Computation of a Class of Continued Fraction Constants

1.6 Hausdorff dimension and constrained CF –expansions. Consider some integer n and a subset Aof N n ∗ . Denote by R A the Cantor set of reals in I whosecontinued fraction expansion is restricted to A,R A := {x ∈ I ; x = [q 1 , q 2 , . . . , q k , . . .], ∀k ≥ 0,(q kn+1 , q kn+2 , q kn+n ) ∈ A}.As soon as A is different from N n ∗ , the Cantor set R A haszero Lebesgue measure, and the Hausdorff dimensionprovides a precise description of it. In particular,the probability that a rational with numerator anddenominator less than N belongs to R A is Θ(N 2sA−2 ),so that the expected time to obtain a rational A–constrained with numerator and denominator less thanN is Θ(N 2−2s A). When A is finite, the reals of R Aare interesting since they are all badly approximable byrationals [31].If, furthermore, the set A contains more than oneelement, the Hausdorff dimension of R A , denoted bys A , is a real number of ]0, 1[, and is proven to be theunique real s ∈]0, 1[ for which the dominant eigenvaluefunction λ A (s) of the transfer operator G s,A equals 1.This is why the Hausdorff dimension belongs to the classof spectral constants.The Hausdorff dimension of the set R A relative toA := {1, 2} has been intensively studied. In 1941,Good [15] showed that 0.5194 ≤ s {1,2} ≤ 0.5433 andin 1982, Bumby [5][6] improved these estimates andobtains s {1,2} = 0.5313 ± 10 −4 . In 1996, Hensley [19]provided a polynomial–time algorithm in the case of afinite set A and obtained the following estimations {1,2} ≈ 0.5312805062772051416.Finally, in 1999, Jenkinson and Pollicott [22] designeda powerful algorithm which computes s {1,2} up to 25digits. Note that it is not a polynomial–time algorithm.The DFV–method has been applied (heuristically) tothe case of a general set A and seemed to be efficient[32]. We propose a proven polynomial–time algorithmbased on the DFV–heuristics for any subset A 1 × A 2 ×. . . × A n of G n . In the particular case when A ⊂ G, itgives rise to proven numerical values of s A , and it seemsto run faster than Hensley’s algorithm.2 Polynomial time algorithms for StrictlyContracting Dynamical Systems.As explained in Section 1, we wish to prove the DFV–method. The following definition is natural in thiscontext.Definition 1. [Operator with good truncations.] LetD be a disk of center x 0 , and consider an operator Gthat acts on A ∞ (D). Consider the projection π n definedin (1.1) and the truncated operator G n := π n ◦ G.The operator G has good truncations if the followingis true: there is θ < 1 such that, for any simple isolatedeigenvalue λ of the operator G, there exist a constantK > 0, an integer n 0 , and a sequence λ n ∈ Sp G n , forwhich, for any n ≥ n 0 , one has:|λ − λ n | ≤ Kθ n .The constant θ is called the truncature ratio.If moreover the triple [θ, K, n 0 ] is computable, then thetruncations are said to be computable.We are interested in constants that arise in spectralobjects relative to complete Dynamical System. Acomplete Dynamical System is a pair (I, T ) formed withan interval I and a map T : I → I which is piecewisesurjective and of class C 2 . We denote by G the set ofthe inverse branches of T ; then, G k is the set of theinverse branches of T k . It is known that contractionproperties of the inverse branches are essential to obtain“good” properties on the Dynamical System. Usually,what is needed is the existence of a disk D which isstrictly mapped inside itself by all the inverse branchesh ∈ G of the system [i.e., h(D) ⊂ D]. Here, we have tostrengthen this hypothesis. This motivates the followingdefinition.Definition 2. [Strongly contracting dynamical system.]A complete Dynamical System of the intervalI is said to be strongly contracting (SCDS in short)when the set G of the inverse branches fulfills the supplementarycondition : For any subset A ⊂ G, there existx 0 ∈ I and two open disks of same center x 0 , the largedisk D L , and the small disk D S , with D S D L andI ⊂ D L , such that any h ∈ A is an element of A ∞ (D L )which strictly maps D L inside D S [i.e., h(D L ) ⊂ D S ].Remark that (x 0 , R S , R L ) depend on A. The largestpossible ratio R S /R L between the radii R S and R L ofthe two disks is called the A–contraction ratio.A strongly contracting system is said to be extra–contracting (X SCDS in short) if, for any A ⊂ G, thereexist an integer k > 1 and another disk D XL (the extra–large disk) cocentric with D S , with D L ⊂ D XL , suchthat that any h ∈ A k is an element of A ∞ (D XL ) whichstrictly maps D XL inside D S [i.e., h(D XL ) ⊂ D S ].We shall prove in the following that many DynamicalSystems relative to Euclidean algorithms belong to theSCDS–setting, and even in the X SCDS–setting.Definition 3. [Transfer operator] Let (I, T ) beof SCDS–type. Consider an integer n ≥ 1, a subsetA ⊂ G n , a real σ A ≥ 0, a sequence (α h ) h∈A of functions

of A ∞ (D L ) positive on D L ∩R, such that, for any s withRs > σ A , the quantityδ(s, A) := ∑ sup |α h (x)| R(s) < ∞.x∈Dh∈A LThen, the relationG s,A [f] = ∑ h∈Aα s h f ◦ h.defines, for Rs > σ A , an operator G s,A : A ∞ (D S ) →A ∞ (D L ) whose norm ‖G s,A ‖ DS ,D Lis at most δ(s, A).Such an operator is called a transfer operator withconstraint A.If moreover the system (I, T ) is of X SCDS–type, withan integer k > 1, the operator G k s,A maps A ∞(D S ) intoA ∞ (D XL ).Our first result is as follows:Theorem 1. [A transfer operator has good truncations.]In the SCDS–setting, a transfer operator G s,A :A ∞ (D S ) → A ∞ (D L ) satisfies the following:(ii) It is compact, and its spectrum is formed withisolated eigenvalues of finite multiplicity, except perhapsat 0.(ii) for any real s, with s > σ A , the operator G s,A has aunique dominant eigenvalue λ A (s) simple, positive andisolated from the other eigenvalues by a spectral gap.(iii) for any real s, with s > σ A , the operator G s,Ahas good truncations. The truncature ratio θ satisfiesθ ≤ R S /R L where R S and R L are the radii of theoptimal pair of disks (D S , D L ) relative to A.(iv) In the X SCDS–setting, the truncature ratio satisfiesθ ≤ R S /R XL where R S and R XL are the radii ofthe optimal pair of disks (D S , D XL ) relative to A,Now, we prove Theorem 1 and provide explicit constantsfor K, n 0 and θ. The first two assertions are easilyadapted from the works of Mayer [30], and we thenfocus on the the proof of the third assertion, which ismainly based on two results. We first a well-knownresult of functional analysis, which says: “When twooperators are close with respect to the norm, theirspectrums are close too”. The second result showsthat the strongly contraction property entails that thetruncated operators converge in norm to the transferoperator.2.1 Functional analysis. Denote by (B, ‖ ‖) acomplex Banach space and by G an operator which actson B. Denote by Sp G the spectrum of G. Considera fixed eigenvalue λ and a circle C = C(λ, r) (withcenter λ and radius r > 0) that isolates λ from theremainder of the spectrum. This means that r satisfiesr < d(λ, Sp G\{λ}). The constants α C (G) and β C (G),defined by(2.1) α C (G) := sup ‖ (G − zI) −1 ‖ ,z∈C(2.2){}1β C (G) := max2α C (G) , 1 12rαC 2 (G), 8r 2 αC 3 (G) ,play a central rôle in the paper. They first intervene inthe following result, which is fundamental here.Lemma 1. Let G and ˜G be two operators on theBanach space (B, ‖ ‖). Suppose that λ is a simple andisolated eigenvalue of G, with an eigenfunction φ, andconsider an isolating circle C = C(λ, r). Then, if G and˜G satisfy ‖G − ˜G‖ ≤ β C (G), then the operator ˜G hasa unique simple isolated eigenvalue ˜λ in C that satisfies(2.3)|˜λ − λ| ≤ 2r · α C (G) · ‖ ˜G[φ] − G[φ]‖‖φ‖≤2r · α C (G) · ‖ ˜G − G‖.The condition ‖G − ˜G‖ ≤ β C (G) implies, via thedefinition of β C (G), three conditions, with a precise goalfor each of them. The first condition ensures that thecircle C is also included in the resolvant set of ˜G. Thesecond condition implies that ˜λ is the unique eigenvalueof ˜G in C. Finally, with the last condition, it is possibleto relate the two spectral spaces related to λ and ˜λ.This proposition will be also useful for computing theHensley constant (in Section 4.2).2.2 Convergence of truncated operators in theSCDS–setting. Consider any operator G : A ∞ (D S ) →A ∞ (D L ). We recall that the non-zero eigenvalues ofthe operator π n ◦ G and the matrix M n are the same.Then, according to the previous Lemma, it is sufficientto obtain the convergence of π n ◦ G to G (in norm).This is the aim of the following lemma, which requiresthe strong contracting property. A proof restricted tothe framework of Continued Fractions can also be foundin [20] with slightly different functional spaces.Lemma 2. Consider an operator G : A ∞ (D S ) →A ∞ (D L ) with norm ‖G‖ DS ,D L. Then, one has:‖π n ◦ G − G‖ DS ≤ ‖G‖ DS ,D LR LR L − R S(RSR L) n+1.Suppose furthermore that there exists some disk D XL ,with D L ⊂ D XL , and some k–th iterate of G whichmaps A ∞ (D S ) into A ∞ (D XL ). Then, for any eigenfunctionφ of G,‖π n [φ] − [φ]‖ DS‖φ‖ DS≤ ‖φ‖ ( ) n+1D XLR XL RS·.‖φ‖ DS R XL − R S R XL

More precisely, (see [3]), if the cost c is of “moderategrowth”, the variance of the total cost of an executioncan be expressed with the derivatives (of order 1 or 2) ofthe dominant eigenvalue λ s,w of G s,w . If the cost is of“large growth”, then the Hausdorff dimension relativeto this cost also involves the dominant eigenvalue λ s,w([7]). And the DFV–method can be proven to apply,with exponential rate of convergence.The list of these possible applications of Theorem 1is not exhaustive. We now focus on the standardEuclidean dynamical system, where it is possible toprovide estimates for the pair [K, n 0 ].3 Proven Computation of the constants in theEuclidean caseWe come back now to the Euclidean Dynamical Systemand its (usual) transfer operators G s,A relative to α h =|h ′ |. The formulae (2.5) (2.6) which define θ, n 0 and Kinvolve two kinds of constants. The constants R S , R L(and R XL ), are closely related to the definition of theoperators and are often easy to compute. This entailsthat the rate θ is in general easy to compute.Then, we have to find an isolating circle C(λ, r), whichneeds an estimate of λ and a lower bound for thespectral gap around λ. We prove in Lemmas 3 and4 that it is possible, at least when λ = λ A (s) is thedominant eigenvalue,If we use the X SCDS–setting, we wish to obtain anestimate of the eigenfunction φ relative to λ.On the other hand, even when C = C(λ, r) is welldefined,a lower bound for the constant β C (G), givenin (2.2) is not easy to compute. It involves, via thedefinition of α C (G) in (2.1) an upper bound for thenorm of an inverse operator. Such an upper bound is ingeneral hard to compute. However, when G is normal,an explicit an expression for α C (G) are known,(3.1) α C (G) =1d(C, Sp G)(we recall that G is normal when it commutes withits dual G ⋆ ) But the normality is a rare phenomenon,which is difficult to prove. Here, it is not true thatthe transfer operator G s,A is normal on spaces A ∞ (D);however, there exists another functional space (a Hardyspace, denoted by H s,A , which depends on (s, A) whereG s,A is normal (note that this normality phenomenondoes not seem to hold for the two other DynamicalSystems C, O). Even if the two spaces, the Hardy spaceand the space A ∞ (D) are different, the associated normscan be compared, and this provides an upper bound forα C (G s,A ), and a lower bound for β C (G). Finally, theresults of this Section lead to the second main result ofthe paper.Theorem 2. The (standard) Euclidean DynamicalSystem is a system of X SCDS–type. For any A ⊂ G, thetriple [K, n 0 , θ] used for approximating the dominanteigenvalue λ A (s) can be estimated, and there exists aneffective algorithm that computes λ A (s) in polynomial–time.The remainder of the Section is devoted to provingTheorem 2.3.1 Disks D S , D L , D XL and truncature ratio.We recall that we deal with the Euclidean system, wherethe set of inverse branches G is defined in (1.3). Here,we consider a subset A of G and A denotes the set ofindices of A. We denote by m A the minimum of A.Then the disks D S , D L can be chosen asx 0 := 1m A, R S = 1m A, R L = 1m A+ m A2 ,R SR L=22 + m 2 .AFurthermore, the operator G 2 s,A maps A ∞(D S ) into theset of functions which are analytic in the half plane{R(z) > −m A }. We then can choose for the disk D XLany disk of center x 0 and radiusR XL = 1m A+ m A − ɛ,R SR XL=(with ɛ > 0), so that11 + m 2 A − ɛm .A3.2 Estimate for the dominant eigenvalue λ A (s)of G s,A . We use the following classical result previouslyused in [10]:Let G be an operator which acts on the space ofanalytic functions on an interval [a, b]. Furthermore,the operator G is positive (i.e., G[f] > 0 if f > 0) andhas a unique dominant eigenvalue λ isolated from theremainder of the spectrum by a spectral gap. Supposethat there exist two strictly positive constants c 1 and c 2and a function f which is strictly positive and analyticon [a, b] and satisfies c 1 f(x) ≤ G s,A [f](x) ≤ c 2 f(x), forany x ∈ [a, b]. Then, the dominant eigenvalue λ satisfiesc 1 ≤ λ ≤ c 2 .Denote by m A the minimum of A and by M A its supremum(possibly infinite if A is infinite). By convention,if M A = ∞, we put h MA = 0. Each LFT h MA ◦ h mA orh mA ◦ h MA is called an extremal LFT; it has a uniquepositive fixed point, denoted by a A or b A . Then thedisk D A with diameter [a A , b A ] is the smallest diskwhich is mapped into itself by all the elements of A.The application of the previous result with the operatorG s,A and the functions f = 1 for the upper bound

Lemma 5. For any complex s with R(s) >max(σ A , 0),(i) the transfer operator G s,A : H s,A → H s,A isisomorphic to an integral operator; it is normal and selfadjointfor real values of s. Thus, for real values of s,the spectrum of G s,A is real.(ii) the spectrum of G s,A on H s,A and the spectrum ofG s,A on A ∞ (D S ) are the same.(iii) Let D be an intermediary disk of center x 0 andradius R with R S < R < R L and f a function ofA ∞ (D). For any subset A of G, the function G s,A [f]belongs to H s,A .(iv) Define, for any R (with R S < R < R L ), threeconstants κ 1 , κ 2 , κ 3 (which depend on x 0 , s, A, R),(3.6)κ 1 = ζ A (2s, x 0 − R), κ 2 = Γ(2s) · ζ A (2s, 2(x 0 − R)),(3.7)κ 3 = ∑ j≥0( ) ( j RSRj!Γ(2s + j)with γ j = e x0/(j+1) .Then, the following is true:(3.8)(3.9)) 1/2(γ j R S ) 2sγj 2s − 1 + ζ A(2s, 0)Γ(2s) 2‖G s,A [f]‖ D ≤ κ 1 · ‖f‖ DS for f ∈ A ∞ (D)‖f‖ D ≤ κ 2 · ‖f‖ for f ∈ H s,A ,(3.10) ‖G s,A [f]‖ ≤ κ 3 · ‖f‖ Dfor f ∈ A ∞ (D).Before proving Lemma 5, we explain how it provides anestimate for α C (G s,A ). Consider the isolating circle Cof center λ A (s) and radius r A (s) described in Lemma4,and consider a point z ∈ C. The two inclusionsG s,A [A ∞ (D S )] ⊂ A ∞ (D) andG s,A [A ∞ (D)] ⊂ H s,A ,together with the relationz(G s,A − zI) −1 = (G s,A − zI) −1 G s,A − Ientail the following(3.11)|z| ‖(G s,A − zI) −1 ‖ DS ≤ κ 1 · ‖(G s,A − zI) −1 ‖ D + 1,(3.12)|z| ‖(G s,A −zI) −1 ‖ D ≤ κ 2·κ 3·‖(G s,A −zI) −1 ‖ +1.Now, G s,A is normal on H s,A , so that‖(G s,A − zI) −1 ‖ =1d(z, Sp G s,A ). Finally, with the formulae (3.11) and (3.12), theinequality‖(G s,A − zI) −1 ‖ ∞,DS ≤ 1 ( )κ1 · κ 2 · κ 3|z| 2 + 1 + 1r A |z| ,holds. Now, the estimate of λ A (s) (given in Lemma 3)yields that any z ∈ C satisfies|z| ≥ ζ A (2s, b) − r A (s) > 0and an upper bound of α C (G s,A ) follows. Finally, weobtain:Lemma 6. Denote by r A (s) the lower bound of Lemma4. For any intermediary radius R, with R S < R < R Land s > max(0, σ A ), there exist constants κ i defined inLemma 5 (which depend on x 0 , R, s, A), for which(3.13)α C (G s,A ) ≤ κ 1 · κ 2 · κ 3 + r A (s)[1 − r A (s) + ζ A (2s, b A )]r A (s)[ζ A (2s, b A ) − r A (s)] 2 ,Proof of Lemma 5. For (i) and (ii), we refer tothe work of Jenkinson, Gonzalez and Urbański [21],and we mainly deal with (iii) and (iv). The strongcontraction condition implies the inequality (3.8). Theinequality (3.9) is a direct application of the Cauchy-Schwartz inequality with the identityΓ(s)ζ A (s, z) =∫ ∞0t s−1 e −zt dν A (t).The proof of inequality (3.10) is more involved and weonly explain here its main steps. First Hensley [20] hasshown that, for all j ≥ 0, the function G s,A [(X − x 0 ) j ]is an element of H s,A whose integral representationis closely related to generalised Laguerre polynomialsL (2s−1)j . The Laguerre polynomials (L (p)j ) form anorthogonal basis for the weight t p e −t on ]0, ∞[ and theyverify the formulaL (p) Γ(p + 1 + j)j (x) =j!∑( ) jj(−1) k kk=0The function G s,A [(X − x 0 ) j ] then satisfy∫ ∞0G s,A [(X − x 0 ) j ](z) =[t s−1/2 e −tz (−x0 ) j j!Γ(2s + j) ts−1/2 L 2s−1jWe deduce that the norm is given by‖G s,A [(X − x 0 ) j ]‖ 2 =[ (−x0 ) j ] 2 ∫j! ∞Γ(2s + j) 0x kΓ(p + 1 + k) .( tx 0)]dν A (t)t 2s−1 L 2s−1j (t)dν A (x 0 t).

But the Laguerre polynomials are positive and decreasingon [0, 2s/(j + 1)]. Using these properties with somerelations of orthogonality and splitting the integrand∫ ∞into ∫ 2s/(j+1)+ ∫ ∞, we prove the inequality0 0 2s/(j+1)∀j ≥ 1, ‖G s,A [(X − x 0 ) j ]‖ ≤ K jK j+1K j→ R S .withNow, the i–th coefficient c i of the Taylor expansion off ∈ A ∞ (D) at x 0 satisfies R j |c j | ≤ ‖f‖ D and finally‖G s,A [f]‖ ≤ κ 3 · ‖f‖ D with κ 3 = ∑ j≥0K jR jNote that the previous series converges exponentiallyfast.Finally, the constants K and n 0 defined in (2.5) and(2.6)involve the isolating circle C whose center λ A (s) andradius r A (s) together with α C (G s,A ). Since all thesequantities are computable, this proves Theorem 2.4 Application of the DFV-Mehtod to threeconstantsThis section applies the previous results and provides(in polynomial–time, via the DFV-Method) proven numericalvalues for three continued fraction constants:the Gauss-Kuz’min-Wirsing constant, the Hensley constantand the Hausdorff dimension of the Cantor setsR A with A ⊂ G.In the second step of the DFV–method, we deal withcomputations on the matrix M s,A,n . First, we have tobuild the matrix; second we have to find the roots ofdet(M s,A,n − zI n ). Then we conclude:For any subset A ⊂ G, building matrix M s,A,n needsO(n 3 ) multiplications and additions on reals and 2n + 1computations of ζ A (s) functions; Computing Sp M s,A,nneeds at most O(n 4 ) arithmetical operations (with abad method).In this section, we shall prove the last result of thepaper:Theorem 3. For the three following constants –the Gauss-Kuz’min-Wirsing constants, the Henley constants,the Hausdorff dimension relative to constraintsA ⊂ G– it is possible to provide d proven digits inpolynomial–time in d.4.1 Algorithm for the Gauss-Kuz’min-Wirsingconstant. The Gauss-Kuz’min-Wirsing constant γ G isthe unique subdominant eigenvalue of G 1 . It is real.It is possible to estimate a circle C that isolates γ Gtogether with the associated constant α C (G 1 ). Thus,the DFV-method provides proven numerical values forγ G . Only one computation of matrix is needed, and thecomplexity of γ G is of order four. Numerical results aresummarized up in Figure 4.1.4.2 Algorithm for the Hensley constant.The Hensley constant (see 1.8) involves the first twoderivatives of λ(s) at s = 1. Since the first derivativeλ ′ (1) has a closed form λ ′ (1) = −π 2 /(6 log 2), it remainsto compute the second derivative λ ′′ (1). Consider aninterval I h of the form I h := [1 − h, 1 + h] and supposethat an estimate ˜λ of λ satisfiesmax(|λ(1 + h) − ˜λ(1 + h)|, |λ(1 − h) − ˜λ(1)− h)|≤ h2 ɛ3 .Then Taylor’s formulae entail the estimate(1) − ˜λ(1 − h) + ˜λ(1∣+ h) − 2 ∣∣∣∣∣ λ′′ h 2≤ 2ɛ3 + h224 sup |λ (4) |I hIt then suffices to know an upper bound for the fourthderivative λ (4) on the interval I h . The applications → G s is analytic and the derivative G ′ s satisfies‖G ′ s‖ DS ≤ 8 for s ≥ 0.9. Then, ‖G s −G 1 ‖ DS ≤ 8|s−1|.We apply Lemma 3: the circle C of center 1 and radiusr 1 := (1−γ G )/2 is an isolating circle for λ = 1. Then, ifs satisfies |s − 1| < r 2 with r 2 = β C (G)/8, the operatorG s admits a unique simple isolated eigenvalue λ(s) inC which satisfies |λ(s)−1| ≤ r with r := 16 r 1 r 2 α C (G)as soon as |s − 1| ≤ r 2 . Since the application s → G sis analytic, the function s → λ(s) is analytic too. TheCauchy formula, applied in the disk of center 1 andradius r 1 yields the upper boundsup[1−h,1+h]|λ (4) 1 + r| ≤ 4!(r 1 − h) 4 .Then, γ H is computable as soon as the two estimatesfor λ(1 + h) and λ(1 − h) are known (asymptotically towithin twice the required precision). This thus needstwo computations of the step 2 of the DFV–method.Tabular 4.2 summarizes some numerical results.4.3 Algorithm for the Hausdorff dimension.The algorithm uses a classical dichotomy principle andcomputes a sequence of intervals of length 2 −k whichcontain the Hausdorff dimension s A . Consider theinterval [u k−1 , v k−1 ] obtained after (k − 1) steps (it isof length 2 −(k−1) and contains s A . Denote by w k the

digits time proven value10 11s -0.303663002820 1m46 -0.3036630028987326585930 9m54 -0.30366300289873265859744812190140 34m710 -0.303663002898732658597448121901556233110850 1h41 -0.30366300289873265859744812190155623311087735225365Figure 2: Gauss-Kuz’min-Wirsing constantdigits time proven value5 2m30s 0.5160610 7m30s 0.516062408915 41mn 0.51606240889999120 2h33mn 0.51606240889999180681Figure 3: Hensley constantmiddle point of the interval [u k−1 , v k−1 ] and computean estimate ˜λ of λ A (w k ) within 2 −(k+1) . Now, thereare three possible cases:(i) If ˜λ − 1 − 2 −(k+1) ≥ 0 then s A ≥ w k and[u k , v k ] := [w k , v k−1 ].(ii) If ˜λ − 1 + 2 −(k+1) ≤ 0 then s A ≤ w k and[u k , v k ] := [u k−1 , w k ](iii) Else, [u k , v k ] := [w k − 2 −(k+1) , w k + 2 −(k+1) ]This algorithm is just a classical binary splitting. Theproof that s A belongs to [u k , v k ] is based on the strictdecrease of λ together with the inequality |λ A (s) −λ A (s + h)| ≥ h. There are at most O(d) iterations, eachof them of cubic complexity (in d). Thus, the complexityof the algorithm is asymptotically O(d 5 ). Numericalresults are given in Figure 4.3 for the Cantor set R {1,2} .5 ConclusionWe proved here that the DFV-method gives rise to analgorithm that computes any isolated simple eigenvalueof a transfer operator, in polynomial–time, providedthat two conditions are fulfilled: (i) the operator hasgood truncations and (ii) the matrices M n are easy tocompute.However, if we are interested in actual proven numericalvalues, we need evaluating the parameters that intervenein the design of the algorithm. These parametersare in general difficult to compute, but we solve thisdifficulty for the transfer operators relative to the StandardEuclidean Algorithm.The DFV-method can also be used to compute theHausdorff dimension of the Cantor sets R A with A ofthe form A = A 1 × A 2 × . . . × A n , A i ∈ H. However,the computation of the matrix is more involved since itscoefficients deal with more complicated zeta–functions.Finally, the authors of [10] and Sebah used the DFVmethodwith x 0 = 1/2. This particular choice donot enter our framework since no disk of center 1/2is strictly mapped into itself. We can use any diskD L of center 1/2 + δ with radius 1/2 + 2δ diameter[−δ, 1 + 3δ] with δ > 0. This leads to truncature ratioR S /R XL which tends to 1/3 − ɛ as δ tends to zero,which is the convergence rate actually observed by theauthors. However, our method of Section 3.4 does notseem to apply there, and we do not know how to obtainan estimate for the pair [K, n 0 ] in this setting.References[1] M. Ahues, A. Largillier, V. Limaye. Spectralcomputations for bounded operators, Chapman &Hall/CRC (2001).[2] K. I. Babenko. On a problem of Gauss, Soviet Math.Dokl. 19 (1978), 136-140.[3] V. Baladi and B. Vallée. Euclidean algorithms areGaussian, Les Cahiers Du GREYC, Université deCaen (2003)[4] K. Briggs. A Precise Computation of the Gauss-Kuzmin-Wirsing Constant. Preliminary report. 2003July 8.http://research.btexact.com/teralab/documents/wirsing.pdf.[5] R. T. Bumby. Hausdorff dimension of Cantor sets, J.Reine Angew. Math. 331 (1982), 192-206[6] R. T. Bumby. Hausdorff dimension of sets arisingin number theory, Number Theory (New-York, 1983-1984), Lecture Notes in Math., 1135, Springer, 1985,pp. 1-8[7] E. Cesaratto. Thesis, University of Buenos Aires,2003[8] T. Cusik. Continuants with bounded digits, Matematika24 (1977), 166-172[9] T. Cusik. Continuants with bounded digits II, Matematika25 (1978), 107-109[10] H. Daudé, P. Flajolet, B. Vallée. An Average-Case Analysis of the Gaussian Algorithm for LatticeReduction, Combinatorics, Probability and Computing(1997) 6, pp 1-34[11] J. G. Dixon. The number of steps in the Euclideanalgorithm, J. Number Theory, 2 (1970), 414-422

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