www.imstat.org/aihp 2008, Vol. 44, No. 4, 749–770
DOI: 10.1214/07-AIHP140
© Association des Publications de l’Institut Henri Poincaré, 2008
A local limit theorem with speed of convergence for Euclidean algorithms and diophantine costs
Viviane Baladi and Aïcha Hachemi
CNRS-UMR 7586, Institut de Mathématiques Jussieu, Paris, France. E-mail: baladi@math.jussieu.fr; hachemi@math.jussieu.fr Received 4 May 2007; revised 1 August 2007; accepted 23 August 2007
Abstract. For largeN, we consider the ordinary continued fraction ofx=p/qwith 1≤p≤q≤N, or, equivalently, Euclid’s gcd algorithm for two integers 1≤p≤q≤N, putting the uniform distribution on the set ofpandqs. We study the distribution of the total cost of execution of the algorithm for an additive cost functioncon the setZ∗+of possible digits, asymptotically forN→ ∞.
Ifcis nonlattice and satisfies mild growth conditions, the local limit theorem was proved previously by the second named author.
Introducing diophantine conditions on the cost, we are able to control the speed of convergence in the local limit theorem. We use previous estimates of the first author and Vallée, and we adapt to our setting bounds of Dolgopyat and Melbourne on transfer operators. Our diophantine condition is generic (with respect to Lebesgue measure). For smooth enough observables (depending on the diophantine condition) we attain the optimal speed.
Résumé. Nous considérons la fraction continue ordinaire de x=p/q pour 1≤p≤q ≤N, ou, de manière équivalente, l’algorithme de pgcd d’Euclide pour deux entiers 1≤p≤q≤N, avecN grand etpetqdistribués uniformément. Nous étu- dions la distribution du coût total de l’exécution de l’algorithme pour un coût additifcsur l’ensembleZ∗+des “digits” possibles, lorsqueNtend vers l’infini. Le théorème de la limite locale a été démontré par le deuxiéme auteur sicest non réseau et satisfait une condition de croissance modérée. En imposant une condition diophantienne sur le coût, nous parvenons à contrôler la vitesse de convergence dans ce théorème de la limite locale. Pour cela nous utilisons des estimées obtenues par le premier auteur et Vallée, et nous adaptons à notre problème des bornes de Dolgopyat et Melbourne sur les opérateurs de transfert. Notre condition dio- phantienne est générique (par rapport à la mesure de Lebesgue). Pour des observables assez régulières (par rapport à la condition diophantienne), nous obtenons la vitesse optimale.
MSC:11Y16; 60F05; 37C30
Keywords:Euclidean algorithms; Local limit theorem; Diophantine condition; Speed of convergence; Transfer operator; Continued fraction
1. Introduction and statement of results
Every rationalx∈ ]0,1]admits a finite continued fraction expansion
x= 1
m1+1/(m2+1/(· · · +1/mP)), mj=mj(x)∈Z∗+, P =P (x)∈Z∗+. (1) Continued fraction expansions can be viewed as trajectories of the Gauss map
T:(0,1] → [0,1], T (x):=1 x −
1 x
.
(Here,[y]is the integer part ofy∈R∗+.) Indeed, ifx=0 is rational, thenTP(x)=0 for someP =P (x)≥1, which is the depth of the continued fraction, and the digitsmj=mj(x)∈Z∗+appearing in (1) are just
mj(x)= 1
Tj−1(x)
, 1≤j≤P (x).
Clearly, this is equivalent to execution of Euclid’s gcd algorithm: for two integers 1≤p≤q, writeq1=q,p1=p andq1=m1p1+r1withm1=m1(p/q)∈Z∗+and a remainderr1∈Z+so thatr1< p1. Ifr1=0 we are done, and p=gcd(p, q), withP (p/q)=1. Ifr1=0, setp2=r1andq2=p1, and iterate this procedure until the remainderrP
vanishes for someP =P (p/q)≥2. ThenpP =gcd(p, q), andmj=mj(p/q)for 1≤j≤P. Note thatmP (p/q)=1 if and only ifp=q.
We shall callcost any (nonidentically zero) function c:Z∗+→R. Given such c, we associate to each rational x=p/q∈(0,1]the followingtotal cost:
C(x)=
P (x)
j=1
c mj(x)
. (2)
(Note that ifc≡1 then the total cost is just the depth P (x)of the continued fraction.) Our goal is to describe the probabilistic behaviour of the total cost associated to the ordinary (Gauss) continued fraction (1) (and some of its
“fast” variants). Before stating our results, we explain our probabilistic setting, and recall previous works.
ConsiderΩ˜ := {(p, q)∈(Z∗+)2,pq ∈(0,1]}, andΩ:= {(p, q)∈ ˜Ω|gcd(p, q)=1}, and endow the sets Ω˜N:=
{(p, q)∈ ˜Ω |q ≤N}, andΩN := {(p, q)∈Ω |q ≤N}with uniform probabilitiesPN andPN, respectively. We shall state our results forPN, but, as observed e.g. in [1] (see (2.18)), they also hold forPN. Note that if (p, q)∈ ΩN we can writeC(p, q)andP (p, q) instead ofC(p/q)andP (p/q). As usual, the expectationEN(C)denotes
(p,q)∈ΩNPN((p, q))C(p, q)and the varianceVN(C)isEN(C2)−(EN(C))2. We shall use the following conditions on a cost functionc:
cis of moderate growth if there existsν >0 so that
m∈Z∗+eνc(m)m−2+ν<∞, chas strong moments up to orderk≥1 if there existsν >0 so that
m∈Z∗+c(m)km−2+ν<∞. Of course, ifcis of moderate growth then it has strong moments up to arbitrary orderk.
Remark 1.1. Ifc(m)=O(log(m))thencis of moderate growth,while ifc(m)=O(m−ν+1/ k)for somek≥1 and someν>0 thenc has strong moments up to orderk.The terminology comes from the fact thatT has a unique absolutely continuous invariant measureμ1on[0,1],with a positive analytic densityf1(see also Section2),so that, writingc∗(x)=c(m)if x∈(1/(m+1),1/m],we have that
(c∗(x))dμ1(x)is well-defined for positive integers ≤k,i.e.c∗has moments up to orderkifchas strong moments up to orderk. (The converse is not true,whence the terminology “strong” moments.See also[1],and Lemma3.3for the need to useν >0.)
Introducing a dynamical approach (centered around the one-dimensional mapT) and using Ruelle-type transfer operatorsHs =Hs,0 to study this problem (see Section 2 for a definition of the transfer operators), Brigitte Vallée obtained in a series of papers (see e.g. [21] for references) precise results for the asymptotics of the expectationEN(C) and other moments. For example, ifcis of moderate growth, there isμ(c)∈R(withμ(c)=0 if
c∗(x)dμ1(x)=0) so that
Nlim→∞
EN(C)
logN =μ(c). (3)
We refer to the recent work [1] for more information, a historical discussion including references to the work of Heilbronn and Dixon and previous work of Vallée. Among other things it was proved in [1] that ifcis of moderate growth then there existsδ(c)∈R∗+so that
Nlim→∞
VN(C) logN =
δ(c)2
. (4)
Remark 1.2. To get(3)and(4)it suffices to assume thatchas strong moments up to order2.See Lemma3.2.
The article [1] also contains the followingcentral limit theorem[1], Theorem 3: for each costcof moderate growth, there isM1(c)≥1 so that for any integerN≥1, and anyy∈R:
PN
C(p, q)−μ(c)logN δ(c)√
logN ≤y
− 1
√2π y
−∞e−x2/2dx
≤ M1(c)
√logN, (5)
whereμ(c)∈Randδ(c) >0 are the same as in (3). The speed of convergence(logN )−1/2in the above central limit theorem is optimal, as is clear from the saddle point-argument in the proof. This speed is the equivalent in our setting of the speed of convergence in the central limit theorem for independent indentically distributed random variables [10].
(Hensley [14] obtained a central limit theorem forc≡1, more than a decade before [1], but with a O((logN )−1/24) bound on the rate of convergence.)
The basic tool to obtain all limit theorems mentioned here is the transfer operator. The transfer operator allows to implement the characteristic function method, via Levy generating functions. However, there is a key difference betweendiscreteandcontinuousproblems (see also [1], Section 2.3, for an illustration). Discrete and continuous does not refer to time here, but to the probabilities: discrete means that we are performing weighted sums over finite sets of increasing size (just like in this paper). In the continuous setting, such as in the pioneering work of Guivarc’h–Le Jan [12], in a geometric context which also involves the Gauss map, a spectral gap argument à la Nagaev is invoked.
However, the transfer operators which appear for discrete problems involve not only the parameteriτ orwfrom the characteristic function, but also another parameter,s, which ranges in a half-plane containing the pole. This other parameter comes from a Dirichlet series in the present paper and in [1], it can also be viewed as the parameter of an L-function in other settings, see e.g. [19]. Dealing withs of large imaginary parts requires the use of fundamental bounds first proved by Dolgopyat [7,8] in the context of hyperbolic flows.
Specifically, in order to prove (5), methods adapted1from Dolgopyat [8], were used to get bounds on the resolvent (Id−Hs,w)−1of the transfer operatorHs,w, with(s, w)=(1,0), when the complex parametersvaries in a half-plane containing 1, but the complex parameterwis close to zero. We refer to [1] (in particular Theorem 2 there) for more details.
We next discuss local limit theorems. A cost functioncis calledlattice, if there exists(L, L0)∈R∗+×R+, with L0/L∈ [0,1)so that(c−L0)/Lis integer-valued. Ifcis lattice but not constant, the largest possibleLis called the spanofcand the correspondingL0is called theshiftofc. Ifcis constant we take spanL= |c|and shiftL0=0.2 A cost function is callednonlatticeif it is not lattice.
If the cost is lattice withL0=0 and enjoys moderate growth, then the followinglocal limit theorem([1], Theo- rem 4) holds: forx∈RandN∈Z∗+, put
Q(x, N )=μ(c)logN+δ(c)x
logN , (6)
then,3there isM2(c)≥1 so that for everyx∈Rand all integersN≥1
logN·PN
C(p, q)−Q(x, N )
∈
−L 2,L
2
− Le−x2/2 δ(c)√
2π
≤ M2(c)
√logN. (7)
(See [1], Section 5.4, for the caseL0=0.) Again, the constantsμ(c)∈Randδ(c) >0 are the same as in (3), and the speed of convergence is optimal. The proof uses operatorsHs,iτwhere the complex parametersvaries in a half-plane, and the real parameterτ lies in the bounded interval[−π/L,π/L).
Very recently, using Breiman’s method (also known as Stone’s trick, see [11] for a recent application of this method to nonuniformly hyperbolic dynamics) to handle noncompactness issues, the second author of the present
1The key difficulty is that the symbolic alphabet in [8] is finite, while the Gauss map has infinitely many branches.
2The definition of lattice stated in [1] and [13] should be replaced there by this one.
3The factorLmultiplying e−x2/2was inadvertently omitted from [1], pp. 351 and 381.
paper obtained [13], Théorème 3, a local limit theorem: for every nonlattice4cost functioncof moderate growth, for each compact intervalJ, we have, writing|J|for the length ofJ, and forμ(c)∈Randδ(c) >0 as in (3):
Nlim→∞sup
x∈R
logN·PN
C(p, q)−Q(x, N )
∈J
− |J| e−x2/2 δ(c)√
2π
=0. (8) We would like to point out that the compact intervalJ in (8) is arbitrary, while in (7) only the interval(−L/2, L/2]
is allowed. Roughly speaking, the local limit theorem in the lattice case can be viewed as the analogue of a result for a Poincaré section of a flow. (And it is well-known that obtaining results for flows is usually much more difficult, sometimes requiring additional conditions, than proving the corresponding results for their Poincaré maps.)
Remark 1.3. Lemma3.2and the arguments in Section4show that the moderate growth assumption for(5), (7)and(8) can be replaced by the requirement thatchas strong moments up to order3.
The purpose of the present article is to obtain a local limit theorem with control of the speed of convergence in the nonlattice case.
Our proof involves transfer operatorsHs,iτ, withsin a half-plane, andτ∈R. Whenτ is confined to any compact set, or when|s|is large enough, we can exploit the bounds of [1] based on Dolgopyat’s work [8] (see Lemmas 3.3 and 3.4). For largeτ, we must adapt other estimates of Dolgopyat [7], introduced to study decay of correlations for Axiom A flows (see also Naud [16] for a reader-friendly account). These estimates of Dolgopyat [7] require a diophantine condition.5In view of proving decay of correlations for nonuniformly hyperbolic flows, Melbourne [15]
recently generalised the estimates of [7] to infinite alphabets. However, the parameters(s, z)˜ of Melbourne’s operators Rs,z˜ [15], Section 3.3, are not of the same nature as our parameters(s, iτ ): while the real and imaginary parts ofs˜ correspond to˜s= sand˜s=τ, respectively, the imaginary part ofzlies in[0,2π], because it arises from locally constant integer return times. The parameterzis thus a periodic parameter, and the return times are “mute” in a reformulation of Melbourne’s diophantine condition ([15], Proof of Corollary 2.4). In our setting, unbounded values ofs are handled by the arguments from [1], as mentioned above, and we are left to deal with|s|in a compact set. The parameters is thus “artificially” bounded, but is not intrinsically periodic. Because of these differences (note also that Melbourne works with the dynamical distance in an abstract Gibbs–Markov setting, while we require the Euclidean metric), we carry out the modified bounds in detail in Section 2. Since the weights|h|are not locally constant integers, we cannot eliminate them from our diophantine condition (see also Remark 1.4 and the proof of Lemma 1.5).
Since the statement of our sufficient diophantine condition on the cost is unpalatable (although very similar to Melbourne’s [15], Theorem 2.3), we postpone it to Section 2, and we formulate here a stronger (but simpler) condition instead. We need more notation. The countable set of digitsm∈Z∗+is in bijection with the setHof inverse branches ofT, throughm→(y→ y+1m). We may thus view the cost functioncas a function onH. For integerp≥2 and any subsetH0 ofH, writeH10=H0andHp0 = {h◦ ˜h|h∈H0,h˜∈Hp0−1}. Note thatx=h(x)forx∈Hp means that x=Tp(x), i.e.xis periodic. The minimal suchpis called the period ofx. Then, we extendcto a function onHpby setting, forh=hp◦ · · · ◦h2◦h1∈Hp,
c(h)= p
j=1
c(hj). (9)
Recall that a vectorx∈Rd ford ≥1 isdiophantineof exponentη0≥d if there existsM >0 so that for each (q1, . . . , qd)∈Zd\ {0}
pinf∈Z
p−
d
k=1
xkqk
≥ M (maxk|qk|)η0.
4The assumption thatcis nonlattice has been inadvertently omitted from [13], Théorème 3: this assumption is necessary to allow arbitrarily largeL in [13], Lemma 2, as is clear e.g. from the proof of [1], Proposition 1(iii).
5See [2,3] for diophantine conditions in a related, but different, probabilistic context, and [4] for a previous “nonlattice of orderp” condition.
For eachη0> d, the set of diophantine vectors of exponentη0has full Lebesgue measure inRd(see e.g. [5]).
Definition. Letη >2.The costcisstrongly diophantineof exponentηif there existη > η0≥2and four periodic pointsxj ∈(0,1),j =1,2,3,4forT,of respective minimal periodspj≥1,withhij(xj)=xj forhij ∈Hpj,and with pairwise disjoint orbits,so that the following holds:settingcj=c(hij),aj=log|hi
j|, L1j=pjc1−p1cj, L1j=pja1−p1aj, j=2,3,4,
thenL13=0,L12=0,withL12/L13diophantine of exponentη0,and,defining Lj k=L1jL1k−L1jL1k, j=k∈ {2,3,4},
we haveL23=0,and the pair L43
L23
,L42
L23
is diophantine of exponentη0.
Remark 1.4. Our definition uses four periodic orbits,like in[15],with the “intertwining” of thecjand theaj due to the previously discussed fact that theaj are not integers. (See also Remark1.6.)The conditionL13=0withL12/L13 diophantine of exponentη0,involving three periodic orbits,is sufficient to ensure that(Id−Hσ,iτ)−1Lip≤M0|τ|α for some α >0, all |τ| ≥2 and appropriate σ (see proof of Proposition 2.1), this is not enough since we need (Id−Hσ+it,iτ)−1Lip≤M0|τ|α for all|t| ≤t0.In the simpler cases studied by Dolgopyat[7]and Naud[16]it is possible to formulate a(non intertwined)sufficient diophantine condition involving only two periodic orbits(because the alphabet is finite).
Before stating our result, we discuss further the above definition. First, it is easy to see that a strongly diophantine cost is nonlattice. By Remark 1.8, it is generic. In Section 2, we shall give the definition of a diophantine cost of exponentηand we shall prove:
Lemma 1.5. If cis strongly diophantine of exponentη,letting H0⊂H be the smallest set so thathij ∈H0pj for j=1,2,3,4,thencis diophantine of exponentηforH0.
Remark 1.6. Eachxj in the strongly diophantine condition is just the quadratic number associated to the infinite repetition of apj-tuplem=(m1,j, . . . , mpj,j)of positive integers.Also,ajcoincides withlogpj−1
=0 (m,j+xj,)−2, where thexj,=T(xj)are the quadratic numbers associated to the circular permutations ofm. Finallycj is the total cost associated to the rational numberu/vwhose continued fraction has depthpj and is given bym.
Remark 1.7. We give an example.Ifcis such that there are four integersmj>0so that,settingaj=log(1+(m2j− mj
m2j+4)/2),we have
c(m2)=c(m3), a1=a2,
c(m1)−c(m2)
(a1−a3)=
c(m1)−c(m3)
(a1−a2), and both c(mc(m1)−c(m2)
1)−c(m3) and the pair
(c(m1)−c(m4))(a1−a3)−(c(m1)−c(m3))(a1−a4) (c(m1)−c(m2))(a1−a3)−(c(m1)−c(m3))(a1−a2),
(c(m1)−c(m4))(a1−a2)−(c(m1)−c(m2))(a1−a4) (c(m1)−c(m2))(a1−a3)−(c(m1)−c(m3))(a1−a2)
are diophantine of exponentη0< η,thencis strongly diophantine of exponentη. (Just consider the case where all pj=1in the definition.)
Remark 1.8. Fixη >2.Choose four periodic pointsxj=hij(xj)∈(0,1]ofT,with pairwise disjoint orbits(and hij ∈Hpj).Then it is not difficult(see[15],Corollary2.4)to show that for Lebesgue almost every (c1, c2, c3, c4) inR4+,any costcso thatc(hij)=cj is strongly diophantine of exponentη(use Fubini).Therefore,the diophantine condition on the costcdeserves to be called generic ifη >2.
Recall (6). Our main result is the following local limit theorem with speed of convergence:
Theorem 1.9. For any diophantine cost functioncof exponentηand subsetH0,with strong moments up to order3:
There existsε∈(0,1/2]so that for each compact intervalJ⊂Rthere exists a constantMJ >0so that for every x∈Rand all integersN≥1:
logNPN
C(p, q)−Q(x, N )
∈J
− |J| e−x2/2 δ(c)√
2π
≤ MJ
(logN )ε.
There existsr≥1so that for any compactly supportedψ∈Cr(R),there exists a constantMψ>0so that for every x∈Rand all integersN≥1:
logNEN
ψ
C(p, q)−Q(x, N )
− e−x2/2 δ(c)√
2π
ψ (y)dy
≤ Mψ
√logN.
The second claim of the theorem says that we attain the optimal speed in the local limit theorem for smooth enough compactly supported observablesψ. If the cost is nonlattice but not diophantine, we expect that arbitrarily slow convergence can take place in the local limit theorem, in the spirit of [17,18].6
Remark 1.10. The statement of Proposition2.1and the proof of Theorem1.9easily imply that for any α > η
2+log supH0|h|−1 log(1/ρ)
1+log supH0|h|−1 log(1/ρ)
, we may take
ε < (2α)−1, r > α+1, (10)
but these conditions are probably not optimal.The proof of Theorem1.9gives a constantK(c)=K(η,H0)so that,if ψis supported in an intervalJψ,
MJ≤K(c)|J|2, Mψ≤K(c)|Jψ| · ψC[r+1]. (11) Remark 1.11. Ifchas strong moments up to orderk+1≥4,a little more work should yield finite Edgeworth expan- sions[10] (see also[2])of orderkfor compactly supportedψ∈Cr(R). (The remainder term beingO((logN )−k/2).) In this case,the condition on the differentiabilityr ofψwill depend not only on the diophantine exponent ofc,but also on the desired orderkfor the Edgeworth expansion.
The paper is organised as follows: in Section 2, we adapt the estimates of Dolgopyat–Melbourne ([7,15]) to our setting to get bounds (Proposition 2.1) for the norm of the resolvent (Id−Hσ+it,iτ)−1 for large |τ|, bounded|t|, andσ >1−δ(τ )for smallδ(τ ), under the diophantine assumption onc. Lemma 1.5 is also proved in Section 2.
In Section 3, we first recall previous material from [1], in particular the connection between(Id−Hσ+it,iτ)−1 and the moment generating functionEN(eiτ C)of smoothened models (via the Perron formula and bivariate Dirichlet series), as well as estimates onEN(eiτ C)for bounded|τ|. We then deduce from Proposition 2.1 our key estimate
6Note however that lower bounds are much less accessible in our setting and the methods in this paper do not provide such bounds for any examples of non-diophantine lattice costs.
(Corollary 3.5), onEN(eiτ C)for large|τ|. The proof of Theorem 1.9 is carried out in Section 4 by reducing to a study of
RχˆJ(τ )e−iτQ(x,N )EN(eiτ C)dτ, respectively
Rψ (τ )eˆ −iτQ(x,N )EN(eiτ C)dτ (withφˆthe Fourier transform ofφ), and decomposing the integral overτ ∈Rinto four domains, over which we apply the estimates from Section 3. In the Appendix, we describe two other (fast) continued fraction algorithms, thecentered algorithmand theodd algorithm, for which Theorem 1.9 holds, with the same proof, since our arguments only use the fact thatT belongs to the good class from [1] and satisfies the condition UNI from [1].
We have already mentioned the difference between the bounds in Section 3 and those in [15]. With respect to [1] and [13], we can mention two innovations (besides our remark that moderate growth can be replaced by strong moments): first, Lemma 3.5 requires a specific smoothening, adapted to the weak bounds for large|τ|from Section 3;
second, we have to regularise the characteristic function of the interval J by convolution in Section 4.2 in order to control large|τ|.
2. Dolgopyat–Melbourne estimates for Hσ+it,iτ
Let us first introduce some notation. PutI = [0,1], and Lip(I )= {u:I →C| uL∞+Lip(u) <∞}, with Lip(u) the smallest Lipschitz constant ofu. (Ifu is not Lipschitz then we put Lip(u)= ∞.) It is well known (see [1] for references) that there existsρ <1,K≥1 andK≥1 so that for allm∈Z∗+and allh∈Hm
suph≤Kρm, h(x)≤Kh(x), ∀x∈I. (12)
In this section, we focus on|t| ≤t0, for some fixedt0>0, and|τ| ≥2, since other values oft andτ are covered in previous works, as explained in the next section.
Forτ∈Rands=σ+it, witht∈R,σ∈Rwithσ >1/2, put foru∈Lip(I )andx∈I Hs,iτ(u)(x)=
h∈H
eiτ c(h)h(x)su h(x)
. (13)
We have for the sames,τ and eachm≥1, recalling the extensionc:Hm→Rfrom (9), Hms,iτ(u)(x)=
h∈Hm
eiτ c(h)h(x)su h(x)
.
Lettingf1be the fixed point ofH1,0so that
If1(x)dx=1 (in fact,f1(x)=(log 2)−1(1+x)−1), we put for allt∈R andτ∈R
H1+it,iτ(u)=H1+it,iτ(f1u) f1
.
By definition, we haveHm1+it,iτ(u)L∞≤ uL∞, for allm∈Z∗+, and all realt andτ. It is not very difficult to show (see e.g. the proof of [1], Lemma 2) that for eacht∈Rthere isK(t )≥1 so that
LipHm1+it,iτ(u)
≤K(t )uL∞+KρmLip(u), ∀m∈Z∗+,∀τ∈R. (14) Inequalities such as the above are usually called Doeblin–Fortet (in probability) or Lasota–Yorke (in dynamics) in- equalities. It will be convenient to use the following norm on Lip(I ):
uLip=max
uL∞, 1
2 sup|t|≤t0K(t )Lip(u)
.
Indeed, recalling (12) and settingn0= [logK/log(1/ρ)] +1, we have for allτ ∈Rand allt∈ [−t0, t0] Hm1+it,iτ
Lip≤Kρm+1, ∀m≥1, Hm1+it,iτ (15)
Lip≤1, ∀m≥n0.
Finally, we give the definition of a diophantine cost:
Definition. Letη≥2.The costcis diophantine of exponentηfor the finite subsetH0⊂Hif there existsβ0≥1so that,for any sequencesτk∈R,tk∈R,θk∈ [0,2π),withlimk→∞|τk| = ∞butsupk|tk|<∞,and for anyM≥1and β≥β0,there existk≥1andx=hx(x)for somehx∈Hp0,withp≥1minimal,so that
dist τk
βlog|τk|
c(hx)+tk
βlog|τk|
loghx+pθk,2πZ
≥ Mp
|τk|η. It is easy to check that any diophantine cost is nonlattice.
The main result of this section is:
Proposition 2.1. If the cost functioncis diophantine of exponentηforH0⊂H,then,taking α > η
2+log supH0|h|−1 log(1/ρ)
1+log supH0|h|−1 log(1/ρ)
,
there existM0≥1,andξ0∈(0,1)so that for each|τ| ≥2,all|t| ≤t0,and everyσ≥1−ξ0|τ|−α (Id−Hσ+it,iτ)−1
Lip≤M0|τ|α. (No growth assumption is required onc.)
Before we prove the proposition by a modification of the argument of Dolgopyat [7], as adapted by Melbourne [15], Section 3, to the case of infinitely many branches, we need further notation and a couple of preliminary lemmas.
IfH0is a strict subset ofH, we letI0=I0(H0)be the invariant Cantor set forT associated toH0, i.e., I0= {x∈ I|Tm(x)∈I0,∀m∈Z+}forI0=
h∈H0h([0,1]). Then, we set Lip(I0)= {u:I0→C|Lip(u) <∞}. Finally, for τ∈Randu∈L∞(I ), we set forx∈(0,1], denoting byhxthe element ofHso thatx∈hx([0,1)),
Mt,τu(x)=T(x)ite−iτ c(hx)u T (x)
.
We may now state and prove the first lemma, which very roughly says that if the iterates of the transfer operator H1+it,iτ decay too slowly, then the operatorMt,τ has an approximate eigenfunction:
Lemma 2.2. LetI0(H0)be associated to a finite setH0⊂H.Letη1>0andβ0≥1.Then for eachα1> η1(2+
log supH0|h|−1
log(1/ρ) ),there existβ1> β0andK0≥1so that the following is true for each|t| ≤t0and every|τ| ≥2,setting n(τ )= [β1log|τ|]:
Suppose that there existsv0=v0,τ,t ∈Lip(I )withv0Lip≤1so that Hj n(τ )1+it,iτ(v0)(x)≥1− 1
|τ|α1, ∀x∈I0, j=0,1,2. (16) Then there existθτ,t∈ [0,2π)andwτ,t:I0→C,with|wτ,t(x)| =1for allxand
Mn(τ )t,τ wτ,t(x)−eiθτ,twτ,t(x)≤ K0
|τ|η1, ∀x∈I0. (17)
Proof. (See Lemma 3.12 and Section 3.3 in [15] adapted from [7], Section 8.) In this proof, we fixt andτ, and we writenforn(τ ). Lettingv0be as in (16), put forj =0,1,2
vj=Hj n1+it,iτ(v0), sj= |vj|. Our assumption (16) implies that
1− 1
|τ|α1 ≤sj(x)≤1, ∀x∈I0, j=0,1,2. (18)
In particular, we may definewj(x)=vsj(x)
j(x) forx∈I0andj=0,1,2 (and we have|wj| ≡1 onI0). Note for further use that (15) implies that there is a constantK1≥1 (which does not depend ont orτ) so thatwjLip(I0)≤K1for j=0,1,2 (first show thatvjLip(I )is uniformly bounded).
Since s1(x)=w11(x)Hn1+it,iτ(s0w0)(x), andH1,0(1)=1 (with1 the constant function ≡1), the bound (18) for j=1 implies that for allx∈I0
h∈Hn
f1(h(x))
f1(x) h(x)
1−|h(x)|it
w1(x) eiτ c(h)s0 h(x)
w0 h(x)
≤ 1
|τ|α1.
The real part of each term in the above sum is nonnegative. Hence, using also thatf1◦h/f1is bounded from above and from below, uniformly inh∈Hn, we can find a constantK2(which does not depend onτ) so that for eachh∈Hn and everyx∈I0
0≤1−s0 h(x)
|h(x)|iteiτ c(h)w0(h(x)) w1(x)
≤ K2
|h(x)||τ|α1.
Sinces0(h(x))≤1, the above bound implies that for eachh∈Hnand everyx∈I0
0≤1−
|h(x)|iteiτ c(h)w0(h(x)) w1(x)
≤ K2
|h(x)||τ|α1.
Using the fact that for any complex numberzof modulus 1 we have|1−z| =√
2(1− z)1/2, we find a constantK3, independent ofτ, so that for eachh∈Hnand everyx∈I0
w1(x)−h(x)iteiτ c(h)w0
h(x)≤ K3
|h(x)|1/2|τ|α1/2.
From now on, we restrict our attention to branchesh∈Hn0. For such a branch, we have
h(x)≥K4−n, (19) whereK4=suph0∈H0sup|h0|−1>1 depends only onH0. Therefore, recalling thatn= [β1log|τ|], ifα1andβ1> β0
satisfy
α1−β1log(K4) >2η1, (20)
then there is a constant K5(independent ofτ) so that for eachx∈I0, settinghx to be the element ofHn0 so that x∈hx([0,1)),
w1
Tn(x)
−Tn
(x)−iteiτ c(hx)w0(x)≤ K5
|τ|η1. (21)
A similar argument givesK6, independent ofτ, so that if (20) holds andx∈I0 w2
Tn(x)
−Tn
(x)−iteiτ c(hx)w1(x)≤ K6
|τ|η1. (22)
Fix an arbitraryx0∈I0and defineθ0=θ0(τ, t )(recall that thewj depend onτ andtand thatn=n(τ )) in[0,2π) by
eiθ0=w0(x0)Tn
(x0)−iteiτ c(hx0).
Lethx0∈Hn0be so thatx0∈hx0([0,1)). Observe next that (21) and the fact thatTn(x)=Tn(xx0)andhx0=hxx
0 for xx0 =hx0(Tn(x))imply that for allx∈I0
w1
Tn(x)
−eiθ0≤ K5
|τ|η1 +w0(xx0)−w0(x0)+Tn
(xx0)−it−Tn
(x0)−it.
Now, on the one hand, since|h/ h| ≤K, with |h| ≤Kρn, and since|t| ≤t0, and|Tn(x)−Tn(x0)| ≤1, there is a constantK7, independent ofτ andh, so that
hx0
Tn(x)it−hx0
Tn(x0)it≤K7·ρn(τ ),
and on the other hand, since|xx0−x0| ≤ρnandxx0∈I0ifx∈I0, we have w0(xx0)−w0(x0)≤K1·ρn(τ ).
Therefore, if (20) holds, and in addition β1> η1
log(1/ρ), (23)
then we have w1
Tn(x)
−eiθ0≤ K1+K5+K7
|τ|η1 , ∀x∈I0. (24)
Next, definingθ1=θ1(t, τ )∈ [0,2π)by eiθ1=w1(x0)Tn
(x0)−iteiτ c(hx0),
the previous argument gives that if (20) and (23) hold then w2
Tn(x)
−eiθ1≤ K1+K6+K7
|τ|η1 , ∀x∈I0. (25)
Putting together (24) and (25), we find forθt,τ=θ0−θ1and allx∈I0 e−iθt,τw1
Tn(x)
−w2
Tn(x)≤2K1+K5+K6+2K7
|τ|η1 . (26)
Takingα1andβ1so that (20) and (23) hold, and substituting (26) into (22) we see that the functionw=w1satisfies
the conclusion of the lemma forK0=2K1+K5+2K6+2K7.
We need another lemma, which very roughly says that good decay for the iterates ofHj10+it,iτ implies moderate τ-growth for its resolvent:
Lemma 2.3. LetI0(H0)be associated to a finite setH0⊂H.For eachα1>0everyα > α1(1+log suplog(1/ρ)H0|h|−1),and eachβ1>0,there existsM0≥1so that the following hold for each|τ| ≥2and|t| ≤t0:
Suppose that for eachv∈Lip(I )withvLip≤1there existsx0∈I0andj0≤ [3β1log|τ|]so that Hj10+it,iτ(v)(x0)≤1− 1
|τ|α1, (27)
then(Id−H1+it,iτ)−1Lip≤M0|τ|α.
Remark 2.4. The proof of Lemma2.3uses heavily the fact thats=1+itand breaks down ifs=σ+itwithσ <1.
Proof of Lemma 2.3. (See Lemma 3.13 in [15], adapted from [7], Section 7.) In this proof we fixt andτ and write n=n(τ )= [3β1log|τ|]. Letv∈Lip(I )be so thatvLip≤1. It suffices to show that(Id−H1+it,iτ)−1(v)exists and (Id−H1+it,iτ)−1(v)Lip≤M0|τ|α for someα >0 andM0≥1 which do not depend onτ ort.
Forj0=j0(v, τ )≤nas in (27), we put u0=Hj10+it,iτ(v) and u=Hn1+it,iτ(v).
