Jean-Baptiste Boyer

Central

CENTRAL LIMIT THEOREM AND LAW OF THE ITERATED LOGARITHM FOR THE LINEAR RANDOM WALK ON THE TORUS

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Introduction

Let Γ be a subgroup of SL d (Z) and ρ a probablity measure on Γ. The action of Γ on the torus X := T d = R d /Z d allows one to define a random walk, setting, for any x ∈ X,

X 0 = x X n+1 = g n+1 X n
where (g n ) ∈ Γ N is chosen with the law ρ ⊗N . We note P x the measure on X N associated to the random walk starting at x. Date: February 24, 2016.

The Markov operator associated to the walk is the one defined for any non-negative borelian function f on X and any x ∈ X by

P f (x) = G f (gx)dρ(g)
We note ν the Lebesgue measure on X. As ν is Γ-invariant, it is also P -invariant : for any continuous function f on X, X P f dν = X f dν One can prove that for any p ∈ [1, +∞], P is a continuous operator on L p (T d , ν) and P L p = 1.

In the sequel, we will need an hypothesis telling that the support of ρ is big. Let H be a closed subgroup of SL d (R). We say that the action of H on R d is strongly irreducible if H doesn't fix any finite union of proper subspaces of R d and we say that the action is proximal if there is some h ∈ H for which there are an h-invariant line V + h in R d and an h-invariant hyperplane V < h such that R d = V + h ⊕ V < h and the restriction of h to V < h has a spectral radius strictly smaller than the restriction of h to V + h . We say that a borelian probability measure ρ on G = SL d (R) has an exponential moment if for some ε ∈ R * + we have

G g ε dρ(g) < +∞
Under these assumptions (exponential moment and strongly irreducible and proximal action of the closed subgroup generated by the support of ρ), we know that P has a spectral radius strictly smaller than 1 in the orthogonal of the constant functions in L 2 (X, ν) (cf Furmann and Shalom in [FS99] and also Guivarc'h in [START_REF] Guivarc'h | Limit theorems for random walks and products of random matrices, Probability measures on groups: recent directions and trends[END_REF]). We will say in that case that P has a spectral gap in L 2 (X, ν).

In particular under these assumptions, for any function f ∈ L 2 (X, ν), there is a function g ∈ L 2 (X, ν) such that f = g -P g + f dν and the law of large numbers and the central limit theorem are already known for ν-a.e. starting point x ∈ T d (see for instance [GL78], [START_REF] Borodin | Limit theorems for functionals of random walks, Limit Theorems for Functionals of Random Walks[END_REF] and [DL03]) the variance in the central limit theorem beeing (1.1)

σ 2 (f ) = g 2 -(P g) 2 dν
In this article, we are interested in the study of the walk starting at an arbitrary point x ∈ T d .

It is easy to see that the rational points in T d have a finite Γ-orbit since any g ∈ Γ increases the denominator of such a point. So to study the walk starting at a rational point one can use the classical results for Markov chains with a finie number of states.

We define a measurable application ν : X → M 1 (X) (the set of probability measures on X) by ν x = ν (Lebesgue measure on X) if x ∈ Q d /Z d and ν x is the equidistributed measure on Γ ρ x if x ∈ Q d /Z d where Γ ρ is the subgroup of SL d (Z) generated by the support of ρ.

Bourgain, Furmann, Lindenstrauss and Mozes proved the following Theorem 1.1 ( [START_REF] Bourgain | Stationary measures and equidistribution for orbits of nonabelian semigroups on the torus[END_REF]). Let ρ be a probability measure on Γ = SL d (Z) having an exponential moment and whose support generates a strongly irreducible and proximal subgroup.

Then, for any x ∈ X, any continuous function f on X and ρ ⊗N -a.e. (g n ) ∈ Γ N ,

1 n n-1 k=0 f (g k . . . g 1 x) -→ f dν x
This theorem is the law of large numbers for the sequence (f (g n . . . g 1 x)) n∈N and we would like to study the central limit theorem and the law of the iterated logarithm. First, we will look at conditions on the function f for which the variance given by equation 1.1 vanishes. This was studied in [FS99] when the measure ρ is aperiodic : it's support is not contained in a class modulo a proper subgroup of G.

Let G be a locally compact group acting continuously on a topological space X preserving the probability measure ν.

We say that the action of G on X is ν-ergodic if every measurable G-invariant function is constant ν-a.e.

We will prove next Proposition (3.4). Let G be a locally compact group acting continuously and ergodically on a topological space X endowed with a G-invariant probability measure ν.

Then, for any g ∈ L 2 (X, ν), the following assertions are equivalent (1) P g 2 = g 2 (2) There is some subgroup H of G and some γ ∈ G such that g is H-invariant and supp ρ ⊂ Hγ.

Remark 1.2. In particular, there is a non-constant function g ∈ L 2 (X, ν) such that P g 2 = g 2 if and only if there is a subgroup H of G whose action on X is not ergodic and some element γ ∈ G such that supp ρ ⊂ Hγ.

From now on, we fix a norm . on R d which defines a distance on the torus, setting, for any x, y ∈ X, d(x, y) = inf p∈Z d

xyp

Where x (resp. y) is a representative of x (resp. y) in R d . We note C 0,γ (X) the space of γ-hölder continuous functions on X that we endow with the norm : for any f ∈ C 0,γ (X), (1.2)

f γ = sup x∈X |f (x)| + sup x,y∈X x =y |f (x) -f (y)| d(x, y) γ
To prove the central limit theorem and the law of the iterated logarithm, we will see in section 4 that the result of Bourgain, Furmann, Lindenstrauss and Mozes in [START_REF] Bourgain | Stationary measures and equidistribution for orbits of nonabelian semigroups on the torus[END_REF] allows one to have a speed of convergence depending on the diophantine properties of x (when f is hölder continuous). This will give us the Theorem (4.17). Let ρ be a probability measure on Γ = SL d (Z) having an exponential moment and whose support generates a strongly irreducible and proximal subgroup.

Then, for any γ ∈]0, 1] there is β 0 ∈ R * + such that for any B ∈ R * + and β ∈]0, β 0 [ we have that for any irrational point x ∈ X such that the inequality d x, p q e -Bq β has a finite number of solutions p/q ∈ Q d /Z d , we have that for any γ-holder continuous function f on the torus, noting σ 2 (f ) the variance given by equation 1.1 we have that

1 √ n n-1 k=0 f (X k ) L -→ N f dν, σ 2 (f ) (If σ 2 = 0, the law N (µ, σ 2 ) is a Dirac mass at µ). Moreover, if σ 2 (f ) = 0 then, P x -a.e., lim inf n-1 k=0 f (X k ) -f dν 2nσ 2 (f ) ln ln n = -1 and lim sup n-1 k=0 f (X k ) -f dν 2nσ 2 (f ) ln ln n = 1
and if σ 2 (f ) = 0, then for ν-a.e. x ∈ X, the sequence

( n-1 k=0 f (X k ) -f dν) n is bounded in L 2 (P x ).
Remark 1.3. Our condition is satisfied in particular for diophantine points of the torus. Therefore, the set of points where our theorem applies has Lebesgue-measure 1. But, the theorem also works for some Liouville numbers.

Our strategy to prove this result is Gordin's method. For a continuous function on the torus, we call Poisson's equation the equation f = g -P g + f dν where g is some unknown function. If this equation has a continuous solution (we already know, with a spectral gap argument, that a solution exists in L 2 (X, ν)) then, we can write for any x ∈ X and ρ ⊗N -a.e.

(g n ) ∈ Γ N , n-1 k=0 f (X k ) = g(X 0 ) -g(X n ) + n-1 k=0 g(X k+1 ) -P g(X k )
The key remark is that M n = n-1 k=0 g(X k+1 ) -P g(X k ) is a martingale with bounded increments and we can use the classic results for the martingales to prove the central limit theorem and the law of the iterated logarithm.

Here, in general, there cannot exist a continuous function g on the torus such that f = g -P g + f dν because this would imply that f has the same integral against all the stationary measures (in particular, we would have that f (0) = f dν). However, we will prove the theorem by showing that for any holder continuous function f on the torus, we can solve Poisson's equation at points having good diophantine properties (cf. section 4). Moreover, the solution we construct will not be bounded on X but will be dominated by a function u : X → [1, +∞] that we call drift function and that satisfies P u au + b for some a ∈]0, 1[ and b ∈ R. This equation means that if u(x) is large, then, in average, u(gx) is much smaller than u(x). Or in other words, the function g that we construct is not bounded but the walk doesn't spend much time at points x where |g(x)| is large.

The first section of this article consists in a study of drift functions and the proof of the central limit theorem and the law of the iterated logarithm for martingales with difference sequence bounded by drift functions.

In the second section, we study the variance appearing in the central limit theorem and the case where it vanishes.

Finally, in the third section, we solve Poisson's equation for points of the torus having good diophantine properties and we prove theorem 4.17.

Drift functions

In this section, we introduce and study some kind of functions that we call "drift functions" and that allow one to control the sequence (f (X n )) when f is dominated by one. Moreover, we prove the law of large number, the central limit theorem and the law of the iterated logarithm for martingales dominated by drift functions.

2.1. Definitions. In this section, (X n ) is a Markov chain on a standard borelian space X.

Definition 2.1 (Drift function). Let u : X → [1, +∞] be a borelian function and C a borelian subset of X.

We say that (u, C) is a drift function if u is bounded on C and if there is some b ∈ R such that P u u + b1 C In general, we will say that u is a drift function without indicating the set C.

Remark 2.2. These functions are studied by many authors and our main reference is [MT93] (see also [GM96]).

Meyn and Tweedie don't assume that u is bounded on C but that C is a so called petite-set and this allows them to prove that one can find a borelian set C ′ such that (u, C ′ ) is a drift function with our definition.

Remark 2.3. Many authors call Lyapunov function any non negative measurable function v : X → [1, +∞[ such that P v v. So, our drift functions are very close to Lyapunov functions.

As we assume that P u u + b1 C , we can study borelian functions f on X such that

(2.1) |f | u -P u + b1 C
We are going to see that we have a good control on the sequence (P n f ) (or, more specifically, on the series whose general terms involves the P n f ). Therefore, we set, for p ∈ R + ,

E p u := f : X → R f is borelian and ∃M ∀x ∈ X, |f (x)| M (u -P u + b1 C ) 1/p
And, for any f ∈ E p u , we set

f E p u = inf M ∈ R ∀x ∈ X, |f (x)| M (u -P u + b1 C ) 1/p
Remark 2.4. The space (E p u , . E p u ) is a Banach space. In the same way, we set, for any p ∈ [1, +∞[,

F p u := f : X → R f is borelian and ∃M, ∀x ∈ X, |f (x)| M u(x) 1/p
and, for f ∈ F p u ,

f F p u = sup x∈X |f (x)| u(x) 1/p
In next lemma, we use the control given by the drift function to prove that the space E 1 u is a subset of the space of integrable functions against the stationary measures for the Markov chain.

Lemma 2.5. Let u be a drift function and ν a borelian probability measure on X that is P -stationnary and such that ν(u < +∞) = 1.

Then, the identity operator defined from E p u (X) to L p (X, ν) is continuous.

Proof. (cf. lemma 3.8 in [BQ13])

Let f ∈ E p u be a non negative function, x ∈ X and n ∈ N * , then, by definition of

E p u , |f | p (x) f p E p u (u -P u + b)(x)
, and so,

1 n n-1 k=0 P k (|f | p )(x) f p E p u n (u -P n u + nb) f p E p u ( 1 n u(x) + b)
But, according to Chacon-Ornstein's ergodic theorem (see for instance the theorem 3.4 of the third chapter of [START_REF] Krengel | Ergodic theorems[END_REF]), there is a P -invariant function f * on X that is non negative and such that |f | p dν = f * dν and, for ν-a.e. x ∈ X,

1 n n-1 k=0 P k |f | p (x) -→ f * (x)
But, since u is finite ν-a.e., we get that f * (x) b f p E p u for ν-a.e. x ∈ X. And so, f * ∈ L ∞ (X, ν) ⊂ L 1 (X, ν) since we assumed that ν is a probability measure. This proves that, f ∈ L p (X, ν) and that f L p (X,ν) b 1/p f E p u .

2.2. The LLN, the CLT and the LIL for martingales.

In this section, we prove three of the classical results in probability theory for martingales with increments dominated by a drift function.

In particular we will prove that the central limit theorem and the law of the iterated logarithm for martingales can be deduced from a law of large numbers and this will be our corollary 2.13.

Remark 2.6. In this section, we make an assumption such as "f ∈ E p u for some p > 1" very often. The reader shall not be afraid of this assumption because in many examples we can construct families of drift functions and if f is dominated by one, f p will be dominated by an other one.

Before we state and prove corollary 2.13, we state some lemmas that we will also use in the study of the random walk on the torus. First, we extend the law of large numbers for martingales (stated in [Bre60]) for measurable functions f ∈ E p u for some p > 1: this will be our proposition 2.10. To prove it, we will use the following Lemma 2.7. Let u be a drift function, x ∈ X, and α ∈ R + , then

sup n∈N n k=0 P k (u -P u)(x) (k + 1) α u(x)
Proof. We can compute :

n k=0 P k (u -P u) (k + 1) α = n k=0 1 (k + 1) α P k u - n k=0 1 (k + 1) α P k+1 u = n k=1 ( 1 (k + 1) α - 1 k α )P k u + u - 1 (n + 1) α P n+1 u u(x) since u is non negative
Then, we prove that it is the same thing to study

1 n n-1 k=0 f (X n ) -P f (X n ) and 1 n n-1 k=0 f (X n+1 ) -P f (X n ).
Thus, having the law of large numbers, the central limit theorem and the law of the iterated logarithm for martingales, we will get these results for functions f that writes f = g -P g.

Lemma 2.8. Let u be a drift function and p > 1. Then, for any f ∈ E p u , any x ∈ X such that u(x) is finite and any ε ∈]0, p[, f (X n ) n 1/(p-ε) -→ 0 P xa.e. and in L 1 (P x ) Remark 2.9. We will use this lemma with p > 1 and pε = 1 and with p > 2 and pε = 2.

Proof. With the notations of the lemma, let's compute, for any n ∈ N,

E x |f (X n )| p f E p u E x u(X n ) -P u(X n ) + b = f E p u P n (u -P u + b)
And so, assuming without any loss of generality that f E p u = 1, we get

n k=0 E x |f (X k )| p (k + 1) p/(p-ε) n k=0 P k (u -P u) (k + 1) 1+ε/(p-ε) + b n k=0 1 (k + 1) 1+ε/(p-ε) u(x) + b n∈N * 1 n 1+ε/(p-ε)
where we used lemma 2.7 to control the first sum. Thus, for any x ∈ X such that u(x) is finie,

+∞ k=0 E x |f (X k )| (k + 1) 1/(p-ε)
p is finite and this finishes the proof.

Proposition 2.10. Let u be a drift function and p ∈]1, +∞[. For any f ∈ E p u and x ∈ X,

1 n n-1 k=0 f (X k+1 ) -P f (X k ) -→ 0 P x -a.e. and in L p (P x ) Proof. For any n ∈ N * , let M n = n-1 k=0 f (X k+1 ) -P f (X k ). Then, (M n ) is a martingale with EM n = 0 and E x |M n+1 -M n | p = E x |f (X n+1 ) -P f (X n )| p = P n (E x |f (X 1 ) -P f (x)| p ) 2 p-1 P n+1 (|f | p )(x) 2 p-1 f E p u P n+1 (u -P u + b) Thus, +∞ n=1 1 n p E x |M n+1 -M n | p 2 p-1 f E p u +∞ n=1 P n+1 (u -P u + b) n p 2 p-1 f E p u u(x) + b +∞ k=0 1 n p
And so, according to the law of large numbers for martingales (see the theorem 2.18 in [START_REF] Hall | Martingale limit theory and its application[END_REF]), we get that 1 n M n -→ 0 P x -a.e. and in L p (P x ). Lemma 2.11. Let u be a drift function such that P u au + b for some a ∈]0, 1[ and b ∈ R and let p ∈]1, +∞[.

Let ψ : N → R + be a decreasing function converging to 0 at +∞. Then, for any f ∈ E p u ,

1 n n-1 k=0 ψ(k)f (X k ) -→ 0 P x -a.e. and in L p (P x )
Proof. We shall assume without any loss of generality that f E p u = 1. To prove the convergence in L p , we compute

E x 1 n n-1 k=0 ψ(k)f (X k ) p 1/p 1 n n-1 k=0 ψ(k) (E x |f (X k )| p ) 1/p 1 n n-1 k=0 ϕ k P k u(x) 1/p 1 n n-1 k=0 ϕ k (u(x) + b/(1 -a)) 1/p
And we conclude with Cesaro's lemma.

To study the a.e.-convergence, in a first time, we are going to prove the for any x such that u(x) is finite,

(2.2) lim sup n 1 n n-1 k=0 |f (X k )| b(1 + b) 1/p 1 -a 1/p P x -p.s.
First, we remark that for any x ∈ X,

|f (x)| p u(x) -P u(x) + b (1 + b)u(x)
Moreover, for any r ∈]0, 1], we note u r the function defined by u r (x) = u(x) r . And so, using the concavity of the function (t → t r ), we get that

P u r (P u) r (au + b) r a r u r + b
This means that u r 1 1-a r (u r -P u r + b). And so, setting r = 1/p, we obtain

1 n n-1 k=0 |f (X k )| (1 + b) 1/p n n-1 k=0 u 1/p (X k ) (1 + b) 1/p 1 -a 1/p 1 n n-1 k=0 u 1/p (X k ) -P u 1/p (X k ) + b (1 + b) 1/p 1 -a 1/p 1 n u(x) + b + (1 + b) 1/p 1 -a 1/p 1 n n-1 k=0 u 1/p (X k+1 ) -P u 1/p (X k )
Moreover, by definition, u p 1/p = u ∈ E 1 u and so, using proposition 2.10, we get

1 n n-1 k=0 u 1/p (X k+1 ) -P u 1/p (X k ) -→ 0 P x -a.e.
This proves inequality 2.2. Thus, a.e., there is some n 0 ∈ N such that for n n 0 ,

1 n n-1 k=0 |f (X k )| 2 b(1 + b) 1/p 1 -a 1/p
And so, for n such that √ n n 0 , we get

1 n n-1 k=0 ψ(k)|f (X k )| ψ(0) n ⌊ √ n⌋-1 k=0 |f (X k )| + ψ(⌊ √ n⌋) n n-1 k=⌊ √ n⌋ |f (X k )| 2 b(1 + b) 1/p 1 -a 1/p ψ(0) √ n + ψ( √ n)
And, as ψ converges to 0, this finishes the proof of the lemma.

Using the same ideas as in the proof of proposition 2.10, we can prove the Lemma 2.12. Let u be a drift function and p > 2.

Let g ∈ E p u and x ∈ X such that u(x) is finite. Then, for any ε

∈ R * + 1 n n-1 k=0 E x (g(X k+1 ) -P g(X k )) 2 1 |g(X k+1 )-P g(X k )| ε √ n -----→ n→+∞ 0 and +∞ n=1 1 √ n E x |g(X n+1 ) -P g(X n )| 1 |g(X n+1 )-P g(Xn)| ε √ n is finite Finally, there is δ ∈ R * + such that +∞ n=1 1 n 2 E x (g(X n+1 ) -P g(X n )) 4 1 |g(X n+1 )-P g(Xn)| δ √ n is finite.
Proof. Using Markov's inequality, we can compute

E x h(X k+1 , X k ) 2 1 |h(X k+1 ,X k )| ε √ n P k (E (g(X 1 ) -P g(X 0 )) p ) ε p-2 n (p-2)/2 where we noted h(x, y) = g(x) -P g(y). But, E x [((g(X 1 ) -P g(X 0 )) p ] ∈ E 1 u , since we took g in E p u . So, 1 n n k=1 E x h(X k+1 , X k ) 2 1 |h(X k+1 ,X k )| ε √ n C n 1+(p-2)/2 ε p-2 n-1 k=0 P k (u -P u + b) C n 1+(p-2)/2 ε p-2 u(x) + bC n (p-2)/2 ε p-2
And the right side converges to 0 since u(x) is finite.

The two sums that we have to study are bounded by constants times

+∞ n=1 1 n 1+(p-2)/2 E x (|g(X n+1 ) -P g(X n )| p )
and, once again, using that g ∈ E p u , we get that E

x (|g(X n+1 ) -P g(X n )| p ) g E p u P n (u -P u + b)
And we shall conclude with lemma 2.7. Lemma 2.12 is important since it is a first step in the proof of the central limit theorem and the law of large numbers as we will see in next Corollary 2.13. Let u be a drift function and p > 2.

Let

g ∈ E p u and x ∈ X such that u(x) is finite. If 1 n n-1 k=0 P (g 2 )(X k ) -(P g(X k )) 2
converges in L 1 (P x ) and a.e. to some constant σ 2 (g, x), then,

1 √ n n-1 k=0 g(X k+1 ) -P g(X k ) L ---→ n→∞ N (0, σ 2 (g, x))
Where we noted N (0, 0) the Dirac mass at 0.

Moreover, if σ 2 (g, x) = 0 then, lim sup n-1 k=0 g(X k+1 ) -P g(X k ) 2nσ 2 (g, x) ln ln(n) = 1 a.e.
and

lim inf n-1 k=0 g(X k+1 ) -P g(X k ) 2nσ 2 (g, x) ln ln(n) = -1 a.e.
Proof. The central limit theorem comes from Brown's one (cf [Bro71]) since the " Lindeberg condition" is satified when g is dominated by a drift function as we saw in lemma 2.12. The law of the iterated logarithm is given by corollary 4.2 and theorem 4.8 in [HH80] since the assumption is satisfied according to lemma 2.12.

About the nullity of the variance

In this section, we study conditions under which the variance appearing in the central limit theorem and in the law of the iterated logarithm can not vanish. Let G be a locally compact group acting continuously on a topological space X preserving the probability measure ν.

We will always assume that the action of G on X is ν-ergodic : this means that every measurable G-invariant function is constant ν-a.e.

Let ρ be a probability measure on G and P the associated Markov operator on X. For any f ∈ L 2 (X, ν), we have, using Jensen's inequality, that

P f 2 2 = X G f (gx)dρ(g) 2 dν(x) G X |f (gx)| 2 dν(x)dρ(g) X |f (x)| 2 dν(x) = f 2 2
And so, the operator P is continuous on L 2 (X, ν) and P 1. It is clear that P = 1 since P 1 = 1.

In our study of the central limit theorem for some function f on X, the variance will always be given

σ 2 (f ) = g 2 2 -P g 2 2
where g ∈ L 2 (X, ν) is a function that we will have constructed such that ff dν = g -P g (in L 2 (X, ν)). It is therefore important to know if there can be some non-constant function g ∈ L 2 (X, ν) such that P g 2 = g 2 .

This question has been studied by Furman and Shalom in [FS99] where they prove that if the measure ρ is aperiodic (that is to say that it's support is not included in a class of a subgroup of G) then there is no non-constant function f ∈ L 2 (X, ν) such that P f 2 = f 2 . We prove in this section that the existence of such functions is equivalent to the existence of a subgroup H of G that does not act ν-ergodically on X and of some g ∈ G such that supp ρ ⊂ Hg. This will be our proposition 3.4.

If ρ is a borelian probability measure on G, we note ρ the symmetrized measure. It is the probability measure defined for any borelian subset A of G by

ρ(A) = G 1 A (g -1 )dρ(g) Remark 3.1. Since the measure ν is G-invariant, we can compute, for any f 1 , f 2 ∈ L 2 (X, ν), X f 2 P ρ f 1 dν = G X f 1 (gx)f 2 (x)dν(x)dρ(g) = G X f 1 (x)f 2 (g -1 x)dν(x) = X f 1 P ρf 2 dν
So, the operator P ρ is the adjoint operator of P ρ in L 2 (X, ν).

Remark 3.2. In our definition of P ρ , we make the element g act on the left. Thus, if ρ 1 , ρ 2 are borelian probability measures on G, for any f ∈ L 2 (X, ν) and any x ∈ X, we get

P ρ 1 P ρ 2 f (x) = G P ρ 2 f (gx)dρ 1 (g) = G G f (g 2 g 1 x)dρ 1 (g 1 )dρ 2 (g 2 ) = P ρ 2 * ρ 1 f (x)
Thus, P ρ 1 P ρ 2 is the operator associated to the measure ρ 2 * ρ 1 . This inversion doesn't have any consequence in this article (since we always convol a measure with it's powers) but in this section we have to remember that the measure associated to P * P is ρ * ρ.

First, we remark that for any f ∈ L 2 (X, ν),

f 2 2 -P f 2 2 = X f 2 (y) -(P f ) 2 (y)dν(y) = X f (y)(I d -P * P )f (y)dν(y)
where P * is the adjoint operator of P in L 2 (X, ν). Moreover, we saw that f 2 -P f 2 0.

Lemma 3.3. Let G be a group, S ⊂ G and S -1 = {g -1 |g ∈ S}.
Then, the subgroup of G generated by SS -1 is the smallest subgroup H of G such that there is g ∈ G with S ⊂ Hg.

Proof. First, let H be a subgroup of G and g ∈ G. If S ⊂ Hg then SS -1 ⊂ Hgg -1 H = H.
On the other hand, let H be a subgroup of G containing SS -1 and let g ∈ S.

Then, for any h ∈ S, we have that h = hg -1 g. But, hg -1 ∈ H and so h ∈ Hg. This proves that S ⊂ Hg.

What we proved is that for any subgroup H of G, we have the equivalence between "SS -1 ⊂ H" and "there is g ∈ G such that S ⊂ Hg". This proves the lemma since the subgroup of G generated by SS -1 is by definition the smallest subgroup of G containing SS -1 . Proposition 3.4. Let G be a locally compact group acting continuously and ergodically on a topological space X endowed with a G-invariant probability measure ν.

Then, for any f ∈ L 2 (X, ν), the three following assertions are equivalent (1) P f 2 = f 2 (2) For ν-a.e. x ∈ X and ρ * ρ-a.e. g ∈ G, f (gx) = f (x).

(3) There is some subgroup H of G and some g ∈ G such that f is H-invariant and supp ρ ⊂ Hg.

Remark 3.5. There can exist a non constant function f ∈ L 2 (X, ν) such that P f 2 = f 2 only if supp ρ is included in a right-class of a subgroup of G whose action on X is not ν-ergodic.

Proof. First, we remark that

X G |f (gx) -f (x)| 2 d(ρ * ρ)(g)dν(x) = X 2|f (x)| 2 -2ℜ(f (x)P * P f (x))dν(x) = 2 f 2 2 -2ℜ f P * P f dν = 2 f 2 2 -2 P f 2 2
So the first point implies the second one.

The second point implies that the function f is invariant by the subgroup generated by (supp ρ)(supp ρ) -1 . But, according to the previous lemma, this subgroup is precisely the smallest subgroup H of G such that there is g ∈ G with supp ρ ⊂ Hg. And so, the second point implies the third.

Finally, if there is some g in G and a subgroup H such that f is H-invariant and supp ρ ⊂ Hg, then, for ν-a.e. x ∈ X and any γ ∈ supp ρ, f (γx) = f (gx) and so,

P f (x) = G f (γx)dρ(γ) = f (gx) Thus, X |P f (x)| 2 dν(x) = X |f (gx)| 2 dν(x) = X |f (x)| 2 dν(x)
And the third point implies the first one.

Corollary 3.6. Let G be a locally compact group acting continuously and ν-ergodically on a topological space X endowed with a G-invariant probability measure ν.

Let ρ be a borelian probability measure on G. Let g ∈ L 2 (X, ν) such that ν x ∈ X sup n P n g 2 (x) < +∞ = 1 and note f = g-P g. Suppose that g 2 = P g 2 then, for ν-a.e. x ∈ X, the sequence

( n k=0 f (g k . . . g 1 x)) is bounded in L 2 (P x ).
Moreover if g belongs to L ∞ (X) then, for ν-a.e. x ∈ X, we have that the sequence

( n k=0 f (g k . . . g 1 x)) is bounded in L ∞ (P x ). Proof.
According to the previous proposition, if g = P g , there is some γ ∈ G and a subgroup H of G such that supp ρ ⊂ Hγ and g is H-invariant.

So, for ν-a.e. x ∈ X and ρ-a.e. g 1 ∈ G, g(g 1 x) = g(γx). In particular, P g(x) = g(γx) and so, f (x) = g(x)g(γx).

Thus, for ν-a.e. x ∈ X and ρ ⊗n -a.e.

(g i ) ∈ G n , n-1 k=0 f (g k . . . g 1 x) = n-1 k=0 g(g k . . . g 1 x) -g(γg k . . . g 1 x) = g(x) -g(g n . . . g 1 x) + n-1 k=0 g(g k+1 . . . g 1 x) -g(γg k . . . g 1 x) = g(x) -g(g n . . . g 1 x)
This computation proves the corollary when the function g is bounded.

Moreover, we have that

G N n-1 k=0 f (g k . . . g 1 x) 2 dρ ⊗N ((g i )) = g(x) 2 + P n (g 2 )(x) -2g(x)P n g(x) g(x) 2 + P n (g 2 )(x) + 2|g(x)| P n (g 2 )(x) 4 sup n P n (g 2 )(x)
Where we used Jensen's inequality to say that |P n g(x)| P n g 2 (x). This finishes the proof of the corollary.

The following example is an illustration of the previous corollary in an explicit context.

Example 3.7. Let

A = 2 1 1 1 et B = 0 1 -1 0
Then, the subgroup of SL 2 (R) generated by A and B is Zariski-dense and the Lebesgue measure ν on the torus [START_REF] Guivarc'h | Limit theorems for random walks and products of random matrices, Probability measures on groups: recent directions and trends[END_REF] that the operator P associated to ρ has a spectral gap in L 2 (T 2 , ν).

T 2 = R 2 /Z 2 is ergodic. Let ρ = 1 2 δ A + 1 2 δ BA . Guivarc'h proved in
Let . be the distance induced on T 2 = R 2 /Z 2 by the euclidean norm on R 2 . And let g be the function defined for any x ∈ T 2 by g(x) = x .

Then, for any x ∈ T 2 ,

P g(x) = 1 2 Ax + 1 2 BAx = Ax = g(Ax)
and

X |P g(x)| 2 dν(x) = X |g(Ax)| 2 dν(x) = X |g(x)| 2 dν(x)
Moreover, if we note f = g -P g, then, for any x ∈ X, n ∈ N and any (g 1 , . . . g n ) ∈ {A, BA} n , we have that

g(g n+1 . . . g 1 x) = g(Ag n . . . g 1 x)
and so,

n-1 k=0 f (g k . . . g 1 x) = g(x) -g(g n . . . g 1 x) + n-1 k=0 g(g k+1 . . . g 1 x) -g(Ag k . . . g 1 x) = g(x) -g(g n . . . g 1 x)
This proves that for any x ∈ X, the sequence ( n-1 k=0 f (g k . . . g 1 x)) is bounded in L ∞ (P x ).

4. Application to the random walk on the torus In this section, we go back to the random walk on the torus. The law of large numbers is known as a corollary of a theorem in [START_REF] Bourgain | Stationary measures and equidistribution for orbits of nonabelian semigroups on the torus[END_REF] which allow one to have a speed of convergence depending on the diophantine properties of the starting point. We use this to prove the central limit theorem and the law of the iterated logarithm.

Let H be a subgroup of SL d (R). We say that the action of H on R d is strongly irreducible if H doesn't fixe any finite union of proper subspaces of R d and that it is proximal if for some h ∈ H we have a decomposition

R d = V + h ⊕ V < h of R d into an h-invariant line V +
h and an h-invariant hyperplane V < h such that the spectral radius of h restricted to V < h is strictly smaller than the one of h restricted to V + h . We say that the group H is strongly irreducible and proximal if it's action is.

If we also assume that H is a subgroup of SL d (Z), then it's action pass to the quotient X := T d = R d /Z d that we endow with a metric defined by a norm on R d and with Lebesgue's measure ν. Moreover, H is strongly irreducible and proximal then any a ∈ Z d \ {0} has an infinite H-orbit and so, according to the proposition 1.5 in [BM00] the action of H on T d is ν-ergodic (every H-invariant function is constant ν-a.e.).

Let ρ be a probability measure on SL d (Z). We define a random walk on X noting, for x ∈ X,

X 0 = x X n+1 = g n+1 X n
where (g n ) ∈ SL d (Z) N is an iid sequence of random variables of common law ρ.

In this constext, Bourgain, Furmann, Lindenstrauss and Mozes proves the following Theorem ( [START_REF] Bourgain | Stationary measures and equidistribution for orbits of nonabelian semigroups on the torus[END_REF]). Let ρ be a borelian probability measure on SL d (Z) whose support generates a strongly irreducible and proximal group and which has an exponential moment 1 .

Note

λ 1 = SL d (Z) P(R d ) ln gx dν(x)dρ(g) > 0 1 There is ε ∈ R * + such that SL d (Z) g ε dρ(g) is finite
where ν is the unique 2 ρ-stationary probability measure on P(R d ).

Then, for any ε ∈ R * + , there is a constant C such that for any x ∈ T d , any a ∈ Z d \{0}, any t ∈]0, 1/2] and any n ∈ N with n -C ln t, if

| ρ * n * δ x (a)| > 2t a then, x admits a rational approximation p/q ∈ Q d /Z d satisfying d x, p q e -(λ 1 -ε)n and |q| t -C
In particular, this proves that if x is irrational, then, for any a ∈ Z d \ {0} and any t ∈]0, 1/2], there are only finitely many n ∈ N such that | ρ * n * δ x (a)| > 2t a . This proves that for any irrational point x in T d and any a ∈ Z d \ {0},

lim n→+∞ ρ * n * δ x (a) = 0
And so, using Weyl's equidistribution criterion we have that for any continuous function f on T d and any irrational point x ∈ T d , (4.1)

lim n→+∞ P n f (x) = f dν
where ν is the Lebesgue measure on T d . Moreover, the speed of convergence depend on the diophantine properties of x (see corollary C in [START_REF] Bourgain | Stationary measures and equidistribution for orbits of nonabelian semigroups on the torus[END_REF]). In this section, we first want to obtain a more explicit speed of convergence in equation 4.1 in terms of diophantine properties of x. Then, we want to use this speed of convergence to prove the central limit theorem and the law of the iterated logarithm for starting points having good diophantine properties.

In the first sub-section, we state a corollary of the theorem 4 that is easier to deal with. The price we have to pay is that we will only be able to study hölder continuous functions. This will be proposition Proposition (4.6). Let ρ be a borelian probability measure on SL d (Z) whose support generates a strongly irreducible and proximal group and which has an exponential moment.

Then for any γ, δ ∈]0, 1] and any strictly non-decreasing function ϕ : R + → R * + with lim inf ln ϕ(s) ln s > 0

there are constants C, C 0 , C 1 ∈ R * + such that for any x ∈ T d and any n ∈ N, W γ (ρ * n * δ x , ν) Cψ(n)h ϕ (x) δ
where h ϕ is the function defined for any x ∈ T d by

h ϕ (x) = sup p/q∈Q d /Z d 1 ϕ(q)d(x, p/q)
2 The fact that λ1 exists and is strictly non negative and that ν exists and is unique comes from a result in [GR85].

the function ψ is defined by

ψ(t) = ϕ -1 (e C 1 t ) -C 0
and W γ is the Wasserstein distance defined for any probability measure ϑ 1 , ϑ 2 on the torus by

W γ (ϑ 1 , ϑ 2 ) = sup f ∈C 0,γ (T d ) f γ 1 X f dϑ 1 - X f dϑ 2
Where C 0,γ (T d ) and f γ where defined in equation 1.2.

Then, we will prove that there is a function u ϕ that dominates the function h ϕ and such that P u ϕ au ϕ + b for some a ∈]0, 1[ and b ∈ R. This means that in average, u ϕ (gx) is much smaller than u ϕ (x) and this will allow us to prove, using the results of section 2.2, the Theorem (4.17). Let ρ be a borelian probability measure on SL d (Z) whose support generated a strongly irreducible and proximal group and which has an exponential moment.

Then, for any γ ∈]0, 1] there is β 0 ∈ R * + such that for any B ∈ R * + and any β ∈]0, β 0 [ we have that for any irrational point x ∈ T d such that the inequality d x, p q e -Bq β has only finitely many solutions p q ∈ Q d /Z d , we have that for any γ-holder continuous function f on the torus, noting σ 2 (f ) the quantity defined in equation 1.1 we have that

1 √ n n-1 k=0 f (X k ) L -→ N f dν, σ 2 (f ) (If σ 2 = 0, the law N (µ, σ 2 ) is a Dirac mass at µ). Moreover, if σ 2 (f ) = 0 then lim inf n-1 k=0 f (X k ) -f dν 2nσ 2 (f ) ln ln n = -1 et lim sup n-1 k=0 f (X k ) -f dν 2nσ 2 (f ) ln ln n = 1
and if σ 2 (f ) = 0, then for ν-a.e. x ∈ X, the sequence

( n-1 k=0 f (X k ) -f dν) n is bounded in L 2 (P x ).

BFLM's result for holder-continuous functions.

In this section, we start with a few remind on Wasserstein's distance and then we state BFLM's result using this distance.

4.1.1. Wasserstein's distance on the torus. Note X the torus T d = R d /Z d endowed with the metric induced by a norm on R d . If ϑ 1 and ϑ 2 are borelian probability measures on X, a way to measure their distance is to compute the total variation

d var (ϑ 1 , ϑ 2 ) = sup f ∈C 0 (X) f ∞ 1 f dϑ 1 -f dϑ 2
This distance is not adapted to our study since, for instance, when ρ has a finite support, so does the measure ρ * n * δ x and so, for any x ∈ X and any n ∈ N,

d var (ρ * n * δ x , ν) = 2
However, we can compute the distance between ϑ 1 and ϑ 2 seen has linear forms on the space C 0,γ (X) of γ-holder continuous functions on X. Therefore, we make the following Definition 4.1 (Wasserstein's distance).

Let ϑ 1 , ϑ 2 be two borelian probability measures on a compact metric space (X, d).

For any γ ∈]0, 1], we define the γ-distance of Wasserstein between ϑ 1 and ϑ 2 by

W γ (ϑ 1 , ϑ 2 ) = sup f ∈C 0,γ (X) f γ 1 f dϑ 1 -f dϑ 2
Remark 4.2. Sometimes, this distance is also named after Kantorovich and Rubinstein and we refer to [START_REF] Villani | Optimal transport, Grundlehren der Mathematischen Wissenschaften[END_REF] for an overview of it's first properties.

On the torus, Wasserstein's distance between a given measure ϑ and Lebesgue's measure is linked to the decreasing of the Fourier coefficients of ϑ. We make this precise in next Lemma 4.3. For any γ ∈]0, 1], there is a constant C depending only on d and γ such that for any borelian probability measure ϑ on the torus T d and any

t ∈ R * + , if W γ (ϑ, ν) > t then there is a ∈ Z d \ {0} such that | ϑ(a)|
Ct C a where we noted ν the Lebesgue measure on T d .

To prove this lemma, we will need a result of Jackson and Bernstein about the rate at which one can approximate in the uniform norm an holder continuous function by more regular ones.

For r ∈ N * , we define the Sobolev space

H r :=    f ∈ L 2 (T d ) a∈Z d | f (a)| 2 (1 + a ) 2r < +∞    Lemma 4.4 (Jackson, Bernstein). Let γ ∈]0, 1] et r ∈ [1, +∞[.
Then, there is some C ∈ R such that for any function f ∈ C 0,γ (T d ), there is a sequence

(f n ) ∈ H r (T d ) N such that for any n ∈ N * , f dν = f n dν, f -f n ∞ C n γ f γ and f n H r C f ∞ n C Proof. For y ∈ R/Z, we note k m (y) = sin(2πmy) sin(πy) 4
and for a point y = (y 1 , . . . ,

y d ) ∈ T d , we note K m (y) = d i=1 k m (y i ). Finally, we note I m = 1/4 -1/4 k m (y)dy -1 . Define, for x ∈ T d , f m (x) = [-1/4,1/4] d I d m f (x + 2y)K m (y)dy = I d m 2 [-1/2,1/2] d f (y)K m ( y -x 2 )dy
Then, we can compute

|f (x)-f m (x)| = [-1/4,1/4] d I d m (f (x) -f (x + 2y))K m (y)dy I d m 2 γ f γ [-1/4,1/4] d y γ K m (y)dy I d m 2 1+γ f γ [0,1/4] d y γ K m (y)dy I d m 2 1+γ f γ [0,1/4] d (y γ 1 + • • • + y γ d )K m (y)dy dI m 2 1+γ f γ [0,1/4] y γ k m (y)dy
Where we used in last inequality the fact that Thus, J m,γ ≍ m 3-γ and finally,

I d m [0,1/4] d y γ 1 K m (y)dy = I m [0,1/4]
I m 1/4 0 y γ k m (y)dy = J m,γ J m,0 ≍ m -γ
And so, what we proved is that there is some constant C such that for any function

f ∈ C 0,γ (T d ), we have that f -f m ∞ C m γ f γ So, what is left is to prove that (for some maybe bigger constant C) f m H r C f ∞ m rd But, it is clear that for any a ∈ Z d , | f m (a)| f ∞
And, using that f m = f * K m and that K m is a trigonometric polynomial of degree at most Cm 4 for some C as we may see by developping

f m H r =   a∈Z d (1 + a ) 2r | f m (a)| 2   1/2   a Cm 4 (1 + a ) 2r   1/2 f ∞ (1 + Cm 4 ) r (Cm 4 ) d/2 f ∞
Which finishes the proof of the lemma.

Proof of lemma 4.3. By definition of W γ (ϑ, ν), there is a function

f ∈ C 0,γ (T d ) such that f γ 1 and | f dϑ -f dν| t 2 . Let r ∈ N * such that a∈Z d \{0} a (1+ a 2 ) r/2 =: C r is finite. According to lemma 4.4, there is a sequence of functions (f n ) ∈ H r (T d ) N such that f -f n ∞ C n γ and f n H r Cn C . Then, f n dϑ -f n dm f dϑ -f dm - (f -f n )dϑ -(f -f n )dm t 2 -2 f -f n ∞ t 2 - 2C n γ
But,

f n dϑ -f n dm = a∈Z d \{0}
f n (a) ϑ(a)

a∈Z d \{0} f n (a) ϑ(a) a∈Z d \{0} f n H r (1 + a 2 ) r/2 | ϑ(a)| f n H r C r sup a∈Z d \{0} | ϑ(a)| a Cn C C r sup a∈Z d \{0} | ϑ(a)| a
and so, sup

a∈Z d \{0} | ϑ(a)| a t 2 -2C n γ CC r n C 20 So, taking n = ⌊ 8C t 1/γ ⌋ + 1 we have that t 2 -2 C 1 n γ
t/4 and there is some constant C ′ such that sup

a∈Z d \{0} | ϑ(a)| a C ′ t 1+C/γ
and this finishes the proof.

With lemma 4.3, we get a straightforward corollary of theorem 4.

Proposition 4.5 ([BFLM11] with Wasserstein's distance). Let ρ be a borelian probability measure on SL d (Z) whose support generated a strongly irreducible and proximal group and which has an exponential moment.

Then, for any ε ∈ R * + and any γ ∈]0, 1], there is a constant C ∈ R + and t 0 ∈]0, 1/2] such that for any n ∈ N, any t ∈]0, t 0 ] with n -C ln t and any x ∈ T d , if

W γ (ρ * n * δ x , ν) t then there is p/q ∈ Q d /Z d with |q| Ct -C and d(x, p/q) e -(λ 1 -ε)n
The previous proposition proves that if the distance between ρ * n * δ x and ν is large and if t is a function of n, then x is well approximated by rational points : for instance, if t = e -αn for some α ∈ R * + then the p/q produced by the proposition satisfies q Ce αCn and so,

d(x, p/q) e -(λ 1 -ε)n C q (λ 1 -ε)/αC
We are going to reverse this to, given a diophantine condition, find a rate of convergence.

From now on, we fix a strictly non-decreasing function ϕ : R + → R * + . For x ∈ X, we note (4.2)

h ϕ (x) = sup p/q∈Q d /Z d 1 ϕ(q)d(x, p/q)
Thus, a point is M -diophantine if h ϕ (x) is finite with ϕ(t) = t M . We also remark that if ϕ grows faster than any polynomial, then ν(h ϕ < +∞) = 1. Proposition 4.6. Let ρ be a borelian probability measure on SL d (Z) whose support generated a strongly irreducible and proximal group and which has an exponential moment.

Then, for any γ, δ ∈]0, 1] and any strictly non-decreasing function ϕ : R

+ → R * + with lim inf ln ϕ(s) ln s > 0 there are constants C, C 0 , C 1 ∈ R * + such that for any x ∈ T d and any n ∈ N, W γ (ρ * n * δ x , ν) Cψ(n)h ϕ (x) δ
where h ϕ is the function defined in equation 4.2 and ψ is the function defined by

ψ(t) = ϕ -1 (e C 1 t ) -C 0 Remark 4.7.
The assumption on ϕ implies that for some c ∈ R * + we have that for any t ∈ R, ϕ(t) ct c . It is not restrictive at all since according to Dirichlet's theorem on diophantine approximation, if ϕ(t) = o(t 1+1/d ), then the function h ϕ only takes infinite values.

Remark 4.8. If we take ϕ(n) = n D , then we get ψ(n) = e -κn for some κ ∈ R * + and this proves that for a generic diophantine point, the convergence is at exponential speed.

In the sequel, we will have to be sure that the sum of the ψ(n) converges and so, we will take ψ(n) = n -1-α for some α ∈ R * + . This will allow us to study irrational points x ∈ T d such that the inequality d x, p q e -Bq β has only finitely many solutions p q ∈ Q d /Z d where β, B will be constants depending on ρ.

Proof. Let C 0 , C 1 , C ∈ [5, +∞[ whose values will be determined later.

We note C 2 the constant given by proposition 4.5.

Let x ∈ X et n ∈ N. If Cψ(n)h ϕ (x) δ 2, then the inequality is satisfied since P n f -f dm ∞ 2 f ∞ . Thus, we shall assume that Cψ(n)h ϕ (x) δ 2. Let t = C 5 ψ(n)h ϕ (x) δ . Then, t < 1 2 and -C 2 ln t = -C 2 ln C 5 ψ(n)h ϕ (x) δ -C 2 ln (ψ(n)) = C 2 C 0 ln ϕ -1 (e C 1 n )
since Ch ϕ (x)/5 1 because C 5 and h ϕ (x) 1. But, there is a constant C 3 such that for any s ∈ R + , ϕ(s) C 3 s C 3 and so,

ϕ -1 (s) s C 3 1/C 3 Therefore, ln ϕ -1 (e C 1 n ) 1 C 3 (C 1 n -ln C 3 ) and -C ln t n if C 0 is small enough (de- pending on C 1 ).
Thus, we can apply proposition 4.5 to find that if

W γ (ρ * n * δ x , ν) t then there is p/q ∈ Q d /Z d with q C 2 t -C 2 such that d x, p q e -λn
Thus, as we shall assume without any loss of generality that

C 2 5 C C 2 1 and C 0 C 2 1, we get that q C 2 5 Cψ(n)h ϕ (x) δ C 2 C 2 5 Cψ(n) C 2 = C 2 5 C C 2 ϕ -1 (e C 1 n ) C 2 C 0 ϕ -1 (e C 1 n ) and e λn x -p/q -1 ϕ(q)h ϕ (x) ϕ(q) 2 Cψ(n) 1/δ e C 1 n 2 Cψ(n) 1/δ Thus, ψ(n) 2 C e -δ(λ-C 1 )n but, ψ(n) e C 1 n C 3 -C 0
Which leads to a contradiction if C 1 < λ, C 0 is small enough and C is large enough. Thus, there is no n ∈ N and x ∈ X such that Cψ(n)h ϕ (x) δ 2 and

W γ (ρ * n * δ x , ν) 1 5 Cψ(n)h ϕ (x) δ
So, for any n ∈ N and any x ∈ T d ,

W γ (ρ * n * δ x , ν) Cψ(n)h ϕ (x) δ
which is what we intended to prove.

Diophantine control along the walk.

In this section, we are going to prove that if x ∈ X satisfies a diophantine condition, then so does the gx with g ∈ SL d (Z). We will deduce from this a control of the speed of convergence in proposition 4.5 along the walk.

We saw in proposition 4.6 that for any irrational point x of T d , ρ * n * δ x converges for Wasserstein's distance to Lebesgue's measure on the torus. Moreover, the rate depend on the way x can be approximated by rational points of the torus.

To prove the central limit theorem starting at some point x, we will have to control the rate of convergence of ρ * n * δ y for any y of Gx ; the problem being that the function h ϕ that we defined may take arbitrarily large values on Gx.

However, the set of points where h ϕ is finite is invariant under the action of Γ = SL d (Z) as one may see noting that for x ∈ T d , p ∈ Q d /Z d and g ∈ Γ we have

g d(x, g -1 p) d(gx, p) = d(gx, gg -1 p) 1 g -1 d(x, g -1 p)
and g -1 p is a rational point with the same denominator than p (since g -1 has integer coefficients) and this estimation proves that for any g ∈ SL d (Z) and any x ∈ T d ,

h ϕ (gx) g h ϕ (x)
In this section, we are going to prove that we can obtain a control that is far better than this trivial one. We will indeed prove that for any irrational point x of the torus, in average, gx is further from the rationals than x.

To do so, we begin by showing that, in average, gx is further from 0 than x. We will prove this in proposition 4.10 but at first, we will need the next Lemma 4.9. Let ρ be a borelian probability measure on SL d (Z) whose support generates a strongly irreducible and proximal group and which has an exponential moment.

For any δ ∈ R * + and x ∈ T d \ {0}, we note

u δ (x) = 1 d(x, 0) δ Then, there are n 0 ∈ N, δ ∈ R * + , a ∈ [0, 1[ and b ∈ R such that for any x ∈ T d \ {0}, P n 0 u δ (x) au δ (x) + b
Proof. The proof is by going back to R d since our assumptions imply that the first Lyapunov exponent is strictly non negative. Let ε ∈ R * + and x ∈ B(0, ε) ⊂ T d . Choose a point x in B(0, ε) ⊂ R d whose projection on the torus is x. Then, for any n ∈ N,

P n u δ (x) = G d(gx, 0) -δ dρ * n (g) = G 1 g 1 ε d(gx, 0) -δ dρ * n (g) + G 1 g > 1 ε d(gx, 0) -δ dρ * n (g) = G 1 g 1 ε gx -δ dρ * n (g) + G 1 g > 1 ε d(gx, 0) -δ dρ * n (g) G gx -δ dρ * n (g) + G 1 g >1/ε g -1 δ x -δ dρ * n (g) x -δ G e -δ ln gx x dρ * n (g) + G 1 g >1/ε g -1 δ dρ * n (g)
Moreover, there is δ 0 ∈ R * + such that for any δ ∈]0, δ 0 ] there are C, t ∈ R * + such that for any n ∈ N,

sup x∈R d \{0} G e -δ ln gx x dρ * n (g) Ce -tn
(we refer to [BL85] theorem 6.1, for a proof of this result).

And so, we get that for any x ∈ B(0, ε) \ {0},

P n u δ (x) u δ (x) Ce -tn + G 1 g >1/ε g -1 δ dρ * n (g)
Let n 0 be such that Ce -tn 0 1/4 and ε such that

G 1 g >1/ε g -1 δ dρ * n 0 (g) 1/4
(such an ε exists since ρ has an exponential moment). What we get is that for this choice of n 0 and ε, for any x ∈ B(0, ε) \ {0},

P n 0 u δ (x) 1 2 u δ (x)
Moreover, if x is on the complement set of the ball,

P n u δ (x) = G d(gx, 0) -δ dρ * n (g) d(x, 0) -δ G g -1 δ dρ * n (g) ε -δ G g -1 δ dρ * n (g)
and this finishes the proof of the lemma.

From now on, we fix δ ∈ R * + such that the function u δ satisfies P n 0 u δ au δ + b for some

n 0 ∈ N * , a ∈ [0, 1[ and b ∈ R. Let a 1 ∈]a, 1[ be such that a -n 0 1 a 1. Note u 0 = n 0 -1 k=0 a -k 1 P k u δ
Then,

P u 0 = n 0 -1 k=0 a -k 1 P k+1 u δ = a 1 n 0 -1 k=1 a -k 1 P k u δ + a -(n 0 -1) P n 0 u δ a 1 n 0 -1 k=1 a -k 1 P k u δ + a -(n 0 -1) 1 (au δ + b) a 1 u 0 (x) + ba -(n 0 -1) 1
Moreover, as

u δ (x) G g -δ dρ * k (g) P k u δ (x) = G gx -δ dρ * k (g) u δ (x) G g -1 δ dρ * k (g),
the function u 0 that we constructed is also equivalent to d(x, 0) -δ or more specifically, 0 < inf

x∈T d \0 u 0 (x) d(x, 0) -δ < sup x∈T d \{0} u 0 (x) d(x, 0) -δ < +∞
So what we just proved is the following Proposition 4.10. Let ρ be a borelian probability measure on SL d (Z) whose support generates a strongly irreducible and proximal group and which has an exponential moment.

Then, there is

δ ∈ R * + , a ∈ [0, 1[, b ∈ R and a function u 0 on T d , such that 0 < inf x∈T d \0 u 0 (x) d(x, 0) -δ < sup x∈T d \{0} u 0 (x) d(x, 0) -δ < +∞ and P u 0 au 0 + b
Now, we are going to use this function u 0 to construct some other that will allow us to prove that if x is not well approximable by rational points, then, in ρ-average, so are the gx.

What we will do is, for a fixed diophantine condition ϕ, constructing u ϕ such that P u ϕ au ϕ + b and u ϕ is finite on points satisfying the condition ϕ.

For Q ∈ N * , we note X Q the set of primitives elements in 1 Q Z d /Z d that is to say, the set of elements of 1 Q Z d /Z d that doesn't belong to 1 q Z d /Z d for q < Q. Then, X Q is SL d (Z)-invariant : indeed, if p ∈ X Q then gp ∈ 1 Q Z d /Z d
since g has integer coefficients and gp can not belong to 1 q Z d /Z d with q < Q because if it was so, so would p = g -1 gp.

Let ϕ : N → R * + be a strictly non decreasing function. For x ∈ T d \ {0}, we note

u ϕ (x) = Q∈N * 1 ϕ(Q) δ p∈X Q u 0 (x -p)
This function u ϕ is proper (it is non negative and lower semi-continuous) Moreover, it carries the diophantine properties of x. Indeed, by definition of h ϕ (x) (see the previous section), we have that for some constant C that doesn't depend on ϕ,

h ϕ (x) δ Cu ϕ (x)
and reciprocally, if ϕ ′ : R → R * + is an other strictly non decreasing function such that

ϕ ′ (Q) ∈ O(ϕ(Q)Q -(d+2)/δ) ) then, u ϕ (x) Ch ϕ ′ (x) δ Q∈N * Q d ϕ ′ (Q) ϕ(Q) δ and so, if h ϕ ′ (x) is finite, so is u ϕ (x).
Thus, controlling u ϕ (x) is controlling the diophantine properties of x and reciprocally. The aim of this construction is the following Lemma 4.11. Let u 0 be the function constructed in the previous lemma. Let ϕ : N → R * + be a strictly non decreasing function such that

n n d ϕ(n) δ < +∞ For x ∈ T d , note u ϕ (x) = 1 + Q∈N * 1 ϕ(Q) δ p∈X Q u 0 (x -p)
Then, there are a ∈]0, 1[ and b ∈ R such that P u ϕ au ϕ + b Remark 4.12. One has to think of ϕ has growing very fast (we will take ϕ(n) = e Bn β ) so the summability assumption will always be satisfied and multiplying ϕ by a polynomial function doesn't really change the points where u ϕ takes finite values. Therefore, it is almost the same thing to say that u ϕ (x) is finite or that h ϕ (x) is.

Proof. Let's remind that P u 0 au 0 + b for some a ∈]0, 1] and b ∈ R.

noting that, by definition of Wasserstein's distance, for any n ∈ N and any x ∈ X,

P n f (x) -f dν f γ W γ (ρ * n * δ x , ν)
4.3. Central limit theorem and law of the iterated logarithm.

In this section, we use the result of the previous ones to finally prove the central limit theorem and the law of the iterated logarithm for the random walk on the torus.

As we now know with corollary 4.6, holder continuous functions f on the torus writes f = g -P g + f dν where g is dominated by a drift function finite on points badly approximalble by rationals. We are going to prove the the validity of "law of large numbers"-type hypothesis in corollary 2.13 and this will allow us to prove the central limit theorem and the law of the iterated logarithm. We don't know how to prove the law of large numbers for functions of E p u and this is why we will go back to the function f to use the speed of convergence given by our corollary or Bourgain-Furmann-Lindenstrauss-Mozes's theorem.

We will need the following Lemma 4.14. Let ρ be a borelian probability measure on SL d (Z) whose support generates a strongly irreducible and proximal group and which has an exponential moment.

For any γ ∈]0, 1], there is α 0 ∈ R such that for any α ∈]α 0 , +∞[ there is β 0 ∈ R * + such that for any β ∈]0, β 0 [ and any B ∈ R, noting ϕ(q) = e Bq β , we have that for any sequence (f n ) of γ-hölder-continuous functions on the torus and such that f n dν = 0,

E x n-1 k=0 f k (X k ) 4 = O n 3 (ln n) α-1 u ϕ (x) max k∈[0,n] f k 4 γ
Where the involved constant doesn't depend on n, x nor the sequence (f n ).

Remark 4.15. What is hidden behind this lemma is a kind of Burckholder inequality that says that if (Y i ) is a sequence of iid bounded random variables on R of null expectation, then for any r ∈ N,

E n-1 k=0 Y k r ∈ O n r/2
Proof. First, we choose α ∈ R * + and we will see a lower bound on α later. We note ψ(n) = n -α and according to proposition 4.6, there is a constant C such that for any n ∈ N and any x ∈ X,

W γ (ρ * n * δ x , ν) C n α u ϕ (x)
where ϕ(q) = e Bq β . Moreover, for n ∈ N, we note

S n = n-1 k=0 f k (X k ) We can compute E x |S n+1 | 4 = 4 k=0 4 k E x f n (X n ) k S 4-k n = E x |S n | 4 + 4E x f n (X n )S 3 n + 4 k=2 4 k E x (f n (X n )) k S 4-k n So, we note A n := E x f n (X n )S 3 n et B n := 4 k=2 4 k E x (f n (X n )) k S 4-k n
and so, we have, noting p(n) (and even only p to simplify notations) a sequence that we will determine later and such that 0 p(n) n, that

A n = 3 k=0 3 k E x f n (X n )(S n -S p ) k S 3-k p = E x f n (X n )S 3 p + 3E x f n (X n )(S n -S p )S 2 p + 3 k=2 3 k E x f n (X n )(S n -S p ) k S 3-k p
We note each of this terms A 1 n , A 2 n et A 3 n . Then, using the fact that f n dν = 0, and that, according to proposition 4.6,

|P n-p+2 f n (X p-1 )| C (n -p + 2) α f n γ u(X p-1 )
and that

P l u(x) a l u(x) + b 1 -a we get that |A 1 n | = E x P n-p+2 f n (X p-1 )S 3 p C (n -p + 2) α f n γ E x u(X p-1 )|S p | 3 C (n -p + 2) α f n γ S p 3 ∞ P p-1 u(x) C (n -p + 2) α f n γ S p 3 ∞ a p-1 + b 1 -a u(x) = O u(x) max k∈[0,n] f k 4 γ n 3 (n -p + 2) α Moreover, A 2 n = n-1 k=p E x f n (X n )f k (X k )S 2 p = q-1 k=p E x P n-k f n (X k )f k (X k )S 2 p + n-1 k=q E x P k-p (f k P n-k f n )(X p )S 2
p and so, for some sequence q(n) that we will determine later and with p(n) < q(n) < n, we have that

|A 2 n | q-1 k=p C (n -k) α f k ∞ f n γ E x u(X k )S 2 p + n-1 k=q C (k -p) α f k P n-k f n γ E x u(X p )S 2 p + f k P n-k f n dν E x S 2 p n 2 max k∈[0,n] f k 4 γ   q-1 k=p CP k u(x) (n -k) α + n-1 k=q C P n-k γ P p u(x) (k -p) α + P q L 2 0 (X,ν)   = O   n 2 max k∈[0,n] f k 4 γ u(x)   n-p k=n-q+1 1 k α + P n-q γ (q -p) α + P q L 2 0 (X,ν)    
So, with q = n-ln n and p = n-n δ with δ < 1/2, and taking α 0 such that δα 0 > ln P γ , we find that

A 2 n = O n 2 max k∈[0,n] f k 4 γ u(x) +∞ k=ln n+1 1 k α + n ln P γ (n δ -ln n) α + P n-ln n L 2 0 (X,ν) = O n 2 (ln n) α-1 max k∈[0,n] f k 4 γ u(x)
and finally,

|A 3 n | 3 k=2 3 k f n ∞ S n -S p k ∞ E x |S p | 3-k But, S n -S p ∞ n-1 k=p f p ∞ (n -p) sup k∈[0,n-1] f k ∞ so A 3 n = O n 1+2δ max k∈[0,n] f k 4 γ
and we recall that we choose δ < 1/2. So we can take δ = 1/4 and we can assume that δα 0 > 1 to get that

|A 1 n | = O u(x) max k∈[0,n] f k 4 γ n 3-δα
thus we proved that

A n = O n 2 (ln n) α-1 u(x) max k∈[0,n] f k 4
To study B n , remark in a first time that

E x S 2 n = n-1 k=0 P k (f k ) 2 (x) + 2 n-1 k=0 k-1 l=0 P l (f l P k-l f k )(x)
The first term of this sum is dominated by n max k∈[0,n] f k γ and a computation similar to the previous one proves that the second one is bounded by constants times

n 2 (ln n) α-1 u(x) max k∈[0,n] f k 2 .
Therefore,

E x (f n (X n )) 2 S 2 n ∈ O n 2 (ln n) α-1 u(x) max k∈[0,n] f k 4
And this proves that

B n = O n 2 (ln n) α-1 u(x) max k∈[0,n] f k 4 So, E x |S n+1 | 4 = E x |S n | 4 + O n 2 (ln n) α-1 u(x) max k∈[0,n] f k 4
and iterating this relation, we get that

E x |S n+1 | 4 = O n 3 (ln n) α-1 u(x) max k∈[0,n] f k 4
which is what we intended to prove.

We are now ready to prove the convergence of the variance in next Lemma 4.16. Let ρ be a borelian probability measure on SL d (Z) whose support generates a strongly irreducible and proximal group and which has an exponential moment.

Then, for any γ ∈]0, 1] there is β 0 ∈ R * + such that for any β ∈]0, β 0 [ and any B ∈ R, noting ϕ(q) = e Bq β , we have that for any γ-hölder-continuous function f on the torus, noting g the solution to Poisson's equation defined in F 3 uϕ and given by corollary 4.13 we have that for any x ∈ X such that u ϕ (x) is finite,

1 n n-1 k=0 P (g 2 )(X k ) -(P g(X k )) 2 -→ g 2 -(P g) 2 dν P x -a.e. and in L 1 (P x )
Proof. We take at first α 0 equal to the one of the previous lemma and take α > α 0 . It comes with it a constant β 0 such that for any B ∈ R and any β ∈]0, β 0 [, the function ψ(t) given by 4.6 satisfies that sup n n α ψ(n) is finite.

Remark that for any γ-hölder-continuous function f on the torus, the function g given by proposition 4.6 is square-integrable agains Lebesgue's measure. We can see this as a consequence of 2.5 or, more simply, use that under our assumptions, the operator P has a spectral gap L 2 (X, ν) as we already saw in the introduction and so, the function g is a.e.-equal to a square-integrable function. We will actually use this spectral gap in the proof of this lemma.

We assume without any loss of generality that f dν = 0. To prove the lemma, we use that f = g -P g to write

I n (x) : = 1 n n-1 k=0 P (g 2 )(X k ) -(P g(X k )) 2 = 1 n n-1 k=0 P (g 2 )(X k ) -(g(X k ) -f (X k )) 2 = 1 n n-1 k=0 P (g 2 )(X k ) -g 2 (X k ) - 1 n n-1 k=0 (f (X k )) 2 + 2 n n-1 k=0 f (X k )g(X k )
According to proposition 2.10, and using that u is a drift function and that g ∈ F 3 u , we get that for any x such that u(x) is finite, 1 n n-1 k=0 P (g 2 )(X k )g 2 (X k ) -→ 0 in L 1 (P x ) and P xa.e. Moreover, the law of large numbers proves that for any irrational point x of the torus (and so in particular, for any x such that u(x) is finite),

1 n n-1 k=0 (f (X k )) 2 -→ X f 2 dν in L 1 (P x
f ∞ C l 1+α h ϕ (X k ) δ/3 f γ 1 n n-1 k=0 C ′ p(k) α h ϕ (X k ) δ/3 f 2 γ for some constant C ′ .
But, by definition of u ϕ , h Theorem 4.17. Let ρ be a borelian probability measure on SL d (Z) whose support generates a strongly irreducible and proximal group and which has an exponential moment.

Then, for any γ ∈]0, 1] there is β 0 ∈ R * + such that for any B ∈ R * + and β ∈]0, β 0 [ we have that for any irrational point x ∈ X such that the inequality d x, p q e -Bq β has a finite number of solutions p/q ∈ Q d /Z d , we have that for any γ-holder continuous function f on the torus, noting σ 2 (f ) the variance given by equation 1.1 we have that

1 √ n n-1 k=0 f (X k ) L -→ N f dν, σ 2 (f )
(If σ 2 = 0, the law N (µ, σ 2 ) is a Dirac mass at µ). Moreover, if σ 2 (f ) = 0 then, P x -a.e.,

lim inf

n-1 k=0 f (X k )f dν 2nσ 2 (f ) ln ln n = -1 and lim sup n-1 k=0 f (X k )f dν 2nσ 2 (f ) ln ln n = 1 and if σ 2 (f ) = 0, then for ν-a.e. x ∈ X, the sequence ( n-1 k=0 f (X k )f dν) n is bounded in L 2 (P x ).

Proof. This is a direct corollary of lemma 4.16, lemma 2.8 and proposition 2.13.

The condition on σ 2 (f ) comes from corollary 3.6 if we note that since P u ϕ au ϕ + b, we have, for any n ∈ N, P n u ϕ a n u ϕ + b 1a and so, for any x satisfying the diophantine condition, sup n P n (g 2 )(x) u ϕ (x) + b 1-a is finite. And moreover, ν(x|u ϕ (x) < +∞) = 1.

y γ k m (

  that for any t ∈ [0, π/2], 2t π sin(t) t, 4 (sin y) 4 dy And so, L m,γ ≍ m 3-γ

  e 2iπmy e -iπye iπy 4 = e 4iπy e -2iπmye 2iπmy 1have that for a > Cm 4 , K m (a) = 0. And this proves that

  ) and P xa.e. Moreover, if p : N → N is a non decreasing function converging to infinity that we will determine later, we have thatgk )P l f (X k )But, according to lemma 4.14 applied to the sequence of functionsf n = p(n)-1 k=0 f P l ff P l f dν k )P l f (X k )f P l f dν 4 k )P l f (X k )f P l f dν 4 = O n 3 (ln(n)) α 0 u(x) f 8 γ P p(n) γ so, if p(n) ≍ δ 1 ln(ln n)with δ 1 such that δ 1 ln P γ < α 0 , we have that for n large enough,P p(n) γ e δ 1 ln(ln n) P γ = (ln n) δ 1 ln P γ k )P l f (X k )f P l f dν k )P l f (X k )f P l f dν -→ 0 P x -a.e. and in L 1 (P x )Moreover, using the spectral gap in L 2 (X, ν) and Cesaro's lemma, we get that ) f (X k )P l f (X k ) converges to 0. But, using proposition 4

P

  X k ) δ/3 -→ 0 P xa.e. and in L1 (P x ) (g 2 )(X k ) -(P g(X k )) 2 -→f 2 dν + 2 +∞ l=0 f P l f dν P x -a.e. and in L 1 (P x )To conclude, we only have to remark thatf 2 dν + 2 +∞ l=0 f P l f dν = -(g -P g) 2 + 2(g -P g)gdν = 2g 2 -2gP gg 2 + 2gP g -(P g) 2 dν = g 2 -(P g) 2 dνwhich finishes the proof of the lemma.

And so, if we note, for Q ∈ N * and x ∈ T

we have, using that SL d (Z) permutes X Q , that

And so,

We are finally able to solve Poisson's equation for hölder-continuous functions in next Corollary 4.13. Under the hypothesis of proposition 4.6, for any γ ∈]0, 1] and any M ∈ R * + , there is β 0 ∈ R * + such that for any B ∈ R * + and any β ∈]0, β 0 [, there is a constant C such that, noting ϕ(n) = e Bn β , we have that for any x such that u ϕ (x) < +∞ we have that

In particular, for any γ-hölder continuous function f on the torus, there is g ∈ F 3 uϕ (cf section 2.1) such that, f = g -P g + f dν on {u ϕ < +∞} and g F 3 u C f γ Proof. We apply proposition 4.6 noting that in this case, there is a consant C such that for any n ∈ N * ,


Ludovic Claudepierre

Rémi Douvenot

Alexandre Chabory

Christophe Morlaas

Influence of the Lightning Protection of Blades on the Field Scattered by a Windturbine

Keywords: Windturbines, windfarm modelling, VOR, lightning protection

This paper investigates the influence of the lightning protection in windturbine blades for their electromagnetic modelling. Two realistic models of blade with lightning protections are compared with two homogeneous models: metallic and dielectric.

I. INTRODUCTION

The fast-growing number of windturbines constrains manufacturers to consider every free space as a possible area for windfarm implantation. In particular, plains are appropriate for windturbine erections. However, these plains are also commonly chosen for civil aviation system implantation such as radar and VHF Omnidirectional Range (VOR). These latter are essential for the navigation of airplanes. Regulations exist to contain the windturbine proliferation nearby VOR. Nevertheless, a relaxation of this regulation could be possible with a trustful modelling of the effect of the windturbines on the VOR signal.

To achieve such an investigation, the precise interactions between the windturbines and the VOR signal have to be known. Thus, accurate simulations are required. Considering the large size and the dielectric composition of the blades, the real structure can not be simulated with a reasonable computing time. So, several simplified modellings of the blades have been developed these last years with different approximations.

A common approximation is to consider blades as metallic [START_REF] Grevin | On the concept of the radar cross section RCS of distorting objects like wind turbines for the weather radar[END_REF] [START_REF] Angulo | Analysis of the mast contribution to the scattering pattern of wind turbines in the UHF band[END_REF] with their original geometry. Other authors additionally propose a geometrical simplification of the blade, approximating the shape by canonical forms [START_REF] Yucedag | Radar cross section calculation of a wind turbine modeled by PEC canonical structures[END_REF] or considering it as plane and rectangular [START_REF] Casanova | Wind farming interference effects[END_REF]. These modellings are fast to compute but can overestimate the VOR error due to the windturbine.

The present article is based on the dielectric modelling of the blade presented by Morlaas et al. [START_REF] Morlaas | Wind turbine effects on VOR system performance[END_REF] where the blade is approximated by its dielectric spar. This model has been geometrically simplified by two parallel homogeneous dielectric and profiled slabs [START_REF] Morlaas | Propagation model for estimating VOR bearing error in the presence of windturbines -hybridation of parabolic equation with physical optics[END_REF]. However, the lightning protections are not accounted.

This paper proposes to investigate the influence of the lightning protection in the windturbine blades in order to refine the previous modelling. Two typical lightning protection technologies [START_REF]Wind turbines -part 24: Lightning protection[END_REF] have been compared with the two dielectric profiled slabs and the metallic profiled slab modellings. In section II, the geometric considerations and the process for comparing the blade models are presented. The results are exposed and analysed in section III. A short discussion on the relevancy of a Physical Optic solver using this model is performed in section IV. Section V concludes this work.

II. COMPARISON PROCESS OF THE BLADE MODELS

In order to determine the influence of the lightning protection in the modelling of windturbine blades, the electromagnetic behaviours of the three blades constituting the rotor have been investigated. The comparisons are realised between four modellings on FEKO simulations using method of moments (MoM). For all the blade models, only the spar is considered. The spherical coordinates are used in this whole paper.

As illustrated in Figure 1, two different technologies of lightning protections are added to this modelling. The first technology (A) consists in a metallic rod (diameter = 1.6 cm) in the middle of the dielectric structure of the blade. The second one (B) consists in a thin metallic slab (width = 3 cm) on the dielectric surface of the blade. Finally these two modellings are compared to the dielectric model proposed by Morlaas et al. [START_REF] Morlaas | Propagation model for estimating VOR bearing error in the presence of windturbines -hybridation of parabolic equation with physical optics[END_REF] and to a metallic and profiled slab. The three blades constituting the rotor of the windturbines do not have the same orientation. So there are some crosspolarisation effects participating to the global scattering of the rotor. For this reason the whole rotor is presented in this here rather than one only blade results.

As the rotor rotates, various positions need to be simulated (Figure 2). The rotor is a periodic structure with three identical blades spaced from 120 • . Moreover the positions α rot = 0 • and α rot = 60 • are horizontally symmetrical and α rot = 30 • and α rot = 90 • are vertically symmetrical, as described in Figure 2. Therefore, the study is performed on the 2 positions α rot = 0 • and α rot = 30 • . The incident field is a plane

α rot

III. RESULTS

The scattered fields are observed for the four blade models. The incident field is a plane wave, horizontally polarised at frequency 114 MHz. As the windturbines are generally far Firstly, the two kinds of lightning protection have a similar behaviour. Indeed, their scattered fields have similar shapes except for low scattering directions with a normal incidence and α rot = 30 • (Figure 8).

The dielectric models with and without lightning protection have similar behaviours. The main lobes have the same level for these both modellings. However the influence of the lightning protection is significant for the direction of incidence θ inc = 90 • , ϕ inc = 45 • . The field scattered by the blades, only 5 dB below the main lobe, is underestimated of 10 dB with the homogeneous dielectric model. Thus, this model does not describe precisely the behaviour of the blades with lightning protection.

The metallic modelling highly overestimates the scattered field, in particular for the main lobe. This homogeneous model gives values from 10 to 15 dB higher than the model with lightning protection (Figure 5).

To conclude, the lightning protection needs to be taken into account to simulate the windturbine effect on the VOR signal and can not be approximated by an homogeneous model for an accurate simulation.

IV. RELEVANCY OF A PO SOLVER

This dielectric model with lightning protection is projected to be integrated in a simulator using PO method. So, now that the blade model is validated, its relevancy to this method has to be investigated.

PO method is faster than the Method of Moment (MoM) but neglects the interactions between the different objects and the diffraction by the edges. Thus, these approximations are validated through a study highlighting their influence on the field scattered by the rotor. Only the most relevant results are presented here. Compared to the MoM, the PO method globally overestimates the scattered field (Figures 9). The combination of PO and blade with no lightning protection provides a field lower than the reference (MoM), particularly in high scattering direction (Figures 9 (a) and(b)). The combination of PO and metallic blades is not accurate either considering the underestimation of the main lobe for some incidence (Figures 9 (a)).

Finally the model developed in the previous section is appropriate to the PO method.

V. CONCLUSION

In this paper, an investigation on the electromagnetic influence of the lightning protection in the windturbine blades Considering the different orientations of each blade in a windturbine rotor, the three blades have been simulated for various rotation angles of the rotor. Although the influence of the lightning protection is moderate, the homogeneous dielectric model underestimates some significant scattered fields. The homogeneous metallic model is not accurate either and highly overestimates the scattered field.

In order to finally use the dielectric model with lightning protection in a PO solver, the interactions between blades have been analysed. The PO method, compared to the MoM, slightly overestimates the scattered field. So the model is validated for a PO solver.

To conclude, the two homogeneous blade models examined in this paper are not appropriate to simulate with precision a blade including a lightning protection. The metallic model can be used for a coarse approximation of the scattered field. The relevancy of the PO methods to compute the field scattered by a windturbine rotor has been demonstrated.

In further work, with this accurate model, the contribution of the rotor blade to the total field scattered by the windturbine could be properly evaluated and integrated to a VOR error simulator.

Figure 1 :

 1 Figure 1: Two technologies for the lightning protection: (A) a metallic rod inside the structure and (B) a metallic slab on the blade [7].

Figure 2 :Figure 3 :

 23 Figure 2: Sketch of the windturbine with a rotor rotation angle of α rot .

Figure 4 :

 4 Figure 4: The 3 main directions of the scattering field.

  Backscattered field (ϕ obs = 45 • ). Transmitted field (ϕ obs = 225 • ).

Figure 5 :

 5 Figure 5: Field scattered (dB) by the rotor with the dielectric modelling alone (black), with PEC rod (blue), with PEC slab (red), and by the metallic modelling (green), with respect to θ obs ( • ) for a plane wave horizontally polarised and incident along the direction ϕ inc = 45 • , θ inc = 90 • and with α rot = 0 • .

  Backscattered field (ϕ obs = 45 • ). Transmitted field (ϕ obs = 225 • ).

Figure 6 :

 6 Figure 6: Field scattered (dB) by the rotor with the dielectric modelling alone (black), with PEC rod (blue), with PEC slab (red), and by the metallic modelling (green), with respect to θ obs ( • ) for a plane wave horizontally polarised and incident along the direction ϕ inc = 45 • , θ inc = 90 • and with α rot = 30 • .

  Backscattered field (ϕ obs = 90 • ). Transmitted field (ϕ obs = 270 • ).

Figure 7 :

 7 Figure 7: Field scattered (dB) by the rotor with the dielectric modelling alone (black), with PEC rod (blue), with PEC slab (red), and by the metallic modelling (green), with respect to θ obs ( • ) for a plane wave horizontally polarised and incident along the direction ϕ inc = 90 • , θ inc = 90 • and with α rot = 0 • .

  Backscattered field (ϕ obs = 90 • ). Transmitted field (ϕ obs = 270 • ).

Figure 8 :

 8 Figure 8: Field scattered (dB) by the rotor with the dielectric modelling alone (black), with PEC rod (blue), with PEC slab (red), and by the metallic modelling (green), with respect to θ obs ( • ) for a plane wave horizontally polarised and incident along the direction ϕ inc = 90 • , θ inc = 90 • and with α rot = 30 • .

  Backscattered field (ϕ obs = 20 • ). Reflected field (ϕ obs = 160 • ). Transmitted field (ϕ obs = 20 • ).

Figure 9 :

 9 Figure9: Fields scattered (dB) by the 3 blades with a direction of incidence of θ inc = 90 • and ϕ inc = 20 • without lightning protection by PO (black), with a metallic line by MoM (long dotted blue) and by PO (dotted red) , and by metallic blades (green mixed line), with respect to θ obs ( • ) for a plane wave horizontally polarised and incident along the direction ϕ inc = 20 • , θ inc = 90 • and with α rot = 30 • .


A Thain

A Jaber

Y Platon

A Herve

G Peres

B Pasquier

L Evain

G Cambon

F Harly

H Lenquette

A Hervé

B Sinigaglia

B Spitz

Thin Diffraction Grating Technology

Keywords: Thin, Diffraction, Grating, Resonant, ILS, stealth I

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INTRODUCTION

Reflective diffraction gratings consist of a periodically repeating three dimensional structure. They redirect incoming radiation as a function of its angle of incidence and its wavelength. They may also suppress the specular reflection encountered on planer structures. This last property can be exploited to prevent unwanted radio reflections, most notably in the field of radio navigation, where buildings can introduce interference to received signals [START_REF] Jull | Interference-free conducting surfaces[END_REF][START_REF] Jull | The reduction of interference from large reflecting surfaces[END_REF][START_REF] Heath | Total Backscatter from Conducting Rectangular Corrugations[END_REF].

The thickness required for conventional diffraction gratings to effectively suppress specular reflections depends upon the angle of incidence and the wavelength of the incoming radiation. For practical applications in the VHF band structures as thick as 1 m are frequently required. It is possible to build such diffraction gratings (indeed a 60cm grating has been successfully implemented on a building at Toulouse International Airport [START_REF] Thain | A Solution for ILS Disturbance Due to a Building[END_REF][START_REF] Thain | Structure de revetement d'une paroi pour redirection des ondes recues par ladite paroi[END_REF]) but their weight requires the building to possess a reinforced support structure. The striking appearance of conventional diffraction gratings also makes them unsuitable for certain non-industrial buildings.

In this paper we present a new solution to this problem: A thin reflective diffraction grating that exploits resonant structures. A theoretical development is exposed, along with the results of simulation parameter sweeps. Finally, the results of measurements performed on a full size demonstrator are presented for both dry and wet conditions.

II. THICK DIFFRACTION GRATINGS

Traditional, 'thick' diffraction gratings consist of a periodic three dimensional array of elements (Figure 1). For an incident plane wave, with a given angle of incidence, inc θ , and angle of diffraction, diff θ , the same feature on neighbouring periods (A and A') will produce an identical amplitude of scattered waves but the phase will be shifted due to an additional outgoing path difference for point A of The grating equation tells us nothing about the amplitudes of the scattered waves. Such information must come from either specific analytical models or, more generally, from numerical simulation. In order to suppress the specular wave, the guiding principle is that one part of each period must be π radians out of phase with another part of the period; the two contributions possessing a similar amplitude. The π phase shift is usually generated by a path difference due to the varying thickness of the grating. This results in structures that vary between fractions of a wavelength to several wavelengths in thickness.

III. AN APPLICATION FOR RADIO NAVIGATION

One particularly useful application of reflective diffraction gratings in the VHF band is to prevent buildings from generating perturbations to aircraft radio navigation systems.

Building Only remnants of specular reflection

Main reflection is directed back to source Largely unperturbed 'DDM' signal ILS antenna

Figure 2. Diffraction grating solution for ILS interference

In Figure 2 we see the example of the Instrument Landing System (ILS). The ILS localizer guides aircraft laterally along the runway centreline in reduced visibility conditions by emitting a spatially modulated signal. The received signal can be perturbed by unwanted reflections coming from nearby obstacles such as buildings. To avoid incidents of aircraft exiting the runway, planning permission is restricted within these zones, unless it can be demonstrated to aviation authorities that the building does not produce perturbations in excess of thresholds defined by the International Civil Aviation Organisation (ICAO) [7,[START_REF]European guidance material on managing building restricted áreas[END_REF].

A diffraction grating can provide just such a solution. As we see in Figure 2, a specular reflection that would have crossed the runway and perturbed incoming aircraft, can be redirected towards the emitting antenna where it poses no problem.

IV. THIN DIFFRACTION GRATINGS

The VHF frequencies employed for the ILS localizer antenna result in very deep diffraction gratings. There is a need for a thinner solution.

It is not immediately obvious how a thin diffraction grating can generate a sufficient phase shift due to a path difference. We therefore employ a different approach: We create a periodic phase difference directly in the currents induced by the incoming plane wave. This is achieved by the periodic structure displayed in Figure 3, which has been specifically designed to operate with a transverse magnetic incident field [START_REF] Thain | Anti-reflecting lining structure with a diffraction grating using resonant elements[END_REF]. One part of the diffraction grating comprises a planar section that reflects the incoming wave in the y direction with a phase difference of π . Alternating with these sections are resonant circuits formed by folded metallic sheets to produce a rectangular tubular structure possessing a slit. The slit acts as a capacitance and the rectangular tube acts as an inductance. Together they form a parallel LC circuit. By selecting appropriate dimensions for the tube and the slit, this circuit may be designed to resonate at any selected frequency. At resonance the incoming wave is reflected without any phase shift. We may therefore obtain the desired destructive interference.

Although the first design was seen to work well in simulations, it presented a problem for outdoor applications: In practice it is necessary to employ a dielectric spacer to hold the metallic plates in place. There are several materials that can be used for this, such as expanded polystyrene foam, and in itself this poses no problem (although the permittivity of the spacer needs to be taken into account when designing the resonant circuits).

The problem arises when wind-driven rain produces a layer of water on the dielectric structure. Since water is so strongly polarizable ( 90 ≈ r ε within the VHF band [START_REF] Sharkov | Dielectric and emissive properties of terrestrial surfaces[END_REF]) its presence can change the capacitance of the gap capacitor, and de-tune the resonant circuit. This can occur even if the rain layer is galvanically isolated from the circuit.

In order to minimize this effect, a second design was explored (see Figure 4) [START_REF] Thain | Diffraction device intended to be fixed to the external face of a wall[END_REF]. Now the only capacitance is that between the plate A and the plane B (this capacitance was already present in the first design but it did not dominate). The advantage of doing so is that this capacitance is generated within a zone that is screened from any external water layer by the plate A. It is still necessary to keep the rain layer at a further distance of approximately 8 cm, but this can be achieved easily with a dielectric screen. In order to find approximate dimensions for the resonant circuit we follow a development similar to that presented for high impedance surfaces in [START_REF] Sievenpiper | High-impedance electromagnetic surfaces with a forbidden frequency band[END_REF]. We emphasise that this reasoning acts only as a guide, and that the fact that neither the plane B nor the resonant structure uniformly cover the plane implies that the concept of equivalent surface impedance is not strictly valid.

The phase change, φ , of the wave reflected by the resonant structure is

              + - = η η φ Z Z ln Im (3)
where η is the impedance of free space, and Z is the impedance of the LC circuit (of inductance L, capacitance C, at an angular frequency of ω )

2 1 ω LC i Z - = (4)
We can see from the previous two equations that the zero phase condition is satisfied at the circuit resonant frequency, 0 ω .

LC 1 0 = ω (5)
In the first design (dimensions as in Figure 3) the capacitance is approximated by that of coplanar sheets

        = g a w C d arccosh 2 1 π ε (6)
where ε is the permittivity of the surrounding medium.

And in the second design (dimensions as in Figure 4), the capacitance is very roughly approximated as that of a parallel plate capacitance

h wa C d ε = 2 (7)
In both cases the inductance is that of the enclosed loop

w ah L µ = (8)
where µ is the permeability of the surrounding medium.

We fix as a design goal h = 8 cm, since expanded foam panels are readily available in this thickness. As expected, the resonant frequency is independent of w, and values of 10s of cms are sufficient to generate resonances in the desired VHF band.

VI. GRATING DESIGN

In practice the resonant circuits and the gaps are strongly coupled, so in order to find optimal dimensions it is necessary use numerical simulations. The time domain solver of CST Microwave Studio was used. A sample panel measuring 30m by 5m was simulated for angles of incidence ranging from 35° to 70° over a range of frequencies (107-113 MHz) encompassing the ILS band (108-112 MHz). The goal function was the reduction of the specular reflection. The optimum values for the plate width, a, and the periodicity, d, are displayed in Figure 5 and Figure 6. As expected, a is dominated by the required resonant frequency and d very approximately follows Bragg's law. Figure 7 displays the resulting simulated signal reduction, corresponding to the optimal dimensions shown in Figure 5 and Figure 6.

VII. FIELD TRIALS

In order to test the technology in a realistic setting, we built a full scale mock-up of a building façade. It is not a trivial matter to perform scattering measurements in the VHF band on such a large structure. To do so we decided to adapt an existing installation of an ILS localizer antenna.

Originally, we intended to implement a movable test platform that could be rotated to produce any selected angle of incidence. One idea was to use a structure mounted upon a lorry trailer. However, preliminary calculations indicated that the structure needed to be 30 m in width and 5 m in height to produce sufficiently sensitive measurements. For these dimensions, only a fixed structure could be implemented at a reasonable cost. We therefore chose to orient the panel at an angle of incidence that corresponds to an identified real world airport application; 53°.

The panel was built upon a hill at the aerodrome of Cahors. The layout of the test is displayed in Figure 8. A permanent ILS localizer antenna is situated at 297 m beyond the end of the runway. A test vehicle measured the reflected field, following a trajectory along the runway centreline. Ordinarily the ILS antenna possesses the radiation pattern displayed by the blue curve in Figure 9. The panel subtends an angle between 23.5° and 25.9° with respect to broadside, placing it at a very weakly illuminated position. For this reason the antenna excitation was modified by an appropriate placement of attenuators and phase shifters to produce the radiation pattern displayed by the red curve in Figure 9. Note that the null is preserved at 0° which is the position of the test vehicle along the runway axis. The measurements in Figure 9 were performed along the double-dashed line in Figure 8.

BEFORE MODIFICATION AFTER MODIFICATION

Figure 9. Measured radiation pattern of ILS localizer antenna before and after modification

The final form of the panel is displayed in Figure 11. Note that the panel is held in place by scaffolding, anchored with 10 metric tons of concrete blocks. The protective screen possesses the distinctive regulatory colours of vertical red and white stripes, as well as night time lighting to make it visible to the other users of the aerodrome.

Numerous measurements were performed. The black curve of Figure 12 shows the measured perturbations before the screen was built. The small observable oscillations are due to reflections from the surrounding environment. The red curve displays the measured perturbation once the conventional planar panel was built. The panel consisted of steel sheets with a vertical corrugations 5 cm in depth, in order to be representative of a real industrial building. We observe a significant perturbation of 262 µ A peak to peak, allowing even a strongly attenuated reflection to be visible above the noise floor.

Once the thin diffraction grating was installed the perturbation was reduced by 91%, (green curve). We also wanted to test the behaviour of the diffraction grating when subjected to wind-driven rain. These conditions were simulated thanks to the local fire-brigade who directed three jets of water towards the screen producing an irregular water layer of approximately 2 mm in thickness. The resulting perturbations (blue curve in Figure 12) are almost the same as those obtained in dry conditions. It would have been desirable to also measure the retroreflected signal to characterise whether absorption is also taking place on the panel. Unfortunately, this is not possible with our setup as measurements can only be performed on the runway axis along the null of the radiation pattern. There are however very few loss mechanisms at VHF frequencies in our structure (finite aluminium conductivity, loss in expanded polystyrene spacer and polycarbonate rain protection) and simulations showed no substantial differences when lossy or lossless materials were employed.

Concrete blocks

Solar powered beacon

Polycarbonate rain protection

Measurements were also performed for other frequencies within the ILS band. The resulting fractional reduction of the specular reflection is displayed in Figure 13. We note that the solution is relatively broadband compared to the allowed ILS frequencies. We observe a good agreement obtained with our simulations used in the design of the panel (red curve in Figure 13).

In conclusion, the measurements validated the proposed idea of an electromagnetically thin diffraction grating, producing an observed signal reduction of 90% that is more than sufficient for almost all air navigation applications. On a practical basis, the field trial also demonstrated that the diffraction grating could easily be installed on a standard hangar façade. The technology is now commercialized by Airbus.

  to occur, the difference of these two path contributions must be equal to an integer number, m, of wavelengths, λ , giving us the grating equation.

Figure 1 .

 1 Figure 1. A conventional 'thick' diffraction grating

Figure 3 .

 3 Figure 3. A first design for a thin diffraction grating

Figure 4 .

 4 Figure 4. A second design for a thin diffraction grating

FrequencyFigure 5 .

 5 Figure 5. Optimal values of the plate width, a (m).

FrequencyFigure 6 .

 6 Figure 6. Optimal values of the period, d (m).

FrequencyFigure 7 .

 7 Figure 7. The simulated signal reduction (relative to bare plate) corresponding to the above optimal dimensions.

Figure 8 .

 8 Figure 8. The layout of the test.

Figure 11 .Figure 12 .

 1112 Figure 11. The thin diffraction grating behind its protective screen

Figure 13 .

 13 Figure 13. Measured signal reduction over the ILS bandwidth