CALCULATING HAUSDORFF DIMENSION OF

JULIA SETS AND KLEINIAN LIMIT SETS

Oliver Jenkinson and Mark Pollicott

Queen Mary University of London and Manchester University

Abstract. We present a new algorithm for efficiently computing the Hausdorff dimension of sets X invariant under conformal expanding dynamical systems. By locating the periodic points of period up to N, we construct approximations s_{N} which converge to dim(X) superexponentially fast in N. This method can be used to give rigorous estimates for important examples, including hyperbolic Julia sets and limit sets of Schottky and quasifuchsian groups.

0. Introduction

Hausdorff dimension is a fundamental invariant of bi-Lipschitz homeomorphism. The Hausdorff dimension of a set is in general neither a rational number nor easily expressible in terms of well-known or easily computable irrationals, so that effective use of this invariant relies on its accurate computation. In general, however, computation of Hausdorff dimension is a non-trivial problem.

Dynamical systems provide not only a fertile source of examples of fractal sets, but also techniques for studying them (see for example [Pe]). Conformal dynamics is a particularly rich source of examples of sets with non-integer Hausdorff dimension, notably Julia sets of rational maps and limit sets of Kleinian groups.

In this paper we present an algorithm for accurately computing the Hausdorff dimension of a large class of such fractals, namely those for which the underlying dynamics is hyperbolic. For such sets X, our algorithm relies on locating the periodic points of the corresponding dynamical system T : X → X. We construct a sequence of functions Δ_{N} in terms of the (first) derivatives of T, evaluated at the periodic points of period up to N. The largest zeros s_{N} of the functions Δ_{N} provide a sequence of approximations to the Hausdorff dimension dim(X) which converges super-exponentially in N.

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We are very grateful to Mark Muldoon for writing a program [Mul] to determine the periodic points for hyperbolic quadratic maps. We also thank Igor Rivin and André Rocha for advice on programming to calculate the dimension of the Schottky group limit sets. The first author thanks the hospitality of the Institut de Mathématiques de Luminy, the Institut des Hautes Études Scientifiques, Manchester University, the Centro de Investigacion en Matemáticas, and the Erwin Schrödinger Institute (FWF grant number P12250-MAT), where parts of this work were carried out. The second author was partly supported by the Leverhulme trust.

More precisely, if X is contained in a C^{ω} manifold of dimension d ∈ ℕ, then there exist 0 < δ < 1 and C > 0 such that |s_{N} − dim(X)| ≤ Cδ^{N^{1+1/d}}. The constants δ and C both depend on the particular set X. Roughly speaking, for a fixed d, the stronger the hyperbolicity of the dynamics, the faster the sequence s_{N} will converge.

In the systems we consider, the number of points of period up to N increases at an exponential rate. Therefore the functions Δ_{N} can be constructed in at worst exponential time^{1}, so that the periodic point algorithm has, at worst, time complexity O (exp (aN^{\frac{d}{1+d}} )), a > 0. In particular, for any d the algorithm has sub-exponential complexity, whereas previous algorithms such as those of McMullen [McM3], Bodart & Zinsmeister [BZ], Widom, Bensimon, Kadanoff & Shenker [WBKS], Garnett [Gar]), and Saupe [Sau], have at best exponential complexity.

A feature of the periodic point algorithm is that it easily lends itself to rigorous Hausdorff dimension estimates, via explicit bounds on the constants δ and C. (Such rigorous estimates appear to be more difficult to justify for the alternative algorithms cited above). Moreover, the empirical dimension estimates, inferred by comparing consecutive zeros s_{N} (see §7, §8), offer an accuracy beyond that of the rigorous bounds.

Our interest in these problems was motivated by the recent paper of McMullen [McM3], who gave a different algorithm (the eigenvalue algorithm) for computing the Hausdorff dimension for the same class of examples as we consider. The complexity of the periodic point algorithm is lower than that of [McM3], and for sufficiently hyperbolic systems our empirical dimension estimates bear this out (see the examples below, and in §7 , §8). For systems of weaker hyperbolicity, however, the constant δ is closer to 1 and the inertia arising from sub - O(δ^{N^{1+1/d}}) asymptotics means that McMullen’s algorithm is for practical purposes superior.

An attractive feature of the periodic point algorithm is that it is canonical. At no stage is it necessary to make (non-canonical) choices of coordinates or of Markov partitions. One simply locates periodic points of the underlying dynamical system. Consequently the algorithm is particularly easy to implement.

The proof of the algorithm’s rapid convergence is based on ideas from thermodynamic formalism, where Bowen and Ruelle interpreted dimension as an implicit solution of a pressure equation, as well as Grothendieck’s classical work on the Fredholm determinants of nuclear operators.

We first apply our algorithm to computing the dimension of limit sets of two classes of Kleinian group:

1. Schottky groups. Fix 2p disjoint closed discs D_{1}, …, D_{2p} in the plane, and Möbius maps g_{1}, …, g_{p} such that each g_{i} maps the interior of D_{i} to the exterior of D_{p+i}. The corresponding Schottky group is defined as the group generated by g_{1}, …, g_{p}. The associated limit set Λ is a Cantor subset of the union of the interiors of the discs D_{1}, …, D_{2p}.

1It is not inconceivable that algorithms will be developed to construct the functions Δ_{N} in subexponential time, in which case the complexity of our algorithm will be reduced. At present this appears to be possible only for piecewise affine systems, where the limit Δ of the Δ_{N} is actually available in closed form; however this is of only academic interest, since for such systems the Hausdorff dimension is easily computable (see [Pe, Th. 13.3]) without recourse to any algorithm.

We define a map T on this union by T |_{int(D_{i})} = g_{i} and T |_{int(D_{p+i})} = g_{i}^{−1}. A reflection group is a Schottky group with D_{i} = D_{p+i} for all i = 1, …, p.

2. Quasifuchsian groups. Such groups are isomorphic to the fundamental group of a compact Riemann surface, and are obtained by a quasiconformal deformation of a Fuchsian group (a Kleinian group whose limit set is contained in some circle). The limit set Λ of a quasifuchsian group is a simple closed curve. We can associate an expanding map T with the limit set of any Fuchsian group (see [BS]), and the quasiconformal deformation induces an expanding map on Λ.

We show that the Hausdorff dimension of the limit sets Λ of both Schottky and quasifuchsian groups can be efficiently calculated via a knowledge of the derivatives DT^{n}(z), evaluated at periodic points T^{n}z = z.

Theorem 1. (Kleinian groups) Let Γ be a finitely generated non-elementary convex cocompact Schottky or quasifuchsian group, with associated limit set Λ. Let T : Λ → Λ be the associated dynamical system. For each N ≥ 1 we can explicitly define a function Δ_{N}, using only the derivatives DT^{n}(z) evaluated at period-n points z, for 1 ≤ n ≤ N, and associate C > 0 and 0 < δ < 1 such that if s_{N} is the largest real zero of Δ_{N} then

|dim(Λ) − s_{N}| ≤ Cδ^{N^{3/2}}.

Remark. The convex cocompact assumption in Theorem 1 corresponds to Γ not containing parabolic elements.

The functions Δ_{N} are defined as follows. First let

a_{n} = \frac{1}{n} ∑_{T^{n}z=z} \frac{|DT^{n}(z)|^{−s}}{det (I − [DT^{n}(z)]^{−1})},

where the summation is over all period-n points z, DT^{n}(z) denotes the derivative of T^{n} at such a point, and |DT^{n}(z)| denotes the modulus of the real (as opposed to isometry) part of this derivative. Then define Δ_{N} by

Δ_{N}(s) = 1 + ∑_{n=1}^{N} ∑_{{(n_{1},... ,nm) \atop n_{1}+...+nm=n}} \frac{(−1)^{m}}{m!} a_{n_{1}} … a_{n_{m}},

where the second summation is over all ordered m-tuples of positive integers whose sum is n.

Example 1. The following family of reflection groups (see §6 for more details) was considered by McMullen [McM3]. Consider three circles C_{0}, C_{1}, C_{2} ⊂ ℂ of equal radius, arranged symmetrically around S^{1}, each intersecting S^{1} orthogonally, and meeting S^{1} in an arc of length θ, where 0 < θ < 2π/3 (see Figure 2). Let Λ_{θ} ⊂ 𝕊^{1} denote the limit set associated

Figure 1 . Limit set for a quasifuchsian group

to the group Γ_{θ} of transformations given by reflection in these circles. For example, with the value θ = π/6 we show that the dimension of the limit set Λ_{π/6} is

dim(Λ_{π/6}) = 0.18398306124833918694118127344474173288 …

which is empirically accurate to the 38 decimal places given.

Consider next the Julia set J of a rational map f : ℂ̂ → ℂ̂. We shall be interested in those Julia sets J which are hyperbolic (i.e. inf_{z∈J} |(f^{n})′(z)| > 1 for some n). The periodic point algorithm allows the accurate computation of Hausdorff dimension of such Julia sets, or more generally the Julia sets of any hyperbolic holomorphic Markov map (i.e. one whose symbolic dynamics is described by a subshift of finite type; see §1 for the definition). A feature of the approach is that we exploit the real-analyticity of the map f. That is, we will identify ℂ with ℝ^{2}, and think of J as lying in the real section ℝ^{2} of ℂ^{2}, with T (x, y) = (Re (f(x + iy)) , Im (f(x + iy))) an analytic map on some suitable domain in ℂ^{2}.

Theorem 2. (Julia sets) Let J be the Julia set of a hyperbolic holomorphic Markov map f : J → J. For each N ≥ 1 we can explicitly define a function Δ_{N}, using only the derivatives (f^{n})′(z) evaluated at period-n points z, for 1 ≤ n ≤ N, and associate C > 0 and 0 < δ < 1 such that if s_{N} is the largest real zero of Δ_{N} then

|dim(J) − s_{N}| ≤ Cδ^{N^{3/2}}.

This method is effective in computing the dimension of many hyperbolic Julia sets. The Julia sets J_{c} of quadratic polynomials f_{c}(z) = z^{2} + c are of particular interest.

Figure 2. Reflection in three circles

Example 2. For the quadratic map f_{i/4}(z) = z^{2} + \frac{i}{4}, the Hausdorff dimension of the associated Julia set J_{i/4} is given by

dim(J_{i/4}) ≈ 1.0231992890309691251 …,

where the first 16 decimal digits are empirically correct.

More generally, our approach gives super-exponential convergence to the Hausdorff dimension of any compact set X which lies in a C^{ω} d-manifold and is invariant under a dynamical system T : X → X satisfying

(1) Markov dynamics: On a symbolic level the dynamics is faithfully coded by a subshift of finite type;

(2) Hyperbolicity: There exists some C > 1 such that |T′(x)| > C for all x ∈ X;

(3) Conformality: T is a conformal map;

(4) Analyticity: T is real-analytic;

(5) Local maximality: For any sufficiently small open neighbourhood U of the invariant

set X we have X = ∩_{n=0}^{∞}T^{−n}U (such an X is sometimes called a repeller). Note that in dimension two or higher, property (4) is implied by (3).

Theorem 3. (General Case) Let X ⊂ M be a locally maximal compact invariant set for a conformal real-analytic hyperbolic Markov map T : X → X, where M is a C^{ω} manifold of dimension d ∈ ℕ. For each N ≥ 1 we can explicitly define a function Δ_{N}, using only the

Figure 3. The Julia set for f_{i/4}(z) = z^{2} + \frac{i}{4}

derivatives DT^{n}(z) evaluated at period-n points z, for 1 ≤ n ≤ N, and associate C > 0 and 0 < δ < 1 such that if s_{N} is the largest real zero of Δ_{N} then

|dim(X) − s_{N}| ≤ Cδ^{N^{1+1/d}}.

In view of this, we see that stronger results than Theorems 1 and 2 can be proved in the case where the limit set actually lies in a one (real) dimensional submanifold.

Corollary 3.1. (Faster convergence on submanifolds)

(1) Let Λ be the limit set of a classical Fuchsian Schottky group, with associated dynamical system T : Λ → Λ. For each N ≥ 1 we can explicitly define a function Δ_{N}, using only the derivatives DT^{n}(z) evaluated at period-n points z, for 1 ≤ n ≤ N, and associate C > 0 and 0 < δ < 1 such that if s_{N} is the largest real zero of Δ_{N} then

|dim(Λ) − s_{N}| ≤ Cδ^{N^{2}}.

(2) Let f_{c} : J_{c} → J_{c} be the quadratic map f_{c}(z) = z^{2} + c restricted to the Julia set J_{c}, where the real parameter c < −2. For each N ≥ 1 we can explicitly define a function Δ_{N}, using only the derivatives (f_{c}^{n})′(z) evaluated at period-n points z, for 1 ≤ n ≤ N, and associate C > 0 and 0 < δ < 1 such that if s_{N} is the largest real zero of Δ_{N} then

|dim(J_{c}) − s_{N}| ≤ Cδ^{N^{2}}.

Returning to Example 1, we see that since Λ_{θ} ⊂ S^{1} then part (1) of Corollary 3.1 means dim(Λ_{θ}) can be approximated at rate O(δ^{N^{2}}), for some 0 < δ < 1.

In the quadratic family f_{c}, for real parameters c < −2 the Julia set J_{c} is hyperbolic and also lies on the real line. By part (2) of Corollary 3.1 we have the improved convergence rate O(δ^{N^{2}}). For example taking c = −5 the periodic point algorithm yields

dim(J_{−5}) = 0.48479829443816043053839847...

where these 26 decimal places are (empirically) accurate.

All the dynamical systems we consider are conformal. Since every one-dimensional expanding map is conformal, the periodic point algorithm applies to a wide class of fractal sets lying in the real line. Here, however, we need the extra real-analyticity hypothesis (a property which is automatically satisfied by higher-dimensional conformal maps).

Analytic cookie-cutters. Many fractals in the line arise as limit sets of so-called cookiecutters. These are special cases of the iterated function schemes (defined in §1) we consider, and are treated in [Fa2, Ch. 4]. Examples of particular interest, whose Hausdorff dimension can all be calculated via the periodic point algorithm, are the following:

(i) Bounded continued fractions. The classic example is the set E_{2} considered in [JP1]. More generally, the set of reals whose continued fraction expansions have partial quotients constrained to lie in some finite alphabet is a Cantor set.

(ii) Limit sets of Hecke groups (see [McM3]).

(iii) Julia sets for Blaschke products, for example the family f_{t}(z) = z/t−1/z considered in [McM3].

Theorems 1-3 are special cases of Theorem 4, which appears in §3.6, and is formulated in terms of conformal iterated function schemes. These iterated function schemes are introduced in §1. In §2 we review the definition of Hausdorff dimension and Bowen’s pressure formula. §3 is the key section, in which we introduce tools such as the transfer operator, Fredholm determinants, and trace formulae, and prove our main results. In §4 we apply these results to Schottky and quasifuchsian groups, and in §5 to Julia sets of holomorphic maps. In §6 we treat in detail the reflection group family of Example 1, and show in §7 how the periodic point algorithm yields rigorous dimension estimates. In §8 we use the algorithm to obtain numerical estimates on the dimension of Julia sets for quadratic maps. In §9 we survey some other examples to which our algorithm can be applied. In Appendix A we discuss the practical implementation of the algorithm (see also the website [JP2]). In Appendix B we prove a lemma about contraction ratios of iterated function schemes, useful for estimating the constant δ in the theorems.

1. Conformal iterated function schemes

In this section we shall introduce notation in order to formulate results in their most general context. We will then specialise to particular examples in later sections.

The defining property of any conformal map ψ : U ⊂ ℝ^{d} → ℝ^{d} is that at each point z ∈ U, its derivative Dψ(z) is the product of a positive real number, denoted by |Dψ(z)|, and an isometry M_{ψ}(z) ∈ SO(d). The map z ↦ |Dψ(z)| is called the conformal derivative of the map ψ.

Definition. Let U_{1}, …, U_{k} ⊂ ℝ^{d} be a finite collection of subsets of ℝ^{d}, each of which is the closure of its own interior. For k ≥ 2, let A be a k × k aperiodic (i.e. some power of A is strictly positive) matrix with entries either 0 or 1. Assume that for every pair of symbols i, j ∈ {1, …, k} such that A(i, j) = 1 (in this case we say the pair (i, j) is admissible) we have associated a real analytic conformal map ϕ_{ji} : U_{i} → U_{j} such that:

(i) The closure of the image ϕ_{ji}(U_{i}) lies inside U_{j} (i.e. ϕ_{ji}(U_{i}) ⊂ int(U_{j})) and (ii) the map ϕ_{ji} is a strict contraction (i.e. there exists 0 < θ < 1 with |Dϕ_{ji}(z)| ≤ θ

for all z ∈ U_{i}).

The collection of maps {ϕ_{ji} : A(i, j) = 1} is called a (real-analytic conformal) iterated function scheme on ℝ^{d}.

Figure 4 . Regions and contractions

Now suppose M is a real-analytic manifold of dimension d. By analogy with the above, suppose we have subsets U_{1}, …, U_{k} ⊂ M, each of which is the closure of its own interior, a k × k aperiodic 0-1 matrix A, and a collection of C^{ω} conformal maps ϕ_{ji} : U_{i} → U_{j} corresponding to admissible pairs (i, j), and satisfying (i), (ii) above. We call this a (real-analytic conformal) iterated function scheme on M. To say the ϕ_{ji} are C^{ω} and conformal means precisely that there is a C^{ω} structure on M consisting of a a finite collection of connected open sets V_{α} ⊂ M and C^{ω} diffeomorphisms f_{α} : V_{α} → ℝ^{d} such that, if the U_{i} are sufficiently small so that each U_{i} is contained in some V_{α_{i}} (we may always assume this is true, by replacing the collection {U_{i}} by some suitable refinement) then each ψ_{ji} := f_{α_{j}} ∘ ϕ_{ji} ∘ f_{α_{i}}^{−1} is a conformal C^{ω} map from W_{i} := f_{α_{i}}U_{i} ⊂ ℝ^{d} to W_{j} := f_{α_{j}}U_{j} ⊂ ℝ^{d}. Each ψ_{ji} is moreover a strict contraction, so the collection of sets W_{i} and maps ψ_{ji} form a conformal iterated function scheme on subsets of ℝ^{d}, whose limit set has the same Hausdorff dimension as the original limit set in M (since in particular the f_{α} are bi-Lipschitz). In this way we can, and will, always make the simplifying assumption that our iterated function scheme is defined in ℝ^{d}.

An ordered (n + 1)-tuple of symbols \underline{i} = (i_{1}, …, i_{n+1}) ∈ \{1, …, k\}^{n+1} such that A(i_{j}, i_{j+1}) = 1 for j = 1, …, n is called an admissible string. For such a string \underline{i} we let

ϕ\underline{i} = ϕ_{i_{n+1}i_{n}} ∘ … ∘ ϕ_{i_{3}i_{2}} ∘ ϕ_{i_{2}i_{1}}

denote the n-fold composition of the corresponding contractions. We write |\underline{i}| = n + 1.

Definition. The limit set Λ ⊂ ∪_{i=1}^{k}U_{i} of an iterated function scheme is defined by

Λ = ⋂_{n=1}^{∞} \overline{⋃_{|\underline{i}|=n+1} ϕ\underline{i}(U_{i_{1}}).}

If i_{1} = j = i_{n+1} then the contraction ϕ\underline{i} : U_{j} → U_{j} has a unique fixed point, which we denote by z\underline{i}. Let Fix(n) = \{\underline{i} : |\underline{i}| = n + 1, i_{1} = i_{n+1}\} denote the set of length-(n + 1) strings of this form, and Fix(n) = \{z\underline{i} : \underline{i} ∈ Fix(n)\} the set of corresponding fixed points.

The above notion of an iterated function scheme is somewhat more general than the usual definition (see for example [MU1]). More commonly, the contractions ϕ_{ji} : U_{i} → U_{j} are assumed to only depend on the target set U_{j}. This is the case, for example, if the contractions arise as inverse branches of an expanding map, or as cookie-cutter systems. Since this more common scenario is our primary motivation, corresponding to the Julia set and Kleinian limit set cases dealt with later, we now briefly indicate the connection between the two set-ups.

Let T : X → X be a conformal real analytic expanding map where X ⊂ ℝ^{d} is a closed, T - invariant, locally maximal hyperbolic subset. We can choose a Markov partition, consisting of a collection of sets X_{1}, …, X_{k} ⊂ X covering X such that the restriction of T to each X_{j} is injective, and the image T (X_{j}) is a union of the form X_{i_{1}} ∪…∪X_{i_{r}}. We also require that the sets X_{1}, … X_{k} have pairwise disjoint interiors, and that each X_{i} is the closure of its own interior. We can naturally extend T to an analytic expanding map T̂ on the disjoint union ∪_{i}X_{i}. The map T̂ is called a Markov map. We then define a k ×k transition matrix A, where A(i, j) = 1 if T̂X_{j} ∩ X_{i} ≠ ∅, and A(i, j) = 0 otherwise. If A(i, j) = 1 then we define a contracting map ϕ_{ji} : X_{i} → X_{j} by requiring that T̂ ∘ ϕ_{ji} = id|_{X_{i}}. The collection {ϕ_{ji} : A(i, j) = 1} is then an iterated function scheme.

In fact, if T̂X_{j} = X_{i_{1}} ∪ … ∪ X_{i_{r}}, then it is common to define an inverse branch S_{j} : X_{i_{1}} ∪ … ∪ X_{i_{r}} → X_{j} by T ∘ S_{j} = id|_{X_{i_{1}} ∪...∪X_{ir}} (so that ϕ_{ji} = S_{j}|_{X_{i}} for each admissible (i, j)). For an admissible n-tuple \underline{j} = (j_{1}, …, j_{n}) we will also define S\underline{j} = S_{j_{n}} ∘ ⋯ ∘ S_{j_{1}} : X_{i_{1}} ∪ … ∪ X_{i_{r}} → X_{j_{n}}, and note that T̂ ∘ S\underline{j} = id|_{X_{i_{1}} ∪...∪X_{ir}}.

Note that such a contraction S\underline{j} has a unique fixed point. This fixed point is precisely z\underline{i} (i.e. the fixed point for ϕ\underline{i}), where \underline{i} = (j_{n}, j_{1}, …, j_{n}). Thus there is a one-to-one correspondence between the set Fix(n) and the set of all admissible length-n words \underline{j}. Moreover, the set Fix(n) of fixed points z\underline{i} of length-n compositions of contractions is precisely the set of period-n points for the map T̂. Throughout we adopt the convention of counting periodic points with multiplicities corresponding to T̂, rather than T. The precise connection between the periodic points for T and T̂ is well understood, using Markov partitions and Manning’s combinatorial lemma [Ma].

2. Hausdorff dimension and Bowen’s pressure formula

Let us recall the definition of Hausdorff dimension.

Definition. For X ⊂ ℝ^{d}, and s ≥ 0 we let H_{ε}^{s}(X) = inf_{U}{∑_{i} diam(U_{i})^{s}} where the infimum is taken over all open covers U = {U_{i}} such that diam(U_{i}) ≤ ε. We define the s-dimensional Hausdorff measure of X by H^{s}(X) = lim_{ε→0} H_{ε}^{s}(X) and then the Hausdorff dimension of X is defined by

dim(X) = inf{s : H^{s}(X) = 0}.

The sets we consider will always be compact, and invariant under some dynamical system. There are various other notions of dimension (see [Fa1]). For the conformal iterated function schemes we consider, the Hausdorff dimension is equal to the box dimension, for example. For dynamically defined sets, the following definition is useful.

Definition. Given a conformal iterated function scheme, we define its pressure function p : ℝ → ℝ by

p(s) = lim_{n→∞} \frac{1}{n} log ∑_{\underline{i}∈Fix(n)} |Dϕ\underline{i}(z\underline{i})|^{s}.

If our iterated function scheme is derived from an expanding Markov map T, in the way described in §1, then p(s) is just the usual pressure (see [Wal])

P (−s log |T̂′|) = lim_{n→∞} \frac{1}{n} log ∑_{T̂^{n}z=z} |DT̂^{n}(z)|^{−s}

of the function s ↦ −s log |T̂′| with respect to the Markov dynamical system T̂.

Proposition 1. Let Λ be the limit set of a conformal iterated function scheme. Then the Hausdorff dimension dim(Λ) is the unique zero of the strictly decreasing real function s ↦ p(s).

A proof of the above result appears in the article of Bowen [Bo], in the special case of quasifuchsian groups. However, as Ruelle observed in [Ru3] for Julia sets, the proof applies in greater generality (see also [Pe], [PU]).

3. Proof of Main Theorem

A key tool in our analysis is a family of bounded linear transfer operators L_{s} on a Banach space of bounded holomorphic functions. The transfer operator is defined in terms of our iterated function scheme. A key property of L_{s} is that it is a nuclear operator (Proposition 2), so that we can define its Fredholm determinant det(I − zL_{s}), an entire function of z (Proposition 4). The fact that L_{s} is defined as a sum of composition operators means that the traces tr(L_{s}^{n}) can be evaluated in terms of fixed points of our iterated function scheme (Proposition 3), and these traces are then used in §3.6 to give approximations to det(I − zL_{s}).

3.1 Complexification of the iterated function scheme.

We have the natural identification

ℝ^{d} = ℝ^{d} × {0} ⊂ ℝ^{d} × iℝ^{d} = ℂ^{d}.

For each symbol i ∈ {1, …, k}, let us choose some open polydisc D_{i} = D_{i}^{(1)}×…×D_{i}^{(d)} ⊂ ℂ^{d}, where each D_{i}^{(l)} is an open disc in ℂ, such that U_{i} × {0} ⊂ D_{i}.

All our maps are real-analytic and contracting. Therefore by considering a refined Markov partition if necessary, we may assume that for each admissible pair (i, j), the maps ϕ_{ji}, |Dϕ_{ji}(·)| all extend holomorphically to maps D_{i} → D_{j}, such that both

\overline{ϕ_{ji}(D_{i})} ⊂ D_{j},

and

sup_{z∈D_{i}} |Dϕ_{ji}(z)| < 1.  (3.1)

By standard abuse of notation we let ϕ_{ji}, |Dϕ_{ji}(·)| denote these holomorphic extensions to D_{i}. Similarly, we define the maps ϕ\underline{i}, |Dϕ\underline{i}(·)| to be the holomorphic extensions of the corresponding previously defined restrictions to the real section.

Note that if \underline{i} ∈ Fix(n) (i.e. if i_{1} = i_{n+1}) then the contraction ϕ\underline{i} has z\underline{i} as a unique fixed point. In particular z\underline{i} lies in the real section ℝ^{d}.

Let D = ∐_{i=1}^{k} D_{i} denote the disjoint union of our polydiscs.

Now to each polydisc D_{i} = D_{i}^{(1)} × … × D_{i}^{(d)} ⊂ ℂ^{d} in our disjoint union D, we want to associate a concentric open polydisc E_{i} = E_{i}^{(1)} × … × E_{i}^{(d)}, where each E_{i}^{(l)} has radius strictly larger than D_{i}^{(l)}.

By (3.1) we know that for each admisible pair (i, j), |Dϕ_{ji}(·)| extends holomorphically to some open neighbourhood D_{i,j} of D_{i}, with the property that

|Dϕ_{j,i}(·)| < 1 on D_{i,j}.  (3.2)

Moreover, since ϕ_{ji} is a contraction then ϕ_{ji}^{−1}D_{j} contains some open neighbourhood of D_{i}. So defining E_{i} to be the largest polydisc, concentric with D_{i}, and contained in

⋂_{j:A(i,j)=1} ϕ_{ji}^{−1}D_{j} ∩ D_{i,j},

ensures it has the property that each of its component discs has radius strictly larger than the corresponding component disc in D_{i}.

Moreover, we have that ϕ_{ji} extends holomorphically to E_{i}, and the important property that

\overline{ϕ_{ji}E_{i}} ⊂ D_{i} for each admissible pair (i, j).  (3.3)

Let E = ∐_{i=1}^{k} E_{i} denote the disjoint union of the larger polydiscs.

3.2 Transfer operators.

For any open set U, let A_{∞}(U) denote the Banach space of those holomorphic functions on U which are bounded on the closure \overline{U}, equipped with the supremum norm. We will consider such spaces for the various choices U = D_{i}, E_{i}, D, E.

For any s ∈ ℂ, and any admissible pair (i, j), define the weight function w_{s,(j,i)} ∈ A_{∞}(E_{i}) by

w_{s,(j,i)}(z) = |Dϕ_{ji}(z)|^{s}.  (3.4)

Then define the bounded linear operator L_{s,(j,i)} : A_{∞}(D_{j}) → A_{∞}(D_{i}) by

L_{s,(j,i)}g(z) = g(ϕ_{ji}z)w_{s,(j,i)}(z).  (3.5)

Since w_{s,(j,i)} ∈ A_{∞}(E_{i}) then by virtue of the inclusion (3.3) we have the inclusion

L_{s,(j,i)} (A_{∞}(D_{j})) ⊂ A_{∞}(E_{i}).  (3.6)

For a fixed i we sum over all (admissible) composition-type operators L_{s,(j,i)} to form the component transfer operator L_{s,i}. That is, we define

L_{s,i}h(z) = ∑_{j:A(i,j)=1} h(ϕ_{ji}z)w_{s,(j,i)}(z).  (3.7)

Note that L_{s,i} is naturally an operator A_{∞}(∐_{j:A(i,j)=1} D_{j}) → A_{∞}(D_{i}), and clearly also defines an operator A_{∞}(D) → A_{∞}(D_{i}). Indeed, since (3.6) holds for each admissible (i, j) then

L_{s,i} (A_{∞}(D)) ⊂ A_{∞}(E_{i}).  (3.8)

Now we define the transfer operator L_{s} : A_{∞}(D) → A_{∞}(D) by setting

L_{s}h|_{D_{i}} = L_{s,i}h

for each h ∈ A_{∞}(D) and each i ∈ {1, …, k}.

The inclusion (3.8) means that

L_{s} (A_{∞}(D)) ⊂ A_{∞}(E).  (3.9)

Note that if we define M_{s,i} : A_{∞}(D) → A_{∞}(D) by

M_{s,i}h|_{D_{i}} = L_{s,i}h, M_{s,i}h|_{D_{j}} = 0 if j ∈ {1, …, k} \ {i},  (3.10)

then we can write

L_{s} = ∑_{i=1}^{k} M_{s,i}.  (3.11)

The following result is standard (see [Ru2]).

Lemma 1. For real s, the transfer operator L_{s} : A_{∞}(D) → A_{∞}(D) has spectral radius e^{p(s)}. The value e^{p(s)} is the unique eigenvalue of maximum modulus, and is simple and isolated.

Remark. As is well known, the spectral properties of the transfer operator L_{s} depend strongly on the underlying Banach space. When acting on the space C^{α}(Λ) of α-Hölder functions, L_{s} again has spectral radius e^{p(s)}, the unique eigenvalue of maximum modulus, but is not a compact operator. However the operator does have a spectral gap; that is, its essential spectral radius is strictly smaller than e^{p(s)} (see [Ru2]). In §3.3 it will be shown that when acting on A_{∞}(D), the operator L_{s} is nuclear, which in particular implies its compactness.

3.3 Nuclear operators.

Definition. A linear operator L : B → B on a Banach space B is called nuclear if there exist u_{n} ∈ B, l_{n} ∈ B^{∗} (with ||u_{n}|| = 1 and ||l_{n}|| = 1) and ∑_{n=0}^{∞} |ρ_{n}| < +∞ such that

L(v) = ∑_{n=0}^{∞} ρ_{n}l_{n}(v)u_{n}, for all v ∈ B.  (3.12)

L is said to be nuclear of order p if ∑_{n} |ρ_{n}|^{p} < ∞, and nuclear of order zero if this holds for all p > 0.

In this section we will show that the transfer operator L_{s} is nuclear of order zero. First we need some notation. Let us fix i ∈ {1, …, k}. Let z^{(i)} = (z_{1}^{(i)}, …, z_{d}^{(i)}) denote the common centre of the two polydiscs. Let σ_{i}^{(l)} be the radius of the l^{th} component disc D_{i}^{(l)}.

Let Γ_{i} = Γ_{i}^{(1)} × … × Γ_{i}^{(d)} be a polycircle, centred at z^{(i)}, exterior to D_{i}, yet interior to E_{i}, such that the radii τ_{i}^{(l)} of the circles Γ_{i}^{(l)} satisfy γ_{i}^{(l)} := σ_{i}^{(l)}/τ_{i}^{(l)} < 1 (possible by choosing Γ_{i}^{(l)} sufficiently close to the boundary of E_{i}^{(l)}.

We call γ_{i}^{(l)} the contraction ratio in the l^{th} component.

Proposition 2. The transfer operator L_{s} : A_{∞}(D) → A_{∞}(D) is nuclear of order zero.

Poof. Consider the component transfer operator L_{s,i} : A_{∞}(D) → A_{∞}(D_{i}).

For h ∈ A_{∞}(D) we know by (3.8) that not only is L_{s,i}h holomorphic on the polydisc D_{i} = D_{i}^{(1)} × … × D_{i}^{(d)}, but it is actually holomorphic on the larger concentric polydisc E_{i} = E_{i}^{(1)} × … × E_{i}^{(d)}.

Therefore by the multi-dimensional Cauchy formula we can integrate around the polycircle Γ_{i} to obtain the representation

L_{s,i}h(z) = L_{s,i}h(z_{1}, …, z_{d}) = \frac{1}{(2π\sqrt{−1})^{d}} ∫_{Γ_{i}} \frac{L_{s,i}h(ξ_{1}, …, ξ_{d})}{(ξ_{1} − z_{1}) … (ξ_{d} − z_{d})}dξ_{1} … dξ_{d}.

Expanding each

(ξ_{l} − z_{l})^{−1} = (ξ_{l} − z_{l}^{(i)})^{−1} (1 − \frac{z_{l} − z_{l}^{(i)}}{ξ_{l} − z_{l}^{(i)}} )^{−1} = (ξ_{l} − z_{l}^{(i)})^{−1} ∑_{r_{l}=0}^{∞} (\frac{z_{l} − z_{l}^{(i)}}{ξ_{l} − z_{l}^{(i)}} )^{r_{l}}

then gives us

L_{s,i}h(z) = ∑_{r_{1}=0}^{∞} … ∑_{r_{d}=0}^{∞} m_{(r_{1},... ,r_{d})}^{(i)}(h) v_{(r_{1},... ,r_{d})}^{(i)}(z_{1}, …, z_{d}).  (3.13)

Here the linear functionals m_{(r_{1},... ,r_{d})}^{(i)} ∈ A_{∞}(D)^{∗} are given by

m_{(r_{1},... ,r_{d})}^{(i)}(h) = \frac{1}{(2π\sqrt{−1})^{d}} ∫_{Γ_{i}} \frac{L_{s,i}h(ξ_{1}, …, ξ_{d})}{(ξ_{1} − z_{1}^{(i)})^{r_{1}+1} … (ξ_{d} − z_{d}^{(i)})^{r_{d}+1}} dξ_{1} … dξ_{d},

and the functions v_{(r_{1},... ,r_{d})}^{(i)} ∈ A_{∞}(D_{i}) are monomials

v_{(r_{1},... ,r_{d})}^{(i)}(z) = v_{(r_{1},... ,r_{d})}^{(i)}(z_{1}, …, z_{d}) = (z_{1} − z_{1}^{(i)})^{r_{1}} … (z_{d} − z_{d}^{(i)})^{r_{d}}.

For each v_{(r_{1},... ,r_{d})}^{(i)} ∈ A_{∞}(D_{i}) we can naturally associate a function v_{(r_{1},... ,r_{d}),i} ∈ A_{∞}(D) defined by

v_{(r_{1},... ,r_{d}),i}|_{D_{i}} = v_{(r_{1},... ,r_{d})}^{(i)}, v_{(r_{1},... ,r_{d}),i}|_{D_{j}} = 0 for j ∈ {1, …, k} \ {i}.

This then allows us to combine all the component transfer operators L_{s,i}, and express the transfer operator L_{s} : A_{∞}(D) → A_{∞}(D) as

L_{s}h(z) = ∑_{i=1}^{k} ∑_{r_{1}=0}^{∞} … ∑_{r_{d}=0}^{∞} m_{(r_{1},... ,r_{d})}^{(i)}(h) v_{(r_{1},... ,r_{d}),i}(z)  (3.14)

for h ∈ A_{∞}(D), z ∈ D.

Let us normalise the functions and functionals in (3.14) by setting

l_{(r_{1},... ,r_{d})}^{(i)} = m_{(r_{1},... ,r_{d})}^{(i)}/||m_{(r_{1},... ,r_{d})}^{(i)}||_{∞}, u_{(r_{1},... ,r_{d}),i} = v_{(r_{1},... ,r_{d}),i}/||v_{(r_{1},... ,r_{d}),i}||_{∞}.

We then set λ_{(r_{1},... ,r_{d}),i} = ||m_{(r_{1},... ,r_{d})}^{(i)}||_{∞}||v_{(r_{1},... ,r_{d}),i}||_{∞}, so that (3.14) becomes

L_{s}h = ∑_{i=1}^{k} ∑_{r_{1}=0}^{∞} … ∑_{r_{d}=0}^{∞} λ_{(r_{1},... ,r_{d}),i} l_{(r_{1},... ,r_{d})}^{(i)}(h) u_{(r_{1},... ,r_{d}),i}.  (3.15)

We have the estimate

\begin{aligned}λ_{(r_{1},... ,r_{d}),i} & ≤ \frac{||L_{s,i}||_{∞}}{(2π)^{d}τ_{i}^{(1)} … τ_{i}^{(d)}} (γ_{i}^{(1)})^{r_{1}} … (γ_{i}^{(d)})^{r_{d}} \\ & = A (γ_{i}^{(1)})^{r_{1}} … (γ_{i}^{(d)})^{r_{d}} , \end{aligned}

where A = A(d, s, i) only depends on d, s, i, and on the radii of the component discs of D_{i} and E_{i}.

In particular, setting γ_{i} = max{γ_{i}^{(1)}, …, γ_{i}^{(d)}} < 1 we have

λ_{(r_{1},... ,r_{d}),i} ≤ Kγ_{i}^{r_{1}+...+r_{d}}.  (3.16)

(Note at this point that there are certainly more refined ways of choosing γ_{i} if we are interested in optimal estimates).

For each i ∈ {1, …, k}, let the sequence {λ_{n,i}}_{n=0}^{∞} be a non-increasing rearrangement of the multi-sequence {λ_{(r_{1},... ,r_{d}),i}}_{r_{1},... ,r_{d}=0}^{∞}, and let the sequence {μ_{n,i}}_{n=0}^{∞} be a nonincreasing rearrangement of the multi-sequence {γ_{i}^{r_{1}+...+r_{d}}}_{r_{1},... ,r_{d}=0}^{∞}. Then (3.16) implies that

λ_{n,i} ≤ Kμ_{n,i}  (3.17)

for all i ∈ {1, …, k}, n ≥ 0.

Note that there are \binom{N+d}{d} non-negative integer vectors (r_{1}, …, r_{d}) ∈ ℕ_{0}^{d} such that r_{1} + … + r_{d} ≤ N (see [Fri, p. 506]). Therefore the smallest n with μ_{n} = γ_{i}^{N+1} is n = \binom{N+d}{d} = O(N^{d}), and we have

μ_{n,i} = O (γ_{i}^{n^{1/d}}) as n → ∞  (3.18)

Together (3.17) and (3.18) give that

λ_{n,i} = O (γ_{i}^{n^{1/d}}) as n → ∞  (3.19)

Defining γ = max{γ_{1}, …, γ_{k}} < 1 (again note that this estimate is not optimal) gives us, for each i ∈ {1, …, k},

λ_{n,i} = O (γ^{n^{1/d}}) as n → ∞.  (3.20)

So if the sequence {λ_{n}}_{n=0}^{∞} is a non-increasing rearrangement of the multi-sequence {λ_{n,i} : (n, i) ∈ ℕ_{0} × {1, …, k}} then setting α = γ^{1/k} < 1 gives us

λ_{n} = O (α^{n^{1/d}}) as n → ∞.  (3.21)

Now each of the above non-increasing rearrangements induces an obvious rearrangement of the corresponding multi-sequences of functions and functionals, with the new sequences just inheriting the same subindices. In particular we arrive at a sequence {l_{n}}_{n=0}^{∞} of functionals, and a sequence {u_{n}}_{n=0}^{∞} of functions, such that the representation (3.15) becomes L_{s}h = ∑_{n=0}^{∞} λ_{n}l_{n}(h)u_{n}. Then the estimate (3.21) ensures that ∑_{n=0}^{∞} |λ_{n}|^{p} < ∞ for all p > 0, so that the operator L_{s} is nuclear of order zero.

3.4 Trace formulae.

The fact that the operators L_{s} are nuclear of order zero means they have well-defined spectral traces (the sum of all eigenvalues, counted with algebraic multiplicities). The key to our method is the following explicit formula for the traces of the powers L_{s}^{n} in terms of the fixed points of our iterated function scheme.

Proposition 3. If L_{s} : A_{∞}(D) → A_{∞}(D) is the transfer operator associated to a conformal iterated function scheme then

tr(L_{s}^{n}) = ∑_{\underline{i}∈Fix(n)} \frac{|Dϕ\underline{i}(z\underline{i})|^{s}}{det(I − Dϕ\underline{i}(z\underline{i}))},

where Dϕ\underline{i} and |Dϕ\underline{i}(·)| are the d-dimensional and conformal derivatives, respectively, of the map ϕ\underline{i}.

Proof. For each admissible string \underline{i} = (i_{1}, …, i_{n+1}) let us first define composition-like operators L_{s,\underline{i}} : A_{∞}(D_{i_{n+1}}) → A_{∞}(D_{i_{1}}) by

L_{s,\underline{i}}g(z) = g(ϕ\underline{i}z)w_{s,\underline{i}}(z),  (3.22)

where the weight functions w_{s,\underline{i}} ∈ A_{∞}(D_{i_{1}}) are given by

w_{s,\underline{i}}(z) = |Dϕ\underline{i}(z)|^{s}.

For a fixed i_{1} = i, the n^{th} iterate of the component transfer operator L_{s,i} (see (3.7)) is given by

L_{s,i}^{n} = ∑_{{|\underline{i}|=n+1 \atop i_{1}=i}} L_{s,\underline{i}},

where the summation is over those length-(n+1) admissible strings \underline{i} = (i_{1}, …, i_{n+1}) with i_{1} = i.

Then note that the n^{th} iterates of the operators M_{s,i} : A_{∞}(D) → A_{∞}(D) (defined by (3.10)) satisfy

M_{s,i}^{n}u|_{D_{i}} = L_{s,i}^{n}u, M_{s,i}^{n}u|_{D_{j}} = 0 if j ∈ {1, …, k} \ {i},

so we can express

L_{s}^{n} = ∑_{i=1}^{k} M_{s,i}^{n}.  (3.23)

The additivity of the trace means we then have

tr(L_{s}^{n}) = ∑_{i=1}^{k} tr(M_{s,i}^{n}) = ∑_{i=1}^{k} tr(L_{s,i}^{n})

= ∑_{i=1}^{k} ∑_{{|\underline{i}|=n+1 \atop i_{1}=i}} tr(L_{s,\underline{i}})

= ∑_{|\underline{i}|=n+1} tr(L_{s,\underline{i}})

= ∑_{\underline{i}∈Fix(n)} tr(L_{s,\underline{i}}).

(3.24)

The last equality in the above follows because if i_{1} ≠ i_{n+1} then the domain and target spaces of the operator L_{s,\underline{i}} : A_{∞}(D_{i_{n+1}}) → A_{∞}(D_{i_{1}}) are not the same, so it has no eigenvalues.

If \underline{i} ∈ Fix(n), however, we have the following trace formula for the operators L_{s,\underline{i}} in terms of the fixed point z\underline{i} of the composition ϕ\underline{i},

tr(L_{s,\underline{i}}) = \frac{w_{s,\underline{i}}(z\underline{i})}{det(I − Dϕ\underline{i}(z\underline{i}))} = \frac{|Dϕ\underline{i}(z\underline{i})|^{s}}{det(I − Dϕ\underline{i}(z\underline{i}))}.  (3.25)

The above formula (3.25) has its origins in the work of Atiyah & Bott [AB] on the Lefschetz fixed point theorem, and in our context is proved in [May1] (see also [May2]). Note that since ϕ\underline{i} : U_{i_{1}} → U_{i_{n}} is a contraction, then the determinant det(I −Dϕ\underline{i}(\underline{z}\underline{i})) > 0.

Combining (3.24) and (3.25) completes the proof.

3.5 Fredholm determinants.

Definition. For s ∈ ℂ and z ∈ ℂ we define the Fredholm determinant det(I − zL_{s}) of the transfer operator L_{s} by

det(I − zL_{s}) = exp (− ∑_{n=1}^{∞} \frac{z^{n}}{n} tr(L_{s}^{n})) = exp (− ∑_{n=1}^{∞} \frac{z^{n}}{n} ∑_{\underline{i}∈Fix(n)} \frac{|Dϕ\underline{i}(z\underline{i})|^{s}}{det(I − Dϕ\underline{i}(z\underline{i}))}) .

(3.26)

Proposition 4. The Fredholm determinant det(I − zL_{s}) is an entire function of both z and s.

Proof. The proof that det(I − zL_{s}) is an entire function of z follows immediately from the fact that L_{s} is nuclear [Gr1]. For entirety as a function of s we note that the operators L_{s} depend holomorphically on s, and then use [Gr2, p. 346].

Proposition 5. For any s ∈ ℂ, let λ_{r}(s), r = 1, 2, … be an enumeration of the non-zero eigenvalues of L_{s}, counted with algebraic multiplicities. Then

det(I − zL_{s}) = ∏_{r=1}^{∞}(1 − zλ_{r}(s)).

In particular, the set of zeros z of the Fredholm determinant det(I − zL_{s}), counted with algebraic multiplicities, is equal to the set of reciprocals of non-zero eigenvalues of L_{s}, counted with algebraic multiplicities.

Proof. This is classical Fredholm theory, valid for any nuclear operator of order 2/3 [Gr2, p. 355], and in particular for the operators L_{s}, which are nuclear of order zero.

Proposition 6. Given an iterated function scheme, the Hausdorff dimension dim(Λ) of its limit set Λ is the largest real zero of the function s ↦ det(I − L_{s}).

Proof. If s is real then by Proposition 1 and Lemma 1, the operator L_{s} has simple maximal eigenvalue e^{p(s)}, which equals 1 if and only if s = dim(Λ). But Proposition 5 means that 1 is an eigenvalue of L_{s} if and only if s is a zero of det(I − L_{s}).

To see that dim(Λ) is the largest real zero of det(I − L_{s}), note that the spectral radius of L_{s} equals e^{p(s)}, so is a decreasing function of s. In particular, if s > dim(Λ) then the spectral radius of L_{s} is less than 1, so that 1 cannot be an eigenvalue of L_{s}, and hence cannot be a zero of det(I − L_{s}).

By Proposition 4 we can expand det(I − zL_{s}) as a power series in z. The power series coefficients are described by the following lemma, due to Grothendieck [Gr2].

Lemma 2. If L_{s}u(z) = ∑_{n=0}^{∞} λ_{n}l_{n}(u)v_{n} is a nuclear representation (cf. (3.12), (3.15)) for the operator L_{s}, and det(I − zL_{s}) = 1 + ∑_{N=1}^{∞} d_{N}(s)z^{N} is the power series expansion of its Fredholm determinant, then

d_{N}(s) = (−1)^{N} ∑_{k_{1}<...<k_{N}} λ_{k_{1}} … λ_{k_{N}} det (l_{i_{p}}(v_{i_{q}}))_{p,q=1}^{n}  (3.27)

where

det [l_{i_{p}}(v_{i_{q}})]_{p,q=1}^{N} = | \begin{matrix}l_{i_{1}}(v_{i_{1}}) & … & l_{i_{N}} (v_{i_{1}}) \\ ⋮ & & ⋮ \\ l_{i_{N}} (v_{i_{1}}) & … & l_{i_{N}} (v_{i_{N}}) \end{matrix} |

denotes the determinant of the N × N matrix with entries l_{i_{p}}(v_{i_{q}}), 1 ≤ p, q ≤ N, corresponding to functionals l_{i_{p}} and functions v_{i_{q}}.

Proposition 7. There exists 0 < δ < 1 such that for all s ≥ 0 the power series coefficients d_{N}(s) defined by (3.27) satisfy d_{N}(s) = O (δ^{N^{1+1/d}}) as N → ∞.

Proof. The estimate (3.21) means there exist 0 < α < 1, K > 0 such that λ_{n} ≤ Kα^{n^{1/d}} for all n ≥ 0. Moreover, by a theorem of Hadamard [Had], any N × N matrix with entries of modulus at most 1 has determinant with modulus bounded by N^{N/2}.

Therefore Lemma 1 gives us

|d_{N}(s)| ≤ K^{N}N^{N/2} ∑_{k_{1}<...<k_{N}} α^{k_{1}^{1/d}} … α^{k_{N}^{1/d}}.  (3.28)

Now β_{N}(α) := ∑_{k_{1}<...<k_{N}} α^{k_{1}^{1/d}} … α^{k_{N}^{1/d}} is precisely the N^{th} power series coefficient of the infinite product f_{α}(z) = ∏_{n=0}^{∞}(1 + α^{n^{1/d}}z) = 1 + ∑_{N=1}^{∞} β_{N}(α)z^{N}. We would like to show that β_{N}(α) = O(δ^{N^{1+1/d}}) for some 0 < δ < 1, so that the decay of β_{N}(α) dominates the growth of K^{N}N^{N/2}, giving d_{N}(s) = O(δ^{N^{1+1/d}}). For d = 1 the term β_{N}(α) can in fact be expressed in closed form, using an N-fold geometric summation (see §6.3). For general d the following analysis closely follows that of Fried [Fri, p. 506].

For any r > 0, Cauchy’s estimate gives β_{N}(α) ≤ M(r)/r^{N}, where M(r) = M_{α}(r) := max_{|z|=r} |f_{α}(z)|. Let m(r) denote the number of zeros z of f_{α} satisfying |z| ≤ r. Since the zeros of f_{α} are just z = −α^{−n^{1/d}}, for n = 0, 1, …, we can estimate m(r) ≤ a(log_{+} r)^{d}, where a = a_{α} := (− log α)^{−d} > 0. From the infinite product defining f_{α}, it is clear that f_{α} has genus zero (see [Boas, §2.5, 2.7]). Since also f(0) = 1, by [Boas, p. 47] we have log M(r) ≤ N(r) + Q(r), where N(r) = ∫_{0}^{r} \frac{m(t)}{t}dt and Q(r) = r ∫_{r}^{∞} \frac{m(t)}{t^{2}}dt. Our upper bound on m(r) then gives us

log M(r) ≤ a ∫_{1}^{r} \frac{(log t)^{d}}{t} dt + ar ∫_{max\{1,r\}}^{∞} \frac{(log t)^{d}}{t^{2}} dt.

Substituting u = log t, we can evaluate these integrals to give

log M_{α}(r) ≤ a_{α}P (log r),

where P (x) := ∑_{j=0}^{d+1} \frac{d!}{j!}x^{j}. In particular, P is a degree-(d + 1) polynomial whose leading coefficient is 1/(d + 1).

Therefore we have

β_{N}(α) ≤ \frac{M_{α}(r)}{r^{N}} ≤ r^{−N} exp (a_{α}P (log r))  (3.29)

for all r > 0. For each N we will evaluate this inequality at r = r_{N} = exp ((N/a)^{1/d}). Now r_{N}^{−N} = exp (−a^{−1/d}N^{1+1/d}), while the leading term in aP (log r_{N}) = aP ((N/a)^{1/d}) is \frac{1}{d+1}a^{−1/d}N^{1+1/d}. Thus

r_{N}^{−N} exp (aP (log r_{N})) = exp ((−1 + 1/(d + 1))a^{−1/d}N^{1+1/d} + O(N)) = ε^{N^{1+1/d}} exp(O(N)),

where ε = exp (\frac{−d}{d+1}a^{−1/d}) = α^{d/(d+1)} < 1.

From (3.29) we have β_{N}(α) ≤ r_{N}^{−N} exp(aP (log r_{N})) = O(ε_{1}^{N^{1+1/d}}) for any 1 > ε_{1} > ε = α^{d/(d+1)}, so substituting into (3.28) gives d_{N}(s) = O(δ^{N^{1+1/d}}) for any α^{d/(d+1)} < δ < 1.

Remark 1. Fried’s estimate actually corrects a minor error in Grothendieck’s original paper [Gr1] which was replicated in Ruelle’s paper [Ru1].

Remark 2. In dimension d = 1 we can in fact take δ = α = γ^{1/k}, where γ is related (by Lemma 4 in Appendix B) to the contraction ratios of the contraction maps in the iterated function scheme (see §6.3 for more details).

3.6 The approximating functions Δ_{N}.

Grothendieck’s expression (3.27) for the power series coefficients of the Fredholm determinant was useful for estimating their asymptotic decay rate (Proposition 7), and can be used to give rigorous estimates on dimension estimates (see §6).

However there is another expression for the d_{N}(s), in terms of fixed points of our iterated function scheme (which correspond to periodic points of any associated expanding map, see §1). This expression arises by applying the series expansion for exp to the trace formula representation (3.26) of det(I − zL_{s}), and then regrouping powers of z. Explicitly we have

Proposition 8. Let det(I − zL_{s}) = 1 + ∑_{N=1}^{∞} d_{N}(s)z^{N} be the power series expansion of the Fredholm determinant of the transfer operator L_{s}. Then

d_{N}(s) = ∑_{{(n_{1},... ,nm) \atop n_{1}+...+nm=N}} \frac{(−1)^{m}}{m!} ∏_{l=1}^{m} \frac{1}{n_{l}} ∑_{\underline{i}∈Fix(n_{l})} \frac{|Dϕ\underline{i}(z\underline{i})|^{s}}{det(I − Dϕ\underline{i}(z\underline{i}))},  (3.30)

where the summation is over all ordered m-tuples of positive integers whose sum is N.

The above formula (3.30) allows an explicit calculation of any coefficient d_{N}(s), provided we can locate all fixed points ∪_{n=1}^{N}Fix(n) of compositions of ≤ N contractions (or, equivalently, all points of period ≤ N if our system is related to an expanding map T, see §1).

With this in mind, we are now in a position to define the sequence of functions Δ_{N} which allow us to approximate the dimension dim(Λ) of the limit set of our iterated function scheme. In view of Proposition 6, these functions will arise by taking the N^{th} order polynomial truncation of the power series expansion for det(I − zL_{s}), and then setting z = 1.

Definition. Define the function Δ_{N} : ℂ → ℂ by

Δ_{N}(s) = 1 + ∑_{n=1}^{N} d_{n}(s) = 1 + ∑_{n=1}^{N} ∑_{{(n_{1},... ,nm) \atop n_{1}+...+nm=n}} \frac{(−1)^{m}}{m!} ∏_{l=1}^{m} \frac{1}{n_{l}} ∑_{\underline{i}∈Fix(n_{l})} \frac{|Dϕ\underline{i}(z\underline{i})|^{s}}{det(I − Dϕ\underline{i}(z\underline{i}))}.

Note we have that

Δ_{N+1}(s) = Δ_{N}(s) + ∑_{{(n_{1},... ,nm) \atop n_{1}+...+nm=N+1}} \frac{(−1)^{m}}{m!} ∏_{l=1}^{m} \frac{1}{n_{l}} ∑_{\underline{i}∈Fix(n_{l})} \frac{|Dϕ\underline{i}(z\underline{i})|^{s}}{det(I − Dϕ\underline{i}(z\underline{i}))}.

If our iterated function scheme is related to an expanding map T, and Markov map T̂, then the formula for the Δ_{N} is simply

Δ_{N}(s) = 1 + ∑_{n=1}^{N} ∑_{{(n_{1},... ,nm) \atop n_{1}+...+nm=n}} \frac{(−1)^{m}}{m!} ∏_{l=1}^{m} \frac{1}{n_{l}} ∑_{T̂^{n_{l}} z=z} \frac{|DT̂^{n_{l}}(z)|^{−s}}{det (I − [DT̂^{n_{l}}(z)]^{−1})}.

Examples. We can easily compute the first few terms in this expansion. Let us consider the case corresponding to a full shift on 2 symbols.

(a) There are two fixed points T̂(z_{0}) = z_{0} and T̂(z_{1}) = z_{1} and we can write

d_{1}(s) = −\frac{|DT̂(z_{0})|^{−s}}{det(I − [DT̂(z_{0})]^{−1})} − \frac{|DT̂(z_{1})|^{−s}}{det(I − [DT̂(z_{1})]^{−1})}.

(b) There is a single orbit {z_{01}, z_{10}} of prime period 2 and we can write

d_{2}(s) = \frac{1}{2} (\frac{|DT̂(z_{0})|^{−s}}{det(I − [DT̂(z_{0})]^{−1})})^{2} + \frac{1}{2} (\frac{|DT̂(z_{1})|^{−s}}{det(I − [DT̂(z_{1})]^{−1})})^{2}

+ \frac{|DT̂(z_{0})|^{−s}|DT̂(z_{1})|^{−s}}{det(I − [DT̂(z_{0})]^{−1})det(I − [DT̂(z_{1})]^{−1})} − \frac{|DT̂^{2}(z_{01})|^{−s}}{det(I − [DT̂^{2}(z_{01})]^{−1})}

− \frac{1}{2} \frac{|DT̂^{2}(z_{0})|^{−s}}{det(I − [DT̂^{2}(z_{0})]^{−1})} − \frac{1}{2} \frac{|DT̂^{2}(z_{1})|^{−s}}{det(I − [DT̂^{2}(z_{1})]^{−1})}.

Our most general result is the following.

Theorem 4. (The general theorem) Let Λ be the limit set for a real-analytic conformal iterated function scheme. Suppose Λ lies in a d-dimensional real-analytic manifold. Then there exists C > 0 and 0 < δ < 1 such that if 0 ≤ s_{N} ≤ d is the largest real zero for Δ_{N} then

|dim(Λ) − s_{N}| ≤ Cδ^{N^{1+1/d}}

Proof. As described in §1 we may assume that the manifold in question is actually ℝ^{d}, and then use the results already proved in §3.

Now s_{∞} := dim(Λ) is the largest real zero of Δ(s) = det(I − L_{s}), by Proposition 6. Let s_{N} denote the largest real zero of Δ_{N}(s). The definition of Δ_{N} ensures s_{N} → s_{∞}.

By the mean value theorem there exists t_{N} between s_{N} and s_{∞} such that

\begin{aligned}(s_{N} − s_{∞})Δ′_{N}(t_{N}) & = Δ_{N}(s_{N}) − Δ_{N}(s_{∞}) = −Δ_{N}(s_{∞}) \\ & = Δ(s_{∞}) − Δ_{N}(s_{∞}) = ∑_{n=N+1}^{∞} d_{n}(s_{∞}). \end{aligned}

If we can show that Δ′(s_{∞}) > 0, then Δ′_{N}(t_{N}) will be bounded away from zero for N sufficiently large. Using the bound |d_{n}(s)| ≤ Cδ^{n^{1+1/d}} of Proposition 7 we will then have

\begin{aligned}|s_{N} − s_{∞}| & ≤ |\frac{1}{Δ′_{N}(t_{N})} ∑_{n=N+1}^{∞} d_{n}(s_{∞})| ≤ \frac{1}{|Δ′_{N}(t_{N})|} ∑_{n=N+1}^{∞} Cδ^{n^{1+1/d}} \\ & = O(δ^{N^{1+1/d}}), \end{aligned}

as required.

To show that indeed Δ′(s_{∞}) > 0, first note that by Proposition 5 we have the infinite product representation

D(z) = det(I − zL_{s}) = ∏_{r=1}^{∞}(1 − zλ_{r}(s)),

where λ_{r}(s) are the eigenvalues of L_{s}, listed according to algebraic multiplicity, and ordered so that their absolute values are non-increasing. For real values of s the leading eigenvalue λ_{1}(s) is simple and positive, so λ_{1}(s) > |λ_{r}(s)| for all r ≥ 2.

Setting Q(s) = ∏_{r=2}^{∞}(1 − λ_{r}(s)) we have

Δ(s) = (1 − λ_{1}(s))Q(s),

so that

Δ′(s) = −λ′_{1}(s)Q(s) + (1 − λ_{1}(s))Q′(s).

We know that λ_{1}(s_{∞}) = 1, so

Δ′(s_{∞}) = −λ′_{1}(s_{∞})Q(s_{∞}).

Moreover, λ_{1}(s) = e^{p(s)} is a strictly decreasing analytic function of s, so in particular −λ′_{1}(s_{∞}) > 0. Therefore we need to show that Q(s_{∞}) > 0.

For any real value of s (and in particular s = s_{∞}), the coefficients in the power series expansion of D(z) are all real, by (3.30). Therefore non-real zeros of D arise as conjugate pairs, both with the same multiplicity. Multiplying out those factors in the product representation of Q corresponding to conjugate pairs, we see that Q(s_{∞}) is an infinite product of strictly positive terms (since |λ_{r}(s_{∞})| < 1 for each r ≥ 2). The sequence of terms converges to 1, since |λ_{r}(s_{∞})| → 0. Therefore the infinite product converges to a strictly positive value. That is, Q(s_{∞}) > 0.

Proof of Theorem 3. This follows immediately from Theorem 4, using the Markov partition and inverse branches of T to construct an iterated function scheme (see §1).

Remark. For any positive integer q we can consider the Fredholm determinant det(I − z^{q}L_{s}^{q}), which factors as the product ∏_{ω} det(I −ωzL_{s}) over all q^{th} roots of unity ω. Setting z = 1 we see that det(I − L_{s}) is a factor of det(I − L_{s}^{q}), and as before we can show that the largest real zero of det(I − L_{s}^{q}) is again the Hausdorff dimension dim(Λ).

Thus, by analogy with the definition of the Δ_{N}, we may define a sequence Δ_{rq}^{(q)}, r = 1, 2, … of approximations to det(I − L_{s}^{q}) as follows:

Δ_{rq}^{(q)} = 1 + ∑_{j=1}^{r} ∑_{{(n_{1},... ,nm) \atop n_{1}+...+nm=j}} \frac{(−1)^{m}}{m!} ∏_{l=1}^{m} \frac{1}{qn_{l}} ∑_{\underline{i}∈Fix(qn_{l})} \frac{|Dϕ\underline{i}(z\underline{i})|^{s}}{det(I − Dϕ\underline{i}(z\underline{i}))}.

To compute Δ_{rq}^{(q)} we need only work with those points whose period is a multiple of q, up to and including rq. As in Theorem 4, we can show there exists 0 < δ < 1 such that |dim(Λ) − s_{N}^{(q)}| = O(δ^{N^{1+1/d}}), where s_{N}^{(q)} denotes the largest real zero of Δ_{rq}^{(q)}.

In some cases it is advantageous to make a judicious choice of q and use the sequence of functions Δ_{rq}^{(q)} rather than the sequence Δ_{N} (see §7 where Δ_{20}^{(2)} gives a better empirical dimension estimate than Δ_{20}).

4. Schottky groups and Quasifuchsian groups

The most general definition of a classical Schottky group is the following (see [Mas]). Let D_{1}, …, D_{p}, D_{p+1}, …, D_{2p} be 2p closed Euclidean discs of dimension d − 1, lying in ℝ^{d} with pairwise disjoint interiors. Let g_{1}, …, g_{p} be Möbius maps such that each g_{i} exchanges the interior of D_{i} with the exterior of D_{p+i} (i.e. g_{i}(int(D_{i})) = ext(D_{p+i})). The corresponding classical Schottky group Γ is the Kleinian group generated by g_{1}, …, g_{p}.

For our theory to apply, we will always make the additional assumption that the discs themselves (rather than just their interiors) are pairwise disjoint (this corresponds to Γ being cocompact). In this case the associated limit set Λ of Γ is a Cantor subset of the union of the interiors of the discs D_{1}, …, D_{2p}. The same limit set can also be generated by changing Γ to a group generated by reflections in a finite number of circles [PS]. A reflection group is a classical Schottky group such that D_{i} = D_{p+i} for all i = 1, …, p.

Classical Schottky groups are most commonly considered in dimension d = 2. There is a more general notion of (non-classical) Schottky groups, where the circles bounding the discs D_{1}, …, D_{2p} are replaced by Jordan curves.

A quasifuchsian group Γ is a Kleinian group whose domain of discontinuity consists of two invariant components Ω_{1}, Ω_{2}, such that Ω_{1}/Γ is a finite Riemann surface [Mas, IX.B.2]. Every quasifuchsian group can be obtained by a quasiconformal deformation of a Fuchsian group ([Mas, IX.F]). The limit set Λ of a quasifuchsian group is a simple closed curve.

Proof of Theorem 1. First suppose Γ is a Schottky group. We define a map T on the union ∪_{j=1}^{2p}D_{j} by T |_{int(D_{j})} = g_{j} and T |_{int(D_{p+j})} = g_{j}^{−1}, for j = 1, …, p, A Markov partition for this map just consists of the collection of interiors {int(D_{i})}_{i=1}^{2p}. The corresponding 2p × 2p transition matrix A has entries A(i, p + i) = 0 = A(p + i, i) for each i = 1, …, p, and all other entries are 1 (in the reflection group case, the transition matrix has zeros along the leading diagonal, and 1’s elsewhere).

Now T is not quite an expanding map, since the conformal derivative |Dg_{j}(z)| = 1 on the boundary of D_{j}. However, the second iterate of T is expanding. Conformality and real-analyticity are clearly satisfied, so by Theorem 3 we deduce the result for Schottky groups.

Suppose Γ is quasifuchsian, with limit set Λ. Now Γ is quasi-conformally conjugate to some Fuchsian group Γ′ ([Mas, IX.F]). Bowen & Series [BS] proved there exists an expanding Markov map S : S^{1} → S^{1} which faithfully models the action of Γ′, and the quasiconformal deformation conjugates this to an expanding Markov map T : Λ → Λ. Conformality and real-analyticity are clearly satisfied, so the quasifuchsian case of Theorem 1 follows from the general Theorem 3.

5. Julia sets

5.1 Markov partitions and hyperbolicity.

For a holomorphic map f : U → U defined on some domain U ⊂ ℂ of the complex plane, we define its Julia set J to be the closure of the repelling periodic points of f. More precisely,

J = \overline{∪_{n≥1}\{z ∈ ℂ : f^{n}z = z and |(f^{n})′(z)| > 1\}}.

The Julia set is a closed f-invariant set, and is locally maximal. Clearly f : J → J is conformal and real-analytic, so if f : J → J is also Markov and hyperbolic then we can use Theorem 3 to compute the Hausdorff dimension dim(J).

Lemma 3. Markov partitions always exist for expanding maps f : J → J [Ru2, p.146], [Bo1].

The following well-known result (see [St, p. 118]) characterizes hyperbolic rational maps.

Proposition 9. The following are equivalent (1) f : J → J is hyperbolic;

(2) J is disjoint from the orbit of the critical points C = {z : f′(z) = 0} (i.e. J ∩ (∪_{n=0}^{∞}f^{n}(C)) = ∅)

If we restrict to the case of quadratic polynomials f_{c}(z) = z^{2} + c, then there is the following characterisation of hyperbolicity (see [CG, p128]) in terms of the Mandelbrot set M := {c ∈ ℂ : |f_{c}^{n}(0)| ↛ +∞ as n → +∞} .

Proposition 10. Let f_{c}(z) = z^{2} + c be a quadratic map, with Julia set J_{c}. Then f_{c} : J_{c} → J_{c} is hyperbolic if and only if either c lies outside M, or f_{c} has an attracting periodic point z (i.e. f_{c}^{n}z = z for some n, and |(f^{n})′(z)| < 1).

Note that if c ∉ M then the Julia set J_{c} is a Cantor set, while if c ∈ M then J_{c} is connected.

The interior int(M) of the Mandelbrot set is a union of simply connected components, the largest of which is the main cardioid M_{1} = \{w ∈ ℂ : |1 − \sqrt{1 − 4w}| < 1\}. For any c ∈ M_{1} the map f_{c} has the attracting fixed point \frac{1}{2}(1 − \sqrt{1 − 4w)}, so is hyperbolic by Proposition 9.

More generally, a component H of int(M) contains a parameter c for which f_{c} is hyperbolic if and only if f_{c′} is hyperbolic for every c′ ∈ H. We then call H a hyperbolic component [St, p.160]. It is unknown whether every component of int(M) is hyperbolic.

5.2 Trace formula for hyperbolic holomorphic Markov maps.

In §3.4 we proved, in a very general case, a trace formula for iterates of the transfer operator L_{s} in terms of fixed points of an iterated function scheme. Now we use this to derive a simplified formula in the special case of hyperbolic holomorphic Markov maps restricted to their Julia sets.

Proposition 11. Let f : J → J be a hyperbolic holomorphic Markov map, with Julia set J. Let L_{s} be the associated transfer operator. Then

tr(L_{s}^{n}) = ∑_{{f^{n}z=z \atop z∈J}} |(f^{n})′(z)|^{−s} (1 + \frac{1 − 2Re((f^{n})′(z))}{|(f^{n})′(z)|^{2}})^{−1} .

Proof. Identifying ℝ^{2} with ℂ we think of J as lying in the real section ℝ^{2} of ℂ^{2}. Letting T (x, y) = (Re (f(x + iy)) , Im (f(x + iy))), we know that T extends to a holomorphic map on some open neighbourhood U ⊂ ℂ^{2} of J.

Using the Markov partition X_{1}, …, X_{k}, we define a collection of inverse branches {S_{j}}_{j=1}^{k} to T satisfying T ∘ S_{j} = id|_{T X_{j}}. As in §1, we consider the set of all admissible length-n strings \underline{j} = (j_{1}, …, j_{n}), and note that they correspond bijectively to those length-(n + 1) strings in the set Fix(n). For the purposes of this proof let L_{s,\underline{j}} denote the operator L_{s,\underline{i}} (see (3.19)), where \underline{i} = (j_{n}, j_{1}, …, j_{n}). By Proposition 3 we therefore have that tr(L_{s}^{n}) = ∑_{|\underline{j}|=n} tr(L_{s,\underline{j}}), where

tr(L_{s,\underline{j}}) = \frac{|(f^{n})′(z\underline{j})|^{−s}}{det(I − DS\underline{j}(x\underline{j}, y\underline{i}))}.  (5.1)

Here S\underline{j} is as in §1 (i.e. an inverse branch of T^{n}), and (x\underline{j}, y\underline{j}) denotes the unique fixed point of S\underline{j}. Note that (x\underline{j}, y\underline{j}) ∈ ℝ^{2}, and we let z\underline{j} = x\underline{j} + iy\underline{j}.

The numerator in (5.1) follows immediately from Proposition 3, since the conformal derivative |DS\underline{j}(x\underline{j}, y\underline{j})| = |DT^{n}(x\underline{j}, y\underline{j})|^{−1} = |f^{n}(x\underline{j} + iy\underline{j})|^{−1}.

To treat the denominator in (5.1) we note that

det(I − DS\underline{j}(x\underline{j}, y\underline{i})) = \frac{det(DT^{n}(x\underline{j}, y\underline{j}) − I)}{det(DT^{n}(x\underline{j}, y\underline{j}))}.  (5.2)

Let us write T^{n}(x, y) = ((T^{n})_{1}(x, y), (T^{n})_{2}(x, y)), so that

\begin{aligned}DT^{n}(x\underline{j}, y\underline{j}) & = ( \begin{matrix}D_{1}(T^{n})_{1}(x\underline{j}, y\underline{j}) & D_{2}(T^{n})_{1}(x\underline{j}, y\underline{j}) \\ D_{1}(T^{n})_{2}(x\underline{j}, y\underline{j}) & D_{2}(T^{n})_{2}(x\underline{j}, y\underline{j}) \end{matrix} ) \\ & = ( \begin{matrix}D_{1}(T^{n})_{1}(x\underline{j}, y\underline{j}) & D_{2}(T^{n})_{1}(x\underline{j}, y\underline{j}) \\ −D_{2}(T^{n})_{1}(x\underline{j}, y\underline{j}) & D_{1}(T^{n})_{1}(x\underline{j}, y\underline{j}) \end{matrix} ) , \end{aligned}

by the Cauchy-Riemann equations, where D_{i} denotes partial differentiation with respect to the i^{th} coordinate.

Therefore

det(DT^{n}(x\underline{j}, y\underline{j}) − I) = (1 − D_{1}(T^{n})_{1}(x\underline{j}, y\underline{j}))^{2} + (D_{2}(T^{n})_{1}(x\underline{j}, y\underline{j}))^{2}
= 1 − 2D_{1}(T^{n})_{1}(x\underline{j}, y\underline{j}) + (D_{1}(T^{n})_{1}(x\underline{j}, y\underline{j}))^{2} + (D_{2}(T^{n})_{1}(x\underline{j}, y\underline{j}))^{2}
= 1 − 2D_{1}(T^{n})_{1}(x\underline{j}, y\underline{j}) + det(DT^{n}(x\underline{j}, y\underline{j})). (5.3)

Now

D_{1}(T^{n})_{1}(x\underline{j}, y\underline{j}) = Re((f^{n})′(z\underline{j})),  (5.4)

and area considerations mean that

det(DT^{n}(x\underline{j}, y\underline{j})) = |(f^{n})′(z\underline{j})|^{2},  (5.5)

so substituting (5.4), (5.5) into (5.3), and then into (5.2) gives us

det(I − DS\underline{j}(x\underline{j}, y\underline{j})) = 1 + \frac{1 − 2Re((f^{n})′(z\underline{i}))}{|(f^{n})′(z\underline{j})|^{2}}.

Substituting into (5.1) gives

tr(L\underline{j}) = |(f^{n})′(z\underline{j})|^{−s} (1 + \frac{1 − 2Re((f^{n})′(z\underline{j}))}{|(f^{n})′(z\underline{j})|^{2}})^{−1} ,

then using the fact that tr(L_{s}^{n}) = ∑_{|\underline{j}|=n} tr(L_{s,\underline{j}}) gives the result.

In particular, for hyperbolic quadratic maps f_{c}(z) = z^{2} + c we have the following trace formula.

Corollary 11.1. Let f_{c}(z) = z^{2} + c be a quadratic map, with hyperbolic Julia set J_{c}. Let L_{s} be the associated transfer operator. Then

tr(L_{s}^{n}) = ∑_{{f^{n}_{c}z=z \atop z∈Jc}} 2^{−sn}|π(z)|^{−s} (1 + \frac{1 − 2^{n+1}Re(π(z))}{2^{2n}|π(z)|^{2}})^{−1} ,

where π(z) := ∏_{r=0}^{n−1} f_{c}^{r}(z) for period-n points z.

Corollary 11.2. Let f : J → J be a hyperbolic holomorphic map, with Julia set J. The corresponding functions Δ_{N}, whose leading zeros s_{N} give a sequence of approximations to dim(J), are given by the formula

Δ_{N}(s) = 1+∑_{n=1}^{N} ∑_{{(n_{1},... ,nm) \atop n_{1}+...+nm=n}} \frac{(−1)^{m}}{m!} ∏_{l=1}^{m} \frac{1}{n_{l}} ∑_{{f^{n_{l}} z=z \atop z∈J}} |(f^{n_{l}})′(z)|^{−s} (1 + \frac{1 − 2Re((f^{n_{l}})′(z))}{|(f^{n_{l}})′(z)|^{2}})^{−1} .

Proof of Theorem 2. This follows from Theorem 3 upon setting d = 2, and using Corollary 11.2 to express the functions Δ_{N} in terms of derivatives at periodic points.

6. Fuchsian reflection in three symmetric circles

6.1 McMullen’s three circle family.

We now analyse in detail one of the examples considered by McMullen [McM3], a family of symmetric Fuchsian reflection groups.

Let C_{0}, C_{1}, C_{2} be three circles of equal radii arranged symmetrically around the unit circle S^{1}, each intersecting S^{1} orthogonally, and meeting S^{1} in an arc of length θ. Let θ denote the angle at the origin subtended by the two points of intersection.

We do not want the C_{i} to intersect each other, so we demand that 0 < θ < 2π/3. For definiteness let us suppose each C_{i} has radius r = r_{θ} = tan \frac{θ}{2}, and that the circle centres are at the points z_{0} = a, z_{1} = ae^{2πi/3} and z_{2} = ae^{−2πi/3}, where

a = a_{θ} = \sqrt{1 + r^{2}} = sec \frac{θ}{2}.

Since 0 < θ < 2π/3, we see that

0 < r_{θ} < \sqrt{3} and 0 < a_{θ} < 2.

Figure 5. Reflection in three circles

The reflection ρ_{i} : ℂ̂ → ℂ̂ in the circle C_{i} takes the explicit form

\begin{aligned}ρ_{i}(z) & = \frac{r^{2}}{z̄ − z̄_{i}} + z_{i} \\ & = \frac{r^{2}}{|z − z_{i}|^{2}} (z − z_{i}) + z_{i}. \end{aligned}

Note these reflections are anti-holomorphic, and in particular reverse orientation.

Let Γ_{θ} be the group generated by these three reflections, and Λ_{θ} the associated limit set.

Since the reflections ρ_{i} each preserve the unit circle S^{1}, then Λ_{θ} ⊂ S^{1}. However, by a conformal change of coordinates we will transform the limit set to a subset of the real line, and by Theorem 3 with d = 1 our algorithm will give O(δ^{N^{2}}) convergence to dim(Λ_{θ}).

More precisely, let

f(z) = i\frac{z − 1}{z + 1}

be our conformal change of coordinates. Thus if we consider the restriction of each ρ_{j} to S^{1}, then the conjugated map T_{j} := f ∘ ρ_{j}|_{S^{1}} ∘ f^{−1} is a map of the real line to itself.

A calculation shows that T_{j} : ℝ → ℝ is the Möbius map

T_{j}(z) = \frac{i(z_{j} − z̄_{j})z + z_{j} + z̄_{j} − 2}{(z_{j} + z̄_{j} + 2)z + i(z̄_{j} − z_{j})}.

Symmetry considerations mean we have

T_{0}(z) = \frac{a_{θ} − 1}{(a_{θ} + 1)z} = \frac{sec \frac{θ}{2} − 1}{(sec \frac{θ}{2} + 1)z} ,

T_{1}(z) = \frac{−\sqrt{3}a_{θ}z − (2 + a_{θ})}{(2 − a_{θ})z + \sqrt{3}a_{θ}},

T_{2}(z) = \frac{\sqrt{3}a_{θ}z − (2 + a_{θ})}{(2 − a_{θ})z − \sqrt{3}a_{θ}}.

Geometrically, each T_{j} : ℝ → ℝ is inversion in some interval. T_{0} is inversion in the interval I_{0} = (− tan \frac{θ}{4}, tan \frac{θ}{4}). T_{1} and T_{2} are inversions in the intervals I_{1} = (tan(\frac{π}{3} − \frac{θ}{4}), tan(\frac{π}{3} + \frac{θ}{4})) and I_{2} = (− tan(\frac{π}{3} + \frac{θ}{4}), − tan(\frac{π}{3} − \frac{θ}{4})) respectively.

Remark. Since our change of coordinates f is bi-Lipschitz, the Hausdorff dimension of the limit set generated by the T_{j} is the same as that generated by our initial reflection system given by the ρ_{j}.

6.2 Compositions and weight functions.

Now consider the absolute values of the derivatives of the maps T_{0}, T_{1}, T_{2}. We have

v_{0}(z) := |T′_{0}(z)| = \frac{a_{θ} − 1}{(a_{θ} + 1)z^{2}}

Now v_{0} : ℝ \ {0} → ℝ is real analytic on any interval not containing zero. It extends to a map v_{0} : ℂ \ {0} → ℂ, holomorphic on any domain not containing zero. For any real s ≥ 0, the weight function

\begin{aligned}w_{s,0}(z) := v_{0}(z)^{s} & = (\frac{a_{θ} − 1}{(a_{θ} + 1)z^{2}} )^{s} \\ & = exp (s log \frac{a_{θ} − 1}{(a_{θ} + 1)z^{2}} ) \end{aligned}

is holomorphic on any cut-plane ℂ \ l, where l is a half-line emanating from the point y_{0} = 0. In particular, w_{s,0} is holomorphic on both ℂ \ (−∞, 0] and ℂ \ [0, ∞).

A similar analysis applies to the maps v_{1} : ℝ \ {y_{1}} → ℝ and v_{2} : ℝ \ {y_{2}} → ℝ, where

v_{1}(z) := |T′_{1}(z)| = \frac{4(a_{θ}^{2} − 1)}{((2 − a_{θ})z + \sqrt{3}a_{θ})^{2}} = \frac{4r_{θ}^{2}}{((2 − a_{θ})z + \sqrt{3}a_{θ})^{2}}

and

v_{2}(z) := |T′_{2}(z)| = \frac{4(a_{θ}^{2} − 1)}{((2 − a_{θ})z − \sqrt{3}a_{θ})^{2}} = \frac{4r_{θ}^{2}}{((2 − a_{θ})z − \sqrt{3}a_{θ})^{2}}

and where y_{1} = −\sqrt{3}a_{θ}/(2 − a_{θ})\}, y_{2} = \sqrt{3}a_{θ}/(2 − a_{θ})\}.

Having initially restricted v_{1}, v_{2} to appropriate subsets of the real line, we then observe that these restrictions are real-analytic. We then consider the holomorphic extensions to appropriate domains in ℂ. Raising to the s^{th} power, s ≥ 0, we see that the weight function

\begin{aligned}w_{s,1}(z) := v_{1}(z)^{s} & = (\frac{4r_{θ}^{2}}{((2 − a_{θ})z + \sqrt{3}a_{θ})^{2}} )^{s} \\ & = exp (s log (\frac{4r_{θ}^{2}}{(2 − a_{θ})z + \sqrt{3}a_{θ}} )) \end{aligned}

is holomorphic on the cut-plane ℂ \ (−∞, y_{1}], where y_{1} = −\sqrt{3}a_{θ}/(2 − a_{θ}). Similarly, the weight function

\begin{aligned}w_{s,2}(z) := v_{2}(z)^{s} & = (\frac{4r_{θ}^{2}}{((2 − a_{θ})z − \sqrt{3}a_{θ})^{2}} )^{s} \\ & = exp (s log (\frac{4r_{θ}^{2}}{(2 − a_{θ})z − \sqrt{3}a_{θ}} )) \end{aligned}

is holomorphic on the cut-plane ℂ \ [y_{2}, ∞), where y_{2} = \sqrt{3}a_{θ}/(2 − a_{θ}).

Now clearly each T_{j} has a holomorphic extension to the whole of ℂ, with a single pole at y_{j}, so by abuse of notation let us write T_{j} : ℂ → ℂ. If D_{j} denotes the disc which is symmetric about ℝ and interects ℝ in the interval I_{j}, then now T_{j} is inversion in the circle ∂D_{j}.

Figure 6. The circles of inversion in the real line If r_{j} denotes the radius of D_{j}, then we have

r_{0} = tan \frac{θ}{4} , r_{1} = r_{2} = \frac{tan(\frac{π}{3} + \frac{θ}{4}) − tan(\frac{π}{3} − \frac{θ}{4})}{2}.

Note that for 0 < θ < 2π/3, the D_{j} do not intersect each other.

Let D = ∐_{i=0}^{2} D_{i} denote the disjoint union of these discs. Let A_{∞}(D) denote the space of bounded functions h : D̄ → ℂ whose restriction h|_{D_{j}} is a holomorphic function h_{j} : D̄ → ℂ, for each j = 0, 1, 2. This is a Banach space with respect to the supremum norm.

Let E_{0} be the disc centred at 0 whose boundary intersects the real axis at the points ± tan(\frac{θ}{4} − \frac{π}{3} ). Let E_{1} be the disc, concentric with D_{1}, whose boundary intersects ℝ at − tan \frac{θ}{4}. Let E_{2} be the disc, concentric with B_{2}, whose boundary intersects ℝ at tan \frac{θ}{4}.

If the function g is holomorphic on D_{j}, then the composition g ∘ T_{j} is holomorphic on T_{j}^{−1}D_{j} = T_{j}D_{j} = ext(D_{j}). In particular, g ∘ T_{j} is holomorphic on both the discs E_{k}, for k ∈ {0, 1, 2} \ {j}.

The above discussion means that if h = (h_{0}, h_{1}, h_{2}) ∈ A_{∞}(D), then the function

z ↦ h(T_{1}z)w_{s,1}(z) + h(T_{2}z)w_{s,2}(z) = h_{1}(T_{1}z)w_{s,1}(z) + h_{2}(T_{2}z)w_{s,2}(z)  (6.1)

is holomorphic on the disc E_{0}, (since w_{s,1} is holomorphic on the cut plane ℂ \ (−∞, y_{1}], w_{s,2} is holomorphic on the cut plane ℂ \ [y_{2}, ∞), and neither of the cuts intersects E_{0}).

The function

z ↦ h(T_{0}z)w_{s,0}(z) + h(T_{2}z)w_{s,2}(z) = h_{0}(T_{0}z)w_{s,0}(z) + h_{2}(T_{2}z)w_{s,2}(z)  (6.2)

is holomorphic on the disc E_{1}, (since both w_{s,0} and w_{s,2} are holomorphic on the cut plane ℂ \ [0, ∞)).

Similarly, the function

z ↦ h(T_{0}z)w_{s,0}(z) + h(T_{1}z)w_{s,1}(z) = h_{0}(T_{0}z)w_{s,0}(z) + h_{1}(T_{1}z)w_{s,1}(z)  (6.3)

is holomorphic on the disc E_{2}, (since both w_{s,0} and w_{s,1} are holomorphic on the cut plane ℂ \ (−∞, 0]).

6.3 The transfer operator and Fredholm determinant.

Letting (L_{s}u)_{0}(z), (L_{s}u)_{1}(z), (L_{s}u)_{2}(z) denote the functions defined by the formulae in (6.1), (6.2), (6.3) respectively, we can define the transfer operator L_{s} : A_{∞}(D) → A_{∞}(E) by

(L_{s}h)(z) = \{ \begin{matrix}(L_{s}h)_{0}(z) = h(T_{1}z)w_{s,1}(z) + h(T_{2}z)w_{s,2}(z) & if z ∈ E_{0} \\ (L_{s}h)_{1}(z) = h(T_{0}z)w_{s,0}(z) + h(T_{2}z)w_{s,2}(z) & if z ∈ E_{1} \\ (L_{s}h)_{2}(z) = h(T_{0}z)w_{s,0}(z) + h(T_{1}z)w_{s,1}(z) & if z ∈ E_{2}, \end{matrix}  (6.4)

Let us also define the component transfer operators L_{s,j} : A_{∞}(D) → A_{∞}(E_{j}) by L_{s,j}u = (L_{s}u)_{j}.

The crucial point in our analysis is that L_{s}h is holomorphic on a larger domain than h. For j = 0, 1, 2, let r′_{j} denote the radius of the disc E_{j}. So we have

r′_{0} = tan (\frac{π}{3} − \frac{θ}{4}) ,

r′_{1} = r′_{2} = \frac{1}{2} (tan (\frac{π}{3} + \frac{θ}{4}) + tan (\frac{π}{3} − \frac{θ}{4})) − tan \frac{θ}{4}.

Let γ_{j}(θ) = γ_{j} < 1 denote the ratio of the radii of the two discs D_{j}, E_{j} centred at y_{j}. We have the contraction ratios

γ_{0} = \frac{r_{0}}{r′_{0}} = \frac{tan \frac{θ}{4}}{tan(\frac{π}{3} − \frac{θ}{4})},  (6.5)

and

γ_{1} = γ_{2} = \frac{r_{2}}{r′_{2}} = \frac{\frac{1}{2} (tan(\frac{π}{3} + \frac{θ}{4}) − tan(\frac{π}{3} − \frac{θ}{4}))}{\frac{1}{2} (tan(\frac{π}{3} + \frac{θ}{4}) + tan(\frac{π}{3} − \frac{θ}{4})) − tan \frac{θ}{4}}

= \frac{sin \frac{θ}{2}}{\frac{\sqrt{3}}{2} − 2 tan \frac{θ}{4} cos(\frac{π}{3} + \frac{θ}{4}) cos(\frac{π}{3} − \frac{θ}{4}).}

(6.6)

Note that γ_{0}(θ) < γ_{1}(θ) = γ_{2}(θ) for all 0 < θ < 2π/3.

Remark. The contraction ratios γ_{j} are degenerate cross-ratios (where two of the points happen to coincide). Since all fractional linear maps preserve cross-ratios, then we cannot hope to obtain smaller contraction ratios by taking the coordinate change f to be a different fractional linear map.

For 0 < ε < r′_{j}, let Γ_{j,ε} denote the circle centred at y_{j}, of radius r′_{j} − ε.

For each j = 0, 1, 2, and h ∈ A_{∞}(D), and z ∈ D, we have

(L_{s}h)_{j}(z) = \frac{1}{2πi} ∫_{Γ_{j,ε}} \frac{(L_{s}h)_{j}(ξ)}{ξ − z}dξ

= \frac{1}{2πi} ∫_{Γ_{j,ε}} \frac{(L_{s}h)_{j}(ξ)}{ξ − y_{j}} (1 − \frac{z − y_{j}}{ξ − y_{j}} )^{−1} dξ

= ∑_{n=0}^{∞} l_{n}^{(j)}(h)v_{n}^{(j)}(z),

where

v_{n}^{(j)}(z) = (z − y_{j})^{n} and l_{n}^{(j)}(u) = \frac{1}{2πi} ∫_{Γ_{j,ε}} \frac{(L_{s}h)_{j}(ξ)}{(ξ − y_{j})^{n+1}} dξ.  (6.7)

Now define the sequence of functions v_{k} ∈ A_{∞}(D) by

v_{3n+j}|_{D_{j}} = v_{n}^{(j)}, and v_{3n+j}|_{D_{i}} = 0 for i ∈ {0, 1, 2} \ {j},

Figure 7. Three pairs of concentric discs, and corresponding circles of integration

and the sequence of linear functionals l_{k} ∈ A_{∞}(D)^{∗} by

l_{3n+j} = l_{n}^{(j)}.

Then we can write our transfer operator L_{s} as the single infinite series

L_{s}h(z) = ∑_{k=0}^{∞} l_{k}(h)v_{k}(z).  (6.8)

Remark. Notice that for any j ∈ {0, 1, 2}, and any integer m ≥ 0 we have (L_{s}v_{3m+j})_{j}(z) = 0. This implies that for any j ∈ {0, 1, 2}, and any integers m, n ≥ 0 we have

l_{3n+j}(v_{3m+j}) = l_{n}^{(j)}(v_{3m+j}) = \frac{1}{2πi} ∫_{Γ_{j,ε}} \frac{(L_{s}v_{3m+j})_{j}(ξ)}{(ξ − y_{j})^{n+1}}dξ = 0.

Consequently, in the infinite matrix (l_{α}(v_{β}))_{α,β=0}^{∞}, the entries in diagonals of the form {(3m + i, i) : i ≥ 0} and {(i, 3n + i) : i ≥ 0} are all zero. It is therefore possible to improve slightly the Hadamard bound (see proof of Proposition 7) on the determinant of submatrices.

As in §3.3, the infinite series expression (6.8) shows that L_{s} is nuclear of order zero, since the terms λ_{k} = ||l_{k}||_{∞}||v_{k}||_{∞} decay exponentially fast (here the uniform norm is taken on the disjoint union D).

To estimate the exponential decay rate of the λ_{k} we note that for z ∈ A we clearly have |v_{3n+j}(z)| ≤ r_{j}^{n}. By (6.7) we also have

||l_{3n+j}||_{∞} ≤ \frac{||L_{s,j}||_{∞,E_{j}}}{2π(r′_{j} − ε)^{n+1}} .

Note in this estimate || · ||_{∞,E_{j}} denotes the uniform norm with respect to the (larger) disc E_{j}. So for all 0 < ε < r′_{j} we have the estimate

λ_{3n+j} = ||l_{3n+j}||_{∞}||v_{3n+j}||_{∞} ≤ \frac{||L_{s,j}||_{∞,E_{j}}}{2π(r′_{j} − ε)} (\frac{r_{j}}{r′_{j} − ε})^{n} .

Letting ε → 0 gives, for j = 0, 1, 2 and n ≥ 0,

λ_{3n+j} = ||l_{3n+j}||_{∞}||v_{3n+j}||_{∞} ≤ K_{j} (\frac{r_{j}}{r′_{j}} )^{n} = K_{j}γ_{j}^{n},  (6.9)

where

K_{j} = K_{j}(θ) = \frac{||L_{s,j}||_{∞,E_{j}}}{2πr′_{j}} ,  (6.10)

and where the γ_{j} are given by (6.5), (6.6).

Having established the exponential decay rate (6.9), we now use Lemma 4 in Appendix B to combine these and find 0 < γ < 1 and K > 0 such that for all n ≥ 0

λ_{n} ≤ Kγ^{n}.  (6.11)

The exponential decay rate (6.11) corresponds to (3.21) with d = 1, and where γ now denotes α.

Now we proceed as in the proof of Proposition 6 to estimate the decay rate of the power series coefficients d_{N}(s) of the Fredholm determinant det(I − zL_{s}). Since d = 1 we have the Euler formula [E]

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

which is proved by successively summing the geometric series on the lefthand side. From (3.28) we then have

|d_{N}(s)| ≤ \frac{K^{N}N^{N/2}γ^{N(N−1)/2}}{(1 − γ) … (1 − γ^{N})} .  (6.12)

6.4 Estimates on transfer operator norms.

Now we would like to estimate the norms ||L_{s,j}||_{∞,A′_{j}}. This will be particularly important for establishing rigorous dimension estimates, such as in §7, where we want to estimate the constants K_{j} defined by (6.10). By (6.2), (6.3), (6.4) defining the L_{s,j} we clearly have

||L_{s,j}||_{∞,E_{j}} ≤ ∑_{{k=0 \atop k≠j}}^{2} ||w_{s}^{(k)}||_{∞,E_{j}}

for each j = 0, 1, 2.

For j = 0 we have ||w_{s,1}||_{∞,E_{0}} = 1, this value of |w_{s,1}(z)| being attained uniquely at the point z = − tan(\frac{θ}{4} − \frac{π}{3} ) where \overline{E_{0}} intersects the isometric circle ∂D_{1} of the map T_{1}. Similarly we have ||w_{s,2}||_{∞,E_{0}} = 1, this value of |w_{s,2}(z)| being attained uniquely at the point z = tan(\frac{θ}{4} − \frac{π}{3} ) where \overline{E_{0}} intersects the isometric circle ∂D_{2} of the map T_{2}. Therefore we have the bound

||L_{s,0}||_{∞,E_{0}} ≤ 2.  (6.13)

For j = 1 we see that ||w_{s,0}||_{∞,E_{1}} = 1, this value of |w_{s,0}(z)| being attained uniquely at the point z = − tan \frac{θ}{4} where \overline{E_{1}} intersects the isometric circle ∂D_{0} of the map T_{0}. We have that

||w_{s,2}||_{∞,E_{1}} = w_{s,2} (− tan \frac{θ}{4}) = (\frac{4r_{θ}^{2}}{(−(2 − a_{θ}) tan \frac{θ}{4} − \sqrt{3}a_{θ})^{2}} )^{s} .

Note in particular that − tan \frac{θ}{4} lies strictly outside the isometric circle of T_{2}, so (for s > 0) we have ||w_{s,2}||_{∞,E_{1}} < 1.

Therefore we have

||L_{s,1}||_{∞,E_{1}} ≤ 1 + (\frac{4r_{θ}^{2}}{(−(2 − a_{θ}) tan \frac{θ}{4} − \sqrt{3}a_{θ})^{2}} )^{s} .  (6.14)

A similar argument for the case j = 2 gives us

||L_{s,2}||_{∞,E_{2}} ≤ 1 + w_{s,1} (tan \frac{θ}{4}) = 1 + (\frac{4r_{θ}^{2}}{((2 − a_{θ}) tan \frac{θ}{4} + \sqrt{3}a_{θ})^{2}} )^{s} .  (6.15)

(Similarly to before, note that tan \frac{θ}{4} lies strictly outside the isometric circle of T_{1}, so (for s > 0) we have ||w_{s,1}||_{∞,E_{2}} < 1.)

7. Rigorous dimension estimates for limit sets

In this section we consider a particular choice of the parameter θ in our family of Fuchsian reflection groups. We choose θ = π/6, and strive for the best possible rigorous estimate of dim(Λ_{θ}). We use the sequence Δ_{N}(s) of approximations to det(I − L_{s}), and also the sequence Δ_{2r}^{(2)}(s) of approximations to det(I − L_{s}^{2}) (cf. the Remark at the end of §3.6).

Example. θ = π/6.

For θ = π/6, the various radii r_{j} and r′_{j} are

r_{0} = tan π/24 = 0.13165 …
r′_{0} = tan 7π/24 = 1.30322 …
r_{1} = r_{2} = \frac{tan \frac{9π}{24} − tan \frac{7π}{24}}{2} = 0.55549 …
r′_{1} = r′_{2} = \frac{1}{2} (tan \frac{9π}{12} + tan \frac{7π}{12} ) − tan \frac{π}{24} = 1.72706 …

Defining the contraction ratios γ_{i} = r_{i}/r′_{i}, we have γ_{0} = 0.10102 … and γ_{1} = γ_{2} = 0.32164 …

For s ≥ 0 the bounds (6.13), (6.14) and (6.15) give us ||L_{s,j}||_{∞,E_{j}} ≤ 2 for each j = 0, 1, 2.

Remark. The bounds for j = 1 and 2 could be sharpened by obtaining a lower bound on s = dim(Λ). For example, such a lower bound is given by − log 3/ log λ, where λ = max_{0≤j≤2} max_{z∈Λ} |T′_{j}(z)| is the slowest contraction rate on the limit set Λ (see [Fa1], Prop. 9.7). Of course better lower bounds are obtainable from our algorithm itself, so that having used the algorithm once with some crude lower bound on dim(Λ) giving the estimates on ||L_{s,j}||_{∞,E_{j}}, we can then go back and use the improved lower bound to sharpen the estimates on ||L_{s,j}||_{∞,E_{j}}, and continue bootstrapping in this way to obtain successively better estimates on dim(Λ).

Using (6.10) we then have

K_{j} = K_{j}(π/6) = \frac{||L_{s,j}||_{∞,E_{j}}}{2πr′_{j}} ,

so that K_{0} ≤ 0.41482 … and K_{1} = K_{2} ≤ 0.54974 . . ..

N  Largest zero of Δ_{N}
2  0.14633481296007741055454748401454596
3  0.18423440272351767688822531747382350
4  0.18399977929621235204864644797773486
5  0.18398305039516509087579859265399133
6  0.18398305988417009403195596234810316
7  0.18398306122261622100816402885866734
8  0.18398306124841998285455137338908131
9  0.18398306124833255797187772764544302
10  0.18398306124833929946685349025674957
11  0.18398306124833918404985469216386875
12  0.18398306124833918700689278881066430
13  0.18398306124833918693967757277042711
14  0.18398306124833918694121655021916395
15  0.18398306124833918694118046846226018
16  0.18398306124833918694118129222351397
17  0.18398306124833918694118127301338345
18  0.18398306124833918694118127345475071
19  0.18398306124833918694118127344451095
20  0.18398306124833918694118127344474707

Table 1. Successive approximations to dim(Λ_{π/6})

Therefore (see Lemma 4 in Appendix B) we define

γ = ( max_{0≤j≤2} γ_{j})^{1/3} = 0.68515 … and K = ( max_{0≤j≤2} K_{j}) γ^{−2} = 1.17105 …

The estimate (6.12) gives us, for s ≥ 0,

| ∑_{N=21}^{∞} d_{N}(s)| ≤ ∑_{N=21}^{∞} \frac{N^{N/2}K^{N}γ^{N(N−1)/2}}{(1 − γ) … (1 − γ^{n})} < 1.3 × 10^{−18}.

Since Δ_{20}(s) = 1 + ∑_{N=1}^{20} d_{N}(s), then for all 0 ≤ s ≤ 1 we have

|det(I − L_{s}) − Δ_{20}(s)| < 1.3 × 10^{−18}.  (7.1)

Let s_{∞} = dim(Λ_{π/6}) denote the largest zero of det(I −L_{s}), and s_{20} the largest zero of Δ_{20}.

If we can find values s^{−} < s_{20} < s^{+} with Δ_{20}(s^{+}) ≥ 1.3 × 10^{−18} and Δ_{20}(s^{−}) ≤ −1.3 × 10^{−18} then by (7.1) we see that det(I − L_{s^{+}}) > 0 and det(I − L_{s^{−}}) < 0. It then follows by the intermediate value theorem applied to det(I − L_{s}) that s^{−} < s_{∞} < s^{+}. It turns out that the choices s^{−} = 0.183983061248339186 and s^{+} = 0.183983061248339188 (which agree to 17 decimal places) give Δ_{20}(s^{+}) > 8 × 10^{−18} and Δ_{20}(s^{−}) < −7 × 10^{−18}.

We have therefore rigorously proved that

dim(Λ_{π/6}) = 0.183983061248339187 ± 10^{−18}.

Of course this rigorous estimate is more conservative than the empirical estimate we would like to infer from the list of leading zeros s_{N} of Δ_{N} given in Table 1. Here the quality of convergence strongly suggests that

dim(Λ_{π/6}) ≈ 0.183983061248339186941181273444 …,

an accuracy of 30 decimal places.

Indeed for this example we can obtain superior empirical estimates by considering the leading zeros s_{2r}^{(2)} of the functions Δ_{2r}^{(2)} (i.e. approximations to det(I −L_{s}^{2}), cf. §3.6). Again working with points up to period 20 we obtain

s_{20}^{(2)} = 0.18398306124833918694118127344474173288 …

which is empirically correct to the 38 decimal places given.

Figure 8 . Graph of s ↦ det(I − L_{s}) for Λ_{π/6}

Remarks. Figure 8 shows part of the graph of Δ(s) = det(I − L_{s}) corresponding to the limit set Λ_{π/6}. Although the graph is concave in the region shown, it is not concave over the whole real line.

For general systems, Δ(s) is not necessarily locally concave at s = dim(Λ). For example the iterated function scheme given by the contractions \frac{x}{2} and \frac{x}{2} + \frac{1}{2}, with limit set the unit interval (whose Hausdorff dimension is 1), has Δ(s) = ∏_{r=0}^{∞}(1 − 2^{1−r−s}). Note that the zeros of Δ are those integers not greater than 1. Moreover, it is easy to show that the second derivative Δ′′(1) > 0, so that Δ is locally convex (rather than concave) at s = 1.

8. Numerical estimates on the dimension of Julia sets for quadratic polynomials

In this section we consider dimension estimates for Julia sets J_{c} of quadratic maps f_{c}(z) = z^{2} + c.

For practical purposes, our algorithm is most effective in computing the dimension of dim(J_{c}) for c either in the main cardioid of the Mandelbrot set, or c outside of M. In the latter case all periodic points are repelling, while in the former case all periodic points are repelling except for a single attractive fixed point.

First we consider the purely imaginary value c = i/4, which lies in the main cardioid of the Mandelbrot set. Table 2 illustrates the successive approximations s_{N} to dim(J_{i/4}) arising from our algorithm.

If we take the parameter value c = −\frac{3}{2} + \frac{2}{3}i, which lies outside the Mandelbrot set, then the sequence of approximations to the dimension of J_{c} are given in Table 3.

For real values of c which are strictly less than −2, the Julia set J_{c} is a Cantor set completely contained in the real line. For such cases we have, by Corollary 3.1, the faster O(δ^{N^{2}}) convergence rate to dim(J_{c}), as illustrated in Table 4 for the case c = −5.

N  N^{th} approximation to dim(J_{−5})
1  0.4513993584764174609675959101241383349
2  0.4841518684194122992464635900326070715
3  0.4847979587486975778612282908975662571
4  0.4847982943561895699730717563576367090
5  0.4847982944381635057518511943420942957
6  0.4847982944381604305347487891271825909
7  0.4847982944381604305383984765793729512
8  0.4847982944381604305383984781726830747

Table 4. Successive approximations to dim(J_{−5})

CALCULATING HAUSDORFF DIMENSION 37

N  N^{th} approximation to dim(J_{i/4})
3  1.1677078534172827136
4  0.9974580934808979848
5  1.0169164188641603339
6  1.0218764720532313644
7  1.0230776911089017648
8  1.0232246810534996595
9  1.0232072525392922127
10  1.0231992637099065199
11  1.0231993120941968028
12  1.0231992857944621198
13  1.0231992888227184780
14  1.0231992890455073830
15  1.0231992890300189633
16  1.0231992890307255210
17  1.0231992890309781268
18  1.0231992890309686742
19  1.0231992890309691466
20  1.0231992890309691251

Table 2. Successive approximations to dim(J_{i/4})

For larger negative real values of c, the hyperbolicity of f_{c} : J_{c} → J_{c} is more pronounced, so that the constant 0 < δ < 1 in the O(δ^{N^{2}}) estimate is closer to zero, and the convergence to dim(J_{c}) consequently faster. Table 5 illustrates this for c = −20 (cf. [BZ, Tableau 2]).

N  N^{th} approximation to dim(J_{−20})
1  0.31485651652009699091265279629753355933688857812644665851918
2  0.31850483144363986562810164826944017431378984622904321285835
3  0.31850809576591085725942984004207253452015913804880055477625
4  0.31850809575800523882867786043747732330759968092023152922729
5  0.31850809575800524988789850335472906645586111530021825766595
6  0.31850809575800524988789848098884346788677292871828344714065
7  0.31850809575800524988789848098884348414792438297975066097358
8  0.31850809575800524988789848098884348414792438305840652044425

Table 5. Successive approximations to dim(J_{−20})

N  N^{th} approximation to dim(J_{−3/2+2i/3})
1  0.7149355610391974853
2  0.9991996994914223217
3  0.8948837401931045135
4  0.8990693400138277172
5  0.9048525377869365908
6  0.9040847144651654898
7  0.9038472818583009063
8  0.9038738383368002502
9  0.9038748469934538668
10  0.9038745896021979531
11  0.9038745956441220338
12  0.9038745968650866636
13  0.9038745968171929578
14  0.9038745968108846487
15  0.9038745968111623979
16  0.9038745968111848616

Table 3. Successive approximations to dim(J_{−3/2+2i/3})

Remarks.

For c lying outside the Mandelbrot set, J_{c} is a Cantor set, and we have the following asymptotic (see [BZ]):

dim(J_{c}) ∼ \frac{log 2}{log 2 + \frac{1}{2} log |c|} as |c| → ∞.

For c inside the main cardioid of the Mandelbrot set the Julia set is a quasi-circle, with dim(J_{c}) strictly greater than one except when c = 0 [Sul]. For |c| small we have the following second-order asymptotic due to Ruelle [Ru3],

dim(J_{c}) ∼ 1 + \frac{|c|^{2}}{4 log 2}, as |c| → 0.

Higher order asymptotics for small c are contained in [WBKS], [CDM], [AMO].

Ruelle [Ru3] proved that, when restricted to either a hyperbolic component of the Mandelbrot set M, or to the exterior, the map c ↦ dim(J_{c}) is real-analytic. McMullen [McM2] proved that this map is continuous when restricted to the real interval (c_{Feig}, 1/4] (where c_{Feig} ≈ −1.401155 is the Feigenbaum point).

However c ↦ dim(J_{c}) is discontinuous at various points on the boundary of the Mandelbrot set. The case c = 1/4 (for which the Julia set J_{c} contains a parabolic fixed point) was studied by Douady, Sentenac & Zinsmeister [DSZ], who proved this is a point of discontinuity (see also [JKP]). Havard & Zinsmeister [HZ] proved that when restricted to the real line, the left derivative of the map c ↦ dim(J_{c}) at the point c = 1/4 is infinite.

For other approaches to numerical computation of Hausdorff dimension of quadratic map Julia sets, see [McM3], [BZ], [WBKS], [Gar].

9. Final comments

Our method applies without significant changes to other systems:

(1) Blaschke products: These are maps f of the unit disk 𝔻 = {z ∈ ℂ : |z| ≤ 1} defined by

f(z) = ∏_{i=1}^{n} (\frac{a_{i}z + 1}{z + \overline{a}_{i}})

for some finite collection a_{1}, …, a_{n} ∈ ℂ satisfying |a_{i}|^{2} = 2 (i.e. a product of linear fractional transformations each of which preserves the unit disk 𝔻). The dynamics of these transformations combine some of the features of both rational maps and Fuchsian groups (see [DM] for more details). The Julia set J of a Blaschke product is contained in the unit circle, and admits a Markov partition, so in the hyperbolic case we can use Theorem 3 with d = 1 to approximate dim(J) at rate O(δ^{N^{2}}).

(2) Apollonian packings: A form of Schottky group which does not quite fit into the scheme of the previous section is that of the Apollonian circle packing. This is the limit set Λ of a group Γ generated by reflections in four mutually tangent circles C_{1}, …, C_{4}.

There are various estimates on dim(Λ). The best rigorous estimate is 1.300197 < dim(Λ) < 1.314534, due to Boyd [Boyd]. McMullen [McM3] gives the empirical estimate dim(Λ) ≈ 1.305688.

Although the Apollonian circle packing is not hyperbolic, it can readily be approximated by hyperbolic systems. For example taking circles C_{i,ε}, i = 1, …, 4, with each C_{i,ε} concentric with C_{i}, and of radius ε > 0 smaller than the radius of C_{i}, we define the reflection group Γ_{ε} to be generated by reflections in the C_{i,ε}. We can apply our techniques to compute the dimension of the associated limit sets Λ_{ε}, and note that lim_{ε→0} dim(Λ_{ε}) = dim(Λ) by Theorem 1.4 of [McM1].

There are various alternative ways of approximating Λ. In [MU2], for example, Λ is expressed as the limit set of a hyperbolic infinite iterated function scheme. Approximating this by some suitably large, yet finite, iterated function scheme, suggests another approach to computing dim(Λ).

Appendix A: Implementation of the periodic point algorithm

We wish to approximate the Hausdorff dimension of the limit set of a real-analytic conformal iterated function scheme, or the Julia set of a hyperbolic holomorphic Markov map. Here we detail the key steps of the periodic point algorithm, and discuss practical issues concerning its effective implementation.

Step 1. Fix a natural number N.

Step 2. Locate all points of period ≤ N. There are various ways of doing this, and the method chosen may vary so as to exploit features of the underlying dynamical system.

For the limit set of a Kleinian group, all maps in the associated iterated function scheme are fractional linear. Therefore all compositions are also fractional linear, so that periodic points (i.e. fixed points of suitable compositions) are just roots of quadratic polynomials. Thus the key computational task is to compute the coefficients of relevant compositions (and hence the coefficients of relevant quadratic polynomials). Identifying Möbius maps with elements of GL(2, ℂ) reduces this task to one of matrix multiplication.

In the case of quadratic maps f_{c}(z) = z^{2} + c, we use the program [Mul] of M. Muldoon, which locates points of period n by Newton’s method. When applied to finding zeros of f_{c}^{n} − id, Newton’s method is rather unstable, because of its sensitivity to the “initial guess”, so it is preferable to use an n-dimensional version. That is, we seek vectors \underline{z} ∈ ℂ^{n} which are fixed points of F (z_{1}, …, z_{n}) := (f_{c}(z_{n}), f_{c}(z_{1}), …, f_{c}(z_{n−1})). Setting G := I − F, the Newton iteration then takes the form \underline{z}^{(i+1)} = \underline{z}^{(i)} − [DG(\underline{z}^{(i)})]^{−1} G(\underline{z}^{(i)}). Care is still needed in making the “initial guess” \underline{z}^{(0)}, so we interpolate complex numbers 0 = c_{0}, c_{1}, …, c_{r} = c, with each |c_{i+1} − c_{i}| suitably small. The periodic points for f_{0} are known explicitly (they are roots of unity), and each period-n orbit for f_{c_{0}} is an approximate orbit for f_{c_{1}}, so can serve as an initial guess in Newton’s method. This is applied recursively, at each step the true period-n orbit for f_{c_{j}} serving as the initial guess for the corresponding period-n orbit for f_{c_{j+1}}, until eventually the relevant period-n orbit for f_{c} = f_{c_{r}} is determined.

Step 3. Evaluate relevant derivatives at each point of period n ≤ N, and substitute into the trace formula. If z\underline{i} is a fixed point of ϕ\underline{i} = ϕ_{i_{n+1}i_{n}} ∘ … ∘ ϕ_{i_{3}i_{2}} ∘ ϕ_{i_{2}i_{1}} (a composition of contractions defining the iterated function scheme), then this trace formula is (cf. (3.25))

tr(L_{s,\underline{i}}) = \frac{|Dϕ\underline{i}(z\underline{i})|^{s}}{det(I − Dϕ\underline{i}(z\underline{i}))}.

If the dynamical system is given in terms of a hyperbolic holomorphic Markov map f then the formula (cf. proof of Prop. 11) becomes

tr(L\underline{j}) = |(f^{n})′(z\underline{j})|^{−s} (1 + \frac{1 − 2Re((f^{n})′(z\underline{j}))}{|(f^{n})′(z\underline{j})|^{2}})^{−1} .

Step 4. For each n ≤ N, combine the trace evaluations for all period-n points to form the term (cf. (3.30))

d_{n}(s) = ∑_{{(n_{1},... ,nm) \atop n_{1}+...+nm=n}} \frac{(−1)^{m}}{m!} ∏_{l=1}^{m} \frac{1}{n_{l}} ∑_{\underline{i}∈Fix(n_{l})} \frac{|Dϕ\underline{i}(z\underline{i})|^{s}}{det(I − Dϕ\underline{i}(z\underline{i}))}.

Step 5. Sum the d_{n}(s) to form the function Δ_{N}(s) = 1 + ∑_{n=1}^{N} d_{n}(s).

Step 6. Locate the largest real zero s_{N} of Δ_{N}. This can be performed to arbitrary precision by means of an appropriate numerical method (eg. Newton’s method, secant method, gradient descent, etc.). The value s_{N} is an approximation to the true Hausdorff dimension.

Practical Considerations.

A computer is necessary for the effective implementation of this algorithm.^{2} We used a standard UNIX workstation, which was effective in computing points of period up to around N = 20 for the quadratic map and reflection group examples considered here.

Programs were written in Mathematica^{TM} by the first author. For the reflection group example they were complemented by C programs of I. Rivin. For the quadratic map example the periodic points were located using M. Muldoon’s program [Mul].

Further details are available via the website [JP2].

Appendix B: Combining contraction ratios

The following lemma, which describes how to combine different contraction ratios, is useful to obtain estimates on the super-exponential rate of convergence, which in turn can be used to prove rigorous dimension estimates (see §7). Part (a) of the lemma gives an optimal estimate on this convergence rate. Part (b), while giving a more conservative estimate, is often of more practical use, since the constant K is often much smaller than that in part (a).

Lemma 4. Suppose K_{0}, …, K_{p−1} > 0, and γ_{0}, …, γ_{p−1} ∈ (0, 1). Suppose (α_{i}) is a sequence of positive real numbers, such that

α_{pn+j} ≤ K_{j}γ_{j}^{n} for all n ≥ 0, and j = 0, …, p − 1.

Let β_{1} ≥ β_{2} ≥ … be the non-decreasing rearrangement of the sequence (α_{i}). Then for all i ≥ 0 we have

β_{i} ≤ Kγ^{i},

where we can choose 0 < γ < 1 and K > 0 to satisfy either (a)

γ = exp ((∑_{j=0}^{p−1} \frac{1}{log γ_{j}} )^{−1}) ,

and

K = max (γ^{−1−2p+∑_{j=0}^{p−1} \frac{log K_{j}}{log γ_{j}}} , max_{0≤j≤p−1} K_{j}) .

or (b)

γ = ( max_{0≤j≤p−1} γ_{j})^{1/p}

2If cruder dimension estimates are of use then approximations based on points of very low period (eg. fixed points and period-2 points, say) are sometimes feasible “by hand”.

and

K = ( max_{0≤j≤p−1} K_{j}) γ^{1−p}.

Proof. (a) For L > 0, define M(L) to be the number of terms in the sequence (α_{i}) which are greater than or equal to L. We then have β_{M(L)} ≥ L > β_{M(L)+1} for all L > 0.

Define the sequence (δ_{i}) by δ_{pn+j} = K_{j}γ_{j}^{n} for n ≥ 0 and j = 0, …, p − 1. Now if α_{pn+j} ≥ L, then certainly δ_{pn+j} = K_{j}γ_{j}^{n} ≥ L. Thus if we define N(L) to be the number of terms in the sequence (δ_{i}) which are greater than or equal to L, then we clearly have N(L) ≥ M(L).

Now

K_{j}γ_{j}^{n} ≥ L if and only if n ≤ \frac{log L − log K_{j}}{log γ_{j}} ,

so we have

N(L) = ∑_{j=0}^{p−1} ([\frac{log L − log K_{j}}{log γ_{j}} ] + 1) = p + ∑_{j=0}^{p−1} [\frac{log L − log K_{j}}{log γ_{j}} ] ,

where [·] denotes the integer part of a real number.

Let ϵ_{1} ≥ ϵ_{2} ≥ … be the non-increasing rearrangement of the sequence (δ_{i}). Suppose we have that

ϵ_{i} ≤ K′γ^{i},  (B.1)

where K′ > 0 and 0 < γ < 1 are chosen to be optimal, in the sense that lim sup_{i→∞} ϵ_{i}^{1/i} = γ, and K′ = sup_{i≥0} ϵ_{i}/γ^{i}.

Now for all L > 0 we have

ϵ_{N(L)} ≥ L > ϵ_{N(L)+1},  (B.2)

so taking N(L)^{th} roots in this inequality, then taking limit suprema as L → 0 (so that N(L) → ∞), we obtain

γ = lim sup_{L→0} L^{1/N(L)} = lim sup_{L→0} L^{(p+log L ∑ \frac{1}{log γ_{j}} −∑ \frac{log K_{j}}{log γ_{j}} )^{−1}}.

Taking logarithms gives

log γ = lim sup_{L→0} \frac{log L}{p + log L ∑ \frac{1}{log γ_{j}} − ∑ \frac{log K_{j}}{log γ_{j}}} = \frac{1}{∑{p−1 \atop j=0} \frac{1}{log γ_{j}}} ,

so that

γ = exp ((∑_{j=0}^{p−1} \frac{1}{log γ_{j}} )^{−1}) .

Now for any positive integer i which is of the form i = N(L) + 1, for some L > 0, the right hand side of (B.2) gives us

\frac{ϵ_{i}}{γ^{i}} < \frac{L}{γ^{N(L)+1}} .  (B.3)

If the integer i ≥ 0 is not of the form N(L) + 1, then there must exist q ≥ 2 (note also that q ≤ p) consecutive terms in the re-arranged sequence (ϵ_{i}) which are all equal to some L > 0. That is, we have

ϵ_{k+1} = … = ϵ_{i−1} = … = ϵ_{k+q} = L > ϵ_{k+q+1}.

Note that k + q = N(L). Since 0 < γ < 1 then

ϵ_{i}/γ^{i} < ϵ_{k+q}/γ^{k+q} = L/γ^{N(L)} < L/γ^{N(L)+1}.  (B.4)

Combining (B.3), (B.4) we see that

sup_{i≥1} ϵ_{i}/γ^{i} ≤ sup_{L>0} L/γ^{N(L)+1}.  (B.5)

Estimating the denominator in (B.5) we have

γ^{N(L)+1} = γ^{1+p+∑_{j=0}^{p−1}([ \frac{log L−log K_{j}}{log γ_{j}} ]+1)}
≥ γ^{1+2p+∑ \frac{log L−log K_{j}}{log γ_{j}}}
= exp (\frac{1 + 2p}{∑ \frac{1}{log γ_{j}}} ) exp(log L) exp (\frac{− ∑ \frac{log K_{j}}{log γ_{j}}}{∑ \frac{1}{log γ_{j}}} )

= γ^{1+2p}Lγ^{− ∑ \frac{log K_{j}}{log γ_{j}}} .

Therefore we have

sup_{L>0} L/γ^{N(L)+1} ≤ γ^{−1−2p+∑_{j=0}^{p−1} \frac{log K_{j}}{log γ_{j}}} .  (B.6)

So from (B.5) we obtain

sup_{i≥1} ϵ_{i}/γ^{i} ≤ γ^{−1−2p+∑_{j=0}^{p−1} \frac{log K_{j}}{log γ_{j}}} .  (B.7)

Now K′ = sup_{i≥0} ϵ_{i}/γ^{i}, and ϵ_{0}/γ^{0} = ϵ_{0} = max_{0≤j≤p−1} K_{j}. So combining this with (B.7) gives us

K′ = sup_{i≥0} ϵ_{i}/γ^{i} ≤ max (γ^{−1−2p+∑_{j=0}^{p−1} \frac{log K_{j}}{log γ_{j}}} , max_{0≤j≤p−1} K_{j}) .  (B.8)

Now defining K to be the right-hand side of (B.8) we use (B.1) to see that

ϵ_{i} ≤ K′γ^{i} ≤ Kγ^{i}.

Moreover, since each α_{i} ≤ δ_{i} for all i ≥ 0, then rearranging both sequences gives β_{i} ≤ ϵ_{i} for all i ≥ 0, from which we deduce that

β_{i} ≤ ϵ_{i} ≤ Kγ^{i},

as required.

(b) Define the sequence (δ_{i})_{i=0}^{∞} as in the proof of part (a), and define (τ_{i})_{i=0}^{∞} by τ_{pn+k} = (max_{0≤j≤p−1} K_{j})(max_{0≤j≤p−1} γ_{j})^{n} for n ≥ 0 and k = 0, …, p − 1. Then we have α_{i} ≤ δ_{i} ≤ τ_{i} for all i ≥ 0, and moreover

τ_{i} ≤ ( max_{0≤j≤p−1} K_{j})( max_{0≤j≤p−1} γ_{j})^{\frac{i−(p−1)}{p}}

= ( max_{0≤j≤p−1} K_{j})γ^{1−p}γ^{i}.

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O. Jenkinson, School of Mathematical Sciences, Queen Mary, University of London, Mile End Road, London, E1 4NS, UK.

   E-mail address: omj@maths.qmw.ac.uk

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

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