FLAT TRACE STATISTICS OF THE TRANSFER OPERATOR OF A RANDOM PARTIALLY EXPANDING MAP

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FLAT TRACE STATISTICS OF THE TRANSFER OPERATOR OF A RANDOM PARTIALLY

EXPANDING MAP

Luc Gossart

To cite this version:

Luc Gossart. FLAT TRACE STATISTICS OF THE TRANSFER OPERATOR OF A RANDOM

PARTIALLY EXPANDING MAP. 2020. �hal-01984379v2�

A RANDOM PARTIALLY EXPANDING MAP

LUC GOSSART

Institut Fourier, 100, rue des maths BP74 38402 Saint-Martin d’Heres France

Abstract. We consider the skew-product of an expanding mapEon the circle Twith an almost surelyC^krandom perturbationτ=τ0+δτof a deterministic functionτ0:

F :

T×R −→ T×R

(x, y) 7−→ (E(x), y+τ(x)) .

The associated transfer operatorL:u∈C^k(T×R)7→u◦Fcan be decomposed with respect to frequency in theyvariable into a family of operators acting on functions on the circle:

L_ξ:

C^k(T) −→ C^k(T) u 7−→ e^iξτu◦E .

We show that the flat traces of Lⁿ_ξ behave as normal distributions in the semiclassical limitn, ξ→ ∞up to the Ehrenfest timen≤cklogξ.

Acknowledgements

The author thanks Alejandro Rivera for his many pieces of advice regarding prob- abilities, and Jens Wittsten, Masato Tsujii, Gabriel Rivi`ere and S´ebastien Gou¨ezel for interesting discussions about this work, as well as the anonymous referees for their careful reading and constructive suggestions.

Date: January 7, 2020.

2010 Mathematics Subject Classification. 37D30 Partially hyperbolic systems and dominated splittings, 37E10 Maps of the circle, 60F05 Central limit and other weak theorems, 37C30 Zeta functions, (Ruelle-Frobenius) transfer operators, and other functional analytic techniques in dy- namical systems.

1

arXiv:1902.00270v2 [math.DS] 6 Jan 2020

Contents

Acknowledgements 1

1. Introduction 3

1.1. Expanding map 4

1.2. Transfer operator 4

1.3. Reduction of the transfer operator 4

1.4. Spectrum and flat trace 4

1.5. Gaussian random fields 5

1.6. Result 6

1.7. Sketch of proof 6

2. Numerical experiments 8

3. Proof of theorem 1.4 9

3.1. Definition of a Gaussian field satisfying theorem 1.4 10

3.2. New expression for Tr^[(Lⁿ_ξ,τ) 11

3.3. A normal law of large variance on the circle is close to uniform 13

3.4. End of proof 16

4. Discussion 19

Appendix A. Proof of lemma A.1 21

Appendix B. Proof of lemma 1.3 on the link between regularity of a

Gaussian field and variance of the Fourier coefficients 21

Appendix C. Ruelle resonances and Flat trace 22

C.1. Ruelle spectrum 22

C.2. Flat trace 23

Appendix D. Proof of lemma 1.8 25

Appendix E. Topological pressure 25

E.1. Definition 25

E.2. Variational principle 26

E.3. Proof of lemma 3.12 26

References 29

1. Introduction

This paper focuses on the distribution of the flat traces of iterates of the transfer operator of a simple example of partially expanding map. It is motivated by the Bohigas-Gianonni-Schmidt [BGS84] conjecture in quantum chaos (see below).

In chaotic dynamics, the transfer operator is an object of first importance linked to the asymptotics of the correlations. The collection of poles of its resolvent, called Ruelle-Pollicott spectrum, can be defined as the spectrum of the transfer operator in appropriate Banach spaces (see [Rue76] for analytic expanding maps, [Kit99],[BKL02],[BT07],[BT08],[GL06],[FRS08] for the construction of the spaces for Anosov diffeomorphisms.)

The study of the Ruelle spectrum for Anosov flows is more difficult because of the flow direction that is neither contracting nor expanding. Dolgopyat has shown in particular in [Dol98] the exponential decay of correlations for the geodesic flow on negatively curved surfaces, and Liverani [Liv04] generalized this result to allC⁴ contact Anosov flows. His method involved the construction of anisotropic Banach spaces in which the transfer operator is quasicompact, and no longer relies on sym- bolic dynamics that prevented from using advantage of the smoothness of the flow.

Tsujii [Tsu10] constructed appropriate Hilbert spaces for the transfer operator of C^r contact Anosov flows, r ≥ 3 and gave explicit upper bounds for the essential spectral radii in terms ofrand the expansion constants of the flow. Butterley and Liverani [BL07] and later Faure and Sjstrand [FS11] constructed good spaces for Anosov flows, without the contact hypothesis. Weich and Bonthonneau defined in [BW17] Ruelle spectrum for geodesic flow on negatively curved manifolds with a finite number of cusps. Dyatlov and Guillarmou [DG16] handled the case of open hyperbolic systems. A simple example of Anosov flow is the suspension of an Anosov diffeomorphism, or the suspension semi-flow of an expanding map. Polli- cott showed exponential decay of correlations in this setting under a weak condition in [Pol85] and Tsujii constructed suitable spaces for the transfer operator and gave an upper bound on its essential spectral radius in [Tsu08].

In this article we study a closely related discrete time model, the skew product of an expanding map of the circle. It is a particular case of compact group extension [Dol02], which are partially hyperbolic maps, with compact leaves in the neutral direction that are isometric to each other . Dolgopyat showed in [Dol02] that in this case the correlation decrease generically rapidly. In our setting of skew-product of an expanding map of the circle, Faure [Fau11] has shown using semi-classical meth- ods an upper bound on the essential spectral radius of the transfer operator under a condition shown to be generic by Nakano Tsujii and Wittsten [NTW16]. Arnoldi, Faure, and Weich [AFW17] and Faure and Weich [FW17] studied the case of some open partially expanding maps, iteration function schemes, for which they found an explicit bound on the essential spectral radius of the transfer operator in a suitable space, and obtained a Weyl law (upper bound on the number of Ruelle resonances outside the essential spectral radius). Naud [Nau16] studied a model close to the one presented in this paper, in the analytic setting, in which the transfer operator is trace-class, and used the trace formula, in the deterministic and random case to obtain a lower bound on the spectral radius of the transfer operator. In the more general framework of random dynamical systems in which the transfer operator

changes randomly at each iteration, for the skew product of an expanding map of the circle, Nakano and Wittsten [NW15] showed exponential decay of correlations.

Semiclassical analysis describes the link between quantum dynamics and the as- sociated classical dynamics in a symplectic manifold. The transfer operator happens to be a Fourier integral operator and the semi-classical approach has thus shown to be useful. The famous Bohigas-Giannoni-Schmidt [BGS84] conjecture of quan- tum chaos states that for quantum systems whose associated classical dynamic is chaotic, the spectrum of the Hamiltonian shows the same statistics as that of a ran- dom matrix (GUE, GOE or GSE according to the symmetries of the system)(see also [Gut13] and [GVZJ91]). We are interested analogously in investigating the possible links between the Ruelle-Pollicott spectrum and the spectrum of random matrices/operators. At first we try to get informations about the spectrum using a trace formula. More useful results could follow from the use of a global normal form as obtained by Faure-Weich in [FW17].

1.1. Expanding map. Let us consider a smooth orientation preserving expanding mapE :T→Ton the circleT=R/Z, that is, satisfying E⁰>1, of degreel, and let us call

m:= infE⁰>1 and

M := supE⁰.

1.2. Transfer operator. Let us fix a functionτ ∈C^k(T) for somek≥1. We are interested in the partially expanding dynamical system onT×Rdefined by

(1.1) F(x, y) = (E(x), y+τ(x))

We introduce the transfer operator Lτ :

C^k(T×R) −→ C^k(T×R) u 7−→ u◦F .

1.3. Reduction of the transfer operator. Due to the particular form of the mapF, the Fourier modes iny are invariant underLτ: if for someξ∈Rand some v∈C^k(T),

u(x, y) =v(x)e^iξy, then

Lτu(x, y) =e^iξτ(x)v(E(x))e^iξy.

Givenξ≥0 and a functionτ, let us consequently consider the transfer operator Lξ,τ defined on functionsv∈C^k(T) by

∀x∈T,Lξ,τv(x) :=e^iξτ(x)v(E(x)),

1.4. Spectrum and flat trace. In appropriate spaces, the transfer operator has a discrete spectrum outside a small disk, the eigenvalues are called Ruelle resonances.

It is in general not trace-class, but one can define its flat trace (see Appendix C for a more precise discussion about Ruelle resonances, flat trace and their relationship).

Lemma 1.1 (Trace formula, [AB67], [G⁺77]). For any C¹ function τ on T, the flat trace of Lⁿ_ξ,τ is well defined and

(1.2) Tr^[Lⁿ_ξ,τ = X

x,Eⁿ(x)=x

e^iξτxn (Eⁿ)⁰(x)−1,

whereτ_xⁿ denotes the Birkhoff sum: For a functionφ∈C(T)and a pointx∈Twe define

(1.3) φⁿ_x:=

n−1

X

k=0

φ(E^k(x)).

1.5. Gaussian random fields. We define our random functions on the circle by means of their Fourier coefficients. We are only interested in C¹ functions. We will denote by N(0, σ²) (respectively N_C(0, σ²)) the real (respectively complex) centered Gaussian law of varianceσ², with respective densities

1 σ√

2πe^−2σ12x2 and 1

σπe^−σ12|z|2.

With these conventions, a random variable of law N_C(0, σ²) has independent real and imaginary parts of lawN(0,^σ₂²), and the variance of its modulus is consequently σ².

Definition 1.2. We will call centered stationary Gaussian random fields onTthe real random distributionsτ whose Fourier coefficients (cp(τ))_p≥1 are independent complex centered Gaussian random variables, with variances growing at most poly- nomially, such that c0(τ) is a real centered Gaussian variable independent of the cp(τ), p≥1. The negative coefficients are necessarily given by

c−p(τ) =cp(τ).

The Gaussian fields are in general defined as distributions if their Fourier coeffi- cients have variances with polynomial growth and the decay of the variances of the coefficients gives sufficient conditions for the regularity of the field.

Lemma 1.3. If E[|c_p(τ)|²] has a polynomial growth, τ = P

pc_p(τ)e^2iπp· defines almost surely a distribution: almost surely

∀φ=X

cp(φ)e^2iπp·∈ C^∞(T), hτ, φi:=X

p

cp(τ)cp(φ)<∞.

Let k∈N.If for someη >0

(1.4) E

|c_p(τ)|²

=O 1

p^2k+2+η

. Thenτ is almost surelyC^k.

Proof. See appendix B.

In what follows we will always assume that (1.4) is satisfied, at least fork= 1, so that our random fields are random variables on C¹(T). This will ensure the existence of flat traces.

1.6. Result. Ifxis a periodic point, let us write its prime period lx:= min{k≥1, E^k(x) =x}.

Let us define for everyn∈N:

(1.5) An:=



 X

Eⁿ(x)=x

l_x ((Eⁿ)⁰(x)−1)²





−¹₂

Theorem 1.4. Let k∈N^∗. Letτ₀∈ C^k(T). Let δτ =X

p∈Z

cpe^2iπp·

be a centered Gaussian random field, such that E[|cp|²] =O(p^−4−ν) for some ν >0. This way,δτ is a.s. C¹. If

(1.6) ∃ >0,∃C >0,∀p∈Z^∗,E h|cp|²i

≥ C

p^2k+2+, then one has the convergence in law of the flat traces

(1.7) A_nTr^[ Lⁿ_ξ,τ

0+δτ

−→ N_C(0,1) asnandξ go to infinity, under the constraint

(1.8) ∃0< c <1,∀n, ξ, n≤c logξ

logl+ (k+¹₂+₂) logM. Note that condition (1.6) can allow τ to beC^k by Lemma 1.3.

Remark 1.5. The statement implies that the convergence still holds if we multiply δτ by an arbitrarily small numberη >0. For instance forτ0= 0,

AnTr^[ Lⁿ_ξ,0

−→ ∞

at exponential speed, uniformly inξ, but ifδτ is an irregular enough Gaussian field in the sense of (1.6), then for anyη >0 andc <1 holds

AnTr^[ Lⁿ_ξ,η·δτ

−→ N_C(0,1) under condition (1.8).

Remark 1.6. Condition (1.8) means that time nis smaller than a constant times the Ehrenfest time logξ, and this constant decreases with the regularity k of the fieldδτ.

1.7. Sketch of proof. The proof is based around the following arguments:

(1) Note first that the convergence (1.7) is satisfied if all the phases appearing in (1.2) are independent and uniformly distributed.

Remark 1.7. For sake of simplicity, in this sketch of proof, we will state pairwise independence for the phases in (1.2), while in fact we must pack them by orbits, since Birkhoff sumsφⁿ_x are the same on all the orbit, but this changes little to the problem. For instance this simplification would remove the factor l_x in the definition (3.21) of A_n corresponding to this multiplicity.

The convergence can be deduced from the standard proof of the central limit theorem showing pointwise convergence of the characteristic function.

However, here, since the periodic points are dense inT, requiring indepen- dence of the values (δτ(x))_En(x)=xwould lead to very bad regularity of the field (it is not hard to see that it would be almost surely nowhere locally bounded).

(2) We fix a Gaussian fieldδτ =Pc_pe^2iπp·fulfilling the hypothesis of Theorem 1.4 and start by constructing an auxiliary field with the same law and show that it satisfies the convergence (1.7). This is sufficient since the convergence in law only involves the law of the random field.

(3) For eachj≥1, we construct a smooth random fieldδτj, such that for every periodic points x 6= y of periodj, δτj(x) and δτj(y) are independent. If moreover ξis large enough, the variables ξ(δτn)ⁿ_x are Gaussian with large variances, soξ(δτ_n)ⁿ_x mod 2π(and therefore the phasese^iξ(δτn)nx) are close to be uniform. Consequently, since by (1.2) Tr^[(Lⁿ_ξ,δτn) only involves points of periodn, and thus independent almost uniform phases, the convergence (1.7) should hold for Tr^[(Lⁿ_ξ,δτn) under a certain relation betweennand ξ that will be explained in number (8).

(4) An important point is that if the phases (e^iξ(δτn)nx)_{x∈T,En(x)=x} are inde- pendent and close to be uniform, then adding toδτn an independent field will not change this fact, as the following lemma suggests:

Lemma 1.8. Let X, X⁰ be real independent random variables such that e^iX, e^iX0 are uniform onS¹. Let Y, Y⁰ be real random variables such that X andX⁰ are independent of both Y andY⁰. Thene^i(X+Y) ande^i(X0+Y0) are still independent uniform random variables on S¹.

Note that no independence betweenY and Y⁰ is needed. See appendix D for the proof.

(5) Using this analogy, if the fieldsδτ_jare chosen independent, it should follow that the convergence (1.7) holds for Tr^[

Lⁿ_ξ,^P

j≥1δτ_j

for large ξ.

(6) The fields δτj are almost surely smooth. However, because the distance between periodic points decreases exponentially fast with n according to Lemma A.1, the requirementE[δτn(x)δτn(y)] = 0 for allx6=yof periodn leads to an exponential growth with respect to nof kδτnk_Ck renormalized by the standard deviation (independent of the pointx):

kδτnk_Ck≈M^knp

E[|δτn(x)|²].

This can be deduced heuristically from the fact (see Definition 3.1 below) that

E[δτ_n(x)δτ_n(y)] =X

p

E[|cp(δτ_n)|²]e^ip(x−y)=:K_n(x−y)

and the incertitude principle: a localisation ofKn at a scaleM⁻ⁿ implies non negligible coefficientsE[|cp(δτn)|²] forpof orderMⁿ. As a consequence we need

(1.9) p

E[|δτn(x)|²]≤ C M^n(k+12+2) to obtain aC^k functionP

j≥1δτj.

(7) This condition, together with (1.6) implies that the Fourier coefficients ˜cp

ofP

j≥1δτj satisfy

E[|˜cp|²]≤CE[|cp|²].

This allows us to define a field δτ0, that we chose independent from the otherδτj, by

E[|cp(δτ0)|²] =CE[|cp|²]−E[|˜cp|²], so that _C¹P

j≥0δτjhas the same law asδτand still satisfies the convergence (1.7) forξlarge enough from (4) of this sketch.

(8) To get an idea of the origin of the relation (1.8) between n and ξ, let us assume that we want all the arguments ξ(δτ_n)ⁿ_x in Tr^[(Lⁿ_ξ,δτ

n) to go uniformly to infinity in order to get approximate uniformity of the phases and thus convergence towards a Gaussian law. Note that for anyx,

(1.10) P

"

|(δτn)ⁿ_x| pE[|(δτn)ⁿ_x|²] ≤

#

→0= O().

Let (Cn) be a sequence going to infinity.

(1.10) implies P





\

Eⁿ(x)=x

{ξ(δτn)ⁿ_x > Cn}



 = 1−P[∃x, Eⁿ(x) =x, ξ(δτn)ⁿ_x≤Cn]

≥1− X

Eⁿ(x)=x

P[ξ(δτn)ⁿ_x≤Cn]

LemmaA.1= 1−(lⁿ−1)P[ξ(δτn)ⁿ_x ≤Cn]

≥

(1.10)

1−Clⁿ Cn

ξp

E[|(δτn)ⁿ_x|²] ifxdenotes any point and ξ √ ^Cn

E[|(δτn)ⁿ_x|²].We deduce from (1.9) that p

E[|(δτn)ⁿ_x|²]≤ Cn M^n(k+12+2). Thus

P[ξ(δτn)ⁿ_x → ∞uniformly w.r.t. xs.t. Eⁿ(x) =x]→1 forξlⁿM^n(k+12+2),which gives (1.8).

2. Numerical experiments We consider an example with the non linear expanding map (2.1) E(x) = 2x+ 0.9/(2π) sin(2π(x+ 0.4))

plotted on Figure 1. In Figure 2, we have the histogram of the modulus S =

AnTr^[ Lⁿ_ξ,τ

0+δτ

obtained after a sample of 10⁴ random functionsδτ. We com- pare the histogram with the function CSexp(−S²) in red, i.e. the radial distri- bution of a Gaussian function, obtained from the prediction of Theorem 1.4. We tookn= 11, ξ= 2.10⁶, τ0 = cos(2πx). We also observe a good agreement for the (uniform) distribution of the arguments that is not represented here.

Figure 1. Graph of the expanding mapE(x) in Eq.(2.1)

0 0.5 1 1.5 2 2.5 |S|3

0 50 100 150 200 250

Figure 2. In blue, the histogram of S =

A_nTr^[ Lⁿ_ξ,τ

0+δτ

for n = 11, ξ = 2.10⁶, τ0 = cos(2πx) and the sample 10⁴ random functionsδτ. The histogram is well fitted byCSexp(−S²) in red, as predicted by Theorem 1.4

3. Proof of theorem 1.4

A stationary centered Gaussian random field is characterized by its covariance function:

Definition 3.1. Letτ =P

p∈Zc_pe^2iπp·be a stationary centered Gaussian random field. Let us define its covariance functionK by

(3.1) K(x) :=X

p

E[|c_p|²]e^2iπpx. For any couple of points (x, y)∈T²,we have

E[τ(x)τ(y)] =K(x−y).

Proof of the last statement. Since we assume (1.4) to hold for some k≥1, E[τ(x)τ(y)] = X

p,q∈Z

E[c_p(τ)c_q(τ)]e^2iπ(px+qy)

=X

p∈Z

E[|cp|²]e^2iπp(x−y)+E[cp2]e^2iπp(x+y)

from the independence relationships of the Fourier coefficients. Now, if we write cp =xp+iyp

E[c_p²] =E[x²_p]−E[y²_p] + 2iE[x_py_p] = 0.

3.1. Definition of a Gaussian field satisfying theorem 1.4. Let us fix a ran- dom centered Gaussian fieldδτ =P

p∈Zcpe^2iπp· satisfying the hypothesis of The- orem 1.4. Let us define the Gaussian fields mentioned in step (3) of the sketch of proof. Let K_init ∈ C^∞_c (R) be a smooth function supported in

−¹₃,¹₃

, with non negative Fourier transform, satisfying¹

(3.2) K_init(0) = 1.

Let k ≥ 1 be the integer involved in Theorem 1.4 giving the regularity of the field. Let >0 be the constant appearing in Theorem 1.4 and define for any integer j≥1

(3.3) Kj(x) = 1

M^j(2k+1+)Kinit(M^jx).

The Fourier transform ofK_j is given by

(3.4) Kc_j(ξ) = 1

M^j(2k+2+)K[_init ξ

M^j

≥0 The functions Kj, for allj≥1, are supported in

−¹₃,¹₃

and can then be seen as functions on the circleTby trivially periodizing them. Letcp,j, forp≥0, j≥1 be independent centered Gaussian random variables of respective variances ˆKj(2πp), and let us write

δτj=X

p

cp,je^2iπp·, wherec_−p,j:=c_p,j, p≥1.

Lemma 3.2. P

j≥1δτ_j is a centered Gaussian random field Pc˜_pe^2iπp· and E

|˜cp|²

=O E

|cp|² . Proof. We have seen in Eq.(3.4) that

Kc_j(ξ) = 1

M^j(2k+2+)K[_init( ξ M^j).

SinceKinitis smooth, there exists a constant C >0 such that

∀ξ∈R,K[_init(ξ)≤ C hξi^2k+2+2,

1To construct such a function, take a non zero even functiong∈ C_c^∞(R). ghas a real Fourier transform. Theng∗g∈ C_c^∞(R) and its Fourier transform is ˆg²≥0. Moreoverg∗g(0) =R

ˆ g²>0.

with the usual notationhξi=p

1 +ξ²≥ |ξ|. Thus, E

|cp,j|²

= 1

M^j(2k+2+)K[init(2πp M^j)

≤ C M^j2

1

|2πp|^2k+2+2. Consequently, since by independence

E h|˜c_p|²i

=E







X

j≥1

c_p,j

2



=X

j≥1

E

h|c_p,j|²i ,

E h|˜cp|²i

=O

1

|2πp|^2k+2+2

=

(1.6)O(E

|cp|² ).

Thus, fixing a constantC such that

CE

|cp|²

≥E h|˜cp|²i

, we can define a random Gaussian fieldδτ0 =P

p∈Zcp,0e^2iπp· with coefficientscp,0

independent from thec_p,j such that E

|cp,0|²

=CE

|cp|²

−E h|˜cp|²i

. This way _C¹ P

j≥0δτ_j and δτ have the same law. By this we mean that their Fourier coefficients have the same laws. By our hypothesis, the convergence of the Fourier series are almost surely normal, thus for any finite subset {xk}k of T, (_C¹ Pδτj(xk))kand (δτ(xk))khave the same law. Therefore, the laws of Tr^[(Lⁿ_ξ,τ

0+δτ) and Tr^[(Lⁿ_ξ,τ

0+_C¹ Pδτj) are the same, and the convergence of Theorem 1.4 is equiv- alent to

(3.5) A_nTr^[(Lⁿ_ξ,τ

0+P

δτj)−→ N^L _C(0,1)

under condition (1.6). (The constant _C¹ can be ’absorbed’ inξup to the replacement ofτ0byCτ0 that has no consequence.) In the rest of the paper we will show (3.5) and will write

(3.6) τ:=τ0+X

j≥0

δτj.

3.2. New expression for Tr^[(Lⁿ_ξ,τ). We will write the set of periodic orbits of (non primitive) periodnas

(3.7) Per(n) :=

{x, E(x),· · · , Eⁿ⁻¹(x)}, Eⁿ(x) =x, x∈T , and the set of periodic orbits of primitive periodnas

(3.8) Pn:=

{x, E(x),· · · , Eⁿ⁻¹(x)}, n= min{k∈N^∗, E^k(x) =x}, x∈T . This way, Per(n) is the disjoint union

(3.9) Per(n) =a

m|n

Pm.

Let us rewrite the sum Tr^[(Lⁿ_ξ,τ), where τ is given by (3.6). We know from (1.2) that

Tr^[(Lⁿ_ξ,τ) = X

Eⁿ(x)=x

e^iξτxn (Eⁿ)⁰(x)−1

= X

Eⁿ(x)=x

e^iξτxn e^Jxn−1,

whereJ(x) = log(E⁰(x))>0 andJ_xⁿ is the Birkhoff sum as defined in (1.3). If f_Oⁿ stands for the Birkhoff sumf_xⁿ for anyx∈O , let us write

(3.10) Tr^[(Lⁿ_ξ,τ) =X

m|n

m X

O∈Pm

e^iξτOn e^JOn−1. ForO∈Per(n), we can write

τ_Oⁿ = (δτn)ⁿ_O+X

j6=n

(δτj)ⁿ_O+ (τ0)ⁿ_O.

Since the covariance function Kn is supported in

−_3M¹n,_3M¹n

, we deduce from Lemma A.1 that the values taken by δτn at different periodic points of period dividingn, which have lawN(0, K_n(0)) are independent random variables.

Thus, forn∈N,m|nandO∈ Pm, (δτ_n)^m_O is a centered Gaussian random variable of variancemK_n(0), and (δτ_n)ⁿ_O= _mⁿ(δτ_n)^m_O has variance (_mⁿ)²mK_n(0) = ⁿ_m²K_n(0).

Definition 3.3. We say that two families of real random variables (X_Oⁿ)n≥1 O∈Per(n)

and (Y_Oⁿ)n≥1 O∈Per(n)

satisfy condition (C) if

(1) for everym|n, andO∈ Pm,X_Oⁿ has lawN(0,ⁿ_m²K_n(0)),

(2) for every O⁰ 6=O ∈Per(n) and everyO⁰⁰ ∈Per(n), X_Oⁿ is independent of X_Oⁿ0 andY_Oⁿ00.

WritingX_Oⁿ = (δτn)ⁿ_O andY_Oⁿ=P

j6=n(δτj)ⁿ_O+ (τ0)ⁿ_O, we have obtained Lemma 3.4. There exist families of random variables(X_Oⁿ),(Y_Oⁿ) satisfying con- dition (C) of Definition (3.3) such that for every n≥1 andO∈Per(n)

(3.11) τ_Oⁿ =X_Oⁿ +Y_Oⁿ

In order to adapt the proof of lemma 1.8, we want to show that for largeξ, the random variablese^iξ(XOn+YOn), O∈Per(n) are close to be independent and uniform onS¹.

Remark 3.5. We have

(3.12) Tr^[(Lⁿ_ξ,τ) =

(3.10),(3.11)

X

m|n

m X

O∈Pm

e^iξ(XOn+YOn) e^JOn−1 .

Our aim is to approximate the characteristic function of AnTr^[(Lⁿ_ξ,τ) which is the expectation of

(3.13) exp iA_n

µRe(Tr^[(Lⁿ_ξ,τ)) +νIm(Tr^[(Lⁿ_ξ,τ))

= Y

m|n

Y

O∈Pm

exp

i mA_n

e^JOn−1(µcos(ξ(X_Oⁿ+Y_Oⁿ)) +νsin(ξ(X_Oⁿ +Y_Oⁿ)))

for fixed,µ, ν∈R.The right hand side of (3.13) can be written as Y

m|n

Y

O∈Pm

fO

e^iξ(XOn+YOn)

!^m

for some continuous functionsf_O:S¹−→C(depending on µ, ν):

fO(z) = exp

i An

e^JOn−1(µRe(z) +νIm(z))

In the next Lemma we first consider indicator functions onS¹ forfO. Lemma 3.6. Let (X_Oⁿ)_n≥1

O∈P er(n)

and (Y_Oⁿ)_n≥1

O∈P er(n)

be two families of real random variables satisfying satisfying condition (C) of Definition (3.3). Assume thatnand ξsatisfy (1.8). Then there is a constantC >0such that for everyn∈Nand every real numbers (αO)_O∈Per(n),(βO)_O∈Per(n) such that

∀O∈Per(n), 0< β_O−α_O<2π,

for every complex numbers (λ_O)_O∈Per(n), if A_O :=e^i]αO,βO[ ⊂S¹ ⊂C and 1AO : S¹→Cis the characteristic function ofAO, we have

(3.14) E

"

Q

m|n

Q

O∈Pm

λO1^AO e^iξ(XOn+YOn)

!^m#

Q

m|n

Q

O∈Pm

λ^m_Oβ

O−αO

2π

−1

≤ξ^c−1n⁻¹²

→

(1.8)

0

.

Remark 3.7. In this expression, we compare the law of the family of random vari- ables e^iξ(XnO+YOn)

O∈Per(n)to the uniform law on the torus of dimension #Per(n).

The proof of this lemma is given in the next subsection.

3.3. A normal law of large variance on the circle is close to uniform. We will need the following lemma, which evaluates how much the lawN(0,1) mod ¹_t differs from the uniform law on the circleR/(¹_tZ) for large values oft.

Lemma 3.8. There exists a constant C >0 such that for every real numbersα, β, such that 0< β−α <2π and every real numbert≥1,

Z

R

X

k∈Z

1^α+2kπ

t ≤x≤^β+2kπ_t e^−x

2 2 dx

√2π−β−α 2π

≤ C

t (β−α).

Proof. By mean value inequality, if|x−y| ≤1, then

e^−x

2 2 −e^−y

2 2

≤ |x−y|f(y)

2π t

β−α t

Figure 3. As t goes to infinity, the red area converges to ^β−α_2π with speedO(^β−α_t ).

for theL¹function

f(y) := sup

|u−y|≤1

|u|e^−u22

Let us then write foru∈[^α_t,^β_t], uk :=u+^2kπ_t andIk := [uk, uk+1]. We have just seen that fort≥2π, for ally∈Ik,

e⁻

u2 k 2 −e^−y

2 2

≤ C tf(y).

Integrating overy∈Ik of length ^2π_t and summing overk∈Zyields

2π t

X

k∈Z

e⁻

u2 k

2 −√

2π

≤ C t

(The value of the constantCchanges at each line, but it depends neither ont, nor onα, β.) Averaging overu∈[^α_t,^β_t] gives

2π β−α

X

k∈Z

Z ^β_t

α t

exp

−(u−2kπ)² 2

du−√

2π

≤ C t. Consequently,

Z

R

X

k∈Z

1^α+2kπ_t ≤x≤^β+2kπ_t e^−x

2 2 dx

√

2π−β−α 2π

≤C

t (β−α)

Proof of lemma 3.6. Let us denote byEthe expectation E:=E



 Y

m|n

Y

O∈Pm

λ_O1AO

e^iξ(XOn+YOn)

!m

.

If we write respectively PX, PY and PX,Y the probability laws of the variables (ξX_Oⁿ)_O∈Per(n), (ξY_Oⁿ)_O∈Per(n)and (ξX_Oⁿ)_O∈Per(n)∪(ξY_Oⁿ)_O∈Per(n)respectively, then condition (C) of Definition (3.3) implies

(3.15) dPX,Y((x_O)_O∈Per(n),(y_O)_O∈Per(n)) = Y

m|n

Y

O∈P_m

e⁻

xO2 2σ2

n,ξ dxO

σn,ξ

√2π⊗dPY((yO)_O∈Per(n)).

with the varianceσ²_n,ξ :=ξ²ⁿ_m²Kn(0). We have

(3.16) E= Z

R^2#Per(n)

Y

O∈Per(n)

X

k∈Z

λ^m_O1]αO+2kπ,βO+2kπ[(xO+yO)

!

dPX,Y((xO)_O∈Per(n),(yO)_O∈Per(n)).

Thus, writingu_O= _σ^xO

n,ξ forO∈ P_m, (3.17)

E= Z

R^#Per(n)

Y

m|n

Y

O∈Pm

Z

R

X

k∈Z

λ^m_O1ⁱαO−yO+2kπ

σn,ξ ,^{βO−yO+2kπ}

σn,ξ

h(uO)e⁻

u2 O 2 duO

√2π

!

dPY((yO)_O∈Per(n)).

Let us write forO∈Per(n) IO =

Z

R

X

k∈Z

λ^m_O1ⁱαO−yO+2kπ

σn,ξ ,^βO−_σn,ξ^yO+2kπh(uO)e⁻

u2 O 2 duO

√ 2π. Lemma 3.8 yields

I_O=λ^m_Oβ_O−α_O

2π (1 +_O), where

∃C >0,|O| ≤ C σ_n,ξ

≤ C

ξp

nK_n(0).

Let us remark that for every finite family{xk}k⊂R, the expansion of the product and factorization after triangular inequality give

Y

k

(1 +xk)−1

≤Y

k

(1 +|xk|)−1.

Thus,

Q

m|n

Q

O∈Pm

IO

Q

m|n

Q

O∈Pm

λ^m_Oβ

O−αO

2π

−1

=

Y

m|n

Y

O∈Pm

(1 +O)−1

≤

1 + C

ξ(nK_n(0))^1/2

^#Per(n)

−1

!

From Lemma A.1 we have #Per(n)≤lⁿ.

Using hypothesis (1.8) we can bound the prefactor:

1 + C

ξ(nK_n(0))^1/2 lⁿ

−1 =

(3.3),(3.2) 1 + CM^n(k+12+2) ξ√

n

!^ln

−1

≤exp(lⁿCM^n(k+12+2) ξ√

n )−1

≤C⁰lⁿM^n(k+12+2) ξ√

n

for someC⁰>0 for nand ξlarge enough and satisfying (1.8) since (3.18) lⁿCM^n(k+12+2)

ξ√

n ≤

(1.8)

Cξ^c−1n⁻¹² →0.

3.4. End of proof. We can now easily extend the lemma 3.6 from characteristic functions to step functions.

Corollary 3.9. Assume thatnandξsatisfy (1.8). For any families(X_Oⁿ)n≥1 O∈Per(n)

and (Y_Oⁿ)_n≥1

O∈Per(n)

of real random variables satisfying condition (C) of Definition (3.3), there exists C > 0 such that, if (fn,O)_n≥1

O∈Per(n)

is a family of step functions S¹→R, then

(3.19) E



 Y

m|n

Y

O∈Pm

f_n,O^m (e^iξ(XOn+YOn))



−Y

m|n

Y

O∈Pm

Z

f_n,O^m dLeb

≤Cξ^c−1n⁻¹² Y

m|n

Y

O∈Pm

Z

|f_n,O^m |dLeb.

Proof. Let us write eachfn,O as f_n,O=

p_n,O

X

q=1

λ_n,O,q1An,O,q,

where the λn,O,q are complex numbers and the An,O,q,1 ≤ q ≤pn,O are disjoint intervals. We develop (3.19), we use Lemma 3.6 and factorize the result:

E:=E



 Y

m|n

Y

O∈Pm

f_n,O^m (e^iξ(XOn+YOn))





= X

(qO)∈Q

m|n

Q

O∈Pm

{1,···,pn,O}

E



 Y

m|n

Y

O∈P_m

λ^m_n,O,qO1^A_n,O,qO(e^iξ(XOn+YOn))



.

Consequently,

E−Y

m|n

Y

O∈Pm

Z

f_n,O^m dLeb

≤ X

(qO)∈Q

m|n

Q

O∈Pm

{1,···,pn,O}

E



 Y

m|n

Y

O∈P_m

λ^m_n,O,qO1^A_n,O,qO(e^iξ(XOn+YOn))



−

Y

m|n

Y

O∈P_m

λ^m_n,O,qOLeb(An,O,q_O)

≤Cξ^c−1n⁻¹² X

(qO)∈Q

m|n

Q

O∈Pm

{1,···,pn,O}

Y

m|n

Y

O∈Pm

|λn,O,q_O|^mLeb(An,O,q_O)

from the previous lemma.

Hence,

E−Y

m|n

Y

O∈Pm

Z

f_n,O^m dLeb

≤Cξ^c−1n⁻¹² Y

m|n

Y

O∈Pm

Z

|fn,O|^mdLeb.

We can use this result in order to estimate the characteristic function of Tr^[(Lⁿ_ξ,τ), using remark (3.5).

Corollary 3.10. Assume thatnandξsatisfy (1.8). Let(X_Oⁿ)n≥1 O∈Per(n)

and(Y_Oⁿ)n≥1 O∈Per(n)

be two families of real random variables satisfying condition (C) of Definition (3.3).

There existsC >0such that for all (µO, νO)_O∈Per(n)∈R^2#Per(n),

E



 Y

m|n

Y

O∈P_m

e^{imµOcos(ξ(XOn+YOn))+imνOsin(ξ(XOn+YOn))}





−Y

m|n

Y

O∈Pm

Z 2π 0

e^{i(mµOcosθ+mνOsinθ)}dθ 2π

≤Cξ^c−1n⁻¹²,

Proof. Let C be the constant from corollary 3.9. For O ∈ Per(n), let fO be the function defined onS¹by

f_O(e^iθ) =e^{i(µOcosθ+νOsinθ)}.

Each fO is bounded by 1, we can consequently find for eachO ∈Per(n) a family (fj,O)jof step functions uniformly bounded by 1 converging pointwise towardsfO. We have fornfixed, by dominated convergence

Ej :=E



 Y

m|n

Y

O∈P_m

f_j,O^m (e^iξ(XOn+YOn))



−−−→

j→∞ E:=E



 Y

m|n

Y

O∈P_m

f^m(e^iξ(XOn+YOn))





as well as Ij:=Y

m|n

Y

O∈Pm

Z 2π 0

f_j,O^m (e^iθ)dθ 2π −−−→

j→∞ I:=Y

m|n

Y

O∈Pm

Z 2π 0

f_O^m(e^iθ)dθ 2π. It is thus possible to find an integerj0such that both

|E−Ej₀| ≤ξ^c−1n⁻¹² and

|I−I_j0| ≤ξ^c−1n⁻¹² hold.

From corollary 3.9, we know that for alln∈N

|Ej0−I_j0| ≤Cξ^c−1n⁻¹²sup|fj0|.

Thus,

|E−I| ≤ |E−Ej₀|+|Ej₀−Ij₀|+|I−Ij₀|

≤(C+ 2)ξ^c−1n⁻¹².

We can know prove the final proposition :

Proposition 3.11. Let(X_Oⁿ)_n≥1

O∈Per(n)

and(Y_Oⁿ)_n≥1

O∈Per(n)

be two families of real ran- dom variables satisfying condition (C) of Definition (3.3). If condition (1.8) is satisfied then we have the following convergence in law

(3.20) T_n,ξ:=A_nX

m|n

m X

O∈Pm

e^iξ(XOn+YOn) e^JOn−1 −→

n,ξ→∞N_C(0,1), with the amplitudeAn defined in (1.5) by

An=



 X

m|n

m² X

O∈Pm

1 (e^JOn−1)²





−¹₂

(3.21) .

Proof. Let us fix two real numbersξ₁andξ₂and letφ_nbe the characteristic function ofTn,ξ:

φ_n(ξ₁, ξ₂) :=E



exp



iA_n



ξ₁X

m|n

m X

O∈Pm

cos(ξ(X_Oⁿ+Y_Oⁿ)) e^JOn−1 + ξ₂X

m|n

m X

O∈Pm

sin(ξ(X_Oⁿ +Y_Oⁿ)) e^JOn−1











.