www.imstat.org/aihp 2009, Vol. 45, No. 4, 1065–1084
DOI: 10.1214/08-AIHP195
© Association des Publications de l’Institut Henri Poincaré, 2009
Quantitative recurrence in two-dimensional extended processes
Françoise Pène
1and Benoît Saussol
aUniversité Européenne de Bretagne, Université de Brest, laboratoire de Mathématiques, CNRS UMR 6205, France.
E-mail: francoise.pene@univ-brest.fr; benoit.saussol@univ-brest.fr Received 21 January 2008; revised 22 September 2008; accepted 17 October 2008
Abstract. Under some mild condition, a random walk in the plane is recurrent. In particular each trajectory is dense, and a natural question is how much time one needs to approach a given small neighbourhood of the origin. We address this question in the case of some extended dynamical systems similar to planar random walks, includingZ2-extension of mixing subshifts of finite type. We define a pointwise recurrence rate and relate it to the dimension of the process, and establish a result of convergence in distribution of the rescaled return time near the origin.
Résumé. Sous certaines conditions, une marche aléatoire dans le plan est récurrente. En particulier, chaque trajectoire est dense, et il est naturel d’estimer le temps nécessaire pour revenir dans un petit voisinage de l’origine. Nous nous intéressons à cette question dans le cas de systèmes dynamiques étendus similaires à des marches aléatoires planaires, notamment celui desZ2-extension de sous-shifts de type fini mélangeants. Nous déterminons une vitesse de convergence ponctuelle que nous relions à la dimension du processus et nous établissons un résultat de convergence en loi du temps de retour à l’origine, correctement normalisé.
MSC:37B20; 37A50; 60Fxx
Keywords:Return time; Random walk; Subshift of finite type; Recurrence; Local limit theorem
1. Introduction 1.1. Motivation
Let us consider a recurrent random walk or a recurrent dynamical system (with finite orσ-finite invariant measure).
Given an initial condition, say x, we thus know that the process will return back ε close to its starting point x. A basic question iswhen? For finite measure preserving dynamical systems this question has some deep relations to the Hausdorff dimension of the invariant measure. Namely, ifτε(x)represents this time, in many situations
τε(x)≈ 1 εdim
for typical pointsx, where dim is the Hausdorff dimension of the underlying invariant measure. This has been proved for interval maps [15] and rapidly mixing systems [1,14]. Another type of result is the exponential distribution of rescaled return times and the lognormal fluctuations of the return times [4,9].
In this paper we are dealing with systems where the underlying natural measure is indeed infinite. This causes the return time to be non-integrable, in contrast with the finite measure case. However, the systems we are thinking about have in common the property that, in some sense, the behaviours at small scale and at large scale are independent.
The large scale dynamics being some kind of recurrent random walk, and the small scale dynamics a finite measure
1Supported in part by the ANR project TEMI (Théorie Ergodique en mesure infinie).
Table 1
Recurrence forZk-extensions.Bεdenotes the ball of radiusε,νis a Gibbs measure on the base anddis the Hausdorff dimension ofν
Dimension Z0-extension Z1-extension Z2-extension
Scale lim
ε→0 logτε
−logε=d lim ε→0
log√τε
−logε =d lim ε→0
loglogτε
−logε =d
Local law ν(ν(Bε)τε> t )→e−t lim
ε→0ν(ν(Bε)√τε> t )≤1+βt1 ν(ν(Bε)logτε> t )→1+βt1
Lognormal fluctuations εdτε εd√τε εdlogτε
preserving system. We provide results in three different cases: two probabilistic models (with some hypothesis of independence) and an infinite measure dynamical system (a Z2-extension of a finite measure dynamical system).
The first case treated in Section 2 is a toy model designed to give the hint of the general case. Then, in Section 3 we consider the case of planar random walks. Finally, in Section 4 we give a complete analysis of the quantitative behaviour of return times in the case ofZ2-extensions of subshifts of finite type. The recurrence forZ2-extensions of subshifts of finite type essentially comes from Guivarc’h and Hardy’s local limit theorem [8]. It was proven afterwards by Conze [6] and Schmidt [16] that it is also a consequence of the central limit theorem.
1.2. Description of the main result:Z2-extensions of subshifts of finite type
We emphasize that this dimension 2 is at the threshold between recurrent and non-recurrent processes, since in higher dimension these processes are not recurrent (except if degenerate). It makes sense to show how our results behave with respect to the dimension. For completeness, we call the non-extended system itself aZ0-extension. In this non- extended case, the type of results we have in mind (see Table 1) have already been established respectively by Ornstein and Weiss [13], Hirata [9] and Collet, Galves and Schmitt [4]. The case ofZ2-extension is completely done in Sec- tion 4. The case ofZ1-extension can be partly derived following the technique used in the present paper.2The essential difference is that the local limit theorem has the one-dimensional scaling in √1
n, instead of 1n in the two-dimensional case. The following table summarizes the different results as the dimension changes. The first line of results cor- responds to Theorem 8, the second to Theorem 9 and the third to Corollary 10. We refer to Section 4 for precise statements.
2. A toy model in dimension two
We present a toy model designed to posses a lot of independence. It has the advantage of giving the right formulas with elementary proofs.
2.1. Description of the model and statement of the results
Let us consider two sequences of independent identically distributed random variables(Xn)n≥1and(Yn)n≥0indepen- dent one from the other such that:
• the random variableX1is uniformly distributed on{(1,0), (−1,0), (0,1), (0,−1)};
• the random variableY0is uniformly distributed on[0,1]2. Let us notice thatSn:=n
k=1Xk(with the conventionS0:=0) is the symmetric random walk onZ2. We study a kind of random walkMnonR2given byMn=Sn+Yn.
2ForZ1-extensions, the method easily gives the three results presented in the table. However, the result on the local law is not sharp. The determi- nation of the precise law will be the object of a forthcoming paper.
Another representation of our model could be the following. LetS=Rdand consider the systemZ2×S. Attached to each sitei∈Z2of the lattice, there is a local system which lives onSandσnis a i.i.d. sequence ofS-valued random variables with some densityρ, independent of theXn’s. Then we look at the random walk(Sn, σn), thinking atσnas a spin.
We want to study the asymptotic behaviour, as ε goes to zero, of the return time in the open ballB(M0, ε)of radiusεcentered atM0(for the euclidean metric). Let
τε:=min
m≥1: |Mm−M0|< ε . We will prove the following:
Theorem 1. Almost surely,log log−logετε converges to the dimension2of the Lebesgue measure onR2asεgoes to zero.
Theorem 2. For allt≥0we have:
εlim→0P λ
B(M0, ε)
logτε≤t
= 1 1+π/t.
2.2. Proof of the pointwise convergence of the recurrence rate to the dimension First, let us defineR1:=min{m≥1: Sm=0}. According to [7], we know that we have:
P(R1> s)∼ π
logs assgoes to infinity. (1)
We then define for anyp≥0 thepth return timeRpin[0,1)2by induction:
Rp+1:=inf{m > Rp: Sm=0}.
Observe thatRp is thepth return time at the origin of the random walkSn on the lattice, thus the delays between successive return timesRp−Rp−1, settingR0=0, are independent and identically distributed. Consequently:
P(Rp−Rp−1> s)=P(R1> s). (2)
The proof of Theorem 1 follows from these two lemmas.
Lemma 3. Almost surely,log loglognRn→1asn→ ∞.
Proof. It suffices to prove that for any 0< α <1, almost surely, en1−α ≤Rn≤2nen1+α providedn is sufficiently large. By independence and Eq. (2) we have
P
logRn≤n1−α
≤P
∀p≤n,log(Rp−Rp−1)≤n1−α
=P
logR1≤n1−αn . According to the asymptotic formula (1), fornsufficiently large
P
logR1≤n1−αn
≤
1− π
2n1−α n
≤e−πnα/2.
The first inequality follows then from the Borel–Cantelli lemma.
Moreover, according to formulas (2) and (1), we have
n≥1P(log(Rn−Rn−1) > n1+α)) <+∞. Hence, by the Borel–Cantelli lemma, we know that almost surely, for allnsufficiently large, we haveRn−Rn−1≤en1+α. From this
we get the second inequality.
LetTε:=min
≥1:|YR−Y0|< ε . Lemma 4. Almost surely,−logλ(B(YlogTε
0,ε))→1asε→0.
Proof. Since(Y)is an i.i.d. sequence independent of theXks, the random variableTε has a geometric distribution with parameterλε:=λ(B(Y0, ε))=πε2. For anyα >0 we have the simple decomposition
P logTε
−logλε−1 > α
=P
Tε> λ−ε1−α +P
Tε< λ−ε1+α . The first term is directly handled by Markov inequality:
P
Tε> λ−ε1−α
≤λαε,
while the second term may be computed using the geometric distribution:
P
Tε< λ−ε1+α
=1−(1−λε)λ−1+αε
=1−exp λ−ε1+α
log(1−λε)
≤ − λ−ε1+α
log(1−λε)
=O λαε
.
Let us defineεn:=n−1/α. According to the Borel–Cantelli lemma, −logTlogλεn
εn converges almost surely to the constant 1.
The conclusion follows from the facts that(εn)n≥1is a decreasing sequence of real numbers satisfying limn→+∞εn= 0 and limn→+∞ εn
εn+1 =1, andTεis monotone inε.
Proof of Theorem 1. Observe that wheneverB(M0, ε)is contained in(0,1)2, we haveτε=RTε. The theorem follows from Lemmas 3 and 4 since
log logτε
−logε =log logRTε
logTε
logTε
−logλε
logλε
logε →1×1×2
almost surely asε→0.
2.3. Proof of the convergence in distribution of the rescaled return time Proof of Theorem 2. Observe thatlogRlogτε
Tε converges almost surely to 1. Lett >0. By independence ofTεand theRn
we have
Fε(t ):=P λ
B(Y0, ε)
logRTε≤t
=
n≥1
P(Tε=n)P
logRn≤ t λε
.
SinceTεhas a geometric law with parameterλε,Fε(t )is equal toGλε(t )with:
Gδ(t ):=
n≥1
δ(1−δ)n−1P
logRn≤ t δ
.
First, we notice that the independence of the times between the successive returns gives for anyu >0 that P(Rn≤u)≤P
k=max1,...,nRk−Rk−1≤u
=P(R1≤u)n.
Letα <1. Using the inequality above and the equivalence relation (1) we get that for anyδ >0 sufficiently small, Gδ(t )≤
n≥1
δ(1−δ)n−1
1−απδ t
n
= 1
1+απ/t +O(δ).
This implies that lim supε→0Fε(t )≤1+π1/t.
FixA >0 and keeping the same notations observe that we haveFε(t )≥Hλε(t )with:
Hδ(t ):=
1≤n≤A/δ
δ(1−δ)n−1P
logRn≤ t δ
.
Note that the independence gives in addition that for anyu >0 P(Rn≤u)≥P
k=max1,...,nRk−Rk−1≤u/n
=P(R1≤u/n)n.
Letα >1. Using the inequality above and the equivalence relation (1) we get that for sufficiently smallδ >0 Hδ(t )≥
1≤n≤A/δ
δ(1−δ)n−1
1−α π
t /δ−logn n
≥
1≤n≤A/δ
δ(1−δ)n−1
1−α2πδ t
n
.
Evaluating the limit whenδ→0 of the geometric sum and then lettingA→ ∞we end up with lim infδ→0Hδ(t )≥
1
1+π/t, which gives the result.
3. Random walk on the plane
3.1. Almost sure convergence
We now consider a true random walk onR2,Sn=X1+ · · · +Xnwhere theXi’s are i.i.d. random variables distributed with a lawμof zero mean, with (invertible) covariance matrixΣ2and characteristic functionμ(t )ˆ =
eit·xdμ(x).
Letτεbe the first return time of the walk in theε-neighbourhood of the origin:
τε:=min
n≥1:|Sn|< ε .
LetΩ∗= {∀n≥1, Sn=0}. We notice that outsideΩ∗the return timeτε is obviously bounded by the first time n≥1 for whichSn=0. Therefore, we will only consider the asymptotic behaviour ofτεonΩ∗.
Theorem 5. Assume additionally that the distributionμsatisfies the Cramer condition lim sup
|t|→∞
μ(t )ˆ <1.
Then onΩ∗we havelimε→0log logτε
−logε =2almost surely.
We remark that a kind of Cramer’s condition on the law is necessary, since there exist some planar recurrent random walks for which the statement of the theorem is false (the return time being even larger than expected). We discovered after the completion of the proof of this theorem that its statement is contained in Theorem 2 of Cheliotis’s recent paper [5]. For completeness we describe the strategy of our original proof. A key point is a uniform version of the local limit theorem. Indeed we need an estimation of the typeP(|Sn|< ε)∼cεn2, with some uniformity inε(for some c >0). We will follow the classical proof of the local limit theorem (see Theorem 10.17 of [3]) to get the following (see Section 3.3 for its proof):
Lemma 6. There existsa >0,ε0>0,an integerN,a sequenceκn→0such that,for anyn > N,anyε∈(0, ε0)and for anyA∈R2we have
P
Sn∈A+ [−ε, ε]2
−γ ε2 n exp
−Σ−2A·A 2n
≤ ε2κn
n +exp(−a√ n) ε2 , withγ= 2
π√ detΣ2.
Then, this information on the probability of return is strong enough to estimate the first return time to the ε- neighbourhood of the origin.
Proof of Theorem 5. For any α >12, usingεn=1/logαn, we get thatP(|Sn|< εn)is summable. By the Borel–
Cantelli lemma, for almost everyx∈Ω∗, for allnlarge enough, we haveτεn(x) > nand thus:
lim inf
n→∞
log logτεn(x)
−logεn ≥lim inf
n→∞
log logn log logαn=1
α,
which implies by monotonicity and the fact thatαis arbitrary that lim infε→0log logτε(x)
−logε ≥2.
Letα < 12. To control the lim sup, we will take n=nε = exp(ε−1/α). Letγ< γ andm=(logε)4. We use a similar decomposition to that of Dvoretsky and Erdös in [7]. LetAk= {|Sk|< εand∀p=k+1, . . . , n,|Sp−Sk|>
2ε}. TheAk’s are disjoint, hence by independence and invariance, and according to Lemma 6, forεsmall enough, we have:
1≥ n k=m
P(Ak)≥ n k=m
P
|Sk|< ε
P(τ2ε> n−k)≥ n k=m
P
|Sk|< ε
P(τ2ε> n)≥ n k=m
γε2
k P(τ2ε> n).
Hence we haveP(τε > n)≤ cε21logn ≤cε1/α−2, if n is large enough. Let εp =p−2/(α−2). By the Borel–Cantelli lemma we have logτεp≤ε−p1/αalmost surely, hence lim supp→∞log logτ−logεεp
p ≤α1. By monotonicity and the fact thatα
is arbitrary we get the result.
3.2. Limit distribution of return times in neighbourhoods
We introduce a slight modification of the model. Letε >0. LetM0εbe uniformly distributed on the ballB(0, ε)and independent3of the sequence(Sn)introduced in the Section 3.1.
We define the random walkMnε=M0ε+Sn. We are interested in the limiting distribution of the first return time in the ballB(0, ε):
˜ τε=inf
n≥1: Mnε < ε .
Theorem 7. Under the hypotheses of Theorem5,the random variableε2logτ˜εconverges in distribution to a random variableY with distributionP(Y > t )= P(Ω∗)
1+2√
det(Σ−2)P(Ω∗)t (t >0).
Proof. To simplify notations, we omit the indicesε.
Upper bound:FixR >0 and an integerK >0. Let Γ =
∀i=1, . . . , K, Mi=M0and|Mi| ≤R .
3Using if necessary a larger probability space, letUbe a random variable uniformly distributed in the ballB(0,1)and independent of the sequence (Sn). Then one can takeM0ε=εU.
Using the same decomposition as in the proof of Theorem 5, we have P(Γ )=
n k=0
P Γ ∩
|Mk|< ε−2νand∀=k+1, . . . , n,|M| ≥ε−2ν ,
where nis equal to exp(t
ε2)for somet >0. Let ν=ε2 andA=2√12νZ2∩B(0, ε−3ν). Notice that the sets Qa:=(a−ν/√
2, a+ν/√
2)2witha∈Aare pairwise disjoints and contained inB(0, ε−2ν). Letm=(logε)4. Hence we have
P(Γ )≥P Γ ∩
∀=1, . . . , n|M| ≥ε +
n k=m
a∈A
P Γ ∩
Mk∈Qaand∀=k+1, . . . , n, M∈/B(0, ε−2ν)
≥P Γ ∩
∀=1, . . . , n, |M| ≥ε +
n k=m
a∈A
P Γ ∩
Mk∈Qaand∀=k+1, . . . , n, S−Sk∈/B(−a, ε−ν)
≥P
Γ ∩ { ˜τε> n} +
n k=m
a∈A
P
Γ ∩ {Mk∈Qa} P
∀=1, . . . , n, S∈/B(−a, ε−ν)
by independence (sincem > K wheneverεis sufficiently small). Note that P
Γ ∩ {Mk∈Qa}
=
{∀i,xi=x0,|xi|≤R}P(Sk−K∈ −xK+Qa)dP(M0,...,MK)(x0, . . . , xK).
According to Lemma 6 we get
∀k≥m P
Γ ∩ {Mk∈Qa}
≥P(Γ )γν2 2k
for some fixedγ< γ, providedεis sufficiently small. Hence we have n
k=m
a∈A
P
Γ ∩ {Mk∈Qa} P
∀=1, . . . , n, S∈/B(−a, ε−ν)
≥ n k=m
a∈A
P(Γ )γν2 2k P
∀=1, . . . , n, M∈/B(0, ε)|M0∈Qa
≥ n k=m
a∈A
P(Γ )γπε2
k P(M0∈Qa)P
∀=1, . . . , n, M∈/B(0, ε)|M0∈Qa
≥ n k=m
P(Γ )γπε2 k
P
∀=1, . . . , n, M∈/B(0, ε)
−P
ε−4ν≤ |M0| ≤ε
≥P(Γ )γπε2log(n)
1+o(1)
P(τ˜ε> n)−o(1).
Therefore sinceε2logn=t+o(1)we get P(Γ )≥P
Γ ∩ { ˜τε> n}
+γπtP(Γ )P(τ˜ε> n)+o(1)
≥P
{ ˜τε> n}
−P
∃1≤i≤K,|Mi|> R
+γπtP(Γ )P(τ˜ε> n)+o(1).
Hence we have lim sup
ε→0
P
˜ τε>exp
t ε2
≤P(Γ )+P(∃1≤i≤K,|Mi|> R) 1+γπtP(Γ ) . TakingR→ ∞first thenK→ ∞we obtain
lim sup
ε→0
P
˜ τε>exp
t ε2
≤ P(Ω∗) 1+γπtP(Ω∗). This holds for allγ< γ, which gives the upper bound.
Lower bound: We only provide a sketch proof since the arguments are very similar. Let m= (logε)4, n= exp(t /ε2),ν=ε2. We have:
P(Γ )≤P
Γ ∩ { ˜τε> nlogn} +
m k=1
pk+
nlogn−n k=m
pk+
nlogn k=nlogn−n
pk,
wherepk=P(Γ ∩ {|Mk|< ε+2νand∀=k+1, . . . , n,|M| ≥ε+2ν}).
By Theorem 5 we have m
k=1
pk≤P
Γ ∩ {τ3ε≤m}
≤P
Γ \Ω∗ +o(1)
since almost sure convergence implies convergence in probability.
Next, takingA=2√12νZ2∩B(0, ε+3ν),Q¯a:= [a−ν/√
2, a+ν/√
2]2and using the same arguments as for the upper bound we get
nlogn−n k=m
pk≤
nlogn−n k=m
a∈A
P Γ ∩
Mk∈ ¯Qaand∀=k+1, . . . , n, M∈/B(0, ε−2ν)
≤P(Γ )γπε2log(nlogn−n)P(τ˜ε> n)+o(1), where nowγ> γ.
Finally,
nlogn k=nlogn−n
pk≤
nlogn k=nlogn−n
P Γ ∩
|Mk| ≤ε+2ν
≤
nlogn k=nlogn−n
P(Γ )γ4ε2 k =o(1).
Putting these estimates together we end up with P(Γ )≤P
Γ ∩ { ˜τε> n}
+o(1)+P
Γ \Ω∗
+P(Γ )γπtP(τ˜ε> n).
This gives lim inf
ε→0 P
˜ τε>exp
t ε2
≥P(Γ )−P(Γ \Ω∗)+o(1) 1+γπtP(Γ ) .
And the conclusion follows as for the upper bound.
3.3. Proof of Lemma6
Proof. FixA∈R2. Letε >0. Letn≥2. For any 0< a < b, letha,b(t )be the piecewise affine function equal to 1 on [−a, a], to 0 outside[−b, b]and with slopeb−1a in between. LetHa,b(x1, x2)=ha,b(x1)ha,b(x2). Let us notice that we haveHε−ε/n,ε≤1[−ε,ε]2≤Hε,ε+ε/nand thus
E
Hε−ε/n,ε(Sn−A)
≤P
|Sn−A|∞≤ε
≤E
Hε,ε+ε/n(Sn−A) .
For anyH∈L1(R2)we define its Fourier transformHˆ and its inverse Fourier transformHˇ by:
∀u∈R2 H (u)ˆ = 1 2π
R2H (x)exp(−ixu)dx and H (u)ˇ = ˆH (−u).
With this definition, we immediately get that ˆHa,bL∞ = 21πHa,bL1 = (a+2πb)2 and after an easy computation ˆHL1≤(a+b+b−a4 )2.
Let us notice that we have for allx∈R2,Ha,b(x)= ˇˆHa,b(x). Hence with the use of Fubini’s theorem we have E
Ha,b(Sn−A)
=EHˇˆa,b(Sn−A)
=E 1
2π
R2
Hˆa,b(u)exp
iu(Sn−A) du
= 1 2π
R2
Hˆa,b(u)E exp
iu(Sn−A) du
= 1 2π
R2
Hˆa,b(u) E
exp(iuX1)n
e−iuAdu.
Denote the characteristic function ofX1byφ (u)=E[exp(iuX1)]. Because of the hypothesis on the distribution ofX1, we know that, ifuis non-null, then|φ (u)|<1. Let us recall that we have:
φ (u)=1−1
2Σ2u·u+g(u)Σ2u·u, with lim
u→0g(u)=0.
Let us fix someβ >0 such that:
∀u∈B(0, β) φ (u)−
1−1
2Σ2u·u ≤1
4Σ2u·u and such that:
usup2>β
φ (u) <1−1 4αβ2,
whereαis the smallest modulus of the eigenvalues ofΣ2. Now we take(a, b)=(ε, ε+εn)or(a, b)=(ε−εn, ε).
Hence we have:
u2>βn−1/6
Hˆa,b(u) φ (u)n
e−iuAdu
≤ ˆHa,bL1
1−αβ2n−1/3 4
n
≤
3ε+4n ε
2
exp
−αβ2n2/3 4
. (3)
Hence, it remains to estimate the following quantity:
B(0,βn−1/6)
Hˆa,b(u) φ (u)n
e−iuAdu=1 n
B(0,βn1/3)
Hˆa,b
v/√ n
φ v/√
nn
e−ivA/√ndv.
We will compare this quantity to 1n
R2Hˆa,b(0)exp(−12Σ2v·v)e−ivA/√ndv.First we have:
1 n
B(0,βn1/3)
Hˆa,b v/√
n
e−ivA/√n
φ v/√
nn
−exp
−1 2Σ2v·v
dv
≤1 n
B(0,βn1/3)
Hˆa,b
v/√n nexp
−n−1 4n Σ2v·v
1 ng1
v
√n
Σ2v·v
dv
≤ ˆHa,bL∞
n sup
w∈B(0,βn−1/6)
g1(w)
R2exp
−1
8Σ2v·v
Σ2v·v dv
≤4ε2
n sup
w∈B(0,βn−1/6)
g1(w)
R2exp
−1
8Σ2v·v
Σ2v·v dv≤ ε2
nκn, (4)
with limu→0g1(u)=0 and limn→+∞κn=0.
Second we have:
1 n
B(0,βn1/3)
exp
−1
2Σ2v·v Hˆa,b
v/√ n
− ˆHa,b(0) dv
≤1
n sup
w∈B(0,βn−1/6)
∇ ˆHa,b(w)
∞
B(0,βn1/3)
|v|2
√nexp
−1 2Σ2v·v
dv
≤1 n
2ελ(B(0,2ε)) 2π
R2
|v|2
√nexp
−1 2Σ2v·v
dv≤ε2
nκn, (5)
with limn→+∞κn=0.
Third we have:
1 n
R2\B(0,βn1/3)
exp
−1
2Σ2v·v Hˆa,b(0) dv≤ε2
nκn, (6)
with limn→+∞κn=0.
Hence, sinceHˆa,b(0)=21πHa,bL1,to estimateE(Ha,b(Sn−A))we are led to study the quantity Ha,bL1
n4π2
R2exp
−ivA/√ n
exp
−1 2Σ2v·v
dv=Ha,bL1 n4π2
2πexp(−Σ−2A·A/(2n))
√detΣ2 .
To conclude we notice that|Ha,bL1−4ε2| ≤10εn2.
4. Case of Euclidean extensions of subshifts of finite type
4.1. Description of theZ2-extensions of a mixing subshift
Let us fix a finite setAcalled alphabet. Let us consider a matrixMindexed byA×Awith 0–1 entries. We suppose that there exists a positive integern0 such that each component ofMn0 is non-zero. We define the set of allowed sequencesΣas follows:
Σ:=
ω:=(ωn)n∈Z:∀n∈Z, M(ωn, ωn+1)=1 .
Fig. 1. Dynamics of theZ2-extensionFof the shift.
We endowΣwith the metricd given by d
ω, ω
:=e−m,
wheremis the greatest integer such thatωi=ωiwhenever|i|< m. We define the shiftθ:Σ→Σbyθ ((ωn)n∈Z)= (ωn+1)n∈Z. For any functionf:Σ→Rwe denote bySnf=n−1
=0f◦θits ergodic sum. Let us consider an Hölder continuous functionϕ:Σ→Z2. We define theZ2-extensionF of the shiftθby (see Fig. 1)
F:Σ×Z2→Σ×Z2, (x, m)→
θ x, m+ϕ(x) .
We want to know the time needed for a typical orbit starting at(x, m)∈Σ×Z2to returnε-close to the initial point after iterations of the mapF. By translation invariance we can assume that the orbit starts in the cellm=0. More precisely, let
τε(x)=min
n≥1: Fn(x,0)∈B(x, ε)× {0}
. Observe thatFn(x, m)=(θnx, m+Snϕ(x)), thus
τε(x)=min
n≥1: Snϕ(x)=0 andd θnx, x
< ε .
Letν be the Gibbs measure associated to some Hölder continuous potentialh, and denote byσh2the asymptotic variance ofhunder the measureν. Recall thatσh2vanishes if and only ifhis cohomologous to a constant, and in this caseνis the unique measure of maximal entropy.
We know that there exists a positive integerm0such that the functionϕis constant on eachm0-cylinder.
Let us denote byσϕ2the asymptotic covariance matrix ofϕ:
σϕ2= lim
n→+∞Covν
1
√nSnϕ
.
We suppose that
Σϕdν=0.
We add the following hypothesis of non-arithmeticity onϕ: We suppose that, for anyu∈ [−π;π]2\ {(0,0)}, the only solutions(λ, g), withλ∈Candg:Σ→Cmeasurable with|g| =1, of the functional equation
g◦σ g=λeiu·ϕ
is the trivial oneλ=1 andg=const.
Note that this condition implies thatσϕ2is invertible. If this non-arithmeticity condition was not satisfied, then the range ofSnϕ would be essentially contained in a sub-lattice and we could make a change of variable (and eventually of dimension) to reduce our study to the corresponding twistedZa-extension (whereais the rank ofσϕ2). Moreover, we emphasize that this non-arithmeticity condition is equivalent to the fact that all the circle extensionsTudefined by Tu(x, t )=(θ (x), t+u·ϕ(x))are weakly-mixing foru∈ [−π;π]2\ {(0,0)}.
In this context, we prove the following results:
Theorem 8. The sequence of random variables log logτ−logεε converges almost surely asε→0to the Hausdorff dimen- siondof the measureν.
Theorem 9. The sequence of random variables ν(Bε(·))logτε(·)converges in distribution asε→0to a random variable with distribution functiont→1+βtβt1(0;+∞)(t ),withβ:= 1
2π detσϕ2.
Corollary 10. If the measure ν is not the measure of maximal entropy, then the sequence of random variables
log logτ√ ε+dlogε
−logε converges in distribution asε→0to a centered Gaussian random variable of variance2σh2.
In the case whereνis the measure of maximal entropy,then the sequence of random variablesεdlogτεconverges in distribution to a finite mixture of the law found in the previous theorem,that is,there exists some probability vector α=(αn)and positive constantsβnsuch that the sequence of random variablesεdlogτεconverges in distribution to a random variable with distribution function
nαn1+βnβt
nt1(0;+∞)(t ).
Example 11. We provide an example where the function ϕ(x) only depends on the first coordinate x0, that is, ϕ(x)=ϕ(x0).On the shiftΣ= {I, E, N, W, S}Zthe functionϕgiven byϕ(I )=(0,0),ϕ(E)=(1,0),ϕ(N )=(0,1), ϕ(W )=(−1,0)andϕ(S)=(0,−1)fulfills the hypothesis.
The remaining part of the section is devoted to the proof of these results. In Section 4.2 we recall some preliminary results and prove a uniform conditional local limit theorem. In Section 4.3 we prove Theorem 8 and in Section 4.4 we prove Theorem 9 and Corollary 10.
4.2. Spectral analysis of the Perron–Frobenius operator and local limit theorem
In order to exploit the spectral properties of the Perron–Frobenius operator we quotient out the “past.” We define:
Σˆ :=
ω:=(ωn)n∈N: ∀n∈N, M(ωn, ωn+1)=1 , dˆ
(ωn)n≥0, ωn
n≥0
:=e−ˆr(ω,ω)
withr((ωˆ n)n≥0, (ωn)n≥0)=inf{m≥0:ωm=ωm}and θˆ
(ωn)n≥0
=(ωn+1)n≥0.
Let us define the canonical projectionΠ:Σ→ ˆΣbyπ((ωn)n∈Z)=(ωn)n≥0. Letνˆbe the image probability measure (onΣ) ofˆ νbyΠ. There exists a functionψ:Σˆ →Z2such thatψ◦Π=ϕ◦θm0.
Let us denote byP:L2(ν)ˆ →L2(ν)ˆ the Perron–Frobenius operator such that:
∀f, g∈L2(ν)ˆ
ˆ Σ
Pf (x)g(x)dν(x)ˆ =
ˆ Σ
f (x)g◦ ˆθ (x)dν(x).ˆ
Letη∈(0,1). Let us denote byB the set of boundedη-Hölder continuous function g:Σˆ →Cendowed with the usual Hölder norm:
gB:= g∞+sup
x=y
|g(y)−g(x)| d(x, y)ˆ η .
