( j≥1 φ j z j ) p K = l≥1 φ l z l d 2 l,λ (8)
In what follows, λ being fixed with modulus 1 and argument α such that α 2π ∈ R \ Q, we will denote for all l ∈ N \ {0},

D l := -4 sin 2 (πlα) < 0 Then |D l | -1 ≥ 1 4 for all l ≥ 1. Let l ≥ 1, then φ l = 0 if l < κ 0 and if l ≥ κ 0 , φ l = 1 D l K∈ {κ 0 ,..., κ N }, K≤l i K+1 a K [δ K,l + m=1,...,K m≤l-K C m K l-K p 1 =1 • • • l-K pm=1 1 p 1 ! . . . p m ! j 1 1 ,...,j 1 p 1 , . . . j m 1 ,...,j m pm , j 1 1 +•••+j 1 p 1 +... +j m 1 +•••+j m pm =l-K φ j 1 1 . . . φ j 1 p 1 . . . φ j m 1 . . . φ j m pm ] (9)
Another recurrence relation is worth mentioning: let z) and denote by k≥1 ψ k z k the Taylor expansion of Ψ λ and by l ψ K,l z l the Taylor expansion of Ψ K . Then

Ψ λ (z) = K∈{κ 0 ,...,κ N } Ψ K (z), Ψ K (z) = i K+1 a K z K e KΦ λ (
Ψ λ (z) = Φ λ (λ -1 z) + Φ λ (λz) -2Φ λ (z) (10)
which implies that for all l ≥ κ 0 ,

ψ l = D l φ l (11)
Moreover, derivating Ψ K , one sees that ψ K,l = 0 for all l < K, ψ K,K = i K+1 a K , and for all n ≥ K + 1,

(n -K)ψ K,n = n-1 k=1 kφ k ψ K,n-k (12)
Remark: One sees that φ l = 0 ⇒ l ∈ M. This can be shown by recurrence: for l = κ 0 , the property holds. Assume it holds for all κ 0 ≤ l ≤ l -1, for a fixed l > κ 0 . Assume that φ l = 0. If l ∈ {κ 0 , . . . , κ N }, then l ∈ M; otherwise, there exists

K ∈ {κ 0 , . . . , κ N }, m ≤ l -K, p 1 ≤ l -K, . . . , p m ≤ l -K and non vanishing φ j 1 1 , . . . , φ j m pm such that j 1 1 + • • • + j m pm = l -K. By recurrence assumption, j 1 1 , . . . , j m pm ∈ M therefore l ∈ M.
A similar fact holds for every Ψ K : if ψ K,l = 0, l > K, then l -K ∈ M. Indeed ψ K,l = 0 if l < K and that ψ K,K = i K+1 a K so the recurrence property holds for l ≤ K. Assume the property holds up to a fixed l ≥ K. If ψ K,l+1 = 0 then by [START_REF] Cheraghi | A proof of the Marmi-Moussa-Yoccoz conjecture for rotation numbers of high type[END_REF]

there exist k ∈ {1, . . . , n-1} such that φ k = 0 and ψ K,l+1-k = 0 so k ∈ M and l + 1 -k ∈ M, therefore l + 1 ∈ M.
Example: If A is a monomial of order K, then Φ λ (z) only has coefficients indexed by multiples of K. Then there exists a function

Ξ λ : C → C such that for all z, Φ λ (z) = Ξ λ (z K ).
The following lemma will be used to bound the coefficients of Φ from below by one of the terms of the sum determining them.

Lemma 3 If there exists θ ∈ R such that for all K ∈ {κ 0 , . . . , κ N }, the complex number i K-1 a K has argument Kθ, then for all l ∈ M, φ l has argument lθ and for all K ∈ {κ 0 , . . . , κ N }, if ψ K,l = 0 then ψ K,l has argument lθ + π.

Proof: First one proves the part of the statement concerning φ l :

1. If l = κ 0 , then [START_REF] Carminati | Linearization of germs: regular dependence on the multiplier[END_REF] implies that φ l = 1 Dκ 0 i κ 0 +1 a κ 0 which is, by assumption, the product of a number of modulus κ 0 θ with -1 Dκ 0 , the latter being real and positive.

2. Let l ≥ κ 0 + 1. Assume that for all l ∈ M such that l ≤ l -1, φ l has argument l θ. If l differs from all κ i , then by ( 9), φ l is the sum of terms which are the product of a number i K+1 a K D l , which by assumption has argument Kθ, with a number which by recurrence assumption has argument (l -K)θ. If l is one of the κ i , then one has to add the term 1 D l i l+1 a l , which has argument lθ.

As for the statement concerning ψ K,l , since ψ K,K = i K+1 a K which has argument Kθ+π, the property holds for l ≤ K. Assume this property holds up to a fixed l ≥ K. By equation [START_REF] Cheraghi | A proof of the Marmi-Moussa-Yoccoz conjecture for rotation numbers of high type[END_REF], if ψ K,l+1 = 0 then ψ K,l+1 is a sum of terms with argument kθ + (l + 1 -k)θ + π for k ∈ {1, . . . , n -1}, therefore it has argument (l + 1)θ + π.

Remark: For instance, A(x) satisfies the assumption of Lemma 3 if it only has coefficients a K ∈ iR + (then θ = π 2 ), or such that i k+1 a K ∈ R -(then θ = 0), or if A is a monomial (without restriction on θ).

The semigroup M

We shall also need the following lemma on the set M: Lemma 4 Let d be the greatest common divisor of κ 0 , . . . , κ n . There exists

N M such that for all integer m ≥ N M , if m is a multiple of d, then m ∈ M. If κ 0 = 1 then N M = 1. Proof: Let I 0 = [κ 0 , κ 0 + • • • + κ N [∩N and for p ≥ 1, let I p = [p(κ 0 + • • • + κ N ), (p + 1)(κ 0 + • • • + κ N )[∩N Then (I p ) p≥0 is a partition of [κ 0 , +∞[∩N. Moreover, if jd ∈ I p ∩ M for some p ≥ 0, then jd + κ 0 + • • • + κ N ,
which is also a multiple of d, belongs to I p+1 ∩ M; therefore, in order to prove this lemma, it is sufficient to prove that for all jd ∈ I 0 \ M, there exists p ≥ 1 such that jd + p(κ 0 + • • • + κ N ) ∈ M. After a finite number of steps, one obtains an integer P such that all multiples of d belonging to I P are also elements of M. Translating by κ 0 + • • • + κ N , one will deduce that all multiples of d greater than p(κ 0 + • • • + κ N ) are also elements of M. Now, let jd ∈ I 0 \ M if it exists (otherwise the proof is finished). By Bezout's theorem, there are relative integers b 0 , . . . , b N , at least one of which is positive, such that b 0 κ 0 +• • •+b N κ N = jd. Sorting the coefficients b i by sign, one infers that jdi/ b i ≤0 b i κ i is a linear combination of the κ i with non negative coefficients, at least one of which is non zero, therefore jd-

i/ b i ≤0 b i κ i ∈ M. Therefore jd+max(-b 0 , . . . , -b N )(κ 0 +• • •+κ N ) ∈ M.

Lemma 4 has the following corollary:

Corollary 5 There exists

N M such that for all a, b ∈ M, if a -b ≥ N M , then a -b ∈ M. Proof: Let d be the greatest common divisor of κ 0 , . . . , κ N . Since every element of M is a multiple of d, if a, b ∈ M, then a -b is a multiple of d. If moreover a -b ≥ N M ,
where N M was defined in Lemma 4, then a -b ∈ M.

The Brjuno sum and the Brjuno function

Let d be the greatest common divisor of κ 0 , . . . , κ N . Let us consider the continued fraction expansion of dα.

Notations:

Let (q k ) be the sequence of the denominators of the approximants of dα.

Recall the well-known recurrence relation: for all j ≥ 0, q j+2 = a j+2 q j+1 + q j [START_REF] Davie | The critical function for the semistandard map[END_REF] where (a j ) is the sequence of integers given by the continued fraction expansion.

The following lemmas are given in order to relate the Brjuno sum with the small divisors of our linearization problem.

Lemma 6 For all k ≥ 1, there is

1 2q k+1 ≤ |D dq k | 1 2 ≤ 3 q k+1
Proof: For all l ∈ Z,

|D l | = 4| sin(πlα)| 2 = 4[(||lα|| Z + R(||πlα|| Z ))] 2
(where R is the remainder in the Taylor-Lagrange formula) whence

||lα|| ≤ |D l | 1 2 ≤ 3||lα|| (14)
Now for all k ≥ 1,

||q k dα|| Z = min p∈Z |q k dα -p| = q k min p∈Z |dα - p q k | = q k |dα - p k q k | Since 1 2q k q k+1 ≤ |dα - p k q k | ≤ 1 q k q k+1 (15)
(see for instance [START_REF] Marmi | The Brjuno functions and their regularity properties[END_REF] remark 1.6), there is

1 2q k+1 ≤ |D dq k | 1 2 ≤ 3 q k+1 .
Remark: This cannot be extended to generalized continued fractions since the inequality [START_REF] Giorgilli | Improved convergence estimates for the Schröder-Siegel problem[END_REF] does not hold anymore for generalized continued fractions.

The following lemmas come from [START_REF] Davie | The critical function for the semistandard map[END_REF]. They concern the structure of the set of small divisors and will be used in the lower bound on the radius of convergence. We recall them here and apply the second to our setting.

Lemma 7 ([13], lemma 2.2) Let k ∈ N and n ∈ N, n ≥ 1, such that d is a divisor of n. If |D n | 1 2 < 1 q k , then n d ≥ q k and either q k divides n d or n d ≥ q k+1 4 . Lemma 8 ([13], lemma 2.3) For all k ≥ 0, n ≥ 1, let A k (n) = {dq k ≤ j ≤ n, d|j, 1 6q k+1 ≤ |D j | 1 2 < 1 6q k }. Let E = max(dq k , dq k+1 4 ).
Then there is a function g k : N → R + such that:

• g k (n) ≤ (1 + 2dq k E ) n dq k ; • for all n 1 , n 2 ∈ N, g k (n 1 ) + g k (n 2 ) ≤ g k (n 1 + n 2 ); • if n ∈ A k (n), then g k (n) ≥ g k (n -1) + 1.
Remark: This application of Davie's lemma is possible because the set A k satisfies:

if j 1 = dj 1 < j 2 = dj 2 ∈ A k , then either dq k divides j 2 -j 1 or j 2 -j 1 ≥ dq k+1
4 . Indeed, letting p 1 (resp. p 2 ) be the integer closest to j 1 dα (resp. j 2 dα),

||(j 2 -j 1 )α|| ≤ |p 2 -p 1 -(j 2 -j 1 )dα| ≤ ||j 2 dα|| + ||j 1 dα|| ≤ |D j 2 | 1 2 + |D j 1 | 1 2 < 1 3q k
(the last inequality comes from j 1 , j 2 ∈ A k ). Therefore, by [START_REF] Giorgilli | Improved estimates for the convergence radius in the Poincaré-Siegel problem[END_REF],

|D j 2 -j 1 | 1 2 < 1 q k . Lemma 7 then implies that q k divides j 2 -j 1 or j 2 -j 1 ≥ q k+1 4 .
Lemma 9 There exists C 0 , C 1 > 0 such that

l≥0 1 q l ≤ C 0 and l≥0 1 q l ln q l ≤ C 1
Proof: Let us give a bound on l≥0 1 q l ln q l . Because of the recurrence relation ( 13), the sequence (q l ) increases at least as fast as a Fibonacci sequence the first two terms of which are in {κ 0 , . . . , κ N }: denoting by (f k ) the Fibonacci sequence with f 0 = f 1 = 1, one recursively proves that

q k ≥ f k (16)
Indeed, q 0 ≥ 1 and q 1 ≥ 1. Assume that q k-1 ≥ f k-1 and q k ≥ f k , then

q k+1 = a k+1 q k+1-1 + q k+1-2 ≥ q k + q k-1 ≥ (f k + f k-1 ) = f k+1 .
Therefore, since the function t → ln t t decreases on [e, +∞[,

l≥0 1 q l ln q l ≤ ln 2 + l≥0 ln f l f l ≤ C 1
where C 1 is a numerical constant. From the inequality [START_REF] Greene | Hamiltonian maps in the complex plane[END_REF], one also infers that

l≥0 1 q l ≤ l≥0 1 f k ≤ C 0
where C 0 is a numerical constant.

Subsequence of fast increasing denominators

Given the number N M which was defined in Lemma 4, let (n k ) the subsequence containing all indices such that

q n 0 ≥ max(N M + 2, 1 + κ max (2 + 1 d )), q n k +1 ≥ q 2 n k + ζ(κ max , d)q n k + η(κ max , d) where ζ, η are given by ζ(κ max , d) = κ max d + 3κ max + 2, η(κ max , d) = ( κ max d + 2κ max )(κ max + 1)
Lemma 10 For all k ≥ 0, it holds that

q n k ≥ max(N M + 2, κ max ) 2 k . Proof: By definition, q n 0 ≥ max(N M + 2, κ max ). Assume that q n k ≥ max(N M + 2, κ max ) 2 k , then q n k+1 ≥ q n k +1 ≥ q 2 n k ≥ max(N M + 2, κ max ) 2 k+1 .
Corollary 11 For all k ≥ 0, k≥0

1 qn k ≤ 1 max(N M +2,κmax)-1 . Proof: Indeed k≥0 1 q n k ≤ k≥0 1 max(N M + 2, κ max ) 2 k ≤ k≥1 1 max(N M + 2, κ max ) k ≤ 1 max(N M + 2, κ max ) -1 . (17)
Lemma 12 There exists C 2 > 0 such that

B(dα) -C 2 ≤ l≥0 | ln |D qn l | | 2q n l ≤ B(dα) + C 2 Proof: Lemma 6 implies that l≥0 ln q n l +1 -ln 3 q n l ≤ l≥0 | ln |D qn l | | 2q n l ≤ l≥0 ln q n l +1 q n l + l≥0 ln 2 q n l ≤ B(dα) + C 0 ln 2
(where we have also used Lemma 9). Now l≥0 ln q n l +1 q n l = B(dα) -l≥0,q 2 l +ζq l +η>q l+1 ln q l+1 q l (where ζ, η were defined at the beginning of the section). Thus

l≥0 ln q n l +1 q n l ≥ B(dα) - l≥0 ln(q 2 l + ζq l + η) q l ≥ B(dα) - l≥0 ln q l + ln(q l + ζ + η) q l ≥ B(dα) -2 l≥0 ln q l q l - l≥0 ln(ζ + η + 1) q l (18)
By Lemma 9,

l≥0 ln(ζ + η + 1) q l ≤ C 0 ln(ζ + η + 1)
Therefore one can define C 2 = 2C 1 + C 0 (ln 3 + ln(ζ + η + 1)) and C 2 = C 0 ln 2.

Recursively defined lower bound

In this section we introduce a function F which will be used in giving a lower bound on the coefficients of the linearization. More precisely, we shall prove that

lim sup k→+∞ F (q n k ) q n k -C ≤ lim sup k→+∞ ln |φ qn k | q n k
where C is a constant not depending on α.

Then we shall bound lim sup k→+∞ F (qn k ) qn k by means of the Brjuno sum in the Lemma 13 below.

The function F is recursively defined on the set {q n k , k ≥ 0} as follows:

F (q n 0 ) = 0 ∀k ≥ 1, F (q n k ) = p k 0 | ln |D qn 0 || + p k 0 F (q n 0 ) + • • • + p k k-1 | ln |D qn k-1 || + p k k-1 F (q n k-1 ) (19)
where the integers p k i are given by successive euclidean divisions on q n k -κ max :

dq n k -κ max = p k k-1 dq n k-1 + r k-1 , r k-1 < dq n k-1 , r k-1 = p k k-2 dq n k-2 + r k-2 , r k-2 < dq n k-2 , . . . , r 1 = p k 0 dq n 0 + r 0 , r 0 < dq n 0 . ( 20
)
Remark 1 Let i ≥ 0. For all k ≥ i + 1,

p k i ≤ q n i+1 q n i (21)
Indeed, the integer p k i are recursively given by

F (q n k ) q n k + κ max /d ≥ 0≤l≤k | ln |D dqn l || q n l - 0≤l≤k 4 ln(2q n l+1 ) q n l+1
which implies

F (q n k ) q n k + κ max /d ≥ 0≤l≤k | ln |D dqn l || q n l -4 l≥0 ln q l q l - l≥0 4 ln 2 q l
Lemma 9 then implies

F (q n k ) q n k + κ max /d ≥ 0≤l≤k | ln |D dqn l || q n l -4C 1 -4 ln 2C 0
Finally, by Lemma 12,

lim sup k→+∞ F (q n k ) q n k = lim sup k→+∞ F (q n k ) q n k + κ max /d ≥ 2B(dα) -2C 2 -4C 1 -4 ln 2C 0
Thus one can define C 4 = 2C 2 + 4C 1 + 4 ln 2C 0 .

An upper bound on the radius of convergence

In this section, one shall assume the following:

Assumption 1 Assume that there exists θ ∈ R such that for all k = 1, . . . , N with a k = 0, the complex number i k-1 a k has argument kθ.

In this case, one can prove a lower bound on the coefficients φ l of the linearization in order to bound the radius of convergence from above.

One needs the following simple lower bound on all coefficients, including those not corresponding to a small divisor.

Lemma 14 For all r ∈ N, if |φ r | = 0, then |φ r | ≥ min(1,

|aκ min | 4 ) r κ 0 . Proof: For all K such that a K = 0, it holds that |φ K | ≥ K|a K | 4 ≥ |aκ min | 4 . If |aκ min | 4 < 1, then |φ K | ≥ ( |aκ min | 4 ) K κ 0 . If |aκ min | 4 ≥ 1 then |φ K | ≥ 1.
Let r ∈ M, r ≥ 2. Assume that the property holds for all r ≤ r -1. Then, either there exists i ∈ {0, . . . , N } such that r = κ i , and in this case, |φ r | ≥

aκ i 4 ≥ min(1, |aκ min | 4 ) ≥ min(1, |aκ min | 4
) κ i κ 0 , or there exists i ∈ {0, . . . , N } such that

|φ r | ≥ |a κ i | 4 |φ r-κ i | ≥ |a κ min | 4 |φ r-κ i | If |aκ min | 4
< 1, then the recurrence assumption implies that |φ r | ≥ (

|aκ min | 4 ) κ i κ 0 min(1, |aκ min | 4 ) r-κ i κ 0 ≥ min(1, |aκ min | 4 ) r κ 0 . If |aκ min | 4 ≥ 1 then |φ r | ≥ 1.
Lemma 15 Let κ be such that a κ = 0. For all j > κ 0 and p ∈ N * , there is

|φ pj | ≥ 1 p|D pj | . . . |D 2j | |φ j | p |D j | p-1
Proof: Notice that

|φ pj | = 1 |D pj | |ψ pj | = 1 |D pj | K∈{κ 0 ,...,κ N } |ψ K,pj | (26)
(one uses the fact that the ψ K,pj have the same argument for every K). Thus by [START_REF] Cheraghi | A proof of the Marmi-Moussa-Yoccoz conjecture for rotation numbers of high type[END_REF],

|φ pj | ≥ 1 |D pj | K (p -1)j pj -K |φ (p-1)j ||ψ K,j | ≥ (p -1)j |D pj |pj |φ (p-1)j ||ψ j | (27)
Iterating this, one obtains

|φ pj | ≥ 1 p|D pj | . . . |D 2j | |φ j ||ψ j | p-1 = 1 p|D pj | . . . |D 2j | |φ j | p |D j | p-1 . ( 28
)
The following lemma states a better lower bound for the coefficients of the linearization corresponding to a small divisor.

Lemma 16 For all k ≥ 0, there is

|φ dqn k+1 | ≥ 1 |D dqn k+1 | |a κmax | (k + 2)! ( 1 4 ) d(p k+1 0 +•••+p k+1 k ) |D dqn 0 | dp k+1 0 -1 dp k+1 0 . . . |D dqn k | dp k+1 k -1 dp k+1 k • |φ dqn 0 | p k+1 0 . . . |φ dqn k | p k+1 k min(1, |a κ min | 4 ) R k κ 0 ( 29
)
where the integers p k+1 0 , . . . , p k+1 k , R k were defined in equation [START_REF] Marmi | On the standard map critical function[END_REF]. Proof: Taking in the recurrence relation ( 9) the term with l = q n k+1 , K = κ max , m = 1,

p 1 = k + 2, j 1 1 = dp k+1 0 (q n 0 + κ max ), . . . , j 1 k+1 = dp k+1 k (q n k + κ max ) and j 1 k+2 = R k , in order to have j 1 1 + • • • + j 1 k+2 = dq n k+1 -κ max , one obtains |φ dqn k+1 | ≥ 1 |D dqn k+1 | κ max |a κmax | (k + 2)! |φ dp k+1 0 (qn 0 +κmax) | . . . |φ dp k+1 k (qn k +κmax) ||φ R k |
Now apply Lemma 15 to every factor in the right hand side, except the last one, and apply Lemma 14 to the last factor. One obtains

|φ dqn k+1 | ≥ 1 |D dqn k+1 | |a κmax | (k + 2)! ( 1 4 ) d(p k+1 0 +•••+p k+1 k ) |D dqn 0 | dp k+1 0 -1 dp k+1 0 . . . |D dqn k | dp k+1 k -1 dp k+1 k • |φ dqn 0 | p k+1 0 . . . |φ dqn k | p k+1 k min(1, |a κ min | 4 ) R k κ 0 . (30)
The following proposition links the radius of convergence of the linearization to the function F .

Proposition 1 There exists C ≥ 0 such that for all k,

1 dq n k ln |D dqn k φ dqn k | ≥ F (q n k ) dq n k -C
Moreover, one can have C = -2 ln(

|aκ min | 4 ) + 2+N M 1+N M (1 -ln |a κ min |) if |aκ min | 4 < 1 and C = 1 + 1 1+N M otherwise.
Remark: Observe that the coefficients of the linearization increase with the nonlinear part of the system.

Proof: Denote C = -2 ln( |aκ min | 8
) if

|aκ min | 8 < 1 and C = 0 otherwise. Let S 0 = ln 4 q n 0 ; ∀k ≥ 1, S k = 1≤j≤k j ln j + ln(d j q n j ) q n j + 0≤j≤k ln 4 q n j (1 + p j 0 + • • • + p j j-1 + R j κ 0 )
Thus, for all k ≥ 0, S k ≥ 0.

Let k ≥ 0. Let us formulate the following recurrence property:

1 dq n k ln |D dqn k φ dqn k | ≥ F (q n k ) dq n k -S k -C (31)
First note that this property holds for k = 0. Indeed, since by Corollary 5, q n 0 ≥ N M , which implies that dq n 0 ∈ M, then there exists b 0 , b 1 , . . . , b N ≥ 0 such that dq n 0 = N i=0 b i κ i . Let I be such that b I > 0. Then

|φ dqn 0 | ≥ 1 |D dqn 0 | κ I |a κ I ||φ κ I | b I -1 J =I |φ κ J | b J
Now on the other side, for all 0 ≤ i ≤ N ,

|φ κ i | ≥ |a κ i | |D κ i | ≥ |a κ min | 4 ≥ |a κ min | 8 hence, if |aκ min | 8 < 1,
|φ dqn 0 | ≥ 1 |D dqn 0 | |a κ I |( |a κ min | 8 ) dqn 0 -1
and if

|aκ min | 8 ≥ 1, |φ dqn 0 | ≥ 1 |D dqn 0 | |a κ I | Thus if |aκ min | 8 < 1, 1 dq n 0 ln |φ dqn 0 | ≥ | ln |D dqn 0 || dq n 0 + dq n 0 -1 dq n 0 ln( |a κ min | 8 ) + ln(|a κ I |) dq n 0 ≥ | ln |D dqn 0 || dq n 0 -S 0 - C and if |aκ min | 8 ≥ 1, 1 dq n 0 ln |φ dqn 0 | ≥ | ln |D dqn 0 || dq n 0 + ln(|a κ I |) dq n 0 ≥ | ln |D dqn 0 ||
dq n 0 therefore the property (31) holds for k = 0. Now assume that the recurrence property holds for all 0 ≤ k ≤ k, for a fixed k ≥ 0. Lemma 16 implies

ln |D dqn k+1 φ dqn k+1 | ≥ ln |a κmax | (k + 2)! -d((p k+1 0 + • • • + p k+1 k ) + k + 1) ln 4 -ln(dp k+1 0 . . . dp k+1 k ) + p k+1 0 ln |D dqn 0 φ dqn 0 | + • • • + p k+1 k ln |D dqn k φ dqn k | + R k κ 0 ln min(1, |a κ min | 4 ) (32)
By recurrence assumption, one infers

1 dq n k+1 ln |D dqn k+1 φ dqn k+1 | ≥ - (k + 2) ln(k + 2) dq n k+1 + ln |a κmax | dq n k+1 - d(p k+1 0 + • • • + p k+1 k ) + k + 1 dq n k+1 ln 4 + p k+1 0 dq n 0 dq n k+1 ( F (q n 0 ) dq n 0 -S 0 -C) + • • • + p k+1 k dq n k dq n k+1 ( F (q n k ) dq n k -S k -C) + R k dκ 0 q n k+1 ln min(1, |a κ min | 4 ) - ln(dp k+1 0 . . . dp k+1 k ) dq n k+1 (33)
thus by definition of F ,

1 dq n k+1 ln |D dqn k+1 φ dqn k+1 | ≥ F (q n k+1 ) dq n k+1 - (k + 2) ln(k + 2) dq n k+1 + ln |a κmax | dq n k+1 - d(p k+1 0 + • • • + p k+1 k ) + k + 1 q n k+1 ln 4 + R k dκ 0 q n k+1 ln min(1, |a κ min | 4 ) -C - p k+1 0 dq n 0 dq n k+1 S 0 -• • • - p k+1 k dq n k dq n k+1 S k - ln(dp k+1 0 . . . dp k+1 k ) dq n k+1 (34)
Now for all i = 0, . . Therefore, the property (31) holds for all k ≥ 0.

Finally note that the partial sums S k converge, since from one side, by choice of the subsequence q n j and corollary 11,

1≤j≤k j ln j + ln(d j q n j ) q n j ≤ 1≤j≤k 2 √ q n j + C 1 ≤ 1≤j≤k 2 q n j-1 + C 1 ≤ 2 1 + N M + C 1
and from the other side, by Remark 1,

ln 4 q n 0 + k j=1 ln 4 q n j (1 + R j κ 0 + p j 0 + • • • + p j j-1 ) ≤ ln 4 q n 0 + k j=1 ln 4 q n j (1 + q n 0 + q n 1 + κ max q n 0 + κ max + • • • + q n j-1 + κ max q n j-2 + κ max + q n j q n j-1 ) ≤ ln 4( 1 q n 0 + k j=1 (j + 1)q n j-1 q n j + 1 q n j-1
)

≤ ln 4( 1 q n 0 + k j=1 j + 2 q n j-1 )
≤ ln 4( 8 Proof: Let ρ be the radius of convergence of Φ, then ρ -1 = lim sup j→+∞ |φ j |

q n 0 + k-1 j=2 j + 3 q n j ) ≤ ln 4( 8 q n 0 + k-1 j=2 1 q n j-1 ) ≤ 9 ln 4 1 + N M (35)

A lower bound on the radius of convergence

In this section, the second part of the main result is proved. The assumption 1 on the coefficients a K is relaxed.

Theorem 18

The radius of convergence is at least exp(-C -l≥0

2 ln q l+1 dq l ), where C ≥ 0 is defined by and let R its radius of convergence. Expanding w in its Taylor series, w(z) = n≥0 σ n z n , one obtains the following relation between the coefficients σ n (it is the same relation as between the coefficients φ j , only replacing a k by i k+1 and without small divisors):

C = ln(|a κmax |) + r + C 0 if |a κmax | >
σ l = K∈ κ 0 ,..., κ N }, K≤l [δ K,l + m=1,...,K m≤l-K C m K l-K p 1 =1 • • • l-K pm=1 1 p 1 ! . . . p m ! j 1 1 ,...,j 1 p 1 , . . . j m 1 ,...,j m pm , j 1 1 +•••+j 1 p 1 +... +j m 1 +•••+j m pm =l-K σ j 1 1 . . . σ j 1 p 1 . . . σ j m 1 . . . σ j m pm ( 36
) One can recursively show that the σ n are non negative real numbers. Moreover the function w is analytic, therefore lim sup n→+∞ -1 n ln σ n is equal to -ln R. The function g giving the upper bound is defined as follows: for all k ≥ 0, let g k be the function defined by Davie's lemma 8. Then, for all integer κ 0 ≤ j < q n 0 , let

g(j) = j ln(|a κmax |) + 36j if |a κmax | > 1, and • if j ≥ κ 0 and |D j | 1 2 ≥ 1 6 , then
|D -1 j ||a κmax |e g(j-κ 0 ) σ j ≤ 36|a κmax |e g(j-κ 0 ) σ j ≤ e g(j) σ j

• otherwise there exists k ≥ 0 such that 1 6q k+1 ≤ |D j | 1 2 < 1 6q k , and in this case, |D -1 2 j | ≤ 6q k+1 . Moreover, j ∈ M implies that d divides j; then j ∈ A k (j) therefore, by construction of g k ,

g k (j) = g k (j -κ 0 ) + 1 therefore ln |D -1 j | + ln |a κmax | + g(j -κ 0 ) ≤ 2 ln q k+1 + ln 36 + ln |a κmax | + g(j -κ 0 ) ≤ g(j) thus |φ j | ≤ |D -1 j ||a κmax |e g(j-κ 0 ) σ j ≤ e g(j) σ j
The recurrence is finished. Thus, for all j ≥ κ 0 , 8 Appendix

1 j ln |φ j | ≤ 1 j g(j) + 1 j ln σ j (40)

8.1

Here we give a proof of equation ( 6) in two lemmas.

Lemma 19 Let H(z, λ) = (h(z, λ), h 2 (z, λ)) be the linearization. Then h 2 (z, λ) = h(z,λ) h(λ -1 z,λ) .

Proof: Let F (z, λ) = (λz N k=1 e ia k z k , λ N k=1 e ia k z k ) be the function generating the system, and R(z, λ) = (λz, λ). Note that if f (z, λ) = N k=1 e ia k z k then F (z, λ) = (zf (z, λ), f (z, λ)). We shall expand the identity H = F • H • R -1 to infer the desired identity. On one side,

F • H • R -1 (z, λ) = F • H(λ -1 z, λ) = F (h(λ -1 z, λ), h 2 (λ -1 z, λ)) = (h(λ -1 z, λ)f (h(λ -1 z, λ), h 2 (λ -1 z, λ)), f (h(λ -1 z, λ), h 2 (λ -1 z, λ))) (42)
By matching the components of

F • H • R -1 with those of H, one infers that h(z, λ) = h(λ -1 z, λ)f (h(λ -1 z, λ), h 2 (λ -1 z, λ)) and h 2 (z, λ) = f (h(λ -1 z, λ), h 2 (λ -1 z, λ)))
Those two identities imply that for all (z, λ), h 2 (z, λ) = h(z, λ) h(λ -1 z, λ) .

Lemma 20 Let h(z, λ) = ize Φ λ (z) , then equation (6) holds.

Proof: Let F (z, λ) = (λz N k=1 e ia k z k , λ N k=1 e ia k z k ) be the function generating the system, and let R(z, λ) = (λz, λ). By definition, F • H = H • R. Now using Lemma 19, F • H(z, λ) = ( h 2 (z, λ) h(λ -1 z, λ) N k=1 e ia k h(z,λ) k , h(z, λ) h(λ -1 z, λ) N k=1 e ia k h(z,λ) k ) and H • R(z, λ) = (h(λz, λ), h(λz,λ) h(z,λ) ). By matching the components and taking the logarithm modulo 2iπ, one obtains equation (6).

A number is Brjuno if and only if its multiple by an integer is Brjuno

In this section, we will give a proof of the following fact:

Proposition 2 Let α ∈ R \ Q. Let N ∈ N, N ≥ 2.
Then α is a Brjuno number if and only if N α is a Brjuno number. In order to prove this, it will be useful to recall a few classical facts about continued fractions. Let α ∈ R \ Q. Let ( pn qn ) n≥0 be the sequence of best approximants. They have the property that for all rational p q , |α -p q | ≤ 1 2q 2 ⇒ ∃n ≥ 0, p q = p n q n Let (a n ) be the sequence of integers given by the algorithm a 0 = [α], x 0 = {α}, x n+1 = { 1 xn }, a n+1 = [ 1 xn ]. Then q n+1 = a n+1 q n + q n-1 which implies that a n+1 ≤ q n+1 q n (43) Also, there is the following important identity:

|α - p k q k | = 1 q k 1 q k+1 + a k+1 q k (44)
As mentioned in [START_REF] Carminati | Linearization of germs: regular dependence on the multiplier[END_REF], there exists a constant C ≥ 0 such that for all irrational α, ) be the sequence of best approximants of N α.

| n≥0 ln q n+1 q n - ln a n+1 q n | ≤ C (
Lemma 21 If a n+1 > 2N , then there exists k ≥ 0 such that p n N q n = p k q k .

Proof: By (44), one has

|α - p n N q n | ≤ 1 N 1 q 2 n a n+1 < 1 2(N q n ) 2 thus p n N q n
is one if the best approximants of α: there exists k ≥ 0 such that p n N q n = p k q k . Now we can prove the Proposition 2. For all n ≥ 0 such that a n+1 > 2N , let k n be an integer such that p n N q n = p kn q kn . Note that q kn ≤ N q n . By (43), ln a n+1 q n ≤ ln q n+1 q n -ln q n q n (46)

The identity (44) adapted to N α implies that q n q n+1 ≤ 1 |N α-

p n q n | therefore ln a n+1 q n ≤ | ln |N α -p n q n || q n -2 ln q n q n ( 47
)
If a n+1 > 2N , then ln a n+1 q n ≤ N | ln |N α -N p kn q kn || q kn -2N ln q n q n (48)

Again by (44), |α -p kn q kn | ≥ 1 2q kn q kn+1 therefore ln a n+1 q n ≤ N ln q kn+1 q kn + N ln q kn q kn + N (ln N + ln 2) q kn -2N ln q n q n (49)

By taking the sum over n ≥ 0,

(

  this last inequality used Corollary 11). Thus, let C = C + 1 1+N M (2 + 9 ln 4) + C 1 , then C satisfies the statement of this proposition. Theorem 17 The radius of convergence of Φ is bounded from above by exp(-lim sup k→+∞ F (qn k ) dqn k + C) where C ≥ 0 was defined in Proposition 1. It is also bounded from above by exp(-2 d B(dα)+ C M ) where C M = C 4 d + C, with C 4 defined in Lemma 13.

  1 and C = r + C 0 otherwise, with r > 0 only depending on the Fourier modes of the trigonometric polynomial A and C 0 defined in Lemma 9. Proof: Let w be an analytic solution of the functional equation w(z) = k∈{κ 0 ,...,κ N } (ze w(z) ) k

Figure 1 :

 1 Figure 1: The Brjuno function of N α for α ∈ (0, 1). Red: N = 1; green: N = 2; blue: N = 3; purple: N = 4.

  45)Therefore α is Brjuno if and only if n≥0 ln a n+1 qn < +∞.Now let N ≥ 2; let a 0 = [N α], x 0 = {N α}, a n+1 = [ 1 x n ], x n+1 = { 1 x n }. Let ( p n q n

  . , k, p i ≤

	and
	1 dq n k+1	ln |D dqn k+1 φ dqn k+1 | ≥	F (q n k+1 ) dq n k+1	-C -S k+1
		qn i+1 qn i	therefore
		ln(dp k+1 0 dq n k+1 . . . dp k+1 k )	≤	ln(d k+1 qn k+1 qn 0 dq n k+1	)

and for i = 0, . . . , k -2, p k i = E( r i+1 dqn i

), which satisfies [START_REF] Yoccoz | Petits diviseurs en dimension 1[END_REF] by definition of r i+1 .

Lemma 13 There exist C 4 > 0 such that the function F satisfies for all k ≥ 0,

Proof: This can be recursively shown. Assume that for a fixed k ≥ 1 and for all k ≤ k -1, one has

(which holds for k = 1). Then

Now by definition of p k-1 ,

and this quantity is greater than 1 since, by assumption on the subsequence n k ,

+ η where ζ, η were defined at the beginning of Section 4.1. Therefore

(where the last inequality comes from the fact that q n 0 ≥ 1 + κ max (2 + 1 d )). Therefore the property holds for all k ≥ 0. Lemma 6 implies that g(j) = 36j otherwise. Now let j ≥ q n 0 and assume that k is the greatest index such that q k ≤ j; let

The function g is increasing. Moreover for all j 1 , j 2 ≥ κ 0 , as a consequence of Davie's lemma,

Now let us prove that |φ j | ≤ σ j e g(j) for all j ≥ κ 0 . First, if

and |φ κ 0 | ≤ 1 = e g(κ 0 ) otherwise. Assume that this holds for all j ≤ j -1 and consider |φ j |. The relation ( 9) implies

. . .

hence, by recurrence assumption,

1 )+...g(j m pm ) ]

(39) therefore

j ||a κmax |e g(j-κ 0 ) σ j We shall distinguish two cases:

N ln q kn+1 q kn + N ln q kn q kn + N (ln N + ln 2)

If α is a Brjuno number, the right hand side is finite, which implies that N α is a Brjuno number.

It remains to prove that if α is a Brjuno number, then 1 N α is Brjuno. Now the functional identity from [START_REF] Marmi | The Brjuno functions and their regularity properties[END_REF], which holds for α ∈ (0, 1):

implies that any irrational α is Brjuno if and only if 1 α is Brjuno. If α is Brjuno, then 1 α is Brjuno, therefore, by the above, N α is Brjuno, and thus its inverse α N is a Brjuno number.


Richard Arena

Muriel Dal

Pont Legrand

Roger Guesnerie

Expectations in Past and Modern Economic Theory 1

This special issue of the Revue d'Economie Politique is dedicated to the concept and the use of expectations in past and modern economic theory. This topic is not really new. Every economist or person interested by economic activity knows the crucial direct or indirect importance of the forecasts or views that decision makers hold about future prices, sales, incomes, profitability or other key microeconomic or macroeconomic magnitudes. As many commentators (see, for instance, Leijonhufvud 1986:4) mentioned, the central difference between economics and natural sciences precisely lies in the forward-looking predictions and decisions made by economic agents since certainty and determinism do not often prevail in the working of the economic system. Therefore, the analysis of the contents and the impact of expectations on the micro-and the macro-economic variables is complex and often requires sophisticated reflections and tools as well as the use of the contributions of other natural (mathematics, physics) or social (psychology, sociology, philosophy) sciences. This is the reason why expectations provide a basic building block of economic theory. Notably they lie at the core of economic dynamics as they usually determine, not only the behavior of the agents, but also the main properties of the economy related to the evolution of such behavior in time. A central aspect of economic theories is indeed that expectations influence the time path of the economy, and conversely the time path influences expectations. This central theoretical importance of expectations played an increasing role throughout the XIX th century. Expectations were already present during the first half of this century especially in the discussions in France and in Great Britain of empirical issues related to money, credit and public policy however, these developments were partial and related to specific observed mechanisms and phenomena. In France things changed in the end of the XIX th century with the works of Clément Juglar and Léon Walras and for two different reasons. Juglar was the first author to focus on the transition from crises theory to business cycle theories and this transition was supposed to imply "the 1 This paper was written to introduce the various papers that make up a special issue published with the

succession of the same facts, at any point of time but in specific circumstances", (Juglar, 1862: xiii), in different periods and regimes; this changing periodicity applied specifically to expectations which Juglar tried to characterize, even without funding his remarks on a real theory. Walras built a theory of General economic equilibrium referring to static expectations and to-day there is still a debate to understand if this equilibrium was a static one or a rest point of a genuinely dynamic process. After Juglar and Walras, the issue of the role of expectations in the implementation of a general equilibrium naturally emerged and developed. For instance, Alfred Marshall was credited with the notion of 'static expectations' of prices but his distinction between very short, short and long periods led to a more substantial reflection various types of expectations and their relations with different equilibria.

Little by little, several issues were introduced which stressed the theoretical importance of expectations (see [START_REF] Shackle | Expectations in economics[END_REF]: let us refer for instance to (i) the nature of the kinds of evidence or data which play a part in the decision maker's task; (ii) the set of rules used by any decision maker in this task; (iii) the role of elements of judgment, of invention, inspiration, or creativity which enter into this task; (iv) the type of valuation and ordering when the outcomes imagined for any action scheme are plural and mutually exclusive; (v) the type of expectations learning and revising processes.

Obviously, this special issue cannot cope with all the contributions which were developed in the past but had to make some choices. We will first refer here to three views of the relations between expectations and economics which differ from both the Walrasian and the Marshallian ones and were often developed in various contributions.

The first one concerns the Stockholm School of economics ("Taking expectations seriously: a leitmotiv in Stockholm School economics") and was written by Hans-Michael Trautwein. It refers to the contributions of the different authors belonging to this school and shows that they have a common core in their direct or indirect Wicksellian heritage which totally differs from the Walrasian one because of his insistence on economic dynamics especially. The author emphasizes the role played by the starting point of these Swedish contributions, namely Myrdal's 1927 dissertation. This contribution showed how the inclusion of expectations under imperfect foresight helped to transform the static framework of general equilibrium theory into a dynamic one, dealing with three main problems: the transformation of an initial structure of prices into another; the nature of the effects that changes have back in time on prior expectations; the integration of risk into the conceptualization of expectations. It also refers to the other leading members of the Stockholm School (before its founders): Lindahl and Lundberg who criticized Myrdal's views and developed a preliminary approach to rational as well as endogenous expectations; and who made intertemporal and temporary equilibria distinct;

Hammarskjöld who related expectations to entrepreneurial strategies; Ohlin who stressed the importance of expectations in regard to profitability of investments and drew attention to the role of these expectations in relation to to the uses of the concepts and the method of ex ante and ex post in the Stockholm School type of dynamics.; Johansson who related expectations to sectoral disproportionalities; and Svennilson who introduced the concept of intertemporal planning under uncertainty.

The second paper of this special issue is related to Keynes's theory ("Why Keynes' theory of individual decisions has not to be macro-or micro-economically founded?") and was written by Richard Arena and Eric Nasica. It refers to Keynes's economic theory including all his different own contributions and not only the General theory. Therefore, the purpose of this second paper is also to investigate expectations in Keynes' economics -which provides another tradition of the economic analysis during the first half of the XX th century and also differs from Walras pure economics and its static equilibrium. It aims to analyze the impact of taking into account a truly uncertain and complex economic environment on Keynes's methodology. Therefore, it clearly excludes an interpretation of Keynes's writings that would be based on a macroeconomic equilibrium and a self-contained conception of economic rationality. Two main parts are made distinct. In the first part, the paper aims to show that the complex decision-making environment induced by Keynes' vision of the "real world" is central to his approach because it strongly conditions his analysis of individual behaviors; to do this, the paper shows how this environment is used by Keynes to build an original and coherent methodology aiming at analyzing rationality and expectations under uncertainty in a way radically different from 'classical' approaches and above all more adapted to the characteristics of an economy subject to change. It concludes that, even when an ordinal or cardinal measure of probability is impossible, Keynes provides a consistent set of tools for the analysis of the formation of economic expectations and decisions. The second part of the paper tries to show that the complex decision and expectations-forming environment described before is also related to the characterization of the individual actor himself and his economic and social interactions. This form of economic analysis implies a specific methodological conception open to the help to the crucial use of philosophy and psychology.

The third paper of this special issue concerns the room attributed by John Hicks in the 1930s to the development and role of expectations in his representation of the functioning of a capitalist economy ("Great expectations. Hicks on expectations from Theory of wages (1932) to Value and Capital (1939)"). It was written by Jean-Sébastien Lenfant. It covers Hicks' contributions to economic theory in the 1930s, especially the second of these two books. As we know, this book is strongly related to the Walrasian tradition. However, Value and Capital influenced the history of macroeconomics and not only the history of general economic equilibrium. Moreover, contrary to Walras 's contribution, Hicks used the general equilibrium approach to model expectations and assess the dynamics of an economic system through time, hence contributing to business cycle theory. It also tried to generalize some of Keynes's contributions concerning the relation between economic dynamics and unemployment. It was often based on a conception of economic dynamics conceived as a succession of short-term temporary equilibria linked by the confrontation of past decisions and new plans from period to period thanks to the use of expectations. Lenfant's paper also tries to connect Hicks's treatment of expectations in 1939 with what he wrote on this theme in the 1930s. Indeed, Hicks developed a specific conception of the microfoundations of macroeconomics, in which the heterogeneity of expectations and its relations with risk play a crucial role. This is why the paper presents first Hicks's views on expectations in various works prior to Value and Capital ( The theory of wages; "Equilibrium and the trade cycle"; "A suggestion for simplifying the theory of money" and "Wages and interest: the dynamic problem"). It shows that expectations are crucial to Hicks's understanding of the functioning of a capitalist economy. Hicks's framework concentrates on the notions of temporary equilibrium and elasticity of expectations. It finally provides some insights on a more general development of a theory of expectations based on Value and Capital.

In this special issue, after the succession of these three first papers, history of economic thought is still present, but it reveals a different approach related to the passage of time.

If indeed we consider the second half of the XX th century, in economics, till the 1970s, a unique research program tended to prevail. Surprisingly, this program was related to the Walrasian tradition we considered earlier, even if things are more complex than it is too often argued for various reasons.

On one side, the program of the General Economic Equilibrium as it is presented by authors as [START_REF] Arrow | The Existence of an Equilibrium for a Competitive Economy[END_REF] strongly differs from the program of Léon Walras. It indeed totally excludes the parts of the latter dedicated to social as well as applied economics and presents pure economics in a genuine way. On the other side, Arrow and Debreu took expectations and dynamics into account including intertemporal equilibrium unlike Walras's initial static equilibrium conception which did not consider this type of intertemporal issue.

Furthermore, besides this program, in 1961, John Muth made the assumption of 'rational expectations' known and contributed to put it at the center of the post-war and the XXI th century reflections on expectations. The three following papers of this special issue can be considered as a development of this perspective.

The fourth paper of this special issue concerns Muth's contribution to the elaboration of the concept of rational expectations ("When Muth entrepreneurs meet Schrödinger's cat") and was written by Rodolphe Dos Santos Ferreira. This paper connects from its very beginning this concept and the general equilibrium approach considering the former as an 'ingredient' of the latter. Moreover, it connects expectations formation with beliefs and predictions of other's conduct when agents act strategically. With this in mind, the author reminds that Muth built his conception of rational expectations in a simple model of a competitive market for a homogeneous good produced under random shocks where fundamental uncertainty combines with coordination issues. However, the paper does not limit its approach to the traditional Marshallian expectational conception of price equilibrium investigated at the beginning of the text. Its purpose is not mainly however to relate expectations and beliefs but rather to pay attention to the importance of predictions of others' behavior when agents act strategically. Agent coordination therefore is at the center of the approach. This is why the author favors a Cournot equilibrium type of model in which uncertainty is reinforced by the existence of oligopolistic competition. Within this framework, agents are also supposed to identify each competitor's type ('dove' or 'hawk'). This identification is not predetermined however but only revealed at equilibrium. The paper compares this contribution to Schrödinger's cat thought experiment where a hypothetical cat which may be considered simultaneously both alive and dead as a result of its fate being linked to a random event that may or may not occur. It also concludes that the combination of a Cournot model assuming oligopolistic competition with rational expectations may imply some form of attractive indeterminacy which is not a symptom of 'bad modelling'.

The fifth paper of this special issue concerns expectational coordination within an 'eductive' framework ("Understanding expectational coordination as a major intellectual challenge: the 'eductive' guideline"). It was written by Roger Guesnerie. The starting point of this paper is that the concept of rational expectations cannot be considered as the simple and direct extension of the hypothesis of rational choice to expectations since it is rational to have rational expectations only if others also have rational expectations. Therefore, the purpose of the paper is to investigate the theoretical assessment of the validity of the rational expectation hypothesis. A standard assessment of such validity is associated with the success of real time learning when agents have adaptive expectations. The viewpoint taken in this paper, "eductive" learning, which has developed from the end of the XX th century, has a game-theoretical flavor, referring to mental processes associated with 'common knowledge', the success of which reflects a simple intuition, i.e the fact that the elasticity of realizations to expectations is not too high. The first part of the paper introduces the research from s two polar examples: the textbook partial equilibrium model -also considered in the previous paperwhich was used by Muth and the infinite horizon general equilibrium RBC model the genuine origin of which is also the Arrow-Debreu neo-Walrasian model. For both models, the paper stresses the argument and results associated with a collective and instantaneous thought process; and compares this "eductive" viewpoint with the real time learning viewpoint. The second part of the paper provides a random walk of the lessons of the "eductive" approach for understanding the quality of expectational coordination in a variety of contexts, going from partial to general equilibrium. The models and questions under scrutiny concern trade -how does a trade agreement affect expectational coordination? -market performance -do prices robustly transmit expected information? why are the standard justifications of the efficient market hypothesis very weak? -macroeconomics, -how and why is it the case that the Keynesian multiplier affects expectational coordination?, how to revise the Taylor rule for "eductive stability ? -finance-why and when does speculation strongly disturb coordination ?-Industrial organization -how to separate allocational and expectational effects of competition policies ? Finally, the exploration of the plausibility of robust expectational coordination stressing rationalizability and eductive learning is part of a more general landscape of research on expectations which is briefly evoked.

The sixth paper of this special issue is also related to rational expectations and favors its temporal aspect (the 'changing world') ("Rational expectations in a changing world") and was written by Alan Kirman. The purpose of the paper is not to argue that the predominant use of the assumption of rational expectations is unrealistic and inconsistent with the empirical evidence. It is to investigate the compatibility of this assumption with the non-ergodic evolution of economic systems and their complexity. The originality of Kirman's paper is therefore related to his view of economic dynamics which differs from the different frameworks of the general economic equilibrium theory. The first half of the paper focuses on the limits of the assumption of rational expectations. They include a very poor conception of time which is sometimes associated with "comparative statics" and often uses a stochastic conception of expectations based on risk and probability which neglects the past debates related to the notion of uncertainty. Moreover, the predominant conception of equilibrium supposes a considerable degree of knowledge about the future and imposes heroic assumptions as the capacity of people to calculate and understand not only their local situation but that of the economy as a whole.

Here again the paper shows that many past debates are under-estimated (for instance Hayek's and Keynes's contributions). The assumption of rational expectations is questionable in this context. It is incompatible with herd behavior, different algorithms using the same information, wrong beliefs, … without introducing some type of stationarity. The second half of the paper focuses on real phenomena related to economic dynamics and crises in a 'changing world': the emergence of phenomena implying new behavior and new properties; non ergodicity excluding extrapolations from the past; radical uncertainty; complexity and feed-back effects; difficulties related to economic policy. This why for the author an alternative approach of economic expectations and dynamics has to be considered. Several possibilities can be considered: simulated models, animal behavior and agent-based macro-models provide the main options for the present and the future according to the author.

Finally, we turn to the four macroeconomics and economic policy papers. Expectations were considered a key feature of pre-Keynesian economics and they became an important element of early business cycles and growth models. In the context of the great depression, few economists started to question the existence of a full employment equilibrium. In their paper ("Expectations and full Sylvie Rivot traces Milton Friedman's gradual reincorporation of the formation of expectations in his analysis of macroeconomic disequilibria ("Information and expectations in policy-making: Friedman's changing approaches to macroecononomic dynamics"). In his early work the focus was on static expectations; in the mid-1960s Friedman began to analyze how agents gradually adjusted their forecasts to the new informational context and finally introduced forward-looking behavior only in the 1970s. This chronology mirrors how Friedman's thinking evolved but also reveals the strong anchoring of his understanding of economic policy in his conception of decision theory. Rivot demonstrates how Friedman's policy writings were related to his early work with Savage, and led him to propose the idea of subjective uncertainty rather than mechanical adaptative expectations. Rivot's paper contrasts the quite lengthy incorporation of those elements with their tenuous relationship with Friedman's views on monetary policy. Almost 50 years after the rational expectations hypothesis (REH), Rivot reveals the ambiguous reception of this concept. Friedman never completely subscribed to the REH -mostly because it meant that "the same unit, relatively brief, is relevant to anticipations for all variables" (1982: 145 fn. 9). In fact, Friedman and Schwartz considered that the core adjustment process dynamic relied precisely on delays in the revisions to agents' expectations (ibid: 65).

Aurelien Goutsmedt investigates how and why explanations of the 1970s US stagflation changed over the years ("From the stagflation to the great inflation: explaining the US economy of the 1970s"). The paper contrasts the dominant explanation of stagflation developed in the 1970s as based mainly on supply-shocks and inflation inertia, with the explanations that emerged in the 1990s. Those later approaches were developed with little reference to the earlier explanations. Goutsmedt proposes a consistent history of this progressive drift in the explanations of stagflation. The originality of this paper is its mixing of the history of economic analysis with quantitative assessments. The author creates an original dataset which gathers all the references to the US stagflation episode .

He uses bibliometric tools to identify communities of papers (and authors) and to provide an accurate overview of the approaches to stagflation that predominated in different periods. Goutsmedt shows that despite their increasing importance in the field, before the 1990s the leading monetarism (Friedman) and New Classical School (Lucas and Kydland and Prescott) authors were rarely cited in works on stagflation. It was not until the late 1990s when a new strand of work emerged that the focus changed and their contributions were referenced more systematically in attempts to explain stagflation. This new strand of the literature proposed various explanations which centered on the role of monetary policy and the major contribution of expectations. The paper provides several interesting results which directly question common narratives that run through the recent history of macroeconomics, and especially the nature and timing of revolution driven by the REH. While the 1970s tend to be depicted as the time of a major revolution in the discipline, and the 1990s as years of consensus, the author shows that the methodological battles in the 1970s contributed only marginally to the crucial stagflation debate, and that the apparent macroeconomic consensus in the 1990s hides some divisions.

There is no doubt that those findings will work to stimulate the new (still emerging) interest in an approach which combines the history of economic thought with bibliometric tools.

Starting from the observation that REH analysis not only assumes an unrealistic degree of agent rationality but also does not explain how equilibrium is achieved, George Evans's paper ("Theories of learning and economic policy") focus attention on boundedly rational approaches in which Rational Expectations are assessed through the lenses of adaptive learning: economic agents predict key aggregate variables using econometric forecast rules, the parameters of which are updated over time.

The subject is introduced in the first Section by considering a forward-looking univariate linear model of asset pricing. Attention is focused on the connections between the perceived law of motion and the actual law of motion. A sequence of temporary equilibria is then generated in which agents use least square learning, to revise their expectations, a sequence hopefully converging asymptotically to the (here unique) Rational Expectations Equilibrium (or a noisy version of it if agents use constant gains learning processes, rather than declining gains occurring with least square learning). This line of research, also associated with the name of Sargent and some of his co-authors, has benefited from George Evans' previous important contributions, and particularly of his joint work with Seppo Honkapohja, (other co-authors are Branch, McGough, Mitra, and Guesnerie in papers comparing Estability of learning and eductive stability).

The present paper put emphasis on a number of chapters of economic policies. Monetary policy is first examined in a canonical New Keynesian model, with forward-looking Taylor-type interest rate rules.

  employment: Hansen, Samuelson and Lange"), Michael Assous, Olivier Bruno, Vincent Carret and Muriel Dal Pont Legrand examine this debate which started in the 1930s when Hansen and Samuelson, inspired by Roy Harrod, proposed a new understanding of the accelerator-multiplier relation. Alvin Hansen argued that a full employment equilibrium may not exist for a positive interest rate in a context of secular stagnation. His statement was challenged fiercely by Arthur Pigou but entry to the debate of Paul Samuelson, a former student of Hansen's, moved discussion from existence of a full employment equilibrium to the issue of the stability of this equilibrium. The Hansen-Samuelson-Lange connection highlighted the possibility of economic collapse as a third possibility based on neither growth nor business cycles theories. The authors built on the emerging evidence from the correspondence between Samuelson and Oskar Lange and show that Samuelson believed expectations were a fundamentally important factor which could generate the instability of full employment equilibrium: its stability depends on how the expectations are formed. Although Samuelson's analysis of that instability is implicit, his correspondence with Lange reveals that he had urged Lange to examine it in more depth. Lange presented his results in his 1944 Monograph published by the Cowles Commission. Although his approach resonated with Hicks's analysis of the role of expectations in equilibrium stability, Lange proposed a different method involving revisions to expectations throughout the adjustment process. Revisiting this debate in light of the Samuelson-Lange correspondence, Assous, Bruno, Carret and Dal Pont Legrand show that the way in which Samuelson tackled the problem of wage and price flexibility contributed to the progressive exclusion of expectations and reminds us that (once ?), expectations were at the core of macroeconomic instability.

Such a rule, even if meeting the Taylor condition on the response of interest rates to inflation, should not respond too aggressively to expected output shocks, in order to generate convergence of learning to the required equilibrium. Neo-Fisherian policies are examined, and the expectational analysis associated with adaptive learning suggests that such policies are likely to lead to instability and recession. Increasing the inflation target has been proposed to cope with the situation of zero lower bound (ZLB). Adaptive learning considerations stress the possibility of persistent learning dynamics, so that increases in the inflation target can lead to temporarily self-fulfilling but unstable learning dynamics. The learning viewpoint is essential for the analysis of policies the discussion of which requires non-linear models with multiple steady states. For example, the learning dimension sheds an original light on hyperinflation episodes and on the connections between ZLB situations and stagnation, and in such a class of cases, the reflection on the design of economic policies differs when expectations are not too far from the targeted steady state or outside its basin of attraction.

Echoing other papers in this special issue, this paper underlines the importance of the nature of expectations for equilibrium analysis and demonstrates that an equilibrium concept cannot be evaluated independently of an analysis addressing how agents would come to coordinate on the socalled equilibrium. To assess the relevance of rational expectation construct, the adaptive learning approach, grounded in temporary equilibrium and bounded rationality, triggers relevant stability insights, as suggested by its strong policy implications.

As guest editors of this Special Issue, we want to express our sincere thanks to the three editors of Revue d'Economie Politique, namely Tarik Tazdaït, Bertrand Wigniolle and Bertrand Cretez. We are grateful also to the anonymous referees for their comments and valuable suggestions on different papers included in this Special Issue.


New insights into oviposition preference of 5 Trichogramma species

Joel Meunier based on reviews by Kévin Tougeron and Eveline C. Verhulst

A recommendation of:

Up and to the light: intra-and interspecific variability of photo-and geotactic oviposition preferences in genus Trichogramma [START_REF] Burte | Up and to the light: intra-and interspecific variability of photo-and geo-tactic oviposition preferences in genus Trichogramma[END_REF]Calcagno, V.(2021), bioRxiv, 2021.03.30.437671, ver. 4

Recommendation

Insects exhibit a great diversity of life-history traits that often vary not only between species but also between populations of the same species [START_REF] Flatt | Mechanisms of life history evolution: the genetics and physiology of life history traits and trade-offs[END_REF]. A better understanding of the variation in these traits can be of paramount importance when it comes to species of economic and agricultural interest [START_REF] Wilby | Natural enemy diversity and pest control: patterns of pest emergence with agricultural intensification[END_REF]. In particular, the control of the development and expansion of agricultural pests generally requires a good understanding of the parameters that favour the reproduction of these pests and/or the reproduction of the species used to control them [START_REF] Bianchi | Habitat functionality for the ecosystem service of pest control: reproduction and feeding sites of pests and natural enemies[END_REF][START_REF] Gäde | Insect peptide hormones: a selective review of their physiology and potential application for pest control[END_REF].

Parasitoid wasps of the genus Trichogramma are a classic example of insects involved in pest control [START_REF] Smith | Biological Control with Trichogramma : Advances, Successes, and Potential of Their Use[END_REF]. This genus comprises over 200 species worldwide, which have been used to control populations of a wide range of lepidopteran pests since the 1900s [START_REF] Flanders | Mass production of egg parasites of the genus Trichogramma[END_REF][START_REF] Hassan | The mass rearing and utilization of Trichogramma to control lepidopterous pests: Achievements and outlook[END_REF]. Despite its common use, the egg-laying preference of this genus is only partially known. For example, all Trichogramma species are often thought to have positive phototaxis (or negative geotaxis) (e.g. [START_REF] Brower | Response of Trichogramma pretiosum and T. evanescens to Whitelight, Blacklight or NoLight Suction Traps[END_REF][START_REF] Van Atta | Effects of UV-B on Environmental Preference and Egg Parasitization by Trichogramma Wasps (Hymenoptera: Trichogrammatidae)[END_REF], but comprehensive studies simultaneously testing this (or other) parameter among Trichogramma species and populations remain rare. This is exactly the aim of the present study [START_REF] Burte | Up and to the light: intra-and interspecific variability of photo-and geo-tactic oviposition preferences in genus Trichogramma[END_REF]. Using a new experimental approach based on automatic image analysis, the authors compared the photo-and geo-tactic oviposition preference among 5 Trichogramma species from 25 populations. Their results first confirm that most Trichogramma species and populations prefer light to shade, and higher to lower positions for oviposition. Interestingly, they also reveal that the levels of preference for light and gravity show inter-and intraspecific variation (probably due to local adaptation to different strata) and that both preferences tend to relax over time.

Overall, this study provides important information for improving the use of Trichogramma species as biological agents. For example, it may help to establish breeding lines adapted to the microhabitat and/or growing parts of plants on which agricultural pests lay eggs most. Similarly, it suggests that the use of multiple strains with different microhabitat selection preferences could lead to better coverage of host plants, as well as a reduction in intraspecific competition in the preferred parts. Finally, this study provides a new methodology to efficiently and automatically study oviposition preferences in Trichogramma, which could be used in other insects with a particularly small size.

Reviews

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Reviewed by Eveline C. Verhulst, 02 Aug 2021 21:01

I once again enjoyed reading the preprint "up and to the light: intraand interspecific variability of photo-and geotactic oviposition preference in genus Trichogramma" by Burte et al. The abstract now reads well, and the reformulated sentences are now clear to me. The figures have better readability. The explanation of the delineation in line 425-426 add clarity. Figure S3 and Table S1 add to the interpretation of the data by the reader. Overall I agree with the changes made and the explanation given in the reply by the authors. I found some small typo's and other textual errors.

Line 89 remove from developing in. Line 181 ...impact on their tube temperature Line 395 double period Line 559 "Even though sampled for most of them only a few years before experiments..." suggested change to "Even though most populations were sampled only a few years before the experiments..." Line 567 " that have over 20 years, and the "young" ones that have only a couple of years" of lab adaptation or something similar should be added.

Reviewed by Kévin Tougeron, 13 Jul 2021 09:15

I really appreciated the changes made to the manuscript. In particular, it now has a more comprehensive introduction and theoretical discussion, which gives additional value to the reading. I think the legend for the species is missing on the left PCA (Figure 5a) L131: It should read "Palearctic" L556: Remove extra column ";" I encourage the authors to carefully go through their article one more time to spot minor typo errors.

Revision round #1 11 May 2021

Author's Reply

Download author's reply (PDF file)Download tracked changes file

Decision round #1

Dear Dr Calcagno, I have received the detailed comments of two reviewers on your manuscript entitled "Up and to the light: intra-and interspecific variability of photo-and geo-tactic oviposition preferences in genus Trichogramma". As you will see, both reviewers liked your study. I agree with them, in that your manuscript is pleasant to read and provides useful, novel and interesting data about the egg-laying behaviour of Trichogramma wasps.

Having said that, the reviewers raise numerous points that need to be addressed to improve the clarity of the text and strengthen the conclusions. Among them, I would like to stress the importance to provide more details about the lab rearing conditions of the tested populations. This would allow you to exclude that the reported results are simple by-products of standard laboratory conditions selecting for "up and to the light" egg-laying sites, thus strengthening the relevance of your results.

I look forward to reading your detailed responses to the numerous and detailed reviewers' comments, and the associated changes in the manuscript.

Best, Joel Meunier

Preprint DOI: https://doi.org/10.1101/2021.03.30.437671

Reviewed by Kévin Tougeron, 09 Apr 2021 15:04

Review for PCI Zoology: Up and to the light: intra-and interspecific variability of photo-and geotactic oviposition preferences in genus Trichogramma

General comments:

I do not have many comments to make. I really appreciated reading this article. The protocol used to unravel the main effects of photo-and geotactic components is smart and allows obtaining good sample sizes.

Introduction: This part is well written but the end of the introduction would benefit from more detailed explanations. Especially in the last paragraph, it would be interesting that the authors write their hypotheses regarding expected inter-populational variations, and interspecific differences.

Results: I don't want to be a dogmatist, but the authors should be careful not discussing too much the results in this section. Of course, anything that helps understanding the results is welcome, but some paragraphs (e.g., P10L6-16 or P12L22-33) would better fit in the discussion section. Discussion: In this section, I appreciated the efforts that the authors put in trying to explain interspecific differences in the observed patterns. It is a difficult task given the poor knowledge on natural history of some species. Would it be possible to discuss more about how other species-or population-dependent life history traits, such as size, mass, energetic reserves, flight capacities, longevity, egg-load, could affect the patterns observed in this study? In addition, what could be the links between microhabitat selection and microclimatic conditions found on/under the leaves? Finally, adding more details on the relevance of your work regarding inundative biocontrol strategies would be great.

Specific comments:

P2L6: Please write "Trichogramma brassicae" in full here, as it is the first time you mention the species in the text.

P2L8: I think it's a bold statement to say that the ecology of trichograms is not well understood. Please temper this statement a little bit. However, I totally agree that the genetic, physiological, behavioral diversity of trichograms is not well understood. P2L24-30: This paragraph is particularly well written and it gives a good idea of the importance of the work done in your study (and yet to be done). I would just suggest adding a short sentence to make clear that we also want to select particular parasitoid strains that fit the environment they will be released in. One cannot release a parasitoid strain that has poor cold resistance in Québec, for instance (see Gariepy, V., Boivin, G., & Brodeur, J. ( 2015). Why two species of parasitoids showed promise in the laboratory but failed to control the soybean aphid under field conditions. Biological Control, 80, 1-7. https://doi.org/10.1016/j.biocontrol.2014.09.006) P3L22: Please write the genus name in full when it begins a sentence P4L5: remove the "s" after "population" P7L17: How closely? Please mention the R² of 95% here as well, before referring to Fig S1. P13L2: Please provide the % variance explained by each of the two axes. Moreover, please provide (as a supplementary figure) the contribution of each variable to each dimension. In addition, would it be possible to try representing the PCA with confidence ellipses around each group (either species, or strata)? It would allow seeing more clearly how groups discriminate on the graph. Of course, I understand it's not possible for n < 3. P13L20-22: Of course, but I still think it's worth of discussion; what could be the interplay, in nature, between geotactism and phototactism? Your results suggest additive effects, so maybe the neurological and sensorial mechanisms are totally dissociated between both attraction components? This is discussed a little bit in the following paragraph, but maybe some more references from the literature can be found regarding additive vs. synergic effects in other animal groups (or even in plants?). P16L4-15: In this last paragraph, or elsewhere in the discussion, it could also be interesting to mention that life-history traits and behaviors often differ a lot between natural and industrial parasitoid strains. In a biological control perspective, it is, as you mention, extremely important to optimize the match between parasitoid strain and the environment (and, of course, the target pest species). Just to give you an example, but other exist, we explored this issue in Tougeron, K. et al. (2020). Trans-generational effects on diapause and life-history-traits of an aphid parasitoid. Journal of Insect Physiology, 121, 104001. https://doi.org/10.1016/j.jinsphys. 2019.104001 andin Saeed, M. M. et al. (2020). Transgenerational phenotypic plasticity of diapause induction and related fitness cost in a commercial strain of the parasitoid Aphidius ervi Haliday. Insect Science, 1744-7917.12794. https://doi.org/10.1111/1744-7917.12794).

Reviewed by Eveline C. Verhulst, 10 May 2021 21:20

I have enjoyed reading the preprint up and to the light: intra-and interspecific variability of photoand geotactic oviposition preference in genus Trichogramma by Burte et al. The oviposition preference of Trichogramma wasps is important for successful biocontrol applications as the biocontrol agent (Trichogramma) and pests (mostly Lepidoptera) can be matched and the massrearing of the agent can be made more efficient. As the Trichogramma genus is a species rich genus, and the number of cryptic species is high, it is important to study phenotypic traits on a large scale to be able to study many species/populations but reduce the time spend. The authors present a high-throughput system for recording and automatically analysing the location of oviposition of many wasps at the same time during a number of days. Using this system they tested the whether 25 populations of 5 species of Trichogramma have a preference to oviposit more to the top of the tube or whether they oviposit in the light vs in the dark.

Introduction

The introduction generally reads pleasantly, however a clear rationale for the study does not entirely emerge from the introduction. A bit more background on the relation between the Lepidoptera host species and Trichogramma species would be helpful, e.g. does the oviposition site of a certain Lepidoptera species have a relation with the preference of the Trichogramma species for light and geotaxis?

In addition, what was the reason for studying the 5 Trichogramma species chosen here? Availability? Or are they hypothesized to be clearly distinct in their phenotypes? The introduction does not give a clear direction for the species tested, only somewhat for the rationale of looking at oviposition. Paragraph starting at line 32 is a bit vague also with respect to the traits that are "adaptively important". It is a bit difficult to figure from the text exactly which traits are meant, and "adaptively important" is a strange sentence construction. Please rewrite this to make it more concrete what is meant here.

Line 42-43 has an awkward sentence construction.

Line 44: to what other purposes is referred here?

Material and methods

It is unclear how many replicates were done per population, only later in the results I could read that it was 40 tubes per population. Please add this information to the M&M.

Was the light intensity equal along the 2*20 grid of tubes? It is not indicated that this is measured.

Page 4

Table 1: Does >2014 mean that it was sampled after 2014? Or do the authors mean <2014? Line 11: disposed >> distributed or glued Line 13-14: I assume that the Trichogramma inocula were glued to the cardboard also to enable vertical positioning of the tubes. What type of glue was used? And were the honey droplets soaked up by the cardboard? Else it would not have stayed in place in the vertical position? Line15: is >> in Line 16: disposed >> placed

Page 5

Line 18: There is no "H" treatment? I guess this should be "P". I'm not an expert in statistics so I cannot properly evaluate the method described in lines 5-20. But I wonder why the square root of nbT is used to weigh against? Please give a rationale for this choice.

Results

Line 32: Add in the text that the failed or male-only hatching refers to the eight parasitized Trichogramma inocula placed in the middle of the card strip.

Page 9

Figure 2: The asterix and axis font could be a bit bigger, and the dark frames are very hard to see compared to the boxes without frame. Maybe a colored pattern in the bars would help. Line 11: are the parasitism rates referred to here overall or per patch (disk of eggs?) Because even if the overall parasitism rate was low, if the preferred patch was fully parasitized it could explain the plasticity observed. However it is not clear from the text exactly what could be the case here.

On page 11, paragraph starting at line 17 also refers to the depletion effect, and also here I would have appreciated to see the distribution of parasitism along the patches.

Page 12

Line 4: Please give the percentage of explaned variance per component.

Line 5: It is stated that the first PC1 axis separates populations with strong preference from those with weak preference or no preference, however populations on the right hand side of PC1 could also be contrasting in their preference for P and G, as is the case for pr002.

Line 18: I don't agree with the statement entirely, it is true that T. principium is attracted to the light based on the date, but only one of the two populations of T. brassiceae, so the phrasing of this sentence should be a bit different.

In addition, the significant difference of P score across species is not something I see reflected in

Discussion

Page 14

Line 5: away >> a way Line 8-10: The statement here may be true for isa1729, but population f5-11 does not seem to suffer from this non-additive effect because of the strength of the preference for the single-cues? So the generality of the statement here is not accurate. Please explain.

Line 23: remove "numbers".

Line 24: Overall low parasitism rate is not informative as it has to do with the percentage parasitism at the prefered site, but this is not given in Fig. S2.

Line 42: Please explain thigmotaxis Page 15 Line 9-11: Sentence construction is a bit awkward. Please change within >> when Line 28: Why would there be a stronger association of light preference with population from the herbaceous layer , and why would this make sense from an adaptive perspective?

  P9L12: Please remind what P and G treatments are, in the figure caption, so the figure is easier to understand.

Page 6 Figure 1 :

 61 Figure 1: In the figure and caption the pictures are referred to as J1 to J5 but in the text these are D1 to D5. Please change in the Figure and caption to D as well. Page 7 Line 7: dark Line 30-33 is a hard to read sentence. I also would advice to add the formula's used to calculate the different values plotted and used in the statistical analyses. Line 41-42: I don't understand what is meant here. Page 8

Fig. 5 ?

 5 Figure 5b: indicate that grey circles are populations without stratum data.

https://doi.org/10.1101/2021.03.30.437671

  peer-reviewed and recommended by Peer Community in Zoology

	Submitted: 02 April 2021, Recommended: 17 August 2021

General comments

The authors take care not to stretch their interpretations beyond the results they obtained and in general do a good job of discussing the limitations of their research. As this is partly a methodology paper, introducing the method is important and although the authors do a good job overall, I do have some questions and remarks (see below). The strains / species / populations used in the study are all established in the lab for multiple years, some longer than others. However the authors do not report how the lines were maintained in the lab for all those years? Where they horizontal or vertical in the incubators? Was the light distribution equal? In terms of adaptation to the original stratum, is it not expected that any preference based on stratum height would have been lost after so many years in the lab?

In the manuscript results are often given for a certain notable population or species in terms of phenotype, but the data for all populations is not given in the supplementary. The entire raw dataset and associated information are not available online yet, as no link is given to them. It is hard to evaluate all results in this way. For example, the preference scores per day (Fig. 3) are only given for one population, and Fig. 3b is not informative for the different populations / species or treatment but only gives an overall picture. Also the distribution of parasitism rates in the separate tubes is only given for one population (Fig. S3).

The notion that the observed plasticity of the preference over the five days could be due to host shortage is refuted by the authors themselves as the overall parasitism rate is low. However, it is not clear to me if this overall parasitism rate is measured across the entire tube, or is measured for the prefered sites? This should be made more clear in the text, and would also be aided by additional figures in the supplementary or online (see above).

The authors also refer to Trichogramma wasps as trichograms throughout the text. This is a new one for me, and upon searching this name I could only find a medical examination with the same name. I would suggest the authors to change trichograms into Trichogramma wasps, unless they have a good reason not to? I also had a hard time connecting the population names to the species throughout the text, especially when refering from the text to the figures and back again. It would help to have the species names in Figure 2 as well (maybe in a 90 deg. angle, between populations label and the population names), and in the text each time a general remark is made about a Trichogramma species result to indicate the color in parenthesis.

Please have continuous line numbering for the entire manuscript, that is easier.

Abstract

The English in the abstract reads a little bit ackward and could use the help of a native English reader to finetune it. "Appropriate" in the third sentence does not clearly refer to biocontrol agent. Line 22-25 should be rephrased as it is a too complex sentence.