Rigorous effective bounds on the Hausdorff dimension of continued fraction Cantor sets: a hundred decimal digits for the dimension of $E_2$

We prove that the algorithm of [13] for approximating the Hausdorff dimension of dynamically defined Cantor sets, using periodic points of the underlying dynamical system, can be used to establish completely rigorous high accuracy bounds on the dimension. The effectiveness of these rigorous estimates is illustrated for Cantor sets consisting of continued fraction expansions with restricted digits. For example the Hausdorff dimension of the set $E_2$ (of those reals whose continued fraction expansion only contains digits 1 and 2) can be rigorously approximated, with an accuracy of over 100 decimal places, using points of period up to 25. The method for establishing rigorous dimension bounds involves the holomorphic extension of mappings associated to the allowed continued fraction digits, an appropriate disc which is contracted by these mappings, and an associated transfer operator acting on the Hilbert Hardy space of analytic functions on this disc. We introduce methods for rigorously bounding the approximation numbers for the transfer operators, showing that this leads to effective estimates on the Taylor coefficients of the associated determinant, and hence to explicit bounds on the Hausdorff dimension.


Introduction
For a finite subset A ⊂ N, let E A denote the set of all x ∈ (0, 1) such that the digits a 1 (x), a 2 (x), . . . in the continued fraction expansion x = [a 1 (x), a 2 (x), a 3 (x), . . .] = 1 a 1 (x) + 1 a 2 (x)+ 1 a 3 (x)+··· all belong to A. Sets of the form E A are said to be of bounded type (see e.g. [14,16]); in particular they are Cantor sets, and study of their Hausdorff dimension has attracted significant attention.
In [13] we outlined a different approach to approximating the Hausdorff dimension of bounded type sets, again using a transfer operator, but exploiting the real analyticity of the maps defining continued fractions to consider the determinant ∆ of the operator, and its approximation in terms of periodic points 1 of an underlying dynamical system. While some highly accurate empirical estimates of Hausdorff dimension were given, for example a 25 decimal digit approximation to dim(E 2 ), these were not rigorously justified. Moreover, although the algorithm was proved to generate a sequence of approximations s n to the Hausdorff dimension (depending on points of period up to n), with convergence rate faster than any exponential, the derived error bounds were sufficiently conservative (see Remark 1 below) that it was unclear whether they could be combined with the computed approximations to yield any effective rigorous estimate.
In the current paper we investigate the possibility of sharpening the approach of [13] so as to obtain rigorous computer-assisted estimates on dim(E A ), with particular focus on E 2 . There are several ingredients in this sharpening. The first step is to locate a disc D in the complex plane with the property that the images of D under the mappings T n (z) = 1/(z + n), n ∈ A, are contained in D. It then turns out to be preferable to consider the transfer operator as acting on a Hilbert space of analytic functions on D, rather than the Banach space of [13]; this facilitates an estimate on the Taylor coefficients of ∆ in terms of the approximation numbers (or singular values) of the operator, which is significantly better than those bounds derived from Banach space methods. The specific Hilbert space used is Hardy space, consisting of those analytic functions on the disc which extend as L 2 functions on the bounding circle. The contraction of D by the mappings T n (z) = 1/(n + z), n ∈ A, prompts the introduction of the contraction ratio, which captures the strength of this contraction, and leads to estimates on the convergence of the approximations to the Hausdorff dimension. The n th Taylor series coefficient of ∆ can be expressed in terms of periodic points of period up to n, and for sufficiently small n these can be evaluated exactly, to arbitrary precision. For larger n, we show it is advantageous to obtain two distinct types of upper bound on the Taylor coefficients: we refer to these as the Euler bound and the computed Taylor bound. The Euler bound is used for all sufficiently large n, while the computed Taylor bound is used for a finite intermediate range of n corresponding to those Taylor coefficients which are deemed to be computationally inaccessible, but where the Euler bound is insufficiently sharp. Intrinsic to the definition of the computed Taylor bounds is the sequence of computed approximation bounds, which we introduce as computationally accessible upper bounds on the approximation numbers of the transfer operator.
As an example of the effectiveness of the resulting method we rigorously justify the first 100 decimal digits of the Hausdorff dimension of E 2 , thereby improving on the rigorous estimates in [4,7,9,10,12]. Specifically, we prove (see Theorem 1) that dim(E 2 ) = 0.53128050627720514162446864736847178549305910901839 87798883978039275295356438313459181095701811852398 . . . , using the periodic points of period up to 25. 1 The periodic points are precisely those numbers in (0, 1) with periodic continued fraction expansion, drawn from digits in A. The reliance on periodic points renders the method canonical, inasmuch as it does not involve any arbitrary choice of coordinates or partition of the space.

Preliminaries
In this section we collect a number of results (see also [13]) which underpin our algorithm for approximating Hausdorff dimension.
2.1. Continued fractions. Let E A denote the set of all x ∈ (0, 1) such that the digits a 1 (x), a 2 (x), . . . in the continued fraction expansion all belong to A. For any i ∈ N we define the map T i by and for a given A ⊂ N, the collection {T i : i ∈ A} is referred to as the corresponding iterated function system. Its limit set, consisting of limit points of sequences Every set E A is invariant under the Gauss map T , defined by and set H δ (E) = lim ε→0 H δ ε (E). The Hausdorff dimension dim(E) is then defined as dim(E) = inf{δ : H δ (E) = 0} .

Pressure formula.
For a continuous function f : E A → R, its pressure P (f ) is given by and if f = −s log |T | then we have the following implicit characterisation of the Hausdorff dimension of E A (see [1,2,6,15]): The function s → P (−s log |T |) is strictly decreasing, with a unique zero at s = dim(E A ).

Transfer operators.
For a given A ⊂ N, and s ∈ R, the transfer operator L A,s , defined by preserves various natural function spaces, for example the Banach space of Lipschitz functions on [0, 1]. On this space it has a simple positive eigenvalue e P (−s log |T |) , which is the unique eigenvalue whose modulus equals its spectral radius, thus by Lemma 1 the Hausdorff dimension of E A is the unique value s ∈ R such that L A,s has spectral radius equal to 1.

2.5.
Determinant. The determinant for L A,s is the entire function defined for z of sufficiently small modulus 2 by and for other z ∈ C by analytic continuation; here the trace tr(L n A,s ) is given (see [13]) by where the point z i , which has period n under T , is the unique fixed point of the n-fold When acting on a suitable space of holomorphic functions, the eigenvalues of L A,s are precisely the reciprocals of the zeros of its determinant. In particular, the zero of minimum modulus for ∆(s, ·) is e −P (−s log |T |) , so the Hausdorff dimension of E A is characterised as the value of s such that 1 is the zero of minimum modulus of ∆(s, ·).
As outlined in [13], this suggests the possibility of expressing ∆(z, s) as a power series ∆(z, s) = 1 + The function D is an entire function of s (see [13]), and solutions s of the equation have the property that the value 1 is an eigenvalue for L A,s ; in particular, the unique zero of D in the interval (0, 1) is precisely dim(E A ), being the unique value of s for which 1 is the eigenvalue of maximum modulus for L A,s . As a result of the trace formula (2), the coefficients δ n (s) are computable 3 in terms of the periodic points of T | E A of period no greater than n, so for some suitable N ∈ N, chosen so that δ 1 (s), . . . , δ N (s) can be computed to a given precision in reasonable time, we can define D N by A solution to the equation D N (s) = 0 (5) 2 The power series ∞ n=1 z n n tr(L n A,s ), and hence the expression (1), is convergent for |z| < e −P (−s log |T |) . 3 By this we mean that for a given s, the δn(s) are computable exactly, to arbitrary precision.
will be an approximate solution to (3), where the quality of this approximation will be related to the smallness of the discarded tail ∞ n=N +1 δ n (s) .
In particular, any rigorous estimate of the closeness of a given approximate solution s N of (5) to the true Hausdorff dimension dim(E A ) will require a rigorous upper bound on the modulus of the tail (6).
Remark 1. In [13] we considered the set E 2 = E {1,2} and, although the empirical estimates of its Hausdorff dimension appeared convincing, the estimate on the tail (6) was not sharp enough to permit any effective rigorous bound. Essentially 4 , the bound in [13] was |δ n (s)| ≤ ε n := CK n n n/2 θ n(n+1) where C = γ ∞ r=1 (1 − γ r ) −1 ≈ 122979405533, K = 45 16π ≈ 0.895247, and θ = 8 9 1/4 ≈ 0.970984. Although the bounding sequence ε n tends to zero, and does so at super-exponential rate O(θ n 2 ), the considerable inertia in this convergence (e.g. the sequence increases for 1 ≤ n ≤ 39 to the value ε 39 ≈ 1.31235 × 10 22 , and remains larger than 1 until n = 85) renders the bound ineffective in practice, in view of the exponentially increasing computation time required to calculate the δ n (s) (as seen in this article, we can feasibly compute several million periodic points, but performing calculations involving more than 2 85 points is out of the question).

Hilbert Hardy space, approximation numbers, approximation bounds
In this section we introduce the Hilbert space upon which the transfer operator acts, then make the connection between approximation numbers for the operator and Taylor coefficients of its determinant, leading to so-called Euler bounds on these Taylor coefficients.
k=0 is square summable. The norm f can then be expressed as In [13] we actually worked with det(I − zL 2 A,s ) rather than det(I − zLA,s), though the methods there lead to very similar bounds for both determinants.

Approximation numbers.
Given a compact operator L : H → H on a Hilbert space H, its i th approximation number s i (L) is defined as The following result exploits our Hilbert space setting, and represents an improvement over analogous Banach space estimates in [13] (where e.g. a multiplicative factor n n/2 reduces the quality of the bound on |δ n (s)|).
, then the n th Taylor coefficient δ n (s) of its determinant can be bounded by Proof. If {λ n (s)} is the eigenvalue sequence for L A,s , ordered by decreasing modulus and counting algebraic multiplicities, then (see e.g. In view of the link between Hausdorff dimension error estimates and the tail (6), together with the bounding of terms in this tail by sums of products of approximation numbers provided by Lemma 2, it will be important to establish upper bounds on the Taylor coefficients δ n (s) for those n where it is not computationally feasible to evaluate exactly via periodic points. We shall derive two distinct types of such upper bound, which we refer to as Euler bounds and computed Taylor bounds. There is an Euler bound on δ n (s) for each n, given as a simple closed form; this bound will be used for all sufficiently large values of n, though for low values of n may be too conservative for our purposes. The finitely many computed Taylor bounds will be on the Taylor coefficients δ P +1 (s), . . . , δ Q (s) where P is the largest integer for which we locate all period-P points, and Q is chosen so that the Euler bounds on |δ n (s)| are sufficiently sharp when n > Q. In view of Lemma 2, the computed Taylor bounds will be derived by first bounding the finitely many approximation numbers s 1 (L A,s ), . . . , s N (L A,s ), for some N ∈ N, by explicitly computable quantities that we call computed approximation bounds. The computations required to derive the computed approximation bounds are not onerous, the main task being the evaluation of numerical integrals defining certain H 2 norms (of the transfer operator images of a chosen orthonormal basis).
We shall approximate L A,s by first projecting H 2 (D) onto the space of polynomials up to a given degree. Let L A,s : H 2 (D) → H 2 (D) be a transfer operator, where D ⊂ C is an open disc of radius centred at c, and {m k } ∞ k=0 is the corresponding orthonormal basis of monomials, given by 3.3. Approximation bounds.
Definition 1. For n ≥ 1, define the n th approximation bound α n (s) to be Proposition 1. For each n ≥ 1, Proof.
The transfer operator L A,s is approximated by the rank-(n − 1) operators and L A,s − L

(n)
A,s can be estimated using the Cauchy-Schwarz inequality as follows: A,s has rank n − 1, it follows that s n (L A,s ) ≤ α n (s), as required.

Contraction ratios. Let
The estimate arising in the following lemma motivates our definition below (see Definition 2) of the contraction ratio associated to a disc D and subset A ⊂ N. Lemma 3. Let D and D be concentric discs, with radii and respectively. If, for i ∈ A, the image T i (D) is contained in D , then for all k ≥ 0, Proof. Let c denote the common centre of the discs D, D . If z ∈ D then We may write and if i is such that T i (D) is contained in the concentric disc D i of radius i then Lemma 3 implies that For our purposes it will be more convenient to work with a slightly simpler (and less sharp) version of (12). This prompts the following definition: Lemma 4. Let A ⊂ N be finite, and D an admissible disc, with contraction ratio h = h A,D . For all k ≥ 0, Proof. If D is as in Definition 2 then = max i∈A i in the notation of (12), and the result follows from (12).

Corollary 1.
Let A ⊂ N be finite, and D an admissible disc, with contraction ratio h = h A,D . For all n ≥ 1, where Proof. Now from Definition 1 and Proposition 1, so Lemma 4 gives and the result follows.

Euler bounds.
We can now derive the Euler bound on the n th Taylor coefficient of the determinant: Proposition 2. Let A ⊂ N be finite, and D an admissible disc, with contraction ratio h = h A,D . If the transfer operator L A,s has determinant det(I − zL A,s ) = 1 + ∞ n=1 δ n (s)z n , then for all n ≥ 1, Proof. By Lemma 2, Henceforth we use the notation so that (17) can be written as and we define the righthand side of (19) (or equivalently of (17)) to be the Euler bound on the n th Taylor coefficient of the determinant.

Computed approximation bounds
For all n ≥ 1, the n th approximation bound is, as noted in Proposition 1, an upper bound on the n th approximation number s n (L A,s ). Each m k is just a normalised monomial (8), and the operator L A,s is available in closed form, so that and we may use numerical integration to compute 5 each Hardy norm L A,s (m k ) as where γ(t) = c + e 2πit . Evaluation of α n (s) involves the tail sum ∞ k=n−1 L A,s (m k ) 2 , and in practice we can bound this by the sum of an exactly computed long finite sum N k=n−1 L A,s (m k ) 2 , for some N n, and a rigorous upper bound on ∞ k=N +1 L A,s (m k ) 2 using (14). More precisely, we have the following definition: and α n,N, Evidently the lower computed approximation bound α n,N,− (s) is a lower bound for α n (s), in view of the positivity of the summands in (9) and (21), while Lemma 5 below establishes that the upper computed approximation bound α n,N,+ (s) is an upper bound for α n (s). Moreover, both α n,N,+ (s) and α n,N,− (s) are readily computable: they are given by finite sums and, as already noted, the summands L A,s (m k ) 2 are computable to arbitrary precision. (23) 5 Numerical integration capability is available in computer packages such as Mathematica, and these norms can be computed to arbitrary precision; although higher precision requires greater computing time, these computations are relatively quick (e.g. for the computations in §6 these integrals were computed with 150 digit accuracy).
Proof. The inequality α n,N,− (s) ≤ α n (s) is immediate from the definitions. To prove that α n (s) ≤ α n,N,+ (s) note that which together with (14) gives and the result follows.

Remark 2.
The upper bound α n,N,+ (s) will be used in the sequel, as a tool in providing rigorous estimates on Hausdorff dimension. In practice N will be chosen so that the values α n,N,− (s) and α n,N,+ (s) are close enough together that the inequality (23) determines α n (s) with precision far higher than that of the desired Hausdorff dimension estimate; in particular, N will be such that the difference α n,N,+ (s) − α n,N,− (s) = O(h N ) is extremely small relative to the size of α n (s).
Combining (15) with (23) immediately gives the exponential bound though the analogous bound for α n,N,+ (s) (which will be more useful to us in the sequel) requires some extra care: Lemma 6. Let s ∈ R. For all n, N ∈ N, with n ≤ N , Proof. Combining (24) with (22) gives and the result follows.
The utility of (25) stems from the fact that in practice N − n will be large, and that for sufficiently small values of n the following more direct analogue of (24) can be used: Proof. Immediate from Lemma 6.

Remark 3.
In practice Q will be of some modest size, dictated by the computational resources at our disposal; specifically, it will be chosen slightly larger than the largest P ∈ N for which it is feasible to compute all periodic points of period ≤ P (e.g. in §6, when estimating the dimension of the set E 2 = E {1,2} , we explicitly compute all periodic points up to period P = 25, and in the proof of Theorem 1 we choose Q = 28). The value N will be chosen to be significantly larger than Q (e.g. in the proof of Theorem 1 we choose N = 600). Since N + 2 − Q is large, h N +2−Q will be extremely small, and J = J Q,N,s will be extremely close to K s ; ideally this closeness ensures that the two constants J Q,N,s and K s are indistinguishable to the chosen level of working precision (e.g. in the proof of Theorem 1, N + 2 − Q = 574 and h ≈ 0.511284, so h N +2−Q ≈ 5.9 × 10 −168 , whereas computations are performed to 150 decimal digit precision).

Computed Taylor bounds
In order to use the computed approximation bounds to provide a rigorous upper bound on the Taylor coefficients of the determinant det(I −zL A,s ), we now fix a further natural number M , satisfying M ≤ N . For any such M , it is convenient to define the sequence (α M n,N,+ (s)) ∞ n=1 to be the one whose n th term equals α n,N,+ (s) until n = M , and whose subsequent terms are given by the exponential upper bound on s n (L A,s ) and α n (s) (cf. (15) where the sum is over those i = (i 1 , . . . , i n ) ∈ N n which satisfy i 1 < i 2 < . . . < i n .
As the name suggests, the Taylor  Note that β M n,N,+ (s) is precisely the n th power series coefficient for the infinite product , and that the sum in (29) is an infinite one; thus we will seek a computationally accessible approximation to β M n,N,+ (s). We expect that β M n,N,+ (s) is well approximated by the n th power series coefficient for the finite product  Remark 6. In §6, for the computations in the proof of Theorem 1, we choose N = 600, M = 400, and Q = 28, using Proposition 3 to obtain the upper bound on |δ n (s)| for P + 1 = 26 ≤ n ≤ 28, having explicitly evaluated δ n (s) for 1 ≤ n ≤ 25 using periodic points of period up to P = 25.

The Hausdorff dimension of E 2
Here we consider the set E 2 , corresponding to the choice A = {1, 2}. We shall suppress the set A from our notation, writing L s instead of L A,s .
The approximation s N to dim(E 2 ), based on periodic points of period up to N , is the zero (in the interval (0, 1)) of the function D N defined by (4); these approximations are tabulated in Table 1 for 18 ≤ n ≤ 25. We note that the 24th and 25th approximations to dim(E 2 ) share the first 129 decimal digits It turns out that we can rigorously justify around three quarters of these decimal digits, proving that the first 100 digits are correct. In fact we prove slightly more than that, by setting s − to be the value Proof. We will show that D(s − ) and D(s + ) take opposite signs, and deduce that dim(E A ), as the zero of D, lies between s − and s + . Let so that ≈ 0.957589818521375342814351002388265920293251603461349541441037951859499 .
The relation (39) ensures that T 1 (c− ) and T 2 (c+ ) are equidistant from c, and this common distance is denoted by The specific choice of c is to ensure that the contraction ratio h = / is minimised, taking the value Having computed the points of period up to P = 25 we can form the functions s → δ n (s) for 1 ≤ n ≤ 25, and evaluate these at s = s − (cf. Table 2 We now aim to show that the approximation D 25 is close enough to D for (40) and (41) to imply, respectively, the negativity of D(s − ) and the positivity of D(s + ). In other words, we seek to bound the tail ∞ n=26 δ n (s), and this will be achieved by bounding the individual Taylor coefficients δ n (s), for n ≥ 26 = P + 1. It will turn out that for n ≥ 29 the cruder Euler bound on δ n (s) is sufficient, while for 26 ≤ n ≤ 28 we will use the Taylor bounds described in §5. More precisely, for P + 1 = 26 ≤ n ≤ 28 = Q we will use the upper computed Taylor bound 6 β M,+ n,N,+ (s) for suitable M, N ∈ N. Henceforth let Q = 28, M = 400, N = 600 (so that in particular Q ≤ M ≤ N , as was assumed throughout §5) and consider the case s = s − .
We first evaluate 7 the H 2 (D) norms of the monomial images L s (m k ) for 0 ≤ k ≤ N = 600. These norms are decreasing in k; Table 3 contains the first few evaluations, for 0 ≤ k ≤ 10, while for k = 600 we have These bounds are decreasing in n; Table 4 contains the first few evaluations, for 1 ≤ n ≤ 10, while for n = 400 we have (42) 7 As described in §4, (20) can be readily evaluated to arbitrary precision using numerical integration; for this particular computation the precision level used was 150 decimal places. 8 Note that h ≈ 0.511284 and N = 600, so h 2(N +1) Combining these bounds with the values taken by αn,N,+(s), it follows that for 1 ≤ n ≤ 400, the approximation bound αn(s) = ( ∞ k=n−1 Ls(m k ) 2 ) 1/2 agrees with both computed approximation bounds αn,N,−(s) and αn,N,+(s) to at least 200 decimal places, a level well beyond the desired precision used in these calculations. 9 The difference β M,+ n,N,+ (s)−β M,− n,N,+ (s) = n−1 l=0 J n−l Q,N,s β M,− l,N,+ (s) h M (n−l) E n−l (h) is smaller than 1.86×10 −210 for 26 ≤ n ≤ 28 = Q, so in fact the upper and lower computed Taylor bounds, and the Taylor bound β M n,N,+ (s), agree to well beyond the 150 decimal place precision used in these computations. 10 See also Table 6 for computations of β M,+ n,N,+ (s) for 1 ≤ n ≤ 28 = Q.
It remains to derive the Euler bounds on the Taylor coefficients δ n (s) for n ≥ 29. For s > 0, the functions w 1,s (z) = 1/(z + 1) 2s and w 2,s (z) = 1/(z + 2) 2s have maximum modulus on D when z = c − , so Combining (47) Combining (48) with (40) then gives It remains to show that D(s + ) is positive. In view of (41), for this it is sufficient to show that | ∞ n=26 δ n (s)| < 10 −101 for s = s + . In fact the stronger inequality (48) (which we have proved for s = s − ) can also be established for s = s + , using the same general method as for s = s − , since the intermediate computed values for the norms L s (m k ) , computed approximation bounds α n,N,+ (s), computed Taylor bounds β M,+ n,N,+ (s), and Euler bounds K n s E n (h), are sufficiently close to those for s = s − = s + − 2/10 101 . Combining (41) with inequality (48) for s = s + gives the required positivity The map s → D(s) is continuous and increasing, so the fact that D(s − ) < 0 < D(s + ) implies that its unique zero (which is equal to the dimension) is contained in (s − , s + ).
Remark 7. If, as in Theorem 1, our aim is to rigorously justify 100 decimal places of the computed approximation s P to the Hausdorff dimension, then roughly speaking P should be chosen so that the modulus of the tail ∞ n=P +1 δ n (s) can be shown to be somewhat smaller than 10 −100 for s ≈ s P . Since |δ n (s)| is bounded above by the upper computed Taylor bound β M,+ n,N,+ (s), the fact that β M,+ 26,N,+ (s) < 7.1 × 10 −103 (see Table 6) for suitably large M, N , together with the rapid decay (as a function of n) of these bounds, suggests that we may choose P = 25, i.e. that it suffices to explicitly locate the periodic points of period ≤ 25.
The choice of the value Q is relatively unimportant, as the upper computed Taylor bounds are only slightly more time consuming to compute than the (instantaneously computable) Euler bounds; in the proof of Theorem 1 we chose Q such that the Euler bounds K n s E n (h) were substantially smaller than 10 −100 for n > Q (our choice Q = 28 has this property, as does any larger Q, and indeed the choice Q = 27 may also be feasible, cf. Table 5).
The values M and N are chosen large enough to ensure that the bound (7) on |δ n (s)| is rendered essentially as sharp as possible using our method (see Proposition 1) of bounding approximation numbers by approximation bounds; equally, the values M and N are of course chosen small enough to allow the β M,+ n,N,+ (s) to be evaluated in reasonable time. Table 2. Exact (to the given precision) Taylor coefficients δ n (s) for the determinant det(I − zL s ) = 1 + ∞ n=1 δ n (s)z n for E 2 transfer operator L s with s = s − k L s (m k ) 0 1.0270790783376427840070677716704413443556765790531396305598028764891 1 0.3937848239109563523505359783093188356154137707117445532439663747781 2 0.1714591180108060752265529053281347472947978460219396035391070667691 3 0.0784792797693053045975192814445601433860119013766718128894674834037 4 0.0368985150737907248938351875080596507139356576758391651885254166051 5 0.0176517923866933707140642945427091399723431868286590018130953901715 6 0.0085477463829669713632455215487177327086334690252589671713112735110 7 0.0041762395195693491669377402131475622078401074275749884365926135321 8 0.0020541561464629266556123666395075007822413063382433235450055746854 9 0.0010155981305058227350650668511905652569101368771929481102954501965 10 0.0005041555520431887383182315523421205104649185947907910778866174462 Table 3.  Table 5. Euler bounds K n s E n (h) (on the n th Taylor coefficient of the determinant for the E 2 transfer operator L s ) with s = s −