LMPA Romuald Ernst
email: ernst.r@math.cnrs.fr

Augustin Mouze
email: augustin.mouze@math.univ-lille1.fr

A QUANTITATIVE INTERPRETATION OF THE FREQUENT HYPERCYCLICITY CRITERION

Keywords: 2010 Mathematics Subject Classification. 47A16, 37B50 frequently hypercyclic operators, weighted densities

We give a quantitative interpretation of the Frequent Hypercyclicity Criterion. Actually we show that an operator which satisfies the Frequent Hypercyclicity Criterion is necessarily A-frequently hypercyclic, where A refers to some weighted densities sharper than the natural lower density. In that order, we exhibit different scales of weighted densities that are of interest to quantify the "frequency" measured by the Frequent Hypercyclicity Criterion. Moreover we construct an example of unilateral weighted shift which is frequently hypercyclic but not A-frequently hypercyclic on a particular scale.

Introduction

The notion of frequent hypercyclicity was introduced in the context of linear dynamics by Bayart and Grivaux in 2006 [START_REF] Bayart | Hypercyclicité: le rôle du spectre ponctuel unimodulaire[END_REF], [START_REF] Bayart | Frequently hypercyclic operators[END_REF]. This latter is now a central notion in that field and is highly connected to combinatorics, number theory and ergodic theory. Let X be a metrizable and complete topological vector space and L(X) be the space of continuous linear operators on X. An operator T ∈ L(X) is said to be hypercyclic if there exists x ∈ X such that for any non-empty open set U ⊂ X, the return set {n ≥ 0 : T n x ∈ U } is non-empty or equivalently infinite. Such a vector x is called a hypercyclic vector for T . Furthermore an operator T is called frequently hypercyclic if there exists x ∈ X such that for any non-empty open set U ⊂ X, the set of integers n satisfying T n x ∈ U has positive lower density, i.e. lim inf

N →+∞ #{k ≤ N : T k x ∈ U } N > 0,
where as usual # denotes the cardinality of the corresponding set. Thus the notion of frequent hypercyclicity extends the classical hypercyclicity and appraises how often the orbit of a hypercyclic vector visits every non-empty open set. In the sequel we denote by N the set of positive integers and for any x ∈ X and any subset U ⊂ X we set N (x, U ) := {n ∈ N : T n x ∈ U }. Given a subset E ⊂ N, we define its lower and upper densities respectively by

d(E) = lim inf N →+∞ #{k ≤ N : k ∈ E} N and d(E) = lim sup N →+∞ #{k ≤ N : k ∈ E} N .
In other words, an operator T ∈ L(X) is hypercyclic (resp. frequently hypercyclic) if there exists x ∈ X such that for any non-empty open set U ⊂ X, the set N (x, U ) is non-empty (resp. has positive lower density). To prove that an operator is hypercyclic we have at our disposal the so-called Hypercyclicity Criterion (see [START_REF] Bayart | Dynamics of linear operators[END_REF], [START_REF] Grosse-Erdmann | Linear Chaos[END_REF] and the references therein). In the same spirit, Bayart and Grivaux stated the Frequent Hypercyclicity Criterion, which ensures that an operator is frequently hypercyclic [START_REF] Bayart | Frequently hypercyclic operators[END_REF]. Let us recall it here.

Theorem 1.1. Let T be an operator on a separable Fréchet space X. If there is a dense subset X 0 of X and a map S : X 0 → X 0 such that, for any x ∈ X 0 , We already know that the above result does not characterize frequently hypercyclic operators. Indeed Bayart and Grivaux have exhibited a frequently hypercyclic weighted shift on c 0 that does not satisfy this criterion [START_REF] Bayart | Invariant gaussian measures for operators on banach spaces and linear dynamics[END_REF]. A natural question arises: what does the Frequent Hypercyclicity Criterion really quantify? In order to answer this question, Bès, Menet, Peris and Puig recently generalized the notion of hypercyclic operators by introducing the concept of A-frequent hypercyclicity, where A refers to a family of subsets of N satisfying suitable conditions [START_REF] Bès | Recurrence properties of hypercyclic operators[END_REF]. In particular, A has to satisfy the following separation condition:

A contains a sequence (A k ) of disjoint sets such that for any j ∈ A k , any j ∈ A k , j = j , we have |j -j| ≥ max(k, k ).

In this abstract framework, they also obtain an A-Frequent Hypercyclicity Criterion and prove that the Frequent Hypercyclicity Criterion has very strong consequences in the sense that if T satisfies the Frequent Hypercyclicity Criterion, then T also satisfies the A-Frequent Hypercyclicity Criterion for any suitable family A. Bonilla and Grosse-Erdmann also studied specific notions related to the concept of A-frequent hypercyclicity in the article [START_REF] Bonilla | Upper frequent hypercyclicity and related notions[END_REF]. On the other hand, the notion of frequent hypercyclicity measures the frequency and the length of the intervals when iterates of a hypercyclic vector visits every non-empty open set in a very specific way, that is given by the natural density. Actually, there are many types and notions of densities different from the natural one. Our goal is to give quantified consequences to the Frequent Hypercyclicty Criterion in terms of weighted densities. To that purpose, we consider a special kind of lower weighted densities, generalizing the natural one but sharper than this one, by using the formalism of matrix summability methods. For such a matrix A, we use the concept of A-density and A-frequent hypercyclicity (see Definitions 2.2 and 2.11 below). These general densities were already used in the context of linear dynamics to study frequently universal series [START_REF] Mouze | Polynomial inequalities and universal Taylor series[END_REF]. In the present paper, we show that the Frequent Hypercyclicity Criterion gives a stronger conclusion than frequent hypercyclicity, that we quantify thanks to explicit weighted densities on different scales. We refer the reader to Proposition 3.5 and Theorem 4.12 below. Therefore an operator which satisfies the Frequent Hypercyclicity Criterion is necessarily A-frequently hypercyclic, where A refers to some weighted densities sharper than the natural lower density.

Let us return to the formalism of A-frequent hypercyclicity. For instance A could be a family of subsets with positive given lower weighted density satisfying the aforementioned separation property. However from [START_REF] Bès | Recurrence properties of hypercyclic operators[END_REF], there is an underlying question: does there exist a frequently hypercyclic operator not being A-frequently hypercyclic? We give a positive answer by constructing an unilateral weighted shift on c 0 which is frequently hypercyclic but not A-frequently hypercyclic with respect to some A-densities covered by the criterion (see Theorem 5.4

below).

The paper is organized as follows: in Section 2 we introduce some densities that will be of interest in the sequel and some properties on these densities. Section 3 is devoted to an improvement of the Frequent Hypercyclicity Criterion for a certain scale of weighted densities. In Section 4, we modify this proof in order to obtain a stronger result to the criterion. Finally in Section 5, we exhibit a new example, inspired by [START_REF] Bayart | Difference sets and frequently hypercyclic weighted shifts[END_REF], of an operator which is frequently hypercyclic although it does not satisfy the Frequent Hypercyclicity Criterion. To ensure this latter property, we will show that this operator is not A-frequently hypercyclic for some suitable matrix A.

Densities: preliminary results

In this section, we state some definitions and results we shall need throughout the paper. Let us first introduce the concept of summability matrix and its connections with some kind of densities on subsets of N. Definition 2.1. A summability matrix is an infinite matrix M = (m n,k ) of complex numbers.

Let us recall that, if (x n ) is a sequence and M = (m n,k ) is a summability matrix, then by M x we denote the sequence ((M x) 1 , (M x) 2 , . . . ) where (M x) n = +∞ k=1 m n,k x k . The matrix M is called regular if the convergence of x to c implies the convergence of M x to c. By a well-known result of Toeplitz (see for instance [START_REF] Zygmund | Trigonometric Series[END_REF]), M is regular if and only if the following three conditions hold:

(2.1)

         (i) lim n→+∞ m n,k = 0, for all k ∈ N, (ii) lim n→+∞ k≥1 m n,k = 1, (iii) sup n k≥1 |m n,k | < ∞.
Freedman and Sember showed that every regular summability matrix M with non-negative real coefficients defines a density d M on subsets of N, called lower M -density [START_REF] Freedman | Densities and summability[END_REF]. Definition 2.2. For a regular matrix M = (m n,k ) with non-negative coefficients and a set E ⊂ N, the lower M -density of E, denoted d M (E), is defined by

d M (E) = lim inf n→+∞ +∞ k=1 m n,k 1 E (k),
and the associated upper M -density, denoted by d M (E), is defined by

d M (E) = 1 -d M (N \ E).
Remark 2.3. For a non-negative regular matrix M = (m n,k ), Proposition 3.1 of [START_REF] Freedman | Densities and summability[END_REF] ensures that the upper M -density of any set E ⊂ N is given by

d M (E) = lim sup n→+∞ +∞ k=1 m n,k 1 E (k).
Let (α k ) k≥1 be a non-negative sequence such that n k=1 α k → +∞ as n tends to ∞. Then, we deal with the special case of A-density where we write

A = α k / n j=1 α j , when A = (α n,k ) with α n,k = α k / n j=1 α j for 1 ≤ k ≤ n and α n,k = 0 for k > n. It is easy to check that
A is a non-negative regular summability matrix. In summability theory the transformation given by x = (x n ) → Ax is called the Riesz mean (A, α n ). Here the associated A-density can be viewed as a weighted density with respect to the non-negative weight sequence (α k ) k≥1 .

Definition 2.4. A summability matrix A = α k / n j=1 α j as above will be called an admissible matrix. We define its summatory function ϕ α as follows:

ϕ α : N → R + , ϕ α (n) = k≤n α k . Example 2.5.
(1) If α k = 1, k = 1, 2, . . . , then the summability matrix A is the wellknown Cesàro matrix and d A is the natural lower density.

(2) If α k = 1/k, k = 1, 2, . . . , d A is the so-called lower logarithmic density, which is derived from the well-known logarithmic summability method. We have ϕ α (k) ∼ log(k), as k tends to +∞. (3) The special case α k = k r , r ≥ -1, for k = 1, 2, . . . , generalizes both the natural density (r = 0) and the logarithmic density (r = -1). Clearly we have ϕ α (k) ∼ k r+1 r+1 , as k tends to +∞, when r > -1. (4) If α k = e k r , 0 < r < 1, for k = 1, 2, . . . , an easy calculation gives ϕ α (k) ∼ k 1-r r e k r , as k tends to +∞.

(5) If α k = e k , for k = 1, 2, . . . , then ϕ α (k) ∼ e e-1 e k , as k tends to +∞. (6) If α 1 = 1 and α k = e k/ log r (k) , r > 0, for k = 2, 3, . . . , a summation by parts gives ϕ α (k) ∼ log r (k)e k/ log r (k) , as k tends to +∞. (7) Let h s be the real function defined by h s = log . log (s) , with log

(s) = log • log • • • • log,
where log appears s times. If α k = e k/hs(k) , l ∈ N, s ≥ 2, for k large enough, again a summation by parts gives ϕ α (k) ∼ h s (k)e k/hs(k) , as k tends to +∞.

In the following sections, we shall use the following definitions connected to Example 2.5.

Definition 2.6. We denote by

(1) C r the admissible matrix C r = k r / n j=1 j r , r ≥ -1; (2) A r the admissible matrix A r = e k r / n j=1 e j r , r ≥ 0; (3) B r the admissible matrix B r = (α k / n j=1 α j ) with α 1 = 1 and α k = e k/ log r (k) , for k ≥ 2, r ≥ 0; (4) Let h s be the real function defined by h s = log . log (s) , with log (s) = log • log • • • • log, where log appears s times. We denote by B s the admissible matrix B s = (α k / n j=1 α j ) with α k = e k/hs(k) , for k large enough and s ≥ 2. For any subset E ⊂ N, we can write E as a strictly increasing sequence (n k ) of positive integers. It is well-known that d(E) = lim inf k→+∞ k n k which allows to deduce the following simple fact: d(E) > 0 if and only if the sequence n k k is bounded [START_REF] Grosse-Erdmann | Linear Chaos[END_REF]. The following lemma extends this remark to suitable A-densities. Lemma 2.7. Let (α k ) be a non-negative sequence such that k∈N α k = +∞. Assume that the sequence (α n / n j=1 α j ) converges to zero as n tends to +∞. Let (n k ) be an increasing sequence of integers forming a subset E ⊂ N. Then, we have

d A (E) = lim inf k→+∞ k j=1 α n j n k j=1 α j
, where d A is the A-density given by the summability matrix A = (α k / n j=1 α j ). Proof. Let us consider n k ≤ N < n k+1 , then

k+1 j=1 α n j n k+1 j=1 α j - α n k+1 n k+1 j=1 α j = k j=1 α n j n k+1 j=1 α j ≤ n j ≤N α n j N j=1 α j ≤ k j=1 α n j n k j=1 α j .
Thus, we deduce

d A (E) = lim inf N →+∞ n j ≤N α n j N j=1 α j = lim inf k→+∞ k j=1 α n j n k j=1 α j .
In the present paper, we are mainly interested in sharper A-densities than the classical natural density. From this point of view, the following lemma gives some conditions to ensure that the sequence (α k ) leads to a sharper density.

Lemma 2.8. Let (α k ) and (β k ) be non-negative sequences such that k∈N α k = k∈N β k = +∞. Assume that the sequence (α k /β k ) is eventually decreasing to zero. Let A = (α k / n j=1 α j ) and B = (β k / n j=1 β j ) be the associated admissible matrices. Then, for every subset E ⊂ N, we have

d B (E) ≤ d A (E) ≤ d A (E) ≤ d B (E). Proof. Let E be a subset of N. For every n ≥ 1, let us define Λ α E (n) = n k=1 α k 1 E (k) (resp. Λ β E (n) = n k=1 β k 1 E (k))
. In particular, one may observe that Λ α N = ϕ α . Now, let N ≥ 1 be an integer such that the sequence (α k /β k ) k≥N is decreasing. Then for every n ≥ N + 1, we have

n k=N +1 α k 1 E (k) = n-1 k=N +1 Λ β E (k) α k β k - α k+1 β k+1 + Λ β E (n) α n β n -Λ β E (N ) α N +1 β N +1 = n-1 k=N +1 Λ β E (k) ϕ β (k) ϕ β (k) α k β k - α k+1 β k+1 + Λ β E (n) ϕ β (n) ϕ β (n) α n β n -Λ β E (N ) α N +1 β N +1 .
Moreover, since (α k /β k ) is a non-negative decreasing sequence and α k = +∞, we deduce

d A (E) = lim sup n→+∞ (ϕ α (n)) -1 N k=1 α k 1 E (k) + n k=N +1 α k 1 E (k) = lim sup n→+∞ (ϕ α (n)) -1 n-1 k=N +1 Λ β E (k) ϕ β (k) ϕ β (k) α k β k - α k+1 β k+1 + Λ β E (n) ϕ β (n) ϕ β (n) α n β n ≤ sup k>N Λ β E (k) ϕ β (k) lim sup n→+∞ (ϕ α (n)) -1 n-1 k=N +1 ϕ β (k) α k β k - α k+1 β k+1 + ϕ β (n) α n β n . Since n-1 k=N +1 ϕ β (k) α k β k - α k+1 β k+1 + ϕ β (n) αn βn = ϕ β (N + 1) α N +1 β N +1 + n k=N +2 α k , we get d A (E) ≤ sup k>N Λ β E (k) ϕ β (k) . Hence letting N → +∞, we obtain d A (E) ≤ d B (E).
The other inequality is obtained using the relations

d A (E) = 1 -d A (N \ E) and d B (E) = 1 -d B (N \ E).
From now on, we are interested in densities given by special admissible matrices given in Definition 2.6. In this case, Lemma 2.8 leads to the following inequalities. Lemma 2.9. For every subset E ⊂ N and for any 0

< r ≤ r , 0 < s ≤ s < 1, 1 < t ≤ t , 2 ≤ l ≤ l , we have d A 1 (E) = d B 0 (E) ≤ d B l (E) ≤ d B l (E) ≤ d Bt (E) ≤ d B t (E)
and

d B t (E) ≤ d A s (E) ≤ d As (E) ≤ d C r (E) ≤ d Cr (E) ≤ d(E).
Moreover, observe that a subset E of N possesses a strictly positive natural lower density if and only if it has a strictly positive lower C r -density, for any r > -1.

Lemma 2.10. Let r > -1. Then, for every subset E ⊂ N the following assertions are equivalent

(i) d Cr (E) > 0, (ii) d(E) > 0.
Proof. Let (n k ) ⊂ N be the increasing sequence of elements of E. We divide the proof in two cases.

Case r ≥ 0: Lemma 2.8 gives d Cr (E) ≤ d(E), hence (i) ⇒ (ii). For the other implication, assume that d(E) > 0. This means that the sequence

n j j
is bounded (see Lemma 2.7). We deduce that there exists an integer M ≥ 1 such that for every k

∈ N, k ≤ n k ≤ M k and n k j=1 j r ≤ M k j=1 j r ≤ (M k + 1) r+1 r + 1 ≤ M r+1 (k + 1) r+1 r + 1 .
Using the fact that k j=1 j r ∼ k r+1 r+1 ∼ (k+1) r+1 r+1 , as k tends to +∞ (cf Example 2.5) we deduce that there exists C > 0 such that

n k j=1 j r ≤ CM r+1 k j=1 j r .

Therefore we have

n k j=1 j r ≤ CM r+1 k j=1 (n j ) r and lim inf k→+∞ k j=1 (n j ) r n k j=1 j r
> 0 which is sufficient to conclude thanks to Lemma 2.7.

Case -1 < r < 0: As in the previous case, Lemma 2.8 gives the implication (ii) ⇒ (i) because d Cr (E) ≥ d(E). For the converse, assume that d Cr (E) > 0. According to Lemma 2.7, there exists C > 0 such that

n k j=1 j r ≤ C k j=1 (n j ) r .
Using the inequality j ≤ n j and since r < 0, we get

(n k + 1) r+1 r + 1 - 1 r + 1 ≤ n k j=1 j r ≤ C k j=1 (n j ) r ≤ C k j=1 j r ≤ C k r+1 r + 1 - 1 r + 1 + 1 .
The positivity of r + 1 now ensures that the sequence n k k is bounded and d(E) > 0.

Finally let us also extend the definition of frequently hypercyclic operators using the general notion of A-densities introduced before. Definition 2.11. Let A = α k / n j=1 α j be an admissible matrix. Using the notations of Section 1, an operator T ∈ L(X) is said to be A-frequently hypercyclic if there exists x ∈ X such that for any non-empty open set U ⊂ X, the set N (x, U ) has positive lower A-density.

Frequent hypercyclicity criterion: classical construction

According to Lemma 2.10, a frequently hypercyclic operator is necessarily C r -frequently hypercyclic for any r > -1. So, even if this is obvious, the Frequent Hypercyclicity Criterion allows to obtain the B r -frequent hypercyclicity too. A careful examination of the well-known proof of this criterion leads to a more precise result. Indeed, in the classical proof of the Frequent Hypercyclicity Criterion the following constructive lemma plays a prominent role [10, Lemma 9.5].

Lemma 3.1.

There exist pairwise disjoint subsets A(l, ν), l, ν ≥ 1, of N of positive lower density such that, for any n ∈ A(l, ν) and m ∈ A(k, µ), we have that n ≥ ν and

|n -m| ≥ ν + µ if n = m.
The proof of this result is based on a specific partition of N using the dyadic representation n = +∞ j=0 a j 2 j = (a 0 , a 1 , . . . ) of any positive integer. Actually the authors define the sets I(l, ν), l, ν ≥ 1, as the sets of all n ∈ N whose dyadic representation has the form n = (0, . . . , 0, 1, . . . , 1, 0, * ) with l -1 leading zeros, exactly followed by ν ones, then one zero and an arbitrary remainder. Let δ k = ν, if k ∈ I(l, ν) for some l ≥ 1. Then they construct the following strictly increasing sequence (n k ) of positive integers by setting

n k = 2 k-1 i=1 δ i + δ k , k ≥ 1.
This construction clearly ensures that for any integers i, j, with i = j, the separation condition stated in Lemma 3.1 holds, that is

|n i -n j | ≥ δ i + δ j .
Finally they define the sets A(l, ν) = {n k ; k ∈ I(l, ν)} and they prove that these sets have positive lower density since (n k ) does and the sets I(l, ν) are arithmetic sequences. Actually we are going to prove that the sequence (n k ) has positive lower B 2 -density. To do that, we start by giving an exact formula for this sequence that will allow to obtain easily its asymptotic behavior. We obtain the following result, whose proof will be given later (see Lemma 4.8 below).

Lemma 3.2. If k = 2 n + n-1 i=0 α i 2 i with α i ∈ {0; 1} for every 0 ≤ i ≤ n -1, then n k = 4k -2   i∈I k L i (i + 1)   -δ k ,
where I k stands for the set of integers i such that α i is the first non-zero integer of a block (of consecutive non-zero coefficients) having length L i in the dyadic decomposition of k.

From this lemma, we deduce the following estimate using the same notations.

Proposition 3.3. The sequence (n k ) satisfies the following estimate

n k -4k = O(log 2 (k)).
Moreover this estimate is optimal in the following sense: there exists an increasing subsequence (λ n ) of positive integers such that the sequence n λ j -4λ j log 2 (λ j ) converges to a non-zero real number.

Proof. According to Lemma 3.2, we have, for

k = 2 l + l-1 i=0 α i 2 i with α i ∈ {0; 1} for every 0 ≤ i ≤ l -1, n k = 4k -2   i∈I k L i (i + 1)   -δ k , where I k = {i ∈ N; α i = 0 and α i-1 = 0}, with the conventions α -1 = 0, α l = 1 and for i ∈ I k , L i = min{j; α i+j = 0}. Obviously we deduce n k ≤ 4k -2 log 2 (k) -1.
Notice that we have the equality

n k = 4k -2(log 2 (k) + 1) -1 for k = 2 l .
On the other hand, we can write

i∈I k L i (i + 1) = i 1 <i 2 <•••<im k L i j (i j + 1),
where

I k = {i 1 < i 2 < • • • < i m k }. Observe that we have i n + L in + 1 ≤ i n+1 for n = 1, . . . , m k -1 and L im k = l -i m k + 1. Since i m k ≤ l, we get i 1 <i 2 <•••<im k L i j (i j + 1) ≤   m k -1 j=1 (i j+1 -(i j + 1))(i j + 1)   + (l + 1 -i m k )(i m k + 1) ≤ (l + 1) 2 .

By construction we have

δ k ≤ log 2 (k) + 1. Since log 2 (k) ≤ l ≤ log 2 (k) + 1, we conclude 4k -2(log 2 (k) + 2) 2 -log 2 (k) -1 ≤ n k ≤ 4k -2 log 2 (k) -1
and the estimate n k -4k = O(log 2 k) holds. Finally let us consider λ j = j l=0 2 2l . An easy calculation gives

n λ j = 4λ j -2 j l=0 (2l + 1) -1 = 4λ j -2j 2 -4j -3.
Since λ j = (4 j+1 -1)/3, the sequence

n λ j -4λ j log 2 (λ j )
converges to a non-zero real number.

We now prove that the sequence (n k ) constructed above not only has positive lower density but has also positive lower B 2 -density.

Lemma 3.4. We have d B 2 ((n k )) > 0.
Proof. Using (6) from Example 2.5, we have

d B 2 (n k ) = lim inf k→+∞ k j=1 e n j / log 2 (n j ) log 2 (n k )e n k / log 2 (n k )
.

According to Proposition 3.3, there exists a constant C > 0 such that, for N large enough, 4k) .

k j=N e n j / log 2 (n j ) log 2 (n k )e n k / log 2 (n k ) ≥ k j=N e (4j-C log 2 (j))/ log 2 (4j-C log 2 (j)) log 2 (4k)e 4k/ log 2 (
A summation by parts gives

k j=N e (4j-C log 2 (j))/ log 2 (4j-C log 2 (j)) ∼ log 2 (k) 4 e (4k-C log 2 (k))/ log 2 (4k-C log 2 (k)) , as k → +∞.
Finally, a similar computation as those needed for Example 2.5, yields

k j=N e (4j-C log 2 (j))/ log 2 (4j-C log 2 (j)) log 2 (4k)e 4k/ log 2 (4k) ∼ e -C 4 , as k → +∞,
which finishes the proof.

Lemma 3.4 allows us to show that the Frequent Hypercyclicity Criterion gives a strengthened result. Proposition 3.5. Let T be an operator on a separable Fréchet space X. If there is a dense subset X 0 of X and a map S : X 0 → X 0 such that, for any x ∈ X 0 ,

(i) +∞ n=0 T n x converges unconditionally, (ii) +∞ n=0 S n x converges unconditionally, (iii) T Sx = x, then T is B 2 -frequently hypercyclic.
The proof of this result is the same as the classical proof of the Frequent Hypercyclicity Criterion. Indeed, from Lemma 3.4, we can deduce that the sets A(l, ν) not only have positive lower density but even have positive lower B 2 -density. We won't detail the proof here because we will prove a stronger result in Section 4.

Thanks to Lemma 2.9, one may actually deduce the following corollary proving that the scale defined by matrices A r is not fine enough to exhibit the limit in term of densities of the Frequent Hypercyclicity Criterion.

Corollary 3.6. Under the assumptions of the previous proposition, the operator T is A rfrequently hypercyclic for every 0 ≤ r < 1.

The previous result proves that for any 0 ≤ r < 1, the A r -frequent hypercyclicity phenomenon exists and is even common. On the other hand, one may also notice that the geometric rate of growth (i.e. r = 1) is unreachable in terms of dynamics. More precisely, we have the following result. Proposition 3.7. There is no A 1 -frequently hypercyclic operator.

Proof. We argue by contradiction. Assume that T is a A 1 -frequently hypercyclic operator on a Banach space X and x is a A 1 -frequently hypercyclic vector. Let also U be a non-empty open subset in X. Then, by definition and with Example 2.5, we get 0 < lim inf

N →+∞ k≤N e k N j=1 e j 1 N (x,U ) (k) = lim inf N →+∞ (1 -e -1
)

k≤N e k-N 1 N (x,U ) (k).
Moreover one may remark that asserting that this limit is non-zero implies that the set N (x, U ) has bounded gaps. Indeed, if one suppose that N (x, U ) has unbounded gaps then there exists a sequence (N i ) and a sequence (p i ) tending to +∞ such that for every i ∈ N, {N i -p i + 1; N i -p i + 2; . . .

; N i } ∩ N (x, U ) = ∅. This gives 0 < lim inf i→+∞ (1-e -1
)

k≤N i e k-N i 1 N (x,U ) (k) ≤ lim inf i→+∞ (1-e -1 ) k≤N i -p i e k-N i = lim i→+∞ e -p i -1 = 0
and this contradiction shows that the set N (x, U ) has bounded gaps. Let us denote by M an upper bound of the length of these gaps. It suffices to choose V so far from the origin such that the norm of T forbids T k (U ) from intersecting V for k ≤ M . This means that the orbit of x will never reach the open set V contradicting the A 1 -frequent hypercyclicity of x.

On the other hand, observe that the following result holds.

Lemma 3.8. For every 0 < r < 1, d Br (n k ) = 0.

Proof. We have to estimate the following limit

d Br (n k ) = lim inf k→+∞ k j=1 e n j / log r (n j ) n k+1 -1 j=1 e j/ log r (j)
.

Remark that by definition of δ k , there exists an increasing sequence of integers

(λ k ) k∈N such that n λ k +1 -n λ k -1 = δ λ k = k + 1 ∼ log 2 (λ k ), as k tends to ∞ (consider for example λ k = 2 k+1 -1). Then, d Br (n k ) ≤ lim inf k→+∞   λ k j=1 e n j / log r (n j ) n λ k +1 -1 j=1 e j/ log r (j)   ≤ lim inf k→+∞   n λ k j=1 e j/ log r (j) n λ k +1 -1 j=1 e j/ log r (j)   .
Using the estimate (6) from Example 2.5 and the one from Proposition 3.3, we get:

d Br (n k ) ≤ lim inf k→+∞ log r (n λ k )e n λ k / log r (n λ k ) log r (n λ k +1 -1)e n λ k / log r (n λ k ) ≤ lim inf k→+∞ e n λ k / log r (n λ k )-(n λ k +δ λ k )/ log r (n λ k +δ λ k )
We begin by studying the term in the exponent

n λ k log r (n λ k ) - n λ k + δ λ k log r (n λ k + δ λ k ) = n λ k log r (n λ k + δ λ k )     1 + log 1 + δ λ k n λ k log(n λ k )   r -1 + δ λ k n λ k  
which reduces to the following thanks to a Taylor expansion:

n λ k log r (n λ k + δ λ k ) 1 log(n λ k ) -1 + o δ λ k log 1+r (n λ k )
.

Now combining the estimate δ λ k ∼ log 2 (λ k ) as k tends to ∞ with the one given by Proposition 3.3, we deduce that δ λ k / log 1+r (n λ k )→0, as k tends to ∞. Hence we get

e n λ k / log r (n λ k )-(n λ k +δ λ k )/ log r (n λ k +δ λ k ) ∼ e n λ k log r ( n λ k +δ λ k ) 1 log(n λ k ) -1 -→ k→+∞ 0.
This proves that d Br (n k ) = 0.

Notice that Proposition 3.3 combined with Lemma 3.8 do not allow us to conclude to the B r -frequent hypercyclicity or not in the Frequent Hypercyclicity Criterion for 1 ≤ r < 2.

Further results

In this section, we are going to improve the conclusion of the Frequent Hypercyclicity Criterion given by Proposition 3.5. To do this, we will modify the sequence (n k ) used in the proof of Lemma 3.1 to obtain a new sequence possessing a positive A-density for an admissible matrix A defining a sharper density than the natural density.

Throughout this section, (a n ) will be an increasing sequence of positive integers with a 1 = 1. Using this sequence we define the function f : N → N, by f (j) = m for all j ∈ {a m , . . . , a m+1 -1}. In the spirit of the sequence studied in the previous section, we also define the sequence (n k (f )) by induction:

n 1 (f ) = f (1) = 1 and n k (f ) = n k-1 (f ) + f (δ k-1 ) + f (δ k ) for k ≥ 2.
Clearly we obtain the following equality, for all k ≥ 2, (4.1)

n k (f ) = 2 k-1 i=1 f (δ i ) + f (δ k ).
Let us notice that, if we set a m = m for every m ≥ 1, then the corresponding sequence (n k (f )) is the sequence (n k ) of Section 3. From now on, we will omit the notation f in (n k (f )) for sake of readability. Our purpose is to compute an exact formula for the new sequence (n k ) to understand its asymptotic behavior. First of all, we obtain an expression for the subsequence (n 2 am ).

Lemma 4.1. For all m ∈ N, we have

n 2 am = 2 am+1 m i=1 1 2 a i -1 -2m + 1.
Proof. Set ∆ (m) j = {0 ≤ l ≤ 2 am -1 : δ l = j}. First let us observe that we have, by definition, for every 1 ≤ j ≤ a m ,

n 2 am = 2 2 am -1 k=1 f (δ k ) + f (δ 2 am ) = 2 am j=1 f (j)#∆ (m) j + f (δ 2 am ).
Thus it suffices to compute the cardinal of the set ∆ (m) j . It easily follows that #∆ (m) j = 1 + am-j-1 i=0 2 am-j-i . Indeed, we separate the case when the first block of ones in the dyadic decomposition of l ends on 2 am-1 and the case when the first block of ones ends before. In the first case, we have no choice, there is only one possibility but in the second case we have a certain number i of zeros at the beginning, then the first block of ones, which is of length j, then one zero (because the first block of ones has to be of length j) and then we have 2 am-j-i possible choices as shown below.

( length am+1 0, 0, . . . , 0 length i , 1, 1, . . . , 1, 1, 0 length j+1 , , , . . . , , , 0, 0, . . .).

A quick calculation leads to #∆

(m) j = 1 + am-j-1 i=0 2 am-j-i = 2 am-j . Therefore we get

n 2 am = 2 am j=1 f (j)2 am-j + f (δ 2 am ) = 2 am j=1 f (j)2 am-j + 1.
Now we use the link between the values of f (j) and the position of j compared to the sequence (a m ) to compute the sum:

n 2 am = 2 am j=1 f (j)2 am-j + 1 = 2 am+1 am-1 j=1 f (j)2 -j + 2m + 1.
Let us now split the sum according to the values of f (j):

n 2 am = 2 am+1 m-1 i=1   a i+1 -1 j=a i f (j)2 -j   + 2m + 1 = 2 am+1 m-1 i=1 i   a i+1 -1 j=a i 2 -j   + 2m + 1 = 2 am+1 m-1 i=1 i 1 2 a i -1 - 1 2 a i+1 -1 + 2m + 1 = 2 am+1 m i=1 1 2 a i -1 -2m + 1.
We strengthen the previous lemma as follows.

Lemma 4.2. Let m ∈ N. For every q ∈ N such that q < a m -a m-1 , the following equality holds

n 2 am-q = 2 am-q+1 m-1 j=1 1 2 a j -1 -2(m -1) + 1.
Proof. This proof works along the same lines as the proof of Lemma 4.1. Thus, we adapt the preceding proof. It yields

n 2 am-q = 2 2 am-q -1 j=1 f (δ j ) + f (δ 2 am-q ) = 2 am-q j=1 f (j)2 am-q-j + 1 = 2 am-q+1     m-2 i=1 a i+1 -1 j=a i i2 -j   + am-q j=a m-1 (m -1)2 -j   + 1 = 2 am-q+1 m-1 j=1 1 2 a j -1 -2(m -1) + 1. Lemma 4.3. For k = 2 n + n-2 i=0 α i 2 i with α i ∈ {0; 1}, 0 ≤ i ≤ n -2
, and (α 0 , . . . , α n-2 ) = (0, . . . , 0), we have

n k-2 n = 2 k-1 i=2 n +1 f (δ i ) + f (δ k ). For k = 2 n + 2 n-1 + n-2 i=0 α i 2 i with α i ∈ {0; 1}, 0 ≤ i ≤ n -2, we have n k-2 n = 2 k-1 i=2 n +1 f (δ i ) -f (δ k-2 n ) -2(f (L) -1) + 2f (δ k ),
where L is the length of the block of one's containing the coefficient one of 2 n in the dyadic decomposition of k.

Proof. We begin by proving the first assertion. We have

n k-2 n = 2 k-2 n -1 i=1 f (δ i ) + f (δ k-2 n ).
Since k = 2 n + n-2 i=0 α i 2 i , observe that for all 2 n + 1 ≤ i ≤ k -1 the dyadic decomposition of i contains a one for some 2 l with 0 ≤ l ≤ n -2 and the coefficient of 2 n-1 is zero. Therefore, the first block of ones in the dyadic decomposition of i does not contain the coefficient of 2 n . Thus, for every such 2 n + 1 ≤ i ≤ k -1, we have δ i = δ i-2 n . This proves the first part of the lemma since δ k = δ k-2 n .

To prove the second assertion, we begin by observing that

2 k-1 i=2 n +1 f (δ i ) = 2 k i=2 n +1 f (δ i ) -2f (δ k ).
Then, as above if the index i is such that the coefficient of 2 n does not belong to the first block of ones (in the dyadic decomposition of i) then δ i = δ i-2 n . On the other hand, if the coefficient of 2 n belongs to the first block of ones and since we have a n-1 = 1 then i has to be of the form i = p l=0 2 n-l for p ∈ {1, . . . , L -1} and δ i = δ i-2 n + 1. Now let us pick a particular index i of the form i = p l=0 2 n-l with p ∈ {1, . . . , L -1}. We consider two cases: Case 1: for every j ∈ {2, . . . , f (L)}, we have p + 1 = a j . Then we have f

(δ i ) = f (p + 1) = f (δ i -1) = f (δ i-2 n ).
Case 2: there exists an integer j ∈ {2, . . . , f (L)} with p+1 = a j . Then we get f

(δ i ) = f (a j ) = j = f (δ i -1) + 1 = f (δ i-2 n ) + 1. Finally we deduce 2 k-1 i=2 n +1 f (δ i ) = 2 k i=2 n +1 f (δ i ) -2f (δ k ) = 2 k-2 n i=1 f (δ i ) + (f (L) -1) -2f (δ k ) = n k-2 n + f (δ k-2 n ) + 2(f (L) -1) -2f (δ k ).
From Lemma 4.3 we deduce the following result. Lemma 4.4. Let L be any non-zero integer and q be an integer. If k = L-1 j=0 2 q+j + k with 0 ≤ k < 2 q-1 then either k = 0 and

n k = n k + L-1 j=0 n 2 q+j + 2 L j=2 (f (j) -1) + L, or k = 0 and n k = L-1 j=0 n 2 q+j + 2 L j=2 (f (j) -1) + L -f (L).
Proof. We proceed by induction on L. For L = 1, set k = 2 q + k . First, observe that if k = 0, the result is clear by Lemma 4.3. So assume that 0 < k < 2 q-1 . We divide n k into two sums

n k = 2 2 q -1 i=1 f (δ i ) + f (δ 2 q ) + f (δ 2 q ) + 2 k-1 i=2 q +1 f (δ i ) + f (δ k ) .

It suffices to apply Lemma 4.3 to obtain

n k = n 2 q + f (δ 2 q ) + n k = n 2 q + 1 + n k
and we have the desired conclusion. Now choose L ≥ 2 and suppose that the result holds for every integer l, with 1 ≤ l ≤ L -1. By Lemma 4.3, we get

n k = k-1 i=1 f (δ i ) + f (δ k ) = 2 q+L-1 -1 i=1 f (δ i ) + 2f (δ 2 q+L-1 ) + 2 k-1 i=2 q+L-1 +1 f (δ i ) + f (δ k ) = n 2 q+L-1 + f (δ 2 q+L-1 ) + n k-2 q+L-1 + f (δ k-2 q+L-1 ) + 2(f (L) -1) -f (δ k ).
We have f (δ 2 q+L-1 ) = 1. Moreover suppose that 0 < k < 2 q-1 , then the block of ones containing the coefficient one of 2 q+L-1 is not the first one, thus δ k = δ k = δ k-2 q+L-1 . Using the induction hypothesis, we obtain

n k = n 2 q+L-1 + n k-2 q+L-1 + 2(f (L) -1) + 1 = n k + L-2 j=0 n 2 q+j + 2 L-1 j=2 (f (j) -1) + L -1 + n 2 q+L-1 + 2(f (L) -1) + 1 = n k + L-1 j=0 n 2 q+j + 2 L j=2 (f (j) -1) + L.
On the other hand, in the case k = 0, the induction hypothesis gives

n k = n 2 q+L-1 + 1 + n k-2 q+L-1 + f (L -1) + 2(f (L) -1) -f (L) = L-2 j=0 n 2 q+j + 2 L-1 j=2 (f (j) -1) + (L -1) -f (L -1) + n 2 q+L-1 + 1 + f (L -1) + 2(f (L) -1) -f (L) = L-1 j=0 n 2 q+j + 2 L j=2 (f (j) -1) + L -f (L).
This completes the proof.

From Lemma 4.4, we immediately get the following result since f (1) = 1. Lemma 4.5. Let L 1 , . . . , L r be non-zero integers and q 1 , . . . , q r be integers such that q i +L i < q i+1 for every

1 ≤ i ≤ r -1. For k = r i=1 L i -1 j=0 2 q i +j we have n k = r i=1 L i -1 j=0 n 2 q i +j + 2 r i=1 L i j=1 f (j) - r i=1 L i -f (L 1 ).
We are ready to obtain a general formula for the sequence (n k ). Let us introduce some notations.

Notations 4.6. Let L 1 , . . . , L r be non-zero integers and q 1 , . . . , q r be integers such that q i + L i < q i+1 for every 1 ≤ i ≤ r -1. We define the integers m i , t i , s i and p i as follows:

(1) m i is the greatest integer such that a

m i ≤ q i + L i -1, ( 2
)
t i = q i + L i -1 -a m i , ( 3
) p i = #{l ∈ N : l < m i and q i ≤ a l ≤ q i + L i -1}, ( 4
) s i = L i -1 -t i -(a m i -a m i -p i ).
To understand these notations, we give the following representation. Now we state an explicit formula for the sequence (n k (f )). This will allow us to obtain a good asymptotic formula for this sequence. j=0 2 q i +j we have

n k (f ) = 2k +∞ l=1 1 2 a l -1 - r i=1   p i -1 u=0   a m i -u -a m i -(u+1) j=1 2 a m i -u -j+1     +∞ l=m i -u 1 2 a l -1   +   t i j=0 2 am i +j+1     +∞ l=m i +1 1 2 a l -1   + s i l=1 2 a m i -p i -l+1   +∞ l=m i -p i 1 2 a l -1     -2 r i=1   am i +t i j=a m i -p i -s i f (j) - L i j=1 f (j)   -f (L 1 ).
Proof. We only prove the lemma for t i < L i , the other case being similar but simpler. We use the notations (4.6) to write (4.2)

L i -1 j=0 n 2 q i +j = L i -1 j=0 n 2 am i +t i -j = t i j=0 n 2 am i +j + L i -1-s i -t i j=1 n 2 am i -j + s i j=1 n 2 a m i -p i -j .
It remains to compute these three sums. We begin by the second one, dropping for the moment the index i for sake of readability. Using Lemma 4.2, we write

L-1-s-t j=1 n 2 am-j = p-1 u=0 a m-u -a m-(u+1) j=1 n 2 a m-u -j = p-1 u=0 a m-u -a m-(u+1) j=1   2 a m-u -j+1 m-(u+1) l=1 1 2 a l -1 -2(m -(u + 1)) + 1   .
Thus, we deduce

(4.3) L-1-s-t j=0 n 2 am-j = p-1 u=0   a m-u -a m-(u+1) j=1 2 a m-u -j+1     m-(u+1) l=1 1 2 a l -1   -2 p u=1 (m -u)(a m-(u-1) -a m-u ) + L -s -t.
In the same spirit we compute the first and third sums as follows (4.4)

t j=0 n 2 am +j =   t j=0 2 am+j+1   m l=1 1 2 a l -1 -2m(t + 1) + t + 1 and (4.5) s j=1 n 2 a m-p -j = s l=1 2 a m-p -l+1   m-(p+1) l=1 1 2 a l -1   -2(m -(p + 1))s + s.
Moreover since we have by definition t i < a m i+1 -a m i and s i < a m i -p i -a m i -p i -1 , when we gather equations (4.3), (4.4) and (4.5), we have to compute the following sum (4.6)

p i u=1 (m i -u)(a m i -(u-1) -a m i -u ) + m i (t i + 1) + (m i -(p i + 1))s i = p i u=1 a m i -(u-1) -a m i -u -1 j=0 f (a m i -u + j) + t i j=0 f (a m i + j) + s i j=1 f (a m i -p i -j) = am i +t i j=a m i -p i -s i f (j).
Thus thanks to Lemma 4.5 and the equations (4.2), (4.3), (4.4), (4.5), (4.6), we deduce

n k = r i=1 L i -1 j=0 n 2 q i +j + 2 r i=1 L i j=1 f (j) - r i=1 L i -f (L 1 ) = r i=1   p i -1 u=0   a m i -u -a m i -(u+1) j=1 2 a m i -u -j+1     m i -(u+1) l=1 1 2 a l -1   +   t i j=0 2 am i +j+1   m i l=1 1 2 a l -1 + s i l=1 2 a m i -p i -l+1   m i -(p i +1) l=1 1 2 a l -1   -2 am i +t i j=a m i -p i -s i f (j) + L i   + 2 r i=1 L i j=1 f (j) - r i=1 L i -f (L 1 ).
We remark that a r i=1 L i comes out from the first sum and cancels the term lying in the end of the preceding equality, we also gather the sums over f (j) and we get:

n k = r i=1   p i -1 u=0   a m i -u -a m i -(u+1) j=1 2 a m i -u -j+1     m i -(u+1) l=1 1 2 a l -1   +   t i j=0 2 am i +j+1   m i l=1 1 2 a l -1 + s i l=1 2 a m i -p i -l+1   m i -(p i +1) l=1 1 2 a l -1     + 2 r i=1   L i j=1 f (j) - am i +t i j=a m i -p i -s i f (j)   -f (L 1 ).
Then, we express the partial sums N l=1 1 2 a l -1 as the series minus its remainder of order N which yields:

n k = +∞ l=1 1 2 a l -1 r i=1   p i -1 u=0   a m i -u -a m i -(u+1) j=1 2 a m i -u -j+1   + t i j=0 2 am i +j+1 + s i l=1 2 a m i -p i -l+1   - r i=1   p i -1 u=0   a m i -u -a m i -(u+1) j=1 2 a m i -u -j+1     +∞ l=m i -u 1 2 a l -1   +   t i j=0 2 am i +j+1     +∞ l=m i +1 1 2 a l -1   + s i l=1 2 a m i -p i -l+1   +∞ l=m i -p i 1 2 a l -1     + 2 r i=1   L i j=1 f (j) - am i +t i j=a m i -p i -s i f (j)   -f (L 1 ).
Now it suffices to remark that coming back to notations with q i 's, then k can be expressed in the following way

k = r i=1   p i -1 u=0   a m i -u -a m i -(u+1) j=1 2 a m i -u -j   + t i j=0 2 am i +j + s i l=1 2 a m i -p i -l   .
Thus we obtain:

n k = 2k +∞ l=1 1 2 a l -1 - r i=1   p i -1 u=0   a m i -u -a m i -(u+1) j=1 2 a m i -u -j+1     +∞ l=m i -u 1 2 a l -1   +   t i j=0 2 am i +j+1     +∞ l=m i +1 1 2 a l -1   + s i l=1 2 a m i -p i -l+1   +∞ l=m i -p i 1 2 a l -1     + 2 r i=1   L i j=1 f (j) - am i +t i j=a m i -p i -s i f (j)   -f (L 1 ).
Observe that if we set (a m ) m = (m) m , then f (j) = j for every integer j and Lemma 4.7 takes the following form. Lemma 4.8. In the aforementioned case, if k = 2 n + n-1 i=0 α i 2 i with α i ∈ {0; 1} for every

0 ≤ i ≤ n -1, then n k = 4k -2   i∈I k L i (i + 1)   -δ k ,
where I k stands for the set of integers i such that α i is the first non-zero integer of a block (of consecutive non-zero coefficients) having length L i in the dyadic decomposition of k.

Proof. Using the notations of Lemma 4.7, we have:

t i = s i = 0, L i = p i + 1, m i = i + L i -1.
Therefore we deduce

n k = 4k -4 r i=1 (p i + 1) -2 r i=1   i+L i -1 j=i j - L i (L i + 1) 2   -L 1 = 4k -4 r i=1 L i - r i=1 (2i + L i -1)L i + r i=1 L i (L i + 1) -L 1 = 4k -2 r i=1 L i (i + 1) -L 1 .
This last inequality gives the result since L 1 = δ k .

Lemma 4.8 is exactly Lemma 3.4 announced in the previous section.

Let us return to the general situation, using the notation (n k (f )) again. From Lemma 4.7, we deduce the following estimate on the sequence (n k (f )) for specific choices of functions f. Lemma 4.9. Using the previous notations, assume that a m = 2 2 . . . 2 m , where 2 appears s times (s ≥ 1). Then the associated function f s : N → N is given by f s (j) = m, for j ∈ {a m , . . . , a m+1 -1}, and the following estimate holds:

2k +∞ l=1 1 2 a l -1 -2 log 2 (k)f s ( log 2 (k) )-14 log 2 (k)-8f s ( log 2 (k) ) ≤ n (s) k ≤ 2k +∞ l=1 1 2 a l -1 , with n k (f s ) = n (s) k .
Proof. We need Lemma 4.7 and its notations. The proof of the upper bound is obvious. For the lower bound, observe first that the subadditivity of f s implies that for k =

r i=1 L i -1 j=0 2 q i +j , L i j=1 f s (j) - am i +t i j=a m i -p i -s i f s (j) = L i j=1 (f s (j) -f s (a m i -p i -s i -1 + j)) ≥ -L i f s (a m i -p i -s i ).
In addition, since for every u ≥ 1, we have a u + 2 < a u+2 and +∞ l=q 2 -j = 2 1-q , we obtain:

  t i j=0 2 am i +j+1     +∞ l=m i +1 1 2 a l -1   = 4   t i j=0 2 am i +j     +∞ l=m i +1 1 2 a l   ≤ 4   t i j=0 2 am i +j   1 2 a m i +1 + 1 2 a m i +1 +1 + 1 2 a m i +1 +2 ≤ 7 2 a m i +1 -am i 2 t i +1 -1 ≤ 7 2 a m i +1 -(am i +t i +1) ≤ 7.
In the same spirit, we also have

s i l=1 2 a m i -p i -l+1   +∞ l=m i -p i 1 2 a l -1   = 4 s i l=1 2 a m i -p i -l   +∞ l=m i -p i 1 2 a l   ≤ 4 s i l=1 2 a m i -p i -l 1 2 a m i -p i + 1 2 a m i -p i +1 + 1 2 a m i -p i +2 ≤ 7 s i l=1 2 -l ≤ 7.
The same method gives again

p i -1 u=0   a m i -u -a m i -(u+1) j=1 2 a m i -u -j+1     +∞ l=m i -u 1 2 a l -1   ≤ 7 p i -1 u=0   a m i -u -a m i -(u+1) j=1 2 -j   ≤ 7p i .
Finally we gather these estimates and we obtain

n (s) k ≥ 2k +∞ l=1 1 2 a l -1 -7 r i=1 (2 + p i ) -2 r i=1 L i f s (a m i -p i -s i ) -f s (L 1 ) ≥ 2k +∞ l=1 1 2 a l -1 -7(2r + m r ) -2 r i=1 L i f s (a mr + t r ) -f s (L 1 ).
Using the fact that a mr + t i = q r + L r -1 ≤ log 2 (k) < q r + L r and k = r i=1 L i -1 j=0 2 q i +j , we get

n (s) k ≥ 2k +∞ l=1 1 2 a l -1 -7(2 log 2 (k) + f s ( log 2 (k) )) -2 log 2 (k)f s ( log 2 (k) ) -f s ( log 2 (k) ) ≥ 2k +∞ l=1 1 2 a l -1 -2 log 2 (k)f s ( log 2 (k) ) -14 log 2 (k) -8f s ( log 2 (k) ).
This finishes the proof.

We now prove that the sequence (n (s) k ) constructed above not only has positive lower density but has also positive lower B s -density, for every s ≥ 2. Proof. According to (7) from Example 2.5, we write

d Bs (n (s) k ) = lim inf k→+∞   k j=1 e n (s) j /hs(n (s) j ) h s (n (s) k )e n (s) k /hs(n (s) k )   .
Observe that Lemma 4.9 ensures the existence of two constants C 1 , C 2 > 1 such that for N large enough, k j=N e n (s)

j /hs(n (s) j ) h s (n (s) k )e n (s) k /hs(n (s) k ) ≥ k j=N e (C 1 j-C 2 hs(j))/hs(C 1 j-C 2 hs(j)) h s (C 1 k)e C 1 k/hs(C 1 k) .
A summation by parts gives

k j=N e (C 1 j-C 2 hs(j))/hs(C 1 j-C 2 hs(j)) ∼ h s (k) C 1 e (C 1 k-C 2 hs(k))/hs(C 1 k-C 2 hs(k)) , as k → +∞.
Then, a similar computation as those needed for (7) from Example 2.5 leads to the following estimate

k j=N e (C 1 j-C 2 hs(j))/hs(C 1 j-C 2 hs(j)) h s (C 1 k)e C 1 k/hs(C 1 k) ∼ e -C 2 C 1 , as k → +∞,
which gives the desired conclusion.

This allows to prove the following combinatorial lemma which extends Lemma 3.1.

Lemma 4.11. There exist pairwise disjoint subsets B (s) (l, ν), l, ν ≥ 1, of N having positive B s density such that, for any n ∈ B (s) (l, ν) and m ∈ B (s) (k, µ), we have that n ≥ f l (ν) and

|n -m| ≥ f l (ν) + f l (µ) if n = m.
Proof. We consider the sequence (n

k ) constructed above and also sets I(l, ν) constructed in [START_REF] Grosse-Erdmann | Linear Chaos[END_REF] that we recalled just after Lemma 3.1. We also define B (s) (l, ν) := {n

k ≥ f s (δ k ) = f s (ν). Finally, if n (s) j ∈ B (s) (l, ν) and n (s) m ∈ B (s) (k, µ) with j > m, then n (s) j -n (s) m = f (δ m ) + 2 j-1 i=m+1 f (δ i ) + f (δ j ) ≥ f (µ) + f (ν).
This strengthened version of Lemma 3.1 allows us to give a stronger conclusion to the so-called Frequent Hypercyclicity Criterion whose proof will be only sketched since it is an adaptation of the classical proof given in [10, Theorem 9.9]. Theorem 4.12. Let T be an operator on a separable Fréchet space X. If there is a dense subset X 0 of X and a map S : X 0 → X 0 such that, for any x ∈ X 0 ,

(i) ∞ n=0 T n x converges unconditionally, (ii) ∞ n=0 S n x converges unconditionally, (iii) T Sx = x,
In the same paper, the authors also provide examples of a U-frequently hypercyclic weighted shift which is not frequently hypercyclic and of a frequently hypercyclic weighted shift which is not distributionally chaotic. In what follows, we modify these constructions in order to provide a frequently hypercyclic weighted shift on c 0 (N) which is not A r -frequently hypercyclic for any 0 < r < 1. To that purpose, we will need the following lemma [5, Lemma 1]: Lemma 5.2. There exist a > 1 and ε > 0 such that d ∪ u≥1 I a,4ε u < 1 and, for any integer

u > v ≥ 1, I a,2ε u ∩ I a,2ε v = ∅, I a,2ε u -I a,2ε v ⊂ I a,4ε u , where I a,ε u = [(1 -ε)a u , (1 + ε)a u ].
The philosophy of the previous lemma is that it suffices to choose a very large and at the same time ε very small to obtain the result stated. This allows us to strengthen this lemma demanding also that the following condition holds:

(5.1) 1 -ε 1 + ε a > 1.
From now on, we suppose that a and ε are given by the previous lemma with the additional condition (5.1).

Let also (b p ) be an increasing sequence of integers such that (5.2)

q≥1 (4q + 1)(2q + 1) b q e 2q < ∞ and b p ≥ 8p.
Finally, let (A p ) be any syndetic partition of N and

E p = ∪ u∈Ap (I a,ε u ∩ (b p N + [0, p])
) . Bayart and Ruzsa construct such sets and they prove that these sets have positive lower density [5, Lemma 2]. Then, for the same reasons we have d(E p ) > 0. Further, the following lemma is almost the same as [5, Lemma 3] once again and it still holds in our context:

Lemma 5.3. Let p, q ≥ 1, n ∈ E p , m ∈ E q with n = m. Then |n -m| > max(p, q).
In particular, (E p + [0, p]) ∩ (E q + [0, q]) = ∅ if p = q. Thus, the sequence of sets (E p ) satisfy conditions (a) and (b) from Theorem 5.1.

We now turn to the construction of the weights of the weighted shift we are looking for. For this construction, we also draw our inspiration from constructions made in [START_REF] Bayart | Difference sets and frequently hypercyclic weighted shifts[END_REF]. We set:

w p 0 • • • w p k-1 = 1 if k / ∈ b p N + [-4p, 4p] 2 p if k ∈ b p N + [-2p, 2p]
and for every k ∈ N, 1 2 ≤ w p k ≤ 2. Then for p, q ≥ 1, u ∈ A p and v ∈ A q with u > v we define

w u,v 0 • • • w u,v k-1 = 1 if k / ∈ I a,4ε u max(2 p , 2 q ) if k ∈ I a,ε u -I a,ε v + [0, p]
and for every k ∈ N, 1 2 ≤ w u,v k ≤ 2. We are now able to give the definition of the weight w. This one is constructed in order to satisfy the following equality:

w 0 • • • w n-1 = max p,u,v w p 0 • • • w p n-1 , w u,v 0 • • • w u,v n-1 .
It is clear by construction that for every n ∈ N, 1 2 ≤ w n ≤ 2, so the weighted backward shift B w is bounded and invertible. Moreover this construction satisfies condition (c) in Theorem 5.1.

Since we want to prove that B w is frequently hypercyclic, the only condition left to prove is condition (d) from Theorem 5.1. Thus let p, q ≥ 1, n ∈ E p and m ∈ E q with m > n and t ∈ [0, q]. Then we have two cases:

• If p = q, then m-n+t ∈ b q N+[-q, 2q] and the definition of w ensures that w 0 • • • w m-n+t ≥ 2 q .

• If p = q, then there exists u > v such that n ∈ I a,ε v and m ∈ I a,ε u . Thus, by definition of w,

w 0 • • • w m-n+t ≥ max (2 p ; 2 q ) ≥ 2 p+q 2 ≥ 2 p 2 • 2 q 2 .
Now, one may define M (p) := 2 p 2 and each case above satisfies condition (d) from Theorem 5.1. Thus we have proved that the weighted shift B w is frequently hypercyclic.

We now turn to the A r -frequent hypercyclicity of B w . We are going to prove by contradiction that B w is not A r -frequently hypercyclic for every 0 < r < 1. Let us suppose that B w is A r -frequently hypercyclic and that E ⊂ N is such that d Ar (E) > 0 and lim n→∞,n∈E w 1 • • • w n = +∞. Such a set exists since B w is A r -frequently hypercyclic. Indeed it suffices to consider E = {n ∈ N : B n w (x) -e 0 ≤ 1 2 } where x is a A r -frequently hypercyclic vector.

For every p ≥ 1, we consider the set:

F p = {n ∈ E : w 1 • • • w n > 2 p }.
This set is a cofinite subset of E, so it has the same lower A r -density. We also consider an increasing enumeration (n k ) of A p . Then An classical calculation ensures that we have Recall that this does not require any property on p so we can let p tend to infinity which, thanks to (5.2), implies that d Ar (E) = lim p→∞ d Ar (F p ) = 0, hence we obtain a contradiction. Thus the weighted shift B w is not A r -frequently hypercyclic. From this construction together with Corollary 3.6, we deduce the following result.

Theorem 5.4. There exists a frequently hypercyclic operator being not A r -frequently hypercyclic, for any 0 < r ≤ 1, hence which does not satisfy the Frequent Hypercyclicity Criterion.

S

  n x converges unconditionally, (iii) T Sx = x, then T is frequently hypercyclic.

Lemma 4 . 7 .

 47 Using the notations (4.6), for k = r i=1 L i -1

Lemma 4 . 10 .

 410 We have d Bs ((n(s) k )) > 0.

k

  ; k ∈ I(l, ν)}. These sets are clearly pairwise disjoint since the sets I(l, ν) are, and the sequence is (n (s) k ) increasing. Moreover by definition of the sets I(l, ν), that are arithmetic sequences, and Lemma 4.9, the conclusion of Lemma 4.10 remains true, i.e. the sets B (s) (l, ν) have positive lower B s -density. Then by definition of n (s) k from (4.1), we get n (s)

d 1 rn r 1 rn r 1 r 1 r

 1111 Ar (F p ) ≤ lim inf k→∞ )a n k , n∈Fp e n r + (1+ε)a n k <n≤(1-ε)a n k+1 , n∈∪ q>p( bq N+[-2q,2q]) we have n≤N e n r ∼ 1 r N 1-r e N r , as N tends to ∞, we get d Ar (F p ) ≤ lim inf k→∞ ((1 + ε)a n k ) 1-r e ((1+ε)a n k ) r + (1+ε)a n k <n≤(1-ε)a n k+1 , n∈∪ q>p( bq N+[-2q,2q])e ((1 -ε)a n k+1 ) 1-r e ((1-ε)a n k+1 ) r using inequality (5.1) proves that the first term on the righthand side tends to 0. We now focus on the second term:d Ar (F p ) ≤ lim inf k→∞ a n k <n≤(1-ε)a n k+1 , n∈∪ q>p( bq N+[-2q,2q])e ((1 -ε)a n k+1 ) 1-r e ((1-ε)a n k+1 ) r ((1 -ε)a n k+1 ) 1-r e ((1-ε)a n k+1 ) r .

,

  as k tends to ∞. Thus we obtaind Ar (F p ) ≤ lim inf (1 -ε)a n k+1 ) 1-r e ((1-ε)a n k+1 ) r ε)a n k+1 +2q) r -((1-ε)a n k+1 ) r 2q)e 2q .

then T is B s -frequently hypercyclic, for every s ≥ 2.

Proof. Let (y n ) be a dense sequence from X 0 that is dense in X. Let . denote an F -norm that defines the topology of X. The unconditional convergence of the series (ii) and (iii) allows to find, for every l ∈ N, an integer N l ≥ 1 such that for every j ≤ l and every finite set

Now let (M l ) be an increasing sequence such that f s (M l ) ≥ N l and (f s (M l )) is increasing. We also define

defines a B s -frequently hypercyclic vector for T . From this point, the proof is just an adaptation of the proof of the Frequent Hypercyclicity Criterion from [START_REF] Grosse-Erdmann | Linear Chaos[END_REF] replacing Lemma 3.1 by Lemma 4.11 stated above.

We may also deduce the following corollary using Lemma 2.9.

Corollary 4.13. Under the assumptions of the previous proposition, the operator T is B rfrequently hypercyclic for every r > 1.

A frequently hypercyclic operator which is not A r -frequently hypercyclic

In this final section, we are going to show that there exists frequently hypercyclic operator, that do not belong to the class of A r -frequently hypercyclic operator, for any 0 < r ≤ 1. According to Proposition 3.5 or Theorem 4.12 such an operator cannot satisfy the Frequent Hypercyclicity Criterion. To build it, we are going to use several ideas of the work [START_REF] Bayart | Difference sets and frequently hypercyclic weighted shifts[END_REF], where the authors provide some counterexamples to questions regarding frequent hypercyclicity.

In a recent paper, Bayart and Ruzsa gave a characterization of frequently hypercyclic weighted shifts on the sequence spaces p and c 0 . We recall here their result on c 0 (N) that will be useful in the following [START_REF] Bayart | Difference sets and frequently hypercyclic weighted shifts[END_REF]Therorem 13].

Theorem 5.1. Let w = (ω n ) n∈N be a bounded sequence of positive integers. Then B w is frequently hypercyclic on c 0 (N) if and only if there exist a sequence (M (p)) of positive real numbers tending to +∞ and a sequence (E p ) of subsets of N such that: (a) For any p ≥ 1, d (E p ) > 0; (b) For any p, q ≥ 1, p = q, (E p + [0, p]) ∩ (E q + [0, q]) = ∅; (c) lim n→∞, n∈Ep+[0,p] ω 1 • • • ω n = +∞; (d) For any p, q ≥ 1, for any n ∈ E p and any m ∈ E q with m > n, for any t ∈ {0, . . . , q},


Using GIS tools to assess the urban environment influence on the particle concentration variability in Paris

OBJECTIVES

Particle is one of the most important megacities' air pollution problems, with public health issues.

To assess personal exposure in Paris, we made few measurements of particles concentrations by foot and by bike. The results show that particles concentration and size distribution are spatially highly variable due to different emissions and air mass movements.

At the scale of the city, street geometry (orientation, width…), the built-up environment (density, color, materials…) and the green spaces also strongly affect the spatial distribution of pollutant concentrations.

sara.duche@gmail.com / malika.madelin@univ-paris-diderot.fr ICUC9, 20-24/7/2015, Toulouse, France We rasterized vector layers (street orientation and land cover) to calculate the mean/proportion of SVF, street orientation and different types of land to each pollution data buffer with the raster calculator.

SOME RESULTS

Influence of terrigeneous road coarsers particles

Correlations between vegetation/street orientation and particle concentration are not significant for these days.

The correlations with the buffers of 100 m are higher than other the buffers.

The results with the buffers of 200 m are not significant.

0.65 0.7 0.75 0.8 0.85 0.9 0.95 0 10 20 30 40 50 Integration of a large database, existing and with low cost and miniaturized sensors.

Sky View Factor

Reflexion on spatial scales.

  Sarah DUCHÉ & Malika MADELIN University Paris Diderot, Sorbonne Paris Cité, UMR PRODIG, Paris, France To define urban morphology parameters, which could explain the spatial variability of particles concentration in Parisian street.Using GIS analysis to study what urban morphology indexes affect the spatial distribution of particle concentration at local scale.

  and PM2.5 concentration in Parisian streets to assess cyclists' exposure to air pollution: in June 2014, between 16 pm and 18 pm during anticyclonic condition, wind speed ± 5 m/s, from north. by bike with Dustmate sensor (Turnkey Instruments) & GPS ( measurement period = 30 seconds) the measured data show a different spatial variation between smaller particle (PM2.5) and coarser particle (PM10). The highest levels of PM10 are in parks (for example, Jardin des Plantes), whereas the highest concentration of PM2.measuring point, the correction coefficient between the ratio PM2.5/PM10 and the proportion of pedestrian road is equal at 0,52. 27% of the variance between smaller and coarser particles is explained by the suspension of coarser particles with breezes and from the soil.

(

  mean in 100 m buffer) PM2,5/PM10 (%) R = -0,35 Low correlation between particle size and SVFPERSPECTIVESAutomation with different days, and meteorological and emission conditions.


domain: sdv phys

text:
9!

While many molecular players involved in growth control have been identified in the past 10! decades, it is often unknown how they mechanistically act to induce specific shape changes 11! during development. Plant morphogenesis results from the turgor-induced yielding of the 12! extracellular, load-bearing cell wall. Its mechano-chemical equilibrium appears as a fundamental 13! link between molecular growth regulation and the effective shape evolution of the tissue. Here, 14! we focus on force-driven polymerization of the cell wall as a central process in growth control.

15!

We propose that mechanical forces facilitate the insertion of wall components, in particular 16! pectins, a process that can be modulated through genetic regulation. We formalize this idea in a 17! mathematical model, which we subsequently challenge with published experimental results.

Introduction

19!

The link between gene regulation and growth, i.e. the irreversible spatial expansion of a cell or a 20! tissue, is a major issue in plant and animal developmental biology. However, although many 21!

5!

components. Such a coupling between mechanics and biochemistry, termed force-driven 63! polymerization, has also been described for the assembly of actin bundles in animal systems [START_REF] Kruse | Aster, Vortices and Rotating Spirals in Active Gels of Polar 327! Filaments[END_REF].

64!

Starting from known descriptions of cell wall structure (see [START_REF] Wolf | Growth Control and Cell Wall Signaling in Plants[END_REF] and references therein) we 65! expose how force-driven polymerization could be central in scaling up molecular processes into 66! viscoplastic behavior and ultimately growth of multicellular structures.

67!

The cell wall is a fiber-reinforced hydrogel

68!

The cell wall is composed of a network of rigid cellulose microfibrils embedded in a matrix, 69! mostly composed of water, polysaccharides such as pectins and hemicelluloses, proteins and ions 70! [START_REF] Caffall | The structure, function, and biosynthesis of plant cell 331! wall pectic polysaccharides[END_REF]. This matrix corresponds to a biphasic mixture between a porous solid phase and a liquid 71! one [START_REF] Iijima | Swelling behaviour of calcium pectin hydrogels by 333! thermomechanical analysis in water[END_REF]. Therefore, the cell wall as a whole can be considered as a fiber-reinforced hydrogel.

72!

From a functional point of view, the wall performs two contrasting functions, as it has to resist 73! turgor-induced constraints whereas simultaneously it should be able to expand. Because of their 74! mechanical characteristics [START_REF] Vincent | From cellulose to cell[END_REF][START_REF] Zhang | Visualization of the nanoscale pattern of recently-deposited cellulose 336! microfibrils and matrix materials in never-dried primary walls of the onion epidermis[END_REF], the cellulose microfibrils tethered to each other by shorter 75! hemicellulose chains are major actors in both of these functions (for reviews see: [START_REF] Wolf | Growth Control and Cell Wall Signaling in Plants[END_REF][START_REF] Cosgrove | Re-constructing our models of cellulose and primary cell wall 339! assembly[END_REF] [START_REF] Cosgrove | How do plant cell walls extend?[END_REF].

76!

The available experimental evidence suggests that cellulose microfibrils deposition is involved in 77! growth inhibition rather than in growth promotion per se. Indeed, after drug-induced inhibition of 78! cellulose deposition, cells can continue to grow during several tens of hours [START_REF] Heisler | Alignment between PIN1 Polarity and Microtubule Orientation 342! in the Shoot Apical Meristem Reveals a Tight Coupling between Morphogenesis and Auxin 343! Transport[END_REF][START_REF] Pelletier | A role for pectin de-methylesterification in a developmentally 345! regulated growth acceleration in dark-grown Arabidopsis hypocotyls[END_REF]. Similarly, 79! growth also occurs in certain mutant backgrounds where cellulose synthesis is impaired [START_REF] Fagard | PROCUSTE1 Encodes a Cellulose Synthase Required for Normal 348! Cell Elongation Specifically in Roots and Dark-Grown Hypocotyls of Arabidopsis[END_REF][START_REF] Hématy | A receptor-like kinase mediates the response of Arabidopsis cells 351! to the inhibition of cellulose synthesis[END_REF]. 80! These experimental clues suggest that growth of the cell wall relies on the expansion of the 81! embedding matrix. In other words a scenario has emerged, where cell wall expansion is restricted 82! by the synthesis of cellulose microfibrils and promoted by the synthesis of matrix material. In the 83! following paragraphs we will discuss an important component of this matrix, i.e. pectin. As we 84! ! 6! will see, there is increasing evidence that pectin could play an important role in growth control.

85!

Pectin is an important structural component of the wall matrix

86!

As indicated above, hemicelluloses such as xyloglucans have been classically considered as 87! essential structural components, central to concepts of wall loosening and induction of growth. 88! However, the recently produced Arabidopsis thaliana double mutant (xxt1/xxt2), showing a loss 89! of xyloglucan synthesis, does not show major growth deficiencies [START_REF] Park | A revised architecture of primary cell walls based on 353! ! 18! biomechanical changes induced by substrate-specific endoglucanases[END_REF][START_REF] Cavalier | Disrupting two Arabidopsis thaliana xylosyltransferase genes 356! results in plants deficient in xyloglucan, a major primary cell wall component[END_REF], suggesting that the 90! irreversible expansion of the cell wall does not rely, at least totally, on hemicellulose dynamics 91! only. This has revived the interest in the role of pectins in growth control. Pectins form up to 35 92! % of the primary cell wall of dicots [START_REF] Caffall | The structure, function, and biosynthesis of plant cell 331! wall pectic polysaccharides[END_REF] i.e. the wall surrounding young and fast growing cells.

93!

Their dynamics have been associated with growth control in several cell types, in particular tip-94! growing cells such as pollen tubes and root hairs, for review see [START_REF] Wolf | Growth Control and Cell Wall Signaling in Plants[END_REF] and references therein, see 95! also e.g. [START_REF] Rojas | Chemically Mediated Mechanical Expansion of the Pollen Tube 359! Cell Wall[END_REF], [START_REF] Parre | Pectin and the role of the physical properties of the cell 361! wall in pollen tube growth of Solanum chacoense[END_REF]. Moreover, AFM-based assays on Arabidopsis thaliana, demonstrated that 96! pectin demethyl-esterification was required and tightly regulated during cell wall softening prior 97! to aerial organ outgrowth [START_REF] Braybrook | Mechano-Chemical Aspects of Organ Formation 363! in Arabidopsis thaliana: The Relationship between Auxin and Pectin[END_REF], [START_REF] Peaucelle | The Control of Growth Symmetry Breaking in the Arabidopsis 365! Hypocotyl[END_REF]. In addition, recent evidence indicates the existence of close 98! interactions between cellulose and pectins which could function as tethers between the 99! microfibrils [START_REF] Cosgrove | Re-constructing our models of cellulose and primary cell wall 339! assembly[END_REF][START_REF] Wang | Cellulose-Pectin Spatial Contacts Are Inherent to Never-Dried 367! Arabidopsis Primary Cell Walls: Evidence from Solid-State Nuclear Magnetic Resonance[END_REF]. Dwarf phenotypes in mutants where pectin synthesis or delivery is impaired 100! further illustrate the importance of pectin production for plant growth [START_REF] Bouton | QUASIMODO1 encodes a putative membrane-bound 370! glycosyltransferase required for normal pectin synthesis and cell adhesion in Arabidopsis. 371![END_REF][START_REF] Gendre | Trans-Golgi network localized ECHIDNA/Ypt interacting protein 373! complex is required for the secretion of cell wall polysaccharides in Arabidopsis[END_REF]. Noteworthy, 101! pectins have also been identified as important regulators of growth in more primitive plants such 102! as the green algae [START_REF] Proseus | Calcium pectate chemistry controls growth rate of 376! Chara corallina[END_REF][START_REF] Proseus | Pectate chemistry links cell expansion to wall 378! deposition in Chara corallina[END_REF]. roughly account for 65% of all of them [START_REF] Mohnen | Cell wall polysaccharide synthesis[END_REF]. A fundamental feature of homogalacturonans is 108! their ability to form multiple intermolecular chelation bonds in presence of calcium cations.

109!

These cations-dependent bonds are able to glue together pectin molecules, forming structures 110! called "egg-boxes" (Fig. 1). In a cation-rich environment, these adhesive interactions lead to the 111! formation of a hydrogel [START_REF] Wolf | Homogalacturonan Methyl-Esterification and Plant Development. 382![END_REF]. In addition, pectins are able to interact with other cell wall 112! components including cellulose and hemicellulose [START_REF] Popper | Xyloglucan-pectin linkages are formed intra-384! protoplasmically, contribute to wall-assembly, and remain stable in the cell wall[END_REF].

113!

An interesting feature is that newly synthesized homogalacturonans cannot spontaneously form 114! egg-box structures. Indeed, in order to trap divalent cations, the carboxyl groups of the 115! homogalacturonan chains must be negatively charged. When synthesized, carboxyl residues are 116! shielded by methyl groups, which must be removed to induce gelification. Therefore, two types 117! of pectin molecules can be distinguished: (i) "inactive" methyl-esterifed M-pectin and (ii) 118! "active", demethyl-esterified pectin, able to aggregate into macromolecular assemblies.

119!

Demethyl-esterification is carried on within the cell wall by specialized enzymes, the pectin 120! methyl-esterases (PMEs). In turn, PME-inhibitors (PMEIs) can modulate PME activity. This 121! process is essential, for both the degree and the pattern of demethyl-esterification influence the 122! mechanical/rheological properties of the constituted gel [START_REF] Peaucelle | Pectin-Induced Changes in Cell Wall Mechanics Underlie Organ 387! Initiation in Arabidopsis[END_REF]. Accordingly, modifications in the 123! levels of PME and PMEI expression significantly alter growth rates [START_REF] Peaucelle | Arabidopsis phyllotaxis is controlled by the methyl-esterification 389! status of cell-wall pectins[END_REF]. Note that other charged 124! residues such as acetyl-esters [START_REF] Caffall | The structure, function, and biosynthesis of plant cell 331! wall pectic polysaccharides[END_REF] can also be present on homogalacturonans and their regulation 125! could play a similar role. However, for the sake of simplicity we only consider methyl-ester 126! groups here. Noteworthy, 66 PMEs and 69 PMEIs have been reported in the Arabidopsis 127! genome, attesting for the importance of the process [START_REF] Wolf | Homogalacturonan Methyl-Esterification and Plant Development. 382![END_REF][START_REF] Sénéchal | Homogalacturonan-modifying enzymes: structure, expression, and 391! roles in plants[END_REF]. The PME/PMEI system appears as a 128! key regulator of cell wall expansion:

129! ! 8!
• First, by regulating the amount of "active" demethyl-esterified pectins, within the matrix 130! liquid phase, ready for insertion it influences its viscosity: The more active molecules 131! available, the faster the expansion. 132!

• Secondly, by removing more or less methyl groups on pectin molecules already stuck 133! together, it regulates the strength of the bonds within the matrix solid phase. This can be 134! interpreted as the tuning of the plastic yielding threshold of the cell wall.

135!

In conclusion, in the previous paragraphs we have seen that the cell wall can be considered as a 136! fiber reinforced hydrogel, which is put under tension by the osmotic pressure within the cells. To 137! grow, the cells have to yield to this pressure by expanding their walls in an irreversible manner.

138!

We will next discuss the chemical and physical processes that are at the basis of this expansion, 139! before indicating how molecular regulation can act to generate specific shapes of tissues and 140! organs. To be able to go beyond qualitative descriptions of growth regulation, we will present a 141! description in the form of a mathematical model. Finally we test this model using available 142! quantitative data 143!

Chemical equilibrium of the pectin hydrogel

144!

If single molecules in solution can bind to each other, macromolecular assemblies are generated.

145!

Assuming a high enough initial concentration of those molecules, such a mechanism eventually 146! leads to the formation of a hydrogel, i.e. the percolation of a macromolecular solid phase within a 147! solution of single molecules. The pectin matrix corresponds to such a biphasic system [START_REF] Iijima | Swelling behaviour of calcium pectin hydrogels by 333! thermomechanical analysis in water[END_REF]. When 148! the cell is growing, new matrix material has to be added: new components are exocytosed within 149! the liquid phase and components from the liquid phase have to be transferred into the solid one.

150!

Because the wall is a closed shell of constant thickness (i.e. with no border where an hypothetic 151! ! 9! nucleation mechanism could take place), a way to add new components to the solid phase is by 152! inserting them in between existing ones. Such a mechanism is sketched in Fig. 1a and formalized 153! in Eq.1, where ! !! and ! respectively depict one bond between two molecules in the solid phase 154! and a free molecule in the liquid one. 155!

! ! ! + !!! ⇄ !2! !! (1) 156!
This chemical equilibrium between the liquid and solid phase of the gel can be formalized by 157! standard concepts from thermodynamics. Indeed, phase equilibrium is reached once the free 158! energy (we will drop the "free" for now on) difference ∆! (= ! ! !"#$% ! -! !"#$!"% ) between the 159! two sides of Eq.1 vanishes. This energy difference reads

∆! = ! ! ! -! ! ! • !"!( ! /! !! ) and 160!
encompasses two general classic features. First, the stabilization of molecules by the adhesive 161! interactions between them is accounted for by the first term ! ! < 0 (the negative sign meaning 162! that adhesion stabilizes the system, i.e. lowers its energy). Second, the insertion of one molecule 163! from the liquid phase into the solid one comes with the loss of some degree of freedom. This is 164! expressed by the second term -! ! ! • !"!( ! /! !! ) a standard expression of mixing entropy, where 165! ! ! ! represents the thermal energy of the system (! ! and ! respectively stand for the Boltzmann 166! constant and the absolute temperature), ! depicts the monomer concentration in the liquid phase 167! and ! !! the water concentration in that same phase (≃ !55!"# • ! _ 1 ). In view of the above, the 168!

equilibrium condition ∆!! = !0 provides a value for a corresponding critical free molecule 169! concentration (with ! = 1 /! ! !): 170! ! ! * = ! ! ⋅ !"# !! ! ! (2) 171!
In other terms, the actual concentration c of monomers in solution controls the chemistry of the

172! ! 10! gel: 173! • If ! > ! ! * ! ⇔ ! ∆!! < 0, monomers spontaneously insert into the solid phase, growth 174! occurs. 175! • If ! < ! ! * ! ⇔ ! ∆! > 0, no spontaneous insertion, no growth occurs.

176!

The critical concentration ! ! * only depends on one parameter: the adhesive energy between 177! molecules in the solid phase (! !! in Eq.2). In order to modulate this equilibrium the cell only has 178! to tune the adhesive links between molecules. This can be achieved using enzymatic activities, 179! i.e. through molecular regulation. However, another efficient way to do so is by pulling on those 180! links (Fig. 1), as detailed in the following paragraph. 181!

Influence of mechanical forces on the chemical equilibrium

182!

A tensile force (!) applied on the solid phase can ease the breaking of existing bonds, thus 183! facilitating the insertion of soluble molecules. This shifts the chemical equilibrium towards 184! polymerizationi.e. towards the right-hand side of Eq.1. In this case, the energy difference has 185! to be lowered by the work !! = !! • ! generated by the tensile force (!) over a distance !, 186! characteristic of the deformation of loaded bonds:

∆! ! = ∆! -! • ! . Consequently, the 187! equilibrium condition ∆! ! = 0 yields, this time, a force-dependent expression for the equilibrium 188! concentration ! * : 189! ! * (!) ! = ! ! ! * • ! !!•!•! (3) 190!
This mechanism assures that tensile forces applied on a gel at equilibrium decrease its critical 191! concentration. As a consequence molecules in solution are recruited into the solid phase, (Fig. 1

& 192! ! 11!
2). This transduction mechanism is called force-induced polymerization and has been shown to 193! play an important role in the dynamics of major load-bearing structures such as the actin 194! cytoskeleton [START_REF] Kozlov | Processive capping by formin suggests a force-393! driven mechanism of actin polymerization[END_REF][START_REF] Courtemanche | Tension modulates actin filament polymerization mediated 395! by formin and profilin[END_REF][START_REF] Shemesh | Actin Polymerization upon Processive Capping by 397! Formin: A Model for Slowing and Acceleration[END_REF][START_REF] Shemesh | normalize concentration of 502! free active pectin molecules in the liquid phase. The orange curve is a visual representation of 503! Eq.3 and therefore represents the chemical equilibrium between the solid and the liquid phase of 504! the gel[END_REF]. Note that this mechanism is by essence irreversible, for when an increased 195! force is applied on a given configuration (black arrow on Fig. 2a) and then removed (dashed black 196! arrow on Fig. 2a) the system does not go back to its original position.

! ! ! = !! • !" • ! ! !•!!! ! (4) 205!
where the relative expansion rate (! ! ) is expressed as a function of the tensile force (!), 206! formalizing the out-of-equilibrium insertion reaction. ! > 0 is a proportionality constant and !" 207! represents the excess of free monomer in solution compared to the equilibrium concentration.

208!

Note that in an exhaustive approach the coefficient ! should depend on the elastic energy stored 209! in the solid phase. But in a turgid growing tissue this stored energy should be roughly constant 210! and can be ignored for the sake of simplicity.

211!

Reinforcing the gel: the cellulose microfibrils network steps in

212! ! 12!
Eq.4 depicts the expansion of a hydrogel under mechanical load, applied to the specific case of 213! pectin within the cell wall. A corollary question concerns the integration of the cellulose 214! microfibril network in this description.

215!

The entanglement between the pectin-based matrix and the cellulose fibrils leads to a complex 216! distribution of mechanical constraints at the molecular scale. An accurate mechanical analysis 217! would require a precise structural description of the considered wall and intensive numerical 218! simulations, similar to the ones produced by Kha et al [42] and Puri et al [43] in the case of cell 219! wall models containing only cellulose and hemi-cellulose fibers. Such a study lies outside of the 220! range of this article and we will only provide a qualitative analysis here.

221!

In a wall containing both pectin and cellulose, the force (!) felt by a pectin strand in a given 222! direction should in principle only be a fraction of the total force (! !"! ) exerted on a unit volume of 223! the wall. This fraction would depend, at the very least, on the relative concentration of cellulose 224! fibers compared to pectin strands ( ! !!! ) and the rigidity of those fibers ( ! ! ):

225! ! = !(! !"! , ! !!! , ! ! ) ≤ ! ! !"! .

226!

Controlling both the angular distribution of cellulose fibers and their rigidity appears as a way to 227! regulate the force exerted on pectin strands and consequently the expansion rate (! ! ) as 228! expressed in Eq.4. Following this idea, two stereotypical asymptotic cases can be evoked: 229!

• All the cellulose fibers are oriented perpendicular to the expanding pectin strand (c.f. Fig.

230!

1b-2). In this case, the cellulose fibers and the pectin strand can be considered in series 231! and the whole force exerted on the wall is transmitted to the strand: ! = ! !"! .

232!

• All the cellulose fibers are oriented in the same direction as the expanding pectin strand 233! ! 13! (c.f. Fig 1b-3). In this case, the cellulose fibers and the pectin strand lie in parallel and the 234! force felt by the pectin strand is smaller than the total one: ! ≈ ! !"! 1 + ! !!! ⋅ ! ! where 235! ! !!! and ! ! respectively account for the ratio of relative concentrations of pectin and 236! cellulose and the rigidity ratio between a pectin strand and a cellulose fiber.

237!

The qualitative point we want to make here is that enhancing the deposition of cellulose in a 238! given direction should directly down-regulate the expansion rate of the matrix in this specific 239! direction.

240!

Finally, it is worth mentioning that a detailed thermodynamical description of such a fiber-241! reinforced hydrogel should not only include dissipation between matrix molecules but also 242! between the fibers themselves and between the fibers and the matrix.

243!

Experiments on Chara fit the force-driven polymerization model

244!

We were not aware of quantitative experimental studies on higher plants focused on the 245! relationship between molecular composition of the cell wall, turgor-generated forces and the 246! morpho-dynamics of the whole tissue. Fortunately, such studies have been conducted on the 247! green algae Chara corallina [44,45] and we confronted the results with our model ( Fig. 3).

248!

In the case of isolated Chara cells, the mechanical force (!) exerted on the cell wall is 249! proportional to the pressure differential (∆!) between the inside and the outside of the wall. 250! Combined with Eq.4, this yields an exponential relationship between the pressure differential 251! exerted on a cell wall and its expansion rate. All other parameters kept constant, the ratio between 252! two relative growth rates ( ! ! /! ! ! ) as a function of the corresponding ratio of pressure 253!

differentials (∆!/∆! ! ) reads: 254! ! 14! ! ! ! ! ! = !"# C ! ⋅ ∆! ! ⋅ ∆! ∆! ! -1
(5) 255! see Fig. 3a for a comparison between experimental points and the fitting function.

256!

If we assume that the insertion of demethyl-esterified pectins is the molecular mechanism behind 257! cell wall expansion, the growing rate is an affine function of the concentration of demethyl-258! esterified pectin molecules in the solute phase. Fig. 3b depicts this behavior and qualitatively 259! matches experimental evidence of this proportionality exposed on Fig. 3

Discussion, perspectives & conclusion

269!

Our ambition was to demonstrate that a fundamental molecular mechanism (force-driven 270! polymerization) applied to a key actor (pectin) can be at the heart of a complex integrated 271! behavior such as morphogenesis.

272!

Although force-driven insertion appears as an appealing mechanism to explain cell wall 273! expansion at the molecular scale, it is worth noting that it might not be the only one possible. For 274! ! 15! instance, Rojas et al [START_REF] Rojas | Chemically Mediated Mechanical Expansion of the Pollen Tube 359! Cell Wall[END_REF] proposed a detailed study of pollen tube expansion using an apposition 275! mechanism: Newly demethyl-esterified pectin molecules are not inserted into the solid phase but 276! rather directly polymerized as a new layer next to the old one. This new layer modifies the 277! mechanical properties of the load-bearing cell wall and induces its viscous relaxation. Such a 278! mechanism does not exclude the model we propose here but is rather complementary: Rojas this mechanism provides a gateway to develop models that integrate quantitative geometrical 295! variables and biochemical equations (Fig. 4). In other words, it becomes possible to express 296! molecular activities in terms of geometrical outputs at the level of entire tissues and organs in a 297! ! 16! fully mechanistic manner. For this purpose, Eq.4 can be coupled to other differential equations 298! describing the dynamics of enzymatically-regulated variables. For instance one could assume that 299! the activation energy ! ! in Eq.4 depends on the demethyl-esterification degree of pectin 300! molecules and that pectin methyl esterases regulate its value. The influence of PME activity on 301! growth by coupling Eq.4 could then be explored using an equation describing PME dynamics.

302!

To assess the validity of the model a wide range of experiments can be considered. The first step 303! would be to extend the quantitative analysis produced by Proseus and Boyer to cells and tissues 304! from higher plants: quantitative tracking to follow cells and tissues submitted to various levels of 305! turgor pressure could be a start. In complement, rescue assays could be tested to restore a normal 306! growth phenotype to mutant or drug-treated plants by modulation of mechanical forces. Finally 307! and more speculatively, the use and analysis of biomimetic systems such as artificial walls could 308! also be very useful.

Glossary box:!

435!

Rheology: Study of the deformation of matter at a mesoscopic or macroscopic scale. 436! Viscosity: Rheological/mechanical property that relates the deformation rate of a material to its 437! mechanical load. For a given loading force, the more viscous a material is the slower it will 438! deform. Note that its inverse is called extensibility and is often used in growth modeling.

439!

! Thermal energy: The portion of the energy of a system due to the random relative microscopic 440! movements of its components. Absolute temperature of the system is, by definition, a direct 441! measure of this internal energy. For living systems which temperature is around 300 K the usual 442! value of their thermal energy is: The conservation equation of the number of bonds in the solid phase reads:

ℰ !! ∝ ! ! ! = 1 ! ≈ 4 • 10 !!" J, where ! ! = 1.38 ⋅ 10 !!" ! ⋅ 443! ! !!
449! !! ! !" = -! ! ! ! + ! ! (B1-1)

450!

The term of the left hand-side of Eq.B1-1 depicts the local time of the bond 451! density (noted ! ! ). The first term of the right hand-side is proportional to the relative 452! expansion rate of the cell wall (! ! ) and accounts for the fact that the matrix can expand.

453!

The minus sign in front of it insures that spatial expansion of the matrix (! ! > 0) tends to

458!

Because the cell wall is constantly under tension, the spacing between molecules in the 459! solid phase and therefore the bond density can be assumed constant over time (i.e.

460!

!! ! !" = 0 in Eq.B1-1). This assumption implies that the two terms of the right hand-461! side of Eq.B1-1 compensate each other, leading to a direct relationship between the 462! expansion rate of the cell wall and the pectin chemical insertion rate:

463! ! ! ! ! = ! ! (B1-2)

464!

The proper derivation of chemical insertion rate (! ! ) from thermodynamical

103 ! 7 !

 1037 Pectins are produced in the Golgi and exocytosed within the cell wall. They are composed of four 104! different polysaccharides: Homogalacturonan, Rhamnogalacturonan I, Rhamnogalacturonan II 105! and Xylogalacturonan. Different pectin molecules correspond to various combinations of those 106! ! three residues covalently attached together. Hereby, homogalacturonans stand out, for they 107!

197 !

 197 Linking mechanics and biochemistry to geometry: spatial expansion198! of the pectin matrix under tensile force 199! One difference between the growth of living tissues and the plasticity of non-living materials is 200! the mass increase of the former. Therefore, it seemed logical to seek for a growth law based on 201! the equation of mass conservation of the solid phase of the matrix (see Box 1 for details). 202! Formalizing the fact that the local expansion of the matrix is fuelled by the insertion mechanism 203! previously exposed yields (combination of Eq. 2, 3 & B1-4): 204!

  of [45]: putting Chara 260! cells in a pectin-enriched solution increased their growing response to mechanical stress. 261! Another qualitative feature of Chara growth observed by Proseus et al and reproduced by our 262! model is its dependency on temperature: Fig 3c depicts how the pressure-growth rate relationship 263! is affected by a slight decrease of temperature (!), the shift from the plain curve to the dashed 264! one has to be compared to the shift between the two curves exposed on Fig.13 of [44]. These 265! qualitative agreements between our theoretical model and experimental data strengthen the idea 266! that force-driven assembly mechanisms of the cell wall matrix play an important role in cellular 267! growth. 268!

  et al 279! also describe a transduction mechanism between pectin chemistry and cell wall mechanics, 280! focusing their attention on the chemical control of the process. This shift of perspective between 281! their approach and ours (i.e. the question of causality between the chemical and the mechanical 282! states of the cell wall) is a molecular version of the debate between Linthillac and Schöpfer that 283! took place a few years back [46,47]: Is growth controlled by wall-loosening enzymes or by 284! turgor-induced stresses ? Our point is that what matters is the resulting equilibrium between both 285! ends which is formalized throught the critical concentration Eq.3. 286! Force-driven polymerization processes have been proposed to play a major part in several 287! biological functions relying on active gel dynamics and mechano-chemical transduction, from 288! actin filament formation and focal adhesion formation [9], [41], [48] (Kozlov et al) in animal 289! cells to cell wall instability and cell division in bacteria [49]. In our opinion, force-driven 290! polymerization processes could be as central in mechanobiology as Michaelis-Menten 291! mechanisms in biochemistry. 292! Indeed, force-driven polymerization can be seen as a transduction mechanism between a 293! mechanical force and a chemical reaction. From a theoretical perspective, in its full 3D version 294!

••••

  Cellulose microfibrils have historically drained most of the attention of biomechanical 422! studies of the cell wall. Recent results, however, suggest that the soft pectin matrix in 423! which they are embedded might also play a significant role in the physico-chemical 424! equilibrium of growing cells. 425! At the molecular scale, biophysicists have shown how mechanical forces applied on 426! molecular assemblies can modulate their chemical state and therefore initiate specific 427! biological responses. 428! At the tissular scale, numerical simulation tools borrowed from material sciences are 429! increasingly used in developmental biology. Morphogenesis in plant tissues is particularly 430!suited for this kind of approach. 431! Bridging the gap between the molecular and the tissular scales is a major challenge in 432! developmental biology, for it would help relating explicitly shape changes and specific 433!

Box 1 :

 1 and ! = 300!! respectively stand for the Boltzmann constant and the absolute temperature. 444! Work of a force: Considering an object in motion and summited to a force, the work developed 445! by this force corresponds to the scalar product between the force vector and the displacement 446From matter conservation to force-induced expansion 448!

  454! diminish the local bond density. Finally, the second term of the right hand-side (named 455! the source term and denoted ! ! ) grasps the chemical insertion mechanism that transfers 456! molecules from the liquid phase into the solid one when the two phases are out of 457! equilibrium.

  Figures legends

Figure 1 : 25 !Figure 2 :

 1252 Figure 1: Pectin strand dynamics & mechanical load distribution.

Figure 3 :

 3 Figure 3: Growth rate dependency on demethyl-esterified pectin

Figure 4 :

 4 Figure 4: Transducing chemical & mechanical inputs into

  Proceedings of the National Academy of Sciences102, 12383-12388 400! 42 Kha, H. et al. (2010) WallGen, Software to Construct Layered Cellulose-Hemicellulose 401! Networks and Predict Their Small Deformation Mechanics. Plant Physiology 152, 774-786 402! 43 Yi, H. and Puri, V.M. (2012) Architecture-Based Multiscale Computational Modeling of 403! Plant Cell Wall Mechanics to Examine the Hydrogen-Bonding Hypothesis of the Cell Wall

	404!
	405!	Network Structure Model. Plant Physiology 160, 1281-1292
	406!	44 Proseus, T.E. et al. (2000) Turgor, temperature and the growth of plant cells: using Chara
	407!	corallina as a model system. J. Exp. Bot. 51, 1481-1494
	408!	45 Proseus, T.E. and Boyer, J.S. (2007) Tension required for pectate chemistry to control
	409!	growth in Chara corallina. J. Exp. Bot. 58, 4283-4292
	410!	46 Wei, C. and Lintilhac, P.M. (2003) Loss of stability-a new model for stress relaxation in
	411!	plant cell walls. Journal of theoretical biology 224, 305-312
	412!	47 SCHOPFER, P. (2008) Is the loss of stability theory a realistic concept for stress relaxation-
	413!	mediated cell wall expansion during plant growth? Plant Physiology 147(3), 935-936 author
	414!	reply 936-938
	415!	48 Nicolas, A. et al. (2004) Cell mechanosensitivity controls the anisotropy of focal adhesions.
	416!	Proceedings of the National Academy of Sciences 101, 12520-12525
	417!	49 Jiang, H. et al. (2011) Mechanical Control of Bacterial Cell Shape. Biophysj 101, 327-335
	418!	!
	419!	!	!
	309!
		! !	19! 20!

Growth