www.elsevier.com/locate/anihpb
Berry–Esseen theorem and local limit theorem for non uniformly expanding maps
Sébastien Gouëzel
Département de mathématiques et applications, École normale supérieure, 45, rue d’Ulm 75005 Paris, France Received 28 June 2004; accepted 24 September 2004
Available online 29 January 2005
Abstract
In Young towers with sufficiently small tails, the Birkhoff sums of Hölder continuous functions satisfy a central limit theorem with speedO(1/√
n ), and a local limit theorem. This implies the same results for many non uniformly expanding dynamical systems, namely those for which a tower with sufficiently fast returns can be constructed.
2005 Elsevier SAS. All rights reserved.
Résumé
Dans les tours de Young ayant des queues suffisamment petites, les sommes de Birkhoff des fonctions hölderiennes satisfont le théorème central limite avec vitesseO(1/√
n )et le théorème de la limite locale. Par conséquent, de nombreux systèmes dynamiques non uniformément dilatants satisfont les mêmes conclusions : il suffit de pouvoir construire une tour avec des retours à la base suffisamment rapides.
2005 Elsevier SAS. All rights reserved.
MSC: 37A30; 37A50; 37C30; 37E05; 47A56; 60F15
Keywords: Non uniformly expanding maps; Berry–Esseen theorem; Local limit theorem
1. Results 1.1. Introduction
LetT:X→X be a probability preserving transformation andf:X→R. The functionsf ◦Tk, fork∈N, are identically distributed random variables, and it is an important problem in ergodic theory to see whether they satisfy the same kind of limit theorems as independent random variables.
E-mail address: [email protected] (S. Gouëzel).
0246-0203/$ – see front matter 2005 Elsevier SAS. All rights reserved.
doi:10.1016/j.anihpb.2004.09.002
Many results are known whenT is uniformly expanding or uniformly hyperbolic (without or with singularities, in the Markov or non Markov case), andf is Hölder continuous. In this case, it is indeed often possible to construct a space of functions containingf on which the transfer operator associated toT has a spectral gap. Therefore, the spectral perturbation method, introduced by Nagaev in the case of Markov chains, makes it possible to mimic the probabilistic proofs on independent variables. In this way, it is possible to get distributional convergence (to normal laws or stable laws), and more subtle results such as the speed of convergence (also called the Berry- Esseen theorem) or the local limit theorem (see for example [30,17,8,3]). These results, in turn, have important consequences concerning the asymptotic behavior of the system [2,33].
On the other hand, when the system is not uniformly expanding or uniformly hyperbolic, it is not possible to use directly the aforementioned spectral method. Consequently, other methods have been devised to handle the distributional convergence of Birkhoff sums. Among many techniques, the most flexible one is probably the martingale argument of Gordin (see for example [23,26,10,11,36]). Some results have also been obtained on the speed in the central limit theorem, by direct estimates (see [22,27]). However, there is currently no result concerning the local limit theorem, which is not surprising since the proof of this theorem requires a heavy Fourier machinery, even in the probabilistic case, and is not easily accessible to elementary methods.
The aim of this article is to prove the local limit theorem and the Berry–Esseen theorem for Hölder functions in the setting of Young towers [35], where the decay of correlations is not exponential and the transfer operator has no spectral gap. The Young towers are abstract spaces which can be used to model many non uniformly expanding maps, for example the Pomeau–Manneville maps in dimension 1 studied by Liverani, Saussol and Vaienti [24], the Viana map (for which a tower is built in [5]), or the unimodal maps for which the critical point does not return too quickly close to itself [9]. Thus, all these maps also satisfy the local limit theorem, and the central limit theorem with speedO(1/√
n). These results also apply in non uniformly hyperbolic settings, with the techniques of [34].
The proof is spectral: it uses perturbations of transfer operators, as in [17], but applied to first return transfer operators associated to an induced map, as defined by Sarig in [32]. The method is related to [14], with a more systematic use of Banach algebra techniques.
1.2. Results in Young towers
A Young tower [35] is a probability space (X, m) with a partition (Bi,j)i∈I,j <ϕi of X by positive measure subsets, whereI is finite or countable andϕi∈N∗, together with a nonsingular map T:X→X satisfying the following properties.
1. ∀i∈I,∀0j < ϕi−1,T is a measure preserving isomorphism betweenBi,j andBi,j+1. 2. For everyi∈I,T is an isomorphism betweenBi,ϕi−1andB:=
k∈IBk,0.
3. Letϕbe the function equal toϕionBi,0, whenceTϕis a function fromBto itself. Lets(x, y)be the separation time of the pointsxandy∈BunderTϕ, i.e.s(x, y)=inf{n| ∃i=j, (Tϕ)n(x)∈Bi,0, (Tϕ)n(y)∈Bj,0}. AsTϕi is an isomorphism betweenBi,0andB, it is possible to consider the inversegmof its jacobian with respect to the measure m. We assume that there exist constants β <1 and C >0 such that ∀x, y∈Bi,0,
|loggm(x)−loggm(y)|Cβs(x,y). 4. The mapT preserves the measurem.
5. The partition∞
0 T−n((Bi,j))separates the points.
The notion of Young tower has been introduced by Young in [34,35] as a model for non uniformly expanding dynamical systems. The non uniformity is measured by the size of tails m{x ∈B|ϕ(x) > n}: if this quantity is very small, then most points enjoy some expansion before time n, when they first return to the basis. This expansion, in turn, is sufficient to study statistical properties of the system, including decay of correlations. Young has proved that, ifm[ϕ > n] =O(1/nβ)for someβ >1, then the correlations of sufficiently regular functions (see
the definition ofCτ(X)below) decay likeO(1/nβ−1). In particular, ifβ >2, these correlations are summable, and a martingale method can be used to prove that a central limit theorem holds.
We extend the separation timesto the whole tower, by settings(x, y)=0 ifx andy are not in the same set Bi,j, ands(x, y)=s(x, y)+1 otherwise, wherexandyare the next iterates ofxandyinB. For 0< τ <1, set
Cτ(X)=
f:X→R| ∃C >0, ∀x, y∈X, f (x)−f (y)Cτs(x,y) .
This space has a normfτ=inf{C| ∀x, y∈X,|f (x)−f (y)|Cτs(x,y)} + f∞.
The following theorem is well known and can for example be proved using martingale techniques (see [35, Theorem 4]).
Theorem 1.1. Letτ <1. Assume thatm[ϕ > n] =O(1/nβ)withβ >2. Letf ∈Cτ(X)have a vanishing integral.
Then there existsσ20 such that
√1 n
n−1
k=0
f ◦Tk→N(0, σ2).
Moreover,σ2=0 if and only iff is a coboundary, i.e. there exists a measurable functiongsuch thatf =g−g◦T almost everywhere.
The main results of this article are Theorems 1.2 and 1.3. To formulate the first one, we will need the following definition:
Definition. A mapf:X→Ris periodic if there existρ∈R,g:X→Rmeasurable,λ >0 andq:X→Z, such thatf =ρ+g−g◦T+λqalmost everywhere. Otherwise, it is aperiodic.
Theorem 1.2 (local limit theorem). Letτ <1. Assume thatm[ϕ > n] =O(1/nβ)withβ >2. Letf ∈Cτ(X)have a vanishing integral, and letσ2be given by Theorem 1.1.
Assume thatf is aperiodic. This implies in particularσ2>0. Then, for any bounded intervalJ⊂R, for any real sequenceknwithkn/√
n→κ∈R, for anyu∈Cτ(X), for anyv:X→Rmeasurable,
√n m
x∈X|Snf (x)∈J+kn+u(x)+v(Tnx)
→ |J|e−κ2/(2σ2) σ√
2π .
The function on the right is the density ofN(0, σ2): this theorem (foru=v=0 andkn=κ√
n) means that
m 1
√nSnf ∈κ+ J
√n
∼P N(0, σ2)∈κ+ J
√n
.
Hence, it shows thatSnf/√
nbehaves likeN(0, σ2)at the local level (contrary to Theorem 1.1 which deals with the global level). It is important thatf is aperiodic. Otherwise,f could be integer valued, and the theorem could not hold, e.g. forkn=0,u=v=0 andJ= [1/3,2/3].
Forf:X→R, define a functionfBonBby fB(x)=
ϕ(x)−1 k=0
f (Tkx). (1)
In the probabilistic case, the Berry–Esseen theorem, giving the speed of convergence in the central limit theorem, holds under anL3moment condition [12]. In the dynamical setting, we will need the same kind of hypothesis, but on the functionfB. Note that, since|fB|f∞ϕandβ >2, we always havefB∈L2(B).
Theorem 1.3 (speed in the central limit theorem). Letτ <1. Assume thatm[ϕ > n] =O(1/nβ)withβ >2. Let f ∈Cτ(X)have a vanishing integral, andσ2be given by Theorem 1.1.
Assume thatσ2>0, and that there exists 0< δ1 such that
|fB|21|fB|>zdm=O(z−δ)whenz→ ∞. If δ=1, assume also that
fB31|fB|zdm=O(1). Then there existsC >0 such that∀n∈N∗,∀a∈R, m
x 1
√nSnf (x)a
−P
N(0, σ2)a C nδ/2.
WhenfB∈Lpfor some 2< p3, then the conditions of the theorem are satisfied forδ=p−2. In particular, when fB∈L3, we obtain a convergence with speedO(1/√
n ), which is the usual Berry–Esseen theorem. Note also that, for anyf ∈Cτ(X), the conditions of the theorem are satisfied forδ=β−2 if 2< β <3, and forδ=1 if β >3. The formulation we have given is more precise than the usual Berry–Esseen theorem, in view of the applications, where anLpcondition would not be optimal (see for example Theorem 1.5). In fact, the conditions of the theorem onfBcorrespond to necessary and sufficient conditions to get a central limit theorem with speed O(n−δ/2)in the probabilistic (independent identically distributed) setting, as shown in [19, Theorem 3.4.1].
Remark. Using the same methods, it is possible to prove the same results in a more general setting, namely maps for which a first return map is Gibbs–Markov in the sense of [1]. For the sake of simplicity, we will only consider Young towers.
1.3. Applications 1.3.1. General setting
Let (X, d) be a locally compact separable metric space, endowed with a Borel probability measureµ, and T:X→X a nonsingular map for which µ is ergodic. Assume that there exist a bounded subsetB of X with µ(B) >0, a finite or countable partition (mod 0)(Bi)i∈IofB, withµ(Bi) >0, and integersϕi>0 such that:
1. ∀i∈I,Tϕi is an isomorphism betweenBi andB.
2. ∃λ >1 such that,∀i∈I,∀x, y∈Bi,d(Tϕix, Tϕiy)λd(x, y).
3. ∃C >0 such that,∀i∈I,∀x, y∈Bi,∀k < ϕi,d(Tkx, Tky)Cd(Tϕix, Tϕiy).
4. ∃θ >0 andD >0 such that,∀i∈I, the jacobiangµdefined onBi bygµ(x)=dµ/d(µ◦T|ϕBi
i)satisfies: for all x, y∈Bi,|loggµ(x)−loggµ(y)|Dd(Tϕix, Tϕiy)θ.
Denote by ϕ the function on B equal to ϕi on each Bi. If µ{x|ϕ(x) > n}is summable, we can define a space X= {(y, j )|y∈B, j < ϕ(x)}, and a mapT:X→X byT(y, j )=(y, j +1)if j < ϕ(x)−1 and T(y, j )=(Tϕ(y)(y),0)otherwise. Define alsoπ:X→Xbyπ(y, j )=Tj(y). Thenπ◦T=T ◦π.
Setµ=∞
n=0T∗n(µ|B∩ {ϕ > n}): it is a measure of finite mass onX, not necessarilyT-invariant. Young has proved in [35, Theorem 1] that there exists a unique invariant probability measuremonXwhich is absolutely continuous with respect toµ. It is ergodic, and(X, T, µ)is a Young tower in the sense of Section 1.2. The measurem=π∗(m)isT-invariant, absolutely continuous and ergodic.
If f:X→Ris Hölder continuous, thenf:=f ◦π:X→Rbelongs to Cτ(X)for τ close enough to 1.
Moreover, the Birkhoff sumsn−1
k=0f◦Tkandn−1
k=0f◦Tkhave the same distribution with respect respectively tomandm. Hence, Theorems 1.1, 1.2 and 1.3 on the functionfin the Young tower(X, T, m)imply the same results on the functionf in(X, T , m).
To apply these theorems, we have to check their assumptions. The conditionm[ϕ > n] =O(1/nβ)withβ >2 corresponds simply to the requirement
ϕi>n
µ(Bi)=O 1 nβ
for someβ >2.
To apply Theorem 1.2, we additionally have to check that the functionf is aperiodic for T, which can be complicated when the extensionXis not explicitly described. On the other hand, the aperiodicity off may be easier to check, using for example the information at the periodic points. In this case, the following abstract theorem ensures thatfis automatically aperiodic, whence we can apply Theorem 1.2.
Theorem 1.4. LetT:X→X be a probability preserving map on a probability space(X, m). Let(X, m)be a standard probability space, T:X→X an ergodic probability preserving map, and π:X→X a map with countable fibers, such thatm=π∗(m)andT ◦π=π◦T. Letf:X→R. Then
• The functionf is a coboundary forT if and only the functionf ◦π is a coboundary forT.
• The functionf is aperiodic forT if and only if the functionf ◦πis aperiodic forT. 1.3.2. Examples
Recently, many maps have been shown to fit in the previous setting. For example, [5, Theorem 3] shows that the Alves–Viana map, given by
T:
S1×R→S1×R, (ω, x)→
16ω, a−x2+εsin(2π ω)
satisfies these assumptions (for anyβ >2) when 0 is preperiodic for the mapx→a−x2, andεis small enough.
In fact, any map close enough toT in theC3-topology also satisfies them.
In the one-dimensional case, [9] shows that many unimodal maps of the interval also satisfy these hypotheses:
it is sufficient that the returns of the critical point close to itself occur at a slow enough rate.
Finally, we will discuss with more details the case of the Pomeau–Manneville maps, studied among many others by Liverani, Saussol and Vaienti [24]. They form an interesting class of applications, since the influence of the fixed point 0 becomes more and more important whenαincreases. The explicit formula (2) is not important, what matters is only the local behavior around the fixed point. Hence, all the following results can be extended to a much larger class of examples but, for the sake of simplicity, we will only consider the following maps.
Letα∈(0,1/2), and considerT:[0,1] → [0,1]given by T (x)=
x(1+2αxα) if 0x1/2,
2x−1 if 1/2< x1. (2)
This map has a parabolic fixed point at 0, and is expanding elsewhere. It has a unique absolutely continuous invariant probability measurem, whose density is Lipschitz on any interval of the form(ε,1][24, Lemma 2.3].
Theorem 1.5. Let 0< α <1/2, and letf:[0,1] →Rbe a Hölder function with vanishing integral, which cannot be written asg−g◦T. Thenf satisfies a central limit theorem with varianceσ2>0.
• Ifα <1/3, orf (0)=0 and there existsγ > α−1/3 such that|f (x)|Kxγ, then there existsC >0 such that∀n∈N∗,∀a∈R,
m
x 1
√nSnf (x)a
−P
N(0, σ2)a C
√n.
• If 1/3< α <1/2,f (0)=0 and there existsγ >0 such that|f (x)|Kxγ andδ:=α−1γ −2∈(0,1), then there existsC >0 such that∀n∈N∗,∀a∈R,
m
x 1
√nSnf (x)a
−P
N(0, σ2)a C nδ/2.
• If 1/3< α <1/2 andf (0)=0, then there existsC >0 such that∀n∈N∗,∀a∈R, m
x 1
√nSnf (x)a
−P
N(0, σ2)a C n(1/2α)−1.
Moreover, iff is aperiodic, it satisfies the local limit theorem.
Proof. Let x0=1, andxn+1be the preimage ofxn in[0,1/2]. Letyn+1be the preimage ofxn in(1/2,1]: the intervals Bn=(yn+1, yn]form a partition ofB=(1/2,1]and, ifϕn=n, all the hypotheses of Section 1.3 are satisfied. Moreover,m(Bn)∼n1/αC+1 andxn∼nD1/α for constantsC, D >0 [24]. In particular,m(ϕ > n)=O(1/nβ) forβ=1/α >2.
Let f be Hölder on[0,1]. If f (0)=0, thenfB =nf (0)+o(n) onBn. Otherwise, letγ >0 be such that
|f (x)|Kxγ. Reducingγ if necessary, we can assume thatγ < α. Then it is easy to check that|fB|Cn1−γ /α onBn.
Using these estimates, we can check the integrability assumptions of Theorem 1.3 forδ=1 in the first case,
1
α−γ −2 in the second case, andα1−2 in the third case. Hence, Theorem 1.3 implies the desired estimates on the speed in the central limit theorem.
Finally, the local limit theorem is a direct consequence of Theorem 1.2. 2
The aperiodicity assumption is a priori not easy to check, since the periodicity equalityf=g−g◦T+ρ+λq is assumed to hold only almost everywhere. However, under suitable regularity assumptions onf, it is possible to prove that this equality holds everywhere (see e.g. [3] for locally constantf, [15] for Hölderf). For example, ifT is given by (2), thenf =log|T| −
log|T|is aperiodic.
In Section 2, we will prove Theorem 1.4, and show that it is sufficient to prove Theorems 1.2 and 1.3 in mixing Young towers (i.e., such that the return timesϕi satisfy gcd(ϕi)=1). The rest of paper is devoted to the proof of these theorems. In Section 3, we prove an abstract spectral result on perturbations of series of operators. In Section 4, we apply this result to first return transfer operators, to get the key result Theorem 4.6. We then use this estimate in the last two sections to prove respectively the local limit Theorem 1.2 and the Berry–Esseen Theorem 1.3.
2. Preliminary reductions 2.1. Proof of Theorem 1.4
Proof of the coboundary result. Iff is a coboundary, i.e.f =g−g◦T, thenf:=f ◦π can be written as f=g−g◦T, whereg=g◦π. However, the converse is not immediate: iff=g−g◦T, the functiong is a priori not constant on the fibersπ−1(x), which prevents us from writingg=g◦π.
We use the following characterization of coboundaries: Let T be an endomorphism of a probability space (X, m). Then a measurable functionf onXcan be written asg−g◦T if and only if
∀ε >0,∃C >0, ∀n1, m
x∈X|Snf (x)C
ε. (3)
This characterization, due to Schmidt, is proved for example in [4].
Iffis a coboundary, then (3) is satisfied byfinX, whence it is also satisfied byf inX(since this condition only involves distributions). Thus,f can be written asg−g◦T. 2
Proof of the aperiodicity result. Iff is periodic onX, i.e.f =ρ+g−g◦T +λqwhereq is integer-valued, thenf◦π=ρ+(g◦π )−(g◦π )◦T+λ(q◦π ), i.e.f ◦πis periodic. On the other hand, iff◦π=ρ+g− g◦T+λq, it is not necessarily possible to write directlyg=g◦π. The proof of the periodicity off will use
ideas of [4]. We can assume for example thatλ=2π. Replacingmby one of its ergodic components, we can also assume thatmis ergodic.
Since the projection π has countable fibers, there exists a measurable subsetA ofX such that π is an iso- morphism betweenAandX, andm(A) >0. Define a functiong˜ onXbyg(x)˜ =g(x), wherex is the unique preimage ofx inA. Replacingf byf − ˜g+ ˜g◦T −ρ, andgbyg− ˜g◦π, we can assume without loss of generality thatg=0 onAandρ=0.
Forx∈X, letWn(x)be the measure onS1given by Wn(x)=1
n n
k=1
δ(eiSkf (x))
whereδ(y)is the Dirac mass aty. Foru∈C0(S1), it is possible by compactness to find a subsequencenksuch that
S1
udWnk(x)→L(u)(x) weak ∗ inL∞(X).
It is possible to obtain this convergence for a dense countable set of functions inC0(S1), by a diagonal argument.
By passing to a further subsequence, it is also possible to guarantee thatn1n k=1
S1udWnk(x)→L(u)(x)on a set Y⊂Xwithm(Y )=1, by Komlos’ Theorem [21]. By density, we get the same convergence for anyu∈C0(S1).
Forx∈Y, the mapu∈C0(S1)→L(u)(x)∈Ris a nonnegative continuous linear functional sending 1 to 1, thus given by a probability measurePx. Moreover, these measures satisfy PT x(S)=Px(eif (x)S) for any Borel subsetSofS1, sinceWn(T x)(S)=Wn(x)(eif (x)S)±2n.
For someε >0, we will prove that m
x|Px {1}
ε
ε. (4)
Ifx∈A∩T −k(A), then eiSkf◦π(x)=ei(g(x)−g◦Tk(x))=1, i.e.Skf ◦π(x)∈2πZ. Hence,
X
Wn(x) {1}
dm(x)=1 n
n
k=1
X
1
Skf (x)∈2πZ
dm(x)1 n
n
k=1
X
1(A∩T −kA)dm(x)
=
X
1A·
1 n
n
k=1
1A◦Tk
dm(x)→m(A)2>0 by Birkhoff Theorem. Thus, for large enoughn,
XWn(x)({1})3ε >0, whence (n1n
k=1Wnk(x))({1})2ε for large enoughn. Since(n1n
k=1Wnk(x))({1})1, we get m
x
1 n
n
k=1
Wnk(x)
{1}
ε
ε.
Thus, the setC= {x|lim sup(1nn
k=1Wnk(x))({1})ε}satisfiesm(C)ε. Finally,Px({1})εonC, and this proves (4).
Define a measure µ onX×S1, by µ(U×V )=
UPx(V )dm(x). Thenµ is invariant under the action of Tf:(x, y)→(T (x),e−if (x)y), and [4] proves that, for almost every ergodic componentP of µ, there exists a compact subgroupHofS1and a mapω:X→S1such that, denoting bymH the Haar measure ofH,
∀U×V ⊂X×S1, P (U×V )=
U
mH ω(x)V
dm(x).
Moreover, in this case, it is possible to write eif (x)=ω(T x)ω(x) ψ (x), whereψtakes its values inH.
If, for all componentP ofµ, we hadH=S1, thenP =m⊗Leb for allP, whenceµ=m⊗Leb. This is a contradiction, since µ(C× {1}) >0, while(m⊗Leb)(C× {1})=0. Thus, for some choice ofP,H=Z/kZ, whenceψ (x)k=1, and eikf (x)=ω(T x)k/ω(x)k. Thus,f is periodic onX. 2
Remark. The proof only shows that the period of f onX divides the period off ◦π onX, not that they are equal. In fact, it is not hard to construct examples of Young towers where the two periods are different. Moreover, the result is not true without the assumption that the fibers are countable.
2.2. Reduction to the mixing case
In the proofs in Young towers, it is often useful to assume that the tower is mixing, i.e. gcd(ϕi)=1. This restriction may seem technical, but it is important (for example, without it, there is no decay of correlations any more). For limit theorems, however, it is irrelevant: Theorems 1.1, 1.2 and 1.3 for mixing towers imply the same results for general towers.
Proof. We assume that Theorems 1.1, 1.2 and 1.3 are true in any mixing Young tower. Let Xbe a non-mixing Young tower, with N =gcd(ϕi) >1, and let f ∈Cτ(X) be of vanishing integral. For k=0, . . . , N −1, set Zk=
Bi,j for j ≡k modN. Then, for every k, (Zk, TN)is a mixing Young tower, to which we can apply Theorems 1.1, 1.2 and 1.3.
On Zk, we consider the functionfk given byN−1
i=0 f (Tix), i.e.(SNf )|Zk. Then
Zkfk =
Xf =0. Theo- rem 1.1 applied tofkon(Zk, TN)gives a constantσksuch that, onZk,
√1
sNSsNf →N(0, σk2).
Writing an integernassN+rwithr < N, we get that √1
nSnf→N(0, σk2)onZk. Finally, ifx∈Z0,Snf (x)and Snf (Tkx)differ by at most 2kf∞. Thus, √1
nSnf−√1nSnf ◦Tk tends to 0 in probability onZ0, which shows thatσk=σ0. Writingσ for this common number, we get that √1
nSnf →N(0, σ2)onX. Moreover, ifσ=0, then f0is a coboundary forTN, i.e.f0=g−g◦TNwhereg:Z0→Ris measurable. We extendgto the whole tower:
if(x, j )∈X, withj =sN+randr < N, setg(x, j )=g(x, sN )−r−1
i=0f (x, sN+i). It is then easy to check thatf=g−g◦T. This proves Theorem 1.1.
For the local limit theorem, let us assume thatf is aperiodic, and takeu, vandkn as in the assumptions of Theorem 1.2. We show that the functionsfkare also aperiodic. Otherwise, for example,f0=ρ+g−g◦TN+λq, wheregandqare defined onZ0. We extendgandqto the whole tower: for(x, j )∈Xwithj=sN+rand 0< r <
N, setg(x, j )=g(x, sN )−r−1
0 (f (x, sN+i)−Nρ)+λq(x, sN ), andq(x, j )=0. Thenf =Nρ+g−g◦T+λq, which is a contradiction. Thus, all the functionsfkare aperiodic. We can apply to them Theorem 1.2 in the mixing tower(Zk, TN), and get that
√sN m
x∈Zk|SsNf (x)∈J+ksN+u(x)+v(TsNx)
→m(Zk)|J|e−κ2/(2σ2) σ√
2π .
Summing overk, we get the conclusion of Theorem 1.2 forf, and for the times of the formsN. For times of the formsN+rwith 0< r < N, we use the same result forf◦Tr,u−Srf, v◦Tr and the sequenceksN+r, and get that
√sN+r m
x∈X|SsNf ◦Tr(x)∈J+ksN+r+u(x)−Srf (x)+v(TsN+rx)
→ |J|e−κ2/(2σ2) σ√
2π . AsSsNf ◦Tr(x)+Srf (x)=SsN+rf (x), this concludes the proof of Theorem 1.2.
Finally, the central limit theorem with speed is deduced from the same result on eachZk, for times of the form sN. We extend the result to arbitrary times: writingnassN+rwithr < N, we have|SsN+rf−SsNf|rf∞. This introduces an error, of orderO(1/√
n )O(1/nδ/2). 2
Remark. For this proof, it was important to have a strong version of the local limit theorem, involving functionsu andv.
Theorem 1.1 is proved in [35, Theorem 4]. The rest of the paper is devoted to the proof of Theorems 1.2 and 1.3 in mixing Young towers. From this point on,Xwill be a mixing Young tower, i.e. gcd(ϕi)=1.
3. An abstract result
If (an)n∈N and (bn)n∈N are two sequences indexed by N, we denote by (an) (bn) the sequence cn = n
k=0akbn−k. If(an)n∈Zand(bn)n∈Zare two summable sequences indexed byZ, we also define their convolution cn=(an) (bn)bycn=∞
k=−∞akbn−k. Hence, if
anzn and
bnznare series with summable coefficients, the coefficient ofzn in(
akzk)(
bkzk)is given by (an) (bn)(more precisely, it is given by thenth term ((ak) (bk))nof this sequence, but we will often abuse notations and write simply(an) (bn)). Finally, we write D= {z∈C| |z|<1}andD= {z∈C| |z|1}.
The goal of this section is to prove the following theorem:
Theorem 3.1. Letβ >2. LetRn, forn∈N∗, be operators on a Banach spaceE, with∞
k=n+1Rk =O(1/nβ).
SetR(z)=
Rnzn, and assume that 1 is a simple isolated eigenvalue ofR(1), whileI−R(z)is invertible for z∈D− {1}. LetP be the spectral projection associated toR(1)and the eigenvalue 1, and assume thatP R(1)P = µP withµ >0.
LetRn(t )be operators onE (fort in some interval[−α, α]withα >0) such that∞
k=n+1Rk(t )−Rk C|t|/nβ−1. SetR(z, t )=
znRn(t ). Letλ(z, t )be the eigenvalue close to 1 ofR(z, t ), for(z, t )close to(1,0).
We assume thatλ(1, t )=1−M(t )withM(t )∼ct2for some constantcwith Re(c) >0.
Then, for small enought,I −R(z, t )is invertible for allz∈D. Let us denote its inverse by
Tn(t )zn. Then there existα>0,d >0 andC >0 such that, for everyt∈ [−α, α], for everyn∈N∗,
Tn(t )− 1 µ 1− 1
µM(t ) n
P C
nβ−1 +C|t| 1 nβ−1
(1−dt2)n. (5)
In the application of Theorem 3.1 to the proof of Theorems 1.2 and 1.3, the operatorsRn will describe the returns to the basisB, and will be easily understood, as well as their perturbationsRn(t ). On the other hand,Tn
will describe all the iterates at timen, and Tn(t )will be closely related to the characteristic functionE(eit Snf).
Thus, (5) will enable us to describe preciselyE(eit Snf), and this information will be sufficient to get Theorems 1.2 and 1.3.
3.1. Banach algebras and Wiener Lemma
In this paragraph, we define some Banach algebras which will be useful in the following estimates. We postpone the proofs of the properties of these algebras to the Appendix.
A Banach algebra Ais a complex Banach space with an associative multiplication A×A→A such that ABAB, and a neutral element. The set of invertible elements is then an open subset ofA, on which the inversion is continuous.
LetCbe a Banach algebra. Ifγ >1, we writeOγ(C)for the set of formal series∞
n=−∞Anzn, whereAn∈C andAn =O(1/|n|γ)whenn→ ±∞, endowed with the standard product of power series, corresponding to the
convolution of the sequences(An)and(Bn). It admits the naive norm supn∈Z(|n| +1)γAn, for which it is not a Banach algebra. However, there exists a norm, equivalent to the previous one, which makesOγ(C)into a Banach algebra (Proposition A.1). Moreover, this algebra satisfies a Wiener Lemma: ifA(z)=
Anzn∈Oγ(C)is such thatA(z)is invertible for everyz∈S1, thenAis invertible inOγ(C)(Theorem A.3).
We will also use the Banach algebraOγ+(C), given by the set of series
Anzn∈Oγ(C)such thatAn=0 for n <0. It is a closed subalgebra ofOγ(C), and it satisfies also a Wiener Lemma (Theorem A.4).
Notation. Iff:[−α, α] ×Z→R+for someα >0, andCis a Banach algebra, we denote byOC(f (t, n))the set of series∞
n=−∞cn(t )znwherecn:[−α, α] →Cis such that there existsα>0 andC >0 such that
∀t∈ [−α, α], ∀n∈Z, cn(t )Cf (t, n).
We will often omit the subscript inOC. As usual, we will often write
cn(t )zn=O(f (t, n))instead of the more correct formulation
cn(t )zn∈O(f (t, n)). We will also writeO(g(n))for the set of series
Anznwith AnCg(n)for some constantC. This is a particular case of the previous notation, where the functionsf (t, n) are independent oft. Until the end of Section 3, the notationOwill always have this signification.
Remark. The notations O and O should not be confused: there are similarities (which is why we have used the same letter), but the calligraphic notationOindicates additionally a Banach algebra. In this case, we can for example use the continuity of inversion.
With these notations, we can reformulate Theorem A.3 as follows: if
Anzn=O(1/(|n| +1)γ)forγ >1, and
Anznis invertible for everyz∈S1, then(
Anzn)−1=O(1/(|n| +1)γ). The fact thatOγ(C)is a Banach algebra also implies that, forγ >1,
O 1
(|n| +1)γ
O 1
(|n| +1)γ
⊂O 1 (|n| +1)γ
, (6)
i.e., if two series
Anznand
Bnzn(withAn, Bn∈C) satisfy sup
n∈Z
|n| +1γ
An<∞ and sup
n∈Z
|n| +1γ
Bn<∞, then the series
Cnzn:=(
Anzn)(
Bnzn)also satisfies supn∈Z(|n| +1)γCn<∞. 3.2. Preliminary technical estimates
For notational convenience, we will often writet instead of|t|in what follows. Equivalently, the reader may consider that the proofs are written fort0. We will also write 1/|n|γ instead of 1/(|n| +1)γ, discarding the problem atn=0.
Lemma 3.2. Whenγ >1 andd >0,
O 1
|n|γ
O
1n0t2(1−dt2)n
⊂O 1
|n|γ +1n0t2 1−d 2t2
n .
Proof. Forn <0, the coefficient in the convolution is less than∞
k=0t2(1−dt2)k|n1|γ |nC|γ. Forn0, it is less than
n/2
k=−∞
1
|k|γt2(1−dt2)n/2+ n
k=n/2
1
(n/2)γt2(1−dt2)n−kCt2(1−dt2)n/2+ C nγ.
Finally, as√
1−dt2(1−d2t2), we get the conclusion. 2
Lemma 3.3. Letγ >1 andd >0. LetGt(z)=O(t /|n|γ+1n0t3(1−dt2)n), and assume thatF (z)=O(1/|n|γ) is invertible for everyz∈S1. Then[F (z)+Gt(z)]−1=F (z)−1+O(|nt|γ +1n0t3(1−64dt2)n).
Proof. We first assume thatF (z)=1. Setting Ht(z)=
n∈Z t
|n|γzn+
n∈Nt3(1−dt2)nzn, the norm of the coefficients of [1+Gt(z)]−1 is less than the coefficients of [1−Ht(z)]−1. Thus, it is sufficient to consider 1/(1−t K(z)−t3/(1−(1−dt2)z))whereK(z)=
zn/|n|γ. Note that 1
1−t K(z)−t3/(1−(1−dt2)z)= 1 1−t K(z)−t3
1−(1−dt2)z
1−(1−dt2)[1+t3/(1−t K(z)−t3)]z. (7) For small enought,|(1−dt2)[1+t3/(1−t K(z)−t3)]|<1, whence
1−(1−dt2)z
1−(1−dt2)[1+t3/(1−t K(z)−t3)]z
=
1−(1−dt2)z∞
n=0
(1−dt2)n
1+ t3
1−t K(z)−t3 n
zn
=1+ t z(1−dt2) 1−t K(z)−t3t2
∞ n=0
(1−dt2)n
1+ t3
1−t K(z)−t3 n
zn. (8)
We first study the sum. LetA=Oγ(C)be the Banach algebra of the series whose coefficients areO(1/|n|γ). As K(z)∈A, we have 1−t K(z)−t3=1+OA(t ). Since the inversion is Lipschitz on a Banach algebra, we get t3/(1−t K(z)−t3)=t3+OA(t4), whence[1+t3/(1−t K(z)−t3)]nAC(1+2t3)n. Let us estimate the coefficient ofzpint2∞
n=0(1−dt2)n[1+t3/(1−t K(z)−t3)]nzn. This is at most t2
∞ n=0
(1−dt2)n 1+ t3 1−t K(z)−t3
n
−n+p
∞
n=0
t2(1−dt2)n(1+2t3)n
|−n+p|γ
∞
n=0
t2 1−d 2t2
n
1
|−n+p|γ
for t small enough so that(1−dt2)(1+2t3)(1−d2t2). We find the same expression as in the convolution betweenO(1n0t2(1−d2t2)n)andO(1/|n|γ), that we have already estimated in Lemma 3.2. Thus, we get at most O(1/|p|γ+1p0t2(1−d4t2)p).
Ast z(1−dt2)/(1−t K(z)−t3)=O(t /|n|γ), another convolution yields that (8) is 1+O t
|n|γ +1n0t3 1−d 8t2
n .
Multiplying by 1/(1−t K(z)−t3)=1+O(t /|n|γ)gives that (7)=1+O(t /|n|γ +1n0t3(1−16dt2)n). This concludes the proof in the caseF (z)=1.
We now handle the case of an arbitraryF (z). Note that F (z)+Gt(z)−1
=
1+F (z)−1Gt(z)−1
F (z)−1.
The Wiener Lemma A.3 implies that F (z)−1 =O(1/|n|γ), whence Lemma 3.2 gives F (z)−1Gt(z) = O(t /|n|γ+1n0t3(1−d2t2)n). Thus, the caseF (z)=1 yields
1+F (z)−1Gt(z)−1
=1+O t
|n|γ +1n0t3 1− d 32t2
n .
Another convolution withF (z)−1gives the result. 2
