COMPUTING THE DIMENSION OF DYNAMICALLY DEFINED

SETS : E_{2} AND BOUNDED CONTINUED FRACTIONS.

Oliver Jenkinson and Mark Pollicott

Institut de Mathématiques de Luminy and University of Manchester

Abstract. We present a powerful approach to computing the Hausdorff dimension of certain conformally self-similar sets. We illustrate this method for the dimension dim_{H}(E_{2}) of the set E_{2}, consisting of those real numbers whose continued fraction expansions contain only the digits 1 or 2. A very striking feature of this method is that the successive approximations converge to dim_{H}(E_{2}) at a super-exponential rate.

0. Introduction

Given any number 0 < x < 1 we can consider its continued fraction expansion

x = [a_{1}, a_{2}, a_{3}, …] = \frac{1}{a_{1} + _{\frac{1}{a_{2}+ \frac{1}{a_{3}+···}}}} .

where a_{1}, a_{2}, a_{3}, … ∈ ℕ. Consider the set E_{2} consisting of all points in the interval for which the continued fraction expansion has each a_{n} equal to either 1 or 2. This forms a Cantor set in the line, contained in the interval [\frac{1}{2}(\sqrt{3} − 1), \sqrt{3} − 1], of zero Lebesgue measure. Furthermore, it represents sets of numbers with certain diophantine approximatibility conditions [Bu1], [Bu2] and its Hausdorff dimension has other number theoretic significance in terms of the Markoff and Lagrange spectra (see §5, [Cu1], [Cu2], [CuF]).

A number of authors have considered the problem of calculating the Hausdorff dimension dim_{H}(E_{2}) of the set E_{2}. In 1941, Good showed that 0.5194 ≤ dim_{H}(E_{2}) ≤ 0.5433 [Good]. In 1982, Bumby [Bu1], [Bu2] improved these estimates using a thermodynamic approach, showing that 0.5312 ≤ dim_{H}(E_{2}) ≤ 0.5314. However, it appears his method is slowly convergent [Bu2], p. 2. Subsequently, using a modified approach and superior computing power, in 1989 Hensley [Hen1] showed that 0.53128049 ≤ dim_{H}(E_{2}) ≤ 0.53128051. In 1996, he improved this estimate to 0.5312805062772051416 (see [Hen2]). In this note, we shall present an efficient approach to calculating the value of dim_{H}(E_{2}).

More generally, our method is applicable to the computation of the Hausdorff dimension of other dynamically defined sets. One such example arises from Schottky groups. Consider P SL(2, ℂ) acting on ℂ by linear fractional transformations.

Typeset by AMS-TEX

1991 Mathematics Subject Classification. 58F20, 58F11, 11J70, 11J06.

A Schottky group Γ =< γ_{1}, …, γ_{n} > is a free group contained in P SL(2, ℂ) for which, in particular, the sets D(γ_{i}) = {z ∈ ℂ : |γ′_{i}(z)| ≤ 1} are disjoint (cf. [Bowen] and [P-S]). The limit set Λ ⊂ ℂ is the set of accumulation points of the sequence γ_{i_{1}} … γ_{i_{k}}(0), say. The value d = dim_{H}Λ has various important interpretations relating to the corresponding surface of negative curvature. For example it equals the topological entropy of the geodesic flow restricted to its recurrent set, and is elegantly related to the smallest eigenvalue of the Laplacian [Sul]. These applications are beyond the scope of the present paper and we shall return to the analysis of these, and other, examples in a later note. For the present, we shall concentrate on establishing our method in the concrete case of computing the dimension of E_{2}, and closely related sets.

The approach we use is motivated by related calculations for quantities arising in the study of the linearized Feigenbaum renormalization operator [A-A-C], [Pol] and the functional determinant of the Laplacian [Pol-R]. A common feature is an explicit expression as a power series, whose coefficients can readily be computed in terms of closed orbits for an underlying dynamical system and which tend to zero very rapidly, allowing very good approximation by only a few terms.

1. Hausdorff dimension and periodic orbits

We briefly recall the definition of Hausdorff dimension.

Definition. Given a subset E ⊂ ℝ^{n} we can consider a cover U = {U_{i}}_{i} for E by open sets. For δ > 0 we can define H_{ϵ}^{δ}(E) = inf_{U}{∑_{i} diam(U_{i})^{δ}} where the infimum is taken over all open covers U = {U_{i}} such that diam(U_{i}) ≤ ϵ. We define H^{δ}(E) = lim_{ϵ→0} H_{ϵ}^{δ}(E) and, finally,

dim_{H}(E) = inf{δ : H^{δ}(E) = 0}.

For conformal systems, such as those we are considering, the Hausdorff dimension is equal to the box dimension (see [Fal]).

The Cantor set E_{2} is invariant under the continued fraction transformation T : [0, 1] → [0, 1] defined by

T (x) = \{ \begin{matrix}1/x & if x ≠ 0 \\ 0 & if x = 0. \end{matrix}

Observe that in this case T′(x) = −1/x^{2}.

In fact E_{2} can be generated using the two contracting maps T_{1}, T_{2} : [0, 1] → [0, 1] defined by

T_{1}(x) = \frac{1}{1 + x} and T_{2}(x) = \frac{1}{2 + x}

More precisely, we can write E_{2} as the set of limit points of sequences T_{i_{1}} … T_{i_{n}}(0), say, for i_{1}, i_{2}, … ∈ {1, 2}. Note that T_{1}, T_{2} both reverse orientation. The nonlinearity of T_{1}, T_{2} means that E_{2} is not exactly self-affine (unlike the classical ‘middle third’ Cantor set, for example) so its Hausdorff dimension cannot be computed using simple self-similarity arguments. However it does have a form of asymptotic self-similarity, which gives us a formula for dim_{H}(E_{2}) in terms of pressure.

COMPUTING THE DIMENSION OF E_{2} 3

Definition. Given any continuous function f : E_{2} → ℝ we define its pressure P (f) (with respect to T) as

P (f) := lim_{n→+∞} \frac{1}{n} log ( ∑_{{T^{n}x=x \atop x∈E_{2}}} e^{f(x)+f(T x)+...+f(T^{n−1}x)})

The particular choice f(x) = −s log |T′(x)| (restricted to E_{2}) gives the following result.

Lemma 1. There is a unique solution 0 ≤ s ≤ 1 to P (−s log |T′|) = 0, which occurs precisely at s = dim_{H}(E_{2}). [Falconer], [Bed]

Formulae of this general type have a long history. Bowen [Bowen] showed that the dimension of quasi-circles could be studied using thermodynamic ideas. Subsequently, Ruelle [Ru2] applied the approach to hyperbolic rational maps and McCluskey-Manning [Mc-M] studied Smale horseshoes. Our example is covered by the standard theory of “cookie-cutters” (see for example Bedford [Bed]).

Given a finite string \underline{i} = i_{1} … i_{n} ∈ \{1, 2\}^{n} (for simplicity we do not put commas between the i_{j}) we write |\underline{i}| = n. The unique fixed point for the contraction

\{ \begin{matrix}T\underline{i} := T_{i_{1}} ⋯ T_{i_{n}} : [0, 1] → [0, 1] \\ T\underline{i} : z ↦ _{\frac{1}{i_{1}+_{\frac{1}{i_{2}+... \frac{1}{in+z}}}}} \end{matrix}

is the point x\underline{i} with corresponding periodic continued fraction expansion [\overline{i_{1}, …, i_{n}}]. This number is a quadratic surd, i.e. a non-rational root of a quadratic equation with integer entries. The points in the periodic orbit \{x\underline{i}, T x\underline{i}, …, T^{n−1}x\underline{i}\} correspond to quadratic surds with periodic continued fractions [\overline{i_{k+1}, …, i_{n}, i_{1}, …, i_{k}}], for k = 1, …, n.

To compute P (−s log |T′|) we must sum terms of the form (|(T^{|\underline{i}|})′(x\underline{i})|)^{−s}. Since T′(x) = −1/x^{2} then these terms have the following simple form

(|(T^{|\underline{i}|})′(x\underline{i})|)^{−s} = ∏_{r=0}^{|\underline{i}|−1} \frac{1}{T′(T^{r}x\underline{i})^{s}} = ∏_{r=0}^{|\underline{i}|−1}[T^{r}x\underline{i}]^{2s}

We therefore introduce the notation w\underline{i} = ∏_{r=0}^{|\underline{i}|−1} T^{r}x\underline{i}, the product of all terms in the orbit of x\underline{i}. Note that we consider this orbit with multiplicity (i.e. we take the product over |\underline{i}| iterates, even if this involves repeats).

Example. The fixed points of T_{1} and T_{2} are \frac{1}{2}(\sqrt{5} − 1) and \sqrt{2} − 1, respectively, while the string 12 corresponds to the periodic orbit \{\sqrt{3} − 1, \frac{1}{2} (\sqrt{3} − 1)\}. Then,

for example, w_{22} = (\sqrt{2} − 1)^{2} = 3 − 2\sqrt{2}.

We can therefore express pressure as

P (−s log |T′|) = lim_{n→∞} \frac{1}{n} log ( ∑_{|\underline{i}|=n} w2s ; \underline{i}) .

2. The main result

In §3 we will introduce a family of linear operators L_{s}, and functional determinants det(I − zL_{s}) with power series expansion

det(I − zL_{s}) = 1 + ∑_{n=1}^{∞} d_{n}(s)z^{n},

where the power series coefficients will be given by

d_{n}(s) = ∑_{n_{1}+...+n_{r}=n} \frac{(−1)^{r}}{r!} ∏_{j=1}^{r} ( ∑_{|\underline{i}|=n_{j}} \frac{w2s ; \underline{i}}{1 − (−1)^{|\underline{i}|}w2 ; \underline{i}} ) ,  (2.1)

and the summation is over all r-tuples (n_{1}, …, n_{r}) of positive integers satisfying n_{1} + … + n_{r} = n.

Setting D(s, z) = det(I − zL_{s})det(I + zL_{s}) = det(I − z^{2}L_{s}^{2}) we have

D(s, z) = 1 + ∑_{n=1}^{∞} b_{n}(s)z^{n}

where

b_{n}(s) = ∑_{k=1}^{n−1}(−1)^{k}d_{k}(s)d_{n−k}(s).  (2.2)

Note that the odd terms b_{2k+1}(s) vanish, while the even terms b_{2k}(s) are given by

b_{2k}(s) = ∑_{n_{1}+...+n_{r}=k} \frac{(−1)^{r}}{r!} ∏_{j=1}^{r} ( ∑_{|\underline{i}|=2n_{j}} \frac{w2s ; \underline{i}}{1 − (−1)^{|\underline{i}|}w2 ; \underline{i}} ) .  (2.3)

Our main result in this paper is the following.

Theorem 1.

(a) The Hausdorff dimension of E_{2} is the largest solution 0 ≤ s ≤ 1 to the equation

D(s, 1) = 1 + ∑_{n=1}^{∞} b_{n}(s) = 0.

(b) There exist 0 < θ < 1 and K > 0 such that if s_{N} denotes the largest solution 0 ≤ s_{N} ≤ 1 to the equation

1 + ∑_{n=1}^{N} b_{n}(s) = 0,

then |s_{N} − dim_{H}(E_{2})| < Kθ^{N^{2}}.

In particular, part (b) of Theorem 1 shows we have a very efficient method of computing dim_{H}(E_{2}). For example, letting N = 16 and working with the eight terms b_{2}(s), b_{4}(s), …, b_{16}(s) we get a solution s_{16} described in the following result.

Corollary 1.1. The approximation

dim_{H}(E_{2}) ≈ s_{16} = 0.5312805062772051416244686 …

is accurate to 25 decimal places.

Scheme of the proof of Theorem 1. For part (a) our method will be to introduce a family of transfer operators L_{s}, whose spectral radius is e^{P (−s log |T′|)}. Since (−L_{s})^{2} = L_{s}^{2}, then −L_{s} also has spectral radius e^{P (−s log |T′|)}, by the spectral radius formula (see [R-S], p. 192).

By Lemma 1, the Hausdorff dimension of E_{2} is the unique real value of s for which L_{s} has spectral radius equal to 1. The operators L_{s} will be of Perron-Frobenius type (see Lemma 2), so that in fact e^{P (−s log |T′|)} is itself an eigenvalue of L_{s}. In particular for s = dim_{H}(E_{2}), the value 1 will be the leading eigenvalue of L_{s}.

In §3 we will show that when restricted to a suitable Banach space of bounded holomorphic functions, the eigenvalues of L_{s} are precisely the reciprocals of the zeros z of det(I − zL_{s}). The zeros z of D(s, z) = det(I − zL_{s})det(I + zL_{s}) are the reciprocals of the union of the eigenvalues of L_{s} and −L_{s}.

Therefore setting z = 1, we see that s will be a solution of

D(s, 1) = det(I − L_{s})det(I + L_{s}) = det(I − L_{s}^{2}) = 0

if and only if 1 is an eigenvalue of either L_{s} or −L_{s}. By the above discussion we know that s = dim_{H}(E_{2}) is one such solution. In fact it is the largest solution, since for s > dim_{H}(E_{2}), the spectral radius of both L_{s} and −L_{s} is less than 1, since P (−s log |T′|) is decreasing in s, so that 1 cannot be an eigenvalue of L_{s} or −L_{s}. In §3 we will prove the formulae (2.1), (2.3) for the power series coefficients of det(I − zL_{s}) and D(s, z).

Part (b) will follow from the fact that for any fixed 0 ≤ s ≤ 1 the terms b_{n}(s) in the power series expansion for det(I − z^{2}L_{s}^{2}) can be bounded in modulus by cθ^{n^{2}}, where both 0 < θ < 1 and c > 0 can be chosen independent of s. Section 3 will be devoted to a proof of this fact. Having proved the decay rate for the b_{n}(s) we then use the following argument. Let F (s) = 1 + ∑_{n=1}^{∞} b_{n}(s) and F_{N}(s) = 1 + ∑_{n=1}^{N} b_{n}(s). Let s_{∞} = dim_{H}(E_{2}). By the mean value theorem there exists t_{N} between s_{N} and s_{∞} such that

\begin{aligned}(s_{N} − s_{∞})F′_{N}(t_{N}) & = F_{N}(s_{N}) − F_{N}(s_{∞}) = −F_{N}(s_{∞}) \\ & = F (s_{∞}) − F_{N}(s_{∞}) = ∑_{n=N+1}^{∞} b_{n}(s_{∞}), \end{aligned}

so that

|s_{N} − s_{∞}| ≤ |\frac{1}{F′_{N}(t_{N})} ∑_{n=N+1}^{∞} b_{n}(s)| ≤ \frac{1}{|F′_{N}(t_{N})|} ∑_{n=N+1}^{∞} cθ^{n^{2}}

= \frac{cθ^{N^{2}}}{|F′_{N}(t_{N})|} ∑_{r=1}^{∞} θ^{2Nr+r^{2}} ≤ \frac{cθ^{N^{2}+1}}{|F′_{N}(t_{N})|} ∑_{r=1}^{∞} θ^{2Nr}

= \frac{cθ^{N^{2}+1}}{|F′_{N}(t_{N})|} \frac{θ^{2N}}{1 − θ^{2N}} ≤ \frac{cθ^{(N+1)^{2}}}{|F′_{N}(t_{N})|(1 − θ)}

Remark. In fact dim_{H}(E_{2}) is also the largest zero s of det(I − L_{s}), by the above discussion, so that Theorem 1 is also true if we replace D(s, 1) by det(I −L_{s}). Note, however, that the odd terms d_{2k+1}(s) in the power series expansion for det(I −zL_{s}) do not vanish. We obtain superior computational results (see §4) by working with D(s, 1).

3. Proof of the Main Result

Our main technical device will be to re-write D(s, z) in terms of the trace of a certain operator L_{s} defined below. A suitable choice of domain ensures L_{s} has good spectral properties, which implies the estimates on the decay rates of b_{n}(s). Following Mayer [May1], we can fix the disk

D = \{z ∈ ℂ : |z − 1| < \frac{3}{2}\} .

If we consider the family of analytic functions A on D which have a continuous extension to the closed disk, then this forms a Banach space with respect to the uniform norm ||f|| = sup_{x∈D} |f(x)|.

We can define the transfer operator L_{s} : A → A (compare with [May1], [Bu1], [Bu2]) by

L_{s}v(z) = (\frac{1}{z + 1})^{2s} v (\frac{1}{z + 1}) + (\frac{1}{z + 2})^{2s} v (\frac{1}{z + 2})

To interpret this definition appropriately we should write

(\frac{1}{z + 1})^{2s} = exp (2s log (\frac{1}{z + 1})) and

(\frac{1}{z + 2})^{2s} = exp (2s log (\frac{1}{z + 2}))

and observe that for z ∈ D and 0 ≤ s ≤ 1 the logarithm is well-defined.

Note that the functions (z + 1)^{−2s} and (z + 2)^{−2s} are just the holomorphic extensions of the restrictions of |T′_{1}(z)|^{s} = |T′(T_{1}z)|^{−s} and |T′_{2}(z)|^{s} = |T′(T_{2}z)|^{−s} to the unit interval. Thus the operator L_{s} is just the Ruelle-Perron-Frobenius operator with weight function −s log |T′|, and we have (see [Pa-Pol]):

Lemma 2. The value e^{P (−s log |T′|)} is the unique eigenvalue of maximum modulus for the operator L_{s}.

In fact we can go beyond the Perron-Frobenius theory, and derive additional spectral information about L_{s}. We recall that a nuclear operator L : A → A is a linear operator for which there exist functions f_{n} ∈ A with ||f_{n}|| = 1, functionals l_{n} ∈ A^{∗} with ||l_{n}|| = 1, and coefficients λ_{n} ∈ ℂ with ∑_{n=1}^{∞} |λ_{n}| < ∞, such that

Lv = ∑_{n=1}^{∞} (λ_{n}l_{n}(v)) f_{n}.  (3.1)

This definition is due to Grothendieck [Gr]. In particular a nuclear operator is compact. We will show that the transfer operator L_{s} : A → A is nuclear. Our analysis follows closely that of Mayer in [May1].

COMPUTING THE DIMENSION OF E_{2} 7

Figure 1

First observe that the closure of the image disks

T_{1}(D) = {z : |z − 8/7| < 6/7} and T_{2}(D) = {z : |z − 4/9| < 2/9} are still within the original disk D (see Figure 1).

Consider the curve

Γ = {z ∈ ℂ : |z − 10/9| = 1}.

Note that Γ does not intersect T_{1}(D) or T_{2}(D), and ξ^{2s} is well-defined and analytic in the region enclosed by Γ. Using Cauchy’s theorem to write (T_{j}z)^{2s}v(T_{j}z) = \frac{1}{2πi} ∫_{Γ} \frac{ξ^{2s}v(ξ)}{ξ−T_{j}z}dξ for j = 1, 2, we have

\begin{aligned}L_{s}v(z) & = \frac{1}{2πi} ∫_{Γ} ξ^{2s}v(ξ) (\frac{1}{ξ − T_{1}z} + \frac{1}{ξ − T_{2}z} ) dξ \\ & = ∑_{n=0}^{∞} ((T_{1}z − \frac{10}{9})^{n} + (T_{2}z − \frac{10}{9})^{n}) (\frac{1}{2πi} ∫_{Γ} \frac{ξ^{2s}v(ξ)}{(ξ − \frac{10}{9})^{n+1}} dξ) , \end{aligned} (3.2)

where for the last line we have just used the trivial summation of the geometric series

∑_{n=0}^{∞} (\frac{T_{j}z − \frac{10}{9}}{ξ − \frac{10}{9}} )^{n} = (1 − \frac{T_{j}z − \frac{10}{9}}{ξ − \frac{10}{9}} )^{−1} .

The expansion in (3.2) is valid since Γ was chosen to be outside the two image discs T_{1}(D), T_{2}(D) (see Figure 1), so |(T_{j}z − \frac{10}{9})/(ξ − \frac{10}{9})| < 1 for z ∈ D, ξ ∈ Γ.

Now let us write

g_{j,n}(z) := (T_{j}z − \frac{10}{9})^{n} and m_{n}(v) := \frac{1}{2πi} ∫_{Γ} \frac{ξ^{2s}v(ξ)}{(ξ − \frac{10}{9})^{n+1}} dξ.

Note the functions g_{j,n} belong to A, and the functionals m_{n} are linear in v. Introducing the normalisations

f_{2n+j} = g_{j,n}/||g_{j,n}||  ,  l_{2n+j} = m_{n}/||m_{n}||  (3.3)

for n ≥ 0 and j = 1, 2, we can express L_{s} as

L_{s}v(z) = ∑_{n=1}^{∞} (λ_{n}l_{n}(v)) f_{n}(z),

where λ_{2n+j} = ||g_{j,n}||.||m_{n}||. Moreover, we have the following exponential bound for the coefficients λ_{n}.

Lemma 3. λ_{n} ≤ \frac{45}{16π} γ^{n} where γ = \sqrt{\frac{8}{9}}.

Proof. Now λ_{2n+j} = ||g_{j,n}||.||m_{n}||, so we will estimate the norms ||g_{j,n}|| and ||m_{n}||. For ξ ∈ Γ we have |ξ − \frac{10}{9}| = 1, which also means we can bound |ξ^{2s}| ≤ |\frac{19}{9}|^{2s} < 5 for 0 ≤ s ≤ 1. Thus ||m_{n}|| ≤ \frac{5}{2π}. For z ∈ D we have |T_{1}z − \frac{10}{9}|, |T_{2}z − \frac{10}{9}| ≤ \frac{8}{9} so that ||g_{j,n}|| ≤ (\frac{8}{9})^{n}. Thus λ_{2n+j} ≤ \frac{5}{2π} (\frac{8}{9})^{n} = \frac{45}{16π} γ^{2n+2}, for j = 1, 2, and the result follows.

In particular this means ∑ λ_{n} < ∞, so we have proved

Corollary 3.1. The transfer operator L_{s} : A → A is nuclear.

The immediate benefit of Corollary 3.1 is that nuclear operators are trace class. That is, we can define the trace tr(L_{s}) to be the sum of its eigenvalues, counted with multiplicities. Following [Gr], [Ru1], [May1] and [Fried] we then define the Fredholm determinant of (I − zL_{s}) : A → A by

det(I − zL_{s}) = exp (− ∑_{n=1}^{∞} \frac{z^{n}}{n} tr(L_{s}^{n}))  (3.4)

for z ∈ ℂ.

Nuclear operators enjoy many of the properties of finite dimensional matrices. In particular we have

Corollary 3.2. [Gr] The zeros z of det(I − zL_{s}), counted with multiplicities, are precisely the eigenvalues of L_{s}, again counted with (algebraic) multiplicities.

We have the following trace formula.

Lemma 4.

tr(L_{s}^{n}) = ∑_{|\underline{i}|=n} \frac{w2s ; \underline{i}}{1 − (−1)^{|\underline{i}|}w2 ; \underline{i}} .  (3.5)

Proof. Given any string \underline{i} = i_{1} … i_{n}, recall that T\underline{i} = T_{i_{1}} ⋯ T_{i_{n}} denotes the associated local inverse to T^{n}, and x\underline{i} denotes the fixed point of T\underline{i}. We have that

T ′ ; \underline{i}(x\underline{i}) = (−1)^{|\underline{i}|}w2 ; \underline{i}.

Define a linear operator L\underline{i} : A → A by

L\underline{i}v(z) = (T\underline{i}z)^{2s}v(T\underline{i}z).

This allows us to write L_{s}^{n} = ∑_{|\underline{i}|=n} L\underline{i} and since the trace is additive we have tr(L_{s}^{n}) = ∑_{|\underline{i}|=n} tr(L\underline{i}), so it will suffice to compute tr(L\underline{i}) for all |\underline{i}| = n. Now if α is an eigenvalue of L\underline{i}, then the eigenvalue equation L\underline{i}v(z) = αv(z) evaluated at the point z = x\underline{i} gives us w2s ; \underline{i}v(x\underline{i}) = αv(x\underline{i}). So provided the eigenvector v does not vanish at x\underline{i}, then α = w2s ; \underline{i}. If however v(x\underline{i}) = 0, then differentiating the eigenvalue equation and evaluating at the point z = x\underline{i} gives

(−1)^{|\underline{i}|}w2 ; \underline{i}w2s ; \underline{i}v′(x\underline{i}) = αv′(x\underline{i}),

so that if v′(x\underline{i}) ≠ 0 then α = (−1)^{|\underline{i}|}w2 ; \underline{i}w2s ; \underline{i}.

Repeating this argument shows that any eigenvalue of L\underline{i} must be of the form (−1)^{k|\underline{i}|}w2(s+k) ; \underline{i}, for some integer k ≥ 0. In fact all of these values are indeed eigenvalues (see [Kam], Lemma 2), and each is of algebraic multiplicity one (see [May2], Lemma 7.10), so we have

tr(L\underline{i}) = ∑_{k=0}^{∞}(−1)^{k|\underline{i}|}w2(s+k) ; \underline{i} = \frac{w2s ; \underline{i}}{1 − (−1)^{|\underline{i}|}w2 ; \underline{i}} ,

and we deduce that

tr(L_{s}^{n}) = ∑_{|\underline{i}|=n} tr(L\underline{i}) = ∑_{|\underline{i}|=n} \frac{w2s ; \underline{i}}{1 − (−1)^{|\underline{i}|}w2 ; \underline{i}} ,

completing the proof of the lemma.

Substituting the trace formula (3.5) into the expression (3.4), and using the power series expansion for the exponential function, we calculate the power series coefficients d_{n}(s) of det(I − zL_{s}) to be as in (2.1). The power series coefficients b_{n}(s) for D(s, z) are then given by (2.2), (2.3), and the scheme of proof outlined in §2 completes the proof of Theorem 1 (a).

It remains to prove part (b) of Theorem 1. By the outline proof of Theorem 1 given in section 2, we know it suffices to show the coefficients b_{n}(s) in the power series expansion for D(s, z) decay at a rate O(θ^{n^{2}}), for some 0 < θ < 1. The first step towards this is the formal expression (see [Gr]) for the Fredholm determinant,

det(I − zL_{s}) = | \begin{matrix}1 − zλ_{1}l_{1}(f_{1}) & −zλ_{1}l_{1}(f_{2}) & −zλ_{1}l_{1}(f_{3}) & … \\ −zλ_{2}l_{2}(f_{1}) & 1 − zλ_{2}l_{2}(f_{2}) & −zλ_{2}l_{2}(f_{3}) & … \\ −zλ_{3}l_{3}(f_{1}) & −zλ_{3}l_{3}(f_{2}) & 1 − zλ_{3}l_{3}(f_{3}) & … \\ ⋮ & ⋮ & ⋮ & ... \end{matrix} | ,

which leads to the following alternative characterisation of its power series coefficients.

Lemma 5. [Gr], [Fried, p. 506] We can write det(I − zL_{s}) = 1 + ∑_{n=1}^{∞} d_{n}(s)z^{n} where

d_{n}(s) = ∑_{i_{1}<...<i_{n}} λ_{i_{1}} … λ_{i_{n}}Det (l_{i_{j}}(f_{i_{k}}))_{j,k=1}^{n}  (3.8)

and Det [l_{i_{j}}(f_{i_{k}})]_{j,k=1}^{n} denotes the determinant of the n × n matrix with entries l_{i_{j}}(f_{i_{k}}), 1 ≤ j, k ≤ n, where the functionals l_{i_{j}} and functions f_{i_{k}} are as in (3.3).

Proof of Theorem 1 (b).

To estimate the speed at which the terms b_{n}(s) tend to zero we will use expression (3.8) to prove the analogous fact about the coefficients d_{n}(s). We first recall Hadamard’s theorem [Had] that an n × n matrix with entries of modulus at most 1 has determinant with modulus bounded by n^{n/2}.

We next note that for any 0 < γ < 1 we have

∑_{k_{1}<...<k_{n}} γ^{k_{1}+...+k_{n}} = \frac{γ^{n(n+1)/2}}{(1 − γ)(1 − γ^{2}) … (1 − γ^{n})}.  (3.9)

To see this we just compute

∑_{k_{1}<...<k_{n}} γ^{k_{1}+...+k_{n}}

∑_{k_{1}=0}^{∞} ∑_{k_{2}=k_{1}+1}^{∞} ⋯ ∑_{k_{n}=k_{n−1}+1}^{∞} γ^{k_{1}+...+k_{n}}

= ∑_{k_{1}=0}^{∞} ∑_{k_{2}=k_{1}+1}^{∞} ⋯ ∑_{k_{n−1}=k_{n−2}+1}^{∞} γ^{k_{1}+...+k_{n−2}+2k_{n−1}} (\frac{γ}{1 − γ} )

= ∑_{k_{1}=0}^{∞} ∑_{k_{2}=k_{1}}^{∞} ⋯ ∑_{k_{n−2}=k_{n−3}+1}^{∞} γ^{k_{1}+...+k_{n−3}+3k_{n−2}} (\frac{γ^{2+1}}{(1 − γ)(1 − γ^{2})})

= ⋯

= \frac{γ^{n+...+2+1}}{(1 − γ) ⋯ (1 − γ^{n})}

from which (3.9) immediately follows.

In particular, if we choose γ = \sqrt{\frac{8}{9}} as in Lemma 3 then we know λ_{i_{j}} ≤ \frac{45}{16π} γ^{i_{j}+1}. So by (3.9) we can now estimate

\begin{aligned}|d_{n}(s)| & ≤ (\frac{45}{16π} )^{n} \frac{γ^{1+n(n+1)/2}}{(1 − γ)(1 − γ^{2}) … (1 − γ^{n})}n^{n/2} \\ & ≤ C (\frac{45}{16π} )^{n} (\frac{8}{9})^{n(n+1)/4} n^{n/2}, \end{aligned}

where C = γ ∏_{r=1}^{∞}(1 − γ^{r})^{−1} > 0.

In particular we see that the coefficients d_{n}(s) = O(θ^{n^{2}}), where θ = \sqrt{γ} = (\frac{8}{9})^{1/4}.

The same is true for the coefficients b_{n}(s) of the determinant D(s, z), by the following argument. Writing

L_{s}^{2}v = L_{s}(L_{s}v) = ∑_{n=1}^{∞} λ_{n}l_{n} ( ∑_{m=1}^{∞} λ_{m}l_{m}(v)f_{m}) f_{n} = ∑_{n=1}^{∞} (μ_{n}l_{n}(v)) f_{n}, say,

we see that

|μ_{n}| ≤ λ_{n} ∑_{r=1}^{∞} λ_{r} ≤ \frac{45γ^{n+1}}{16π(1 − γ)},

and applying the same estimates as above gives that b_{n}(s) = O(θ^{n^{2}}), as required. This completes the proof of Theorem 1 (b).

4. Some calculations

We now present some details of our calculation of dim_{H}(E_{2}). Of course there is nothing special about the digits 1 and 2 in the definition of E_{2}. More generally we could restict the continued fraction coefficients to any finite alphabet {n_{1}, …, n_{r}}, and consider the resulting Cantor set E_{{n_{1},... ,n_{r}}}. With the appropriate contraction maps T_{i}(x) = 1/(n_{i} + x) we would generate the corresponding periodic orbits, define the Fredholm determinant accordingly, and compute its zeros to obtain the Hausdorff dimension of E_{{n_{1},... ,n_{r}}}. This method works equally well if we further forbid a finite number of strings from our finite alphabet (so that the underlying dynamics is a subshift of finite type), see [Pa-Pol].

The fundamental objects in our calculations are the strings \underline{i} = i_{1} … i_{n} and their corresponding weights w\underline{i}. An elementary yet crucial property of these weights is that they are multiplicative. That is, if \underline{i}′ = (\underline{i})^{r} is the r-fold concatenation of the string \underline{i} then w\underline{i}′ = wr ; \underline{i}.

To illustrate the speed at which the weights w\underline{i} tend to zero as |\underline{i}| grows we list the strings up to period five in Table 1. We only give one string per cyclic permutation class, and do not list those which are concatenation powers of smaller strings (since the weights of these can be computed using the multiplicative property).

We now explicitly compute the first few terms of D(s, z). We have

D(s, z) =det(I − zL_{s})det(I + zL_{s})
= exp (− ∑_{n=1}^{∞} \frac{z^{n}}{n} ∑_{|\underline{i}|=n} (\frac{w2s ; \underline{i}}{1 − (−1)^{|\underline{i}|}w2 ; \underline{i}} + \frac{(−1)^{n}w2s ; \underline{i}}{1 − (−1)^{|\underline{i}|}w2 ; \underline{i}} ))

= exp (− ∑_{n=1}^{∞} C_{n}(s)z^{n})

where

C_{n}(s) = \{ \begin{matrix}0 & if n is odd \\ \frac{2}{n} ∑|\underline{i}|=n \frac{w2s ; \underline{i}}{1−w2 ; \underline{i}} & if n is even. \end{matrix}

We list below the first three non-zero coefficients C_{n}(s).

C_{2}(s) = 2.154701 (0.267949)^{2s} + 1.030330 (0.414214)^{4s} + 1.170820 (0.618034)^{4s}

12 OLIVER JENKINSON AND MARK POLLICOTT

\underline{i}  Orbit  w\underline{i}
1  \{\frac{1}{2}(\sqrt{5} − 1)\}  \frac{1}{2}(\sqrt{5} − 1)
2  \{\sqrt{2} − 1\}  \sqrt{2} − 1
12  \{\sqrt{3} − 1, \frac{1}{2}(\sqrt{3} − 1)\}  2 − \sqrt{3}
112  \{\frac{1}{2}(\sqrt{10} − 2), \frac{1}{3}(\sqrt{10} − 1), \frac{1}{3}(\sqrt{10} − 2)\}  \sqrt{10} − 3
122  \{\frac{1}{6}(\sqrt{85} − 5), \frac{1}{10}(\sqrt{85} − 5), \frac{1}{6}(\sqrt{85} − 7)\}  \frac{1}{2}(\sqrt{85} − 9)
1112  \{\frac{1}{3}(2\sqrt{6} − 3), \frac{2}{5}(\sqrt{6} − 1), \frac{1}{2}(\sqrt{6} − 1), \frac{1}{5}(2\sqrt{6} − 3)\}  5 − 2\sqrt{6}
1122  \{\frac{1}{10}(\sqrt{221} − 9), \frac{1}{14}(\sqrt{221} − 5), \frac{1}{14}(\sqrt{221} − 9), \frac{1}{10}(\sqrt{221} − 11)\}  \frac{1}{2}(15 − \sqrt{221})
1222  \{\frac{2}{7}(\sqrt{30} − 3), \frac{1}{6}(\sqrt{30} − 3), \frac{2}{7}(\sqrt{30} − 4), \frac{1}{4}(\sqrt{30} − 4)\}  11 − 2\sqrt{30}
11112  \{\frac{1}{5}(\sqrt{65} − 5), \frac{1}{8}(\sqrt{65} − 3),
\frac{1}{7}(\sqrt{65} − 4), \frac{1}{7}(\sqrt{65} − 3), \frac{1}{8}(\sqrt{65} − 5)\}  \sqrt{65} − 8
11122  \{\frac{1}{8}(\sqrt{145} − 7), \frac{1}{12}(\sqrt{145} − 5), \frac{1}{10}(\sqrt{145} − 5),
\frac{1}{12}(\sqrt{145} − 7), \frac{1}{8}(\sqrt{145} − 9)\}  \sqrt{145} − 12
11212  \{\frac{1}{14}(\sqrt{533} − 15), \frac{1}{22}(\sqrt{533} − 7), \frac{1}{22}(\sqrt{533} − 15),
\frac{1}{14}(\sqrt{533} − 13), \frac{1}{26}(\sqrt{533} − 13)\}  \frac{1}{2}(\sqrt{533} − 23)
11222  \{\frac{1}{12}(5\sqrt{13} − 11), \frac{1}{17}(5\sqrt{13} − 6), \frac{1}{17}(5\sqrt{13} − 11),
\frac{1}{12}(5\sqrt{13} − 13), \frac{1}{13}(5\sqrt{13} − 13)\}  5\sqrt{13} − 18
12122  \{\frac{1}{11}(\sqrt{290} − 9), \frac{1}{19}(\sqrt{290} − 10), \frac{1}{10}(\sqrt{290} − 10),
\frac{1}{19}(\sqrt{290} − 9), \frac{1}{11}(\sqrt{290} − 13)\}  \sqrt{290} − 13
12222  \{\frac{1}{34}(\sqrt{2813} − 29), \frac{1}{58}(\sqrt{2813} − 29), \frac{1}{34}(\sqrt{2813} − 39),
\frac{1}{38}(\sqrt{2813} − 37), \frac{1}{38} (\sqrt{2813} − 39)\}  \frac{1}{2}(\sqrt{2813} − 53)

Table 1. Weights associated to strings

C_{4}(s) = \frac{1}{2} [4.008316 (0.0455488)^{2s} + 4.018018 (0.0669656)^{2s}

+ 4.041241 (0.101021)^{2s} + 2.010363 (0.267949)^{4s} + 1.000867 (0.414214)^{8s} + 1.021749 (0.618034)^{8s}] ���

COMPUTING THE DIMENSION OF E_{2} 13

C_{6}(s) = \frac{1}{3} [3.000436 (0.109772)^{4s} + 3.002082 (0.162278)^{4s} + 2.00074 (0.267949)^{6s}

+ 1.000026 (0.414214)^{12s} + 1.003115 (0.618034)^{12s}

+ 6 (1.000061 (0.00781298)^{2s} + 1.000132 (0.0114958)^{2s}

+ 1.000149 (0.0121969)^{2s} + 1.000298 (0.0172465)^{2s}

+ 2.000638 (0.0178628)^{2s} + 1.000659 (0.0256579)^{2s}

+ 1.000732 (0.0270468)^{2s} + 1.001486 (0.0385186)^{2s})]

We now use the series expansion of the exponential function to compute the coefficients b_{n}(s) in the power series det(I − z^{2}L_{s}^{2}) = 1 + ∑_{n=1}^{∞} b_{n}(s)z^{n}.

If n is odd then b_{n}(s) is zero, since all the odd terms C_{i}(s) are zero. If n is even then we can express b_{n}(s) in terms of C_{i}(s), i ≤ n. Writing C_{i} = C_{i}(s) the first few non-zero coefficients b_{n}(s) are as follows:

b_{2}(s) = −C_{2}
b_{4}(s) = −C_{4} + \frac{1}{2}C_{2}^{2}
b_{6}(s) = −C_{6} + C_{2}C_{4} − \frac{1}{6}C_{2}^{3}
b_{8}(s) = −C_{8} + \frac{1}{2}C_{4}^{2} + C_{2}C_{6} − \frac{1}{2}C_{2}^{2}C_{4} + \frac{1}{24}C_{2}^{4}
b_{10}(s) = −C_{10} + C_{2}C_{8} + C_{4}C_{6} − \frac{1}{2}C_{2}^{2}C_{6} − \frac{1}{2}C_{2}C_{4}^{2} + \frac{1}{6}C_{2}^{3}C_{4} − \frac{1}{120}C_{2}^{5}
b_{12}(s) = −C_{12} + \frac{1}{2}C_{6}^{2} + C_{2}C_{10} + C_{4}C_{8} − \frac{1}{2}C_{2}^{2}C_{8} − C_{2}C_{4}C_{6}

− \frac{1}{6}C_{4}^{3} + \frac{1}{6}C_{2}^{3}C_{6} + \frac{1}{4}C_{2}^{2}C_{4}^{2} − \frac{1}{24}C_{2}^{4}C_{4} + \frac{1}{720}C_{2}^{6}

b_{14}(s) = −C_{14} + C_{2}C_{12} + C_{4}C_{10} + C_{6}C_{8} − \frac{1}{2}C_{2}^{2}C_{10} − C_{2}C_{4}C_{8} − \frac{1}{2}C_{2}C_{6}^{2}

− \frac{1}{2}C_{4}^{2}C_{6} + \frac{1}{6}C_{2}^{3}C_{8} + \frac{1}{2}C_{2}^{2}C_{4}C_{6} + \frac{1}{6}C_{2}C_{4}^{3}

− \frac{1}{24}C_{2}^{4}C_{6} − \frac{1}{12}C_{2}^{3}C_{4}^{2} + \frac{1}{120}C_{2}^{5}C_{4} − \frac{1}{5040}C_{2}^{7}

b_{16}(s) = − C_{16} + C_{2}C_{14} + C_{4}C_{12} + C_{6}C_{10} + \frac{1}{2}C_{8}^{2} − \frac{1}{2}C_{2}^{2}C_{12} − C_{2}C_{4}C_{10}

− C_{2}C_{6}C_{8} − \frac{1}{2}C_{4}^{2}C_{8} − \frac{1}{2}C_{4}C_{6}^{2} + \frac{1}{6}C_{2}^{3}C_{10} + \frac{1}{2}C_{2}^{2}C_{4}C_{8} + \frac{1}{4}C_{2}^{2}C_{6}^{2}

+ \frac{1}{2}C_{2}C_{4}^{2}C_{6} + \frac{1}{24}C_{4}^{4} − \frac{1}{24}C_{2}^{4}C_{8} − \frac{1}{6}C_{2}^{3}C_{4}C_{6} − \frac{1}{12}C_{2}^{2}C_{4}^{3}

+ \frac{1}{120}C_{2}^{5}C_{6} + \frac{1}{48}C_{2}^{4}C_{4}^{2} − \frac{1}{720}C_{2}^{6}C_{4} + \frac{1}{40320}C_{2}^{8}

Our aim is to estimate the Hausdorff dimension of E_{2}. By Part (b) of Theorem 1 we know the solutions s_{N} to 1 + ∑_{n=1}^{N} b_{n}(s_{N}) = 0 converge to dim_{H}(E_{2}) at a super-exponential rate, as illustrated by the following table.

\begin{matrix}N & Value of s_{N} \\ 2 & 0.5736497519931504418805111910527545373580227988451 \\ 4 & 0.5306685908375786932401360892819672638338139332442 \\ 6 & 0.5312815004119405094246378072799053114028763869533 \\ 8 & 0.5312805060532824073617342947311916959469295389062 \\ 10 & 0.5312805062772122436722740896245350937162598578572 \\ 12 & 0.5312805062772051415924357878618056620528002946593 \\ 14 & 0.5312805062772051416244686680372019574162019023770 \\ 16 & 0.5312805062772051416244686473684698701845034218984 \end{matrix}

In particular, if we solve the degree-16 truncation 1 + ∑_{n=1}^{16} b_{n}(s) = 0, then we derive the approximation

dim_{H}(E_{2}) = 0.5312805062772051416244686

which is correct to the 25 decimal places given.

As noted previously, our algorithm will efficiently compute the Hausdorff dimension of other Cantor sets consisting of numbers whose continued fraction expansions have a constrained alphabet. We list a few calculations below. This should be compared with Table 1 in Hensley [Hen2], p. 16.

Alphabet  Dimension
{1, 2}  0.5312805062772051416244686
{1, 3}  0.4544890776618
{1, 4}  0.41118272477479
{2, 3}  0.3374367808060636363044949103868456
{2, 4}  0.3063127680527840302779083
{3, 4}  0.26373748289742655875986338427545
{10, 11}  0.1469212353907834633111086285159040730670831296767553
{100, 10000}  0.0522465926386588786525884163005081810126762844316811

All dimensions are correct to the given accuracy, and we do not round up the final decimal place. To obtain the dimension of E_{2} = E_{1,2} we used a degree-16 truncation. To obtain the dimension^{1} of E_{2,3} we used a degree-14 truncation. All other values were obtained using a degree-12 truncation. In fact this allowed us to compute the dimension of E_{10,11} and E_{100,10000} to a precision of (at least) 61 and 122 decimal places respectively, though for ease of presentation we do not give these values in full.

The speed of convergence of our solutions is essentially controlled by the smallest derivative at a fixed point of T lying in E_{{n_{1},... ,n_{r}}}. For the set E_{100,10000}, for example, this smallest derivative is roughly −10002, so we have extremely fast convergence. The degree-2 truncation already gives the same estimate as in [Hen2], while the degree-12 truncation gives an accuracy of (at least) 122 decimal places.

1Note our value for the set E_{2,3} disagrees with Hensley’s (which is presumably a misprint, the last three digits of his value for E_{2,3} being the same as the (correct) value for E_{1,4} which appears beneath it in Table 1, [Hen2])

5. The Markoff and Lagrange spectra

We now consider some number-theoretic applications of the improved estimates on Hausdorff dimension arising from our method. These applications are closely related to Bumby’s original motivation for computing dim_{H}(E_{2}).

Given an irrational number α, it is well known (see Hardy & Wright [H-W], Ch. 11) that there exists C > 0 such that infinitely many rationals p/q satisfy |α − p/q| < C/q^{2}. A natural question is to ask how small C can be chosen. We define

C(α) = lim sup_{q→∞} [min_{p∈ℕ} |q^{2}α − pq|] .

By convention (see [CuF]) we take reciprocals of these numbers, and define the Lagrange spectrum to be the set 𝕃 = {1/C(α) : 0 < α < 1}.

A closely related notion is that of the Markoff spectrum 𝕄. Consider quadratic forms f(x, y) = ax^{2} + bxy + cy^{2} (with a, b, c ∈ ℤ) for which d(f) := b^{2} − 4ac > 0. If we denote m(f) = inf|f(x, y)|, then 𝕄 is defined to be the set of all possible values of \sqrt{d(f)}/m(f).

In fact both spectra can also be defined in terms of doubly infinite sequences of positive integers. Given a = (a_{n})_{n∈ℤ} we define

λ_{i}(a) = [a_{i}, a_{i+1}, …] + [0, a_{i−1}, a_{i−2}, …]

where as usual [c_{0}, c_{1}, …] denotes the continued fraction whose partial quotients are c_{0}, c_{1}, . . .. We then define

L(a) = lim sup_{|i|→∞} λ_{i}(a)  ,  M(a) = sup_{i∈ℤ} λ_{i}(a).  (5.1)

Then the Lagrange and Markoff spectra are defined, respectively, as the set of all values of L(a) or of M(a).

It is clear from this definition that the Lagrange spectrum is a subset of the Markoff spectrum, and in fact it can be shown (see [CuF]) that it is a proper subset. Both spectra are closed subsets of ℝ. In fact both spectra contain the interval [μ, +∞), where μ ≈ 4.527829566 (in fact this μ is optimal and is known exactly, a result due to Freiman [Fre], see [CuF], p. 55).

The portions of both spectra below the value 3 are known exactly, and here too they coincide. Markoff [Mar] characterised this portion as the set of numbers \frac{1}{z} \sqrt{9z^{2} − 4}, where x^{2} + y^{2} + z^{2} = 3xyz, and x, y ≤ z are positive integers. Such triples (x, y, z) are known as Markoff triples. It follows that the smallest value in both spectra is \sqrt{5} (in fact this is Hurwitz’s Theorem). The next smallest values (in ascending order) are \sqrt{8}, \sqrt{221}/5, \sqrt{1517}/13 (see [CuF], p. 10 for a more complete list). Since this portion is countable, and its only accumulation point is the value 3 itself, we see that dim_{H}(M ∩ [0, 3]) = dim_{H}(L ∩ [0, 3]) = 0.

It is in the interval (3, μ) where the two spectra differ. The largest known number in 𝕄 but not in 𝕃 is β ≈ 3.293 (the number is known exactly, see [CuF], p. 35). Since the number μ above is optimal, there is a gap (μ_{−}, μ) in both spectra, for some number μ_{−}. That is, both spectra do not intersect this interval. Various other gaps have been identified, the first (historically) being the interval (\sqrt{12}, \sqrt{13}) (see Perron [Per]). Note that if a = (a_{n})_{n∈ℤ} has at least one entry greater than 2 then M(a) ≥ \sqrt{13}, and indeed M(a) = \sqrt{13} if and only if a = (3, 3, 3, …). However, if a has entries only 1’s and 2’s then M(a) ≤ \sqrt{12}, so we have Perron’s result.

This is the first indication of why the Cantor set E_{2} might be interesting in this context. Related Cantor sets are important in the problem of finding the Lebesgue measure and Hausdorff dimension of various portions of the spectra. Let us define 𝕃_{t} = 𝕃 ∩ [0, t] and 𝕄_{t} = 𝕄 ∩ [0, t].

If we consider Λ_{1} ⊂ E_{2} to be those numbers whose continued fraction expansions do not have consecutive triples (i_{k}i_{k+1}i_{k+2}) = (121) then 𝕄\sqrt{10} = 𝕄 ∩ [0, \sqrt{10}] ⊂ Λ_{1} + Λ_{1} [CuF], p. 65. In particular, dim_{H}(𝕄\sqrt{10}) ≤ 2dim_{H}(Λ_{1}), and we can use Theorem 1 to estimate the value of dim_{H}(Λ_{1}).

Similarly, if we consider Λ_{2} ⊂ E_{2} to be those numbers whose continued fraction expansions do not have consecutive quadruples (i_{k}i_{k+1}i_{k+2}i_{k+3}) = (1212) then 𝕄_{\sqrt{689}/8} = 𝕄∩[0, \sqrt{689}/8] ⊂ Λ_{2} +Λ_{2} and dim_{H}(𝕄\sqrt{10}) ≤ 2dim_{H}(Λ_{1}) [CuF], p. 75.

Using degree-16 truncated equations we can estimate^{2}

dim_{H}(Λ_{1}) ≈ 0.4060752878114, dim_{H}(Λ_{2}) ≈ 0.4858259763

Therefore we have the following result.

Theorem 2. We can estimate

dim_{H}(𝕃\sqrt{10}) ≤ dim_{H}(𝕄\sqrt{10}) ≈ 0.8121505756228 and
dim_{H}(𝕃_{\sqrt{689}/8}) ≤ dim_{H}(𝕄_{\sqrt{689}/8}) ≈ 0.9716519526

Both estimates are correct to the number of decimal places given, and the final digit is not rounded up.

In particular the above result implies that 𝕄_{\sqrt{689}/8} has zero Lebesgue measure. Defining

t_{+} = inf{t : 𝕄_{t} has positive Lebesgue measure }.

we see that t_{+} > \sqrt{689}/8. In fact this lower bound was improved by Bumby [Bu1] who showed t_{+} > 3.33437, again by choosing a set Λ ⊂ E_{2} corresponding to some subshift of finite type, and showing that dim_{H}(Λ) < 1/2, (see [Bu1], p. 206 for full details). The increased efficiency of our method should make it possible to improve this lower bound on t_{+} by judicious choice of a set Λ with Hausdorff dimension closer to 1/2.

6. Limit sets for Schottky groups

We end by briefly outlining how our method of calculation can be applied to the more geometric setting of Schottky groups. Consider n pairs of closed disks D_{1}, …, D_{n}, D_{n+1}, …, D_{2n} in the complex plane which are pairwise disjoint. For i = 1, …, n we can choose linear fractional transformations T_{i} : ℂ → ℂ which map the interior of D_{i} to the exterior of D_{n+i} (and consequently T_{i}^{−1} : ℂ → ℂ maps the interior of D_{n+i} to the exterior of D_{i}).

2Our estimate for dim_{H}(Λ_{2}) disagrees with that of Bumby in [Bu1], who claims the value is between 0.469 and 0.470. The reader may convince himself of the validity of our result by solving a low order truncated version of 1 + ∑ b_{n}(s) = 0. For example the degree-4 truncation (which only contains terms corresponding to the two fixed points and the three period-4 orbits) gives the approximation 0.4843

We call the group Γ =< T_{1}, …, T_{n} > of linear fractional transformations a Schottky group. This is isomorphic to a free group on n-generators. We define the limit set Λ ⊂ ℂ associated to Γ to be the limit points of the set Γ(0). The set Λ is a Cantor set in ℂ.

Let A denote the family of real analytic functions f : ∪_{i=1}^{2n}int(D_{i}) → ℂ. We define a transfer operator L_{s} : A → A by

L_{s}h(z) = ∑_{i≠j+n mod 2n} \frac{1}{|T′_{i}(T_{i}^{−1}z)|^{s}} h(T_{i}^{−1}z)), if z ∈ D_{j}

We need to work with real analytic functions to be assured that the terms _{\frac{1}{|T′_{i}(T_{i}^{−1}z)|^{s}}} are well defined.

To apply the theory we have described in the preceeding sections, we may consider the complexification of real analytic functions f : ∪_{i=1}^{2n}int(D_{i}) → ℂ. This means that we consider those functions which extend as complex analytic functions f : D̃ → ℂ, where D̃ ⊃ ∪_{i=1}^{2n}int(D_{i}) is an open set in ℂ^{2}. The transfer operators act in a natural way on the Banach space of analytic functions on D̃, which have a continuous extension to its closure. These are again nuclear operators.

We will pursue these and other related themes in a forthcoming article.

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O. Jenkinson, UPR 9016 CNRS, Institut de Mathématiques de Luminy, 163 avenue de Luminy, case 907, 13288 Marseille, cedex 9, France.

   E-mail address: omj@iml.univ-mrs.fr

M. Pollicott, Department of Mathematics, University of Manchester, Oxford Road, Manchester M13 9PL, UK

   E-mail address: mp@ma.man.ac.uk