Contemporary Mathematics

Coverings of Cantor sets

Frederick P. Gardiner and Linda Keen

Abstract. Let X be the Riemann sphere from which a Cantor set C quasiconformally equivalent to the middle thirds Cantor set has been removed and let G be a Fuchsian group representing X. We prove that the universal covering map has a well defined extension as a projection from the non-conical subset of the limit set Λ(G) onto the boundary C of X.

1. Introduction

Suppose S is a conformally finite Riemann surface represented by a Fuchsian group G acting on the upper half plane ℍ. Because the punctures on S form a finite set and the group is finitely generated, it is natural to extend the projection map π : ℍ → ℍ/G ≡ S to the parabolic fixed points of G by mapping any such fixed point to a puncture. The parabolic fixed points have a dynamical characterization used by Patterson and Sullivan [12, 16] and by Beardon and Maskit [4] as the set of non-conical limit points NC(G); that is, the set of points such that no cone based at the point contains infinitely many points congruent under the group.

Many Riemann surfaces of infinite type arise in the context of holomorphic dynamics as the complement in the Riemann sphere of a Cantor set of bounded geometry, that is, of a set invariant under a dynamical system formed from a collection of expanding maps. Using the dynamics, one can find bounds on the conformal geometry of such surfaces. Studying the relation between the conformal and hyperbolic geometries of these surfaces, it is natural to ask whether the uniformizing map extends to a map of a natural subset of the limit set onto the boundary of the surface. The answer in general is yes and the subset is precisely the non-conical limit set.

In this paper we consider a particular instance of this situation. Our Riemann surface, X, is the complement in the Riemann sphere of the standard middle thirds Cantor set C. We prove there is a natural map from NC(G) to C. The techniques we develop clearly extend to the more general dynamically defined Riemann surfaces

\begin{matrix}c \\ ◯ \end{matrix}0000 (copyright holder)

1991 Mathematics Subject Classification. Primary 20H10, 30F60, Secondary 30C62, 30C80, 32G15.

Key words and phrases. Cantor sets, bounded geometric type, non-conical limit sets, Fuchsian groups, hyperbolic distortion lemma, fundamental domains.

Research partially supported by PSC-CUNY and by NSF.

mentioned above as well as to an even larger class. We plan to carry out this extension in future work.

As a consequence of our constructions, we also prove that there is a canonical lift ω_{0} of X to ℍ and that the map from the boundary of ω_{0} in ∂ℍ to C is Hölder continuous. The crucial point is that there are sequences of geodesics defining the ends of X whose hyperbolic lengths are uniformly bounded.

Let D be the disk centered at z = 1/2 of radius 7/9, D_{0} = E_{0}^{−1}(D), D_{1} = E_{1}^{−1}(D), where E_{0}(z) = 3z, E_{1}(z) = 3z − 2. Then, the middle thirds Cantor set C is the accumulation set of all possible backwards images of points in D under the expanding dynamical system defined by

E(z) = \{ \begin{matrix}E_{0}(z) & if z ∈ D_{0} and \\ E_{1}(z) & if z ∈ D_{1}. \end{matrix} (1)

The map E restricted to C defines a dynamical system topologically conjugate to the shift on two symbols. Using all of the different possible images of the boundary of the disk D under backwards branches of E we obtain a collection of simple closed curves on X. These are round circles in the Euclidean geometry of X. In the free homotopy class of each one of these curves there is a unique hyperbolic geodesic with respect to the hyperbolic metric on the complement of C. We call these circles, or the corresponding hyperbolic geodesics (resp. conformal or hyperbolic) waist curves. Nested sequences of these waist curves characterize the points in C. Employing the conformality of E we obtain upper and lower bounds on the lengths of the conformal waist curves. These upper bounds, in turn, give upper bounds on the lengths of the hyperbolic waist curves that allow us to identify points in C with certain nested sequences of lifts of the hyperbolic waist curves to ℍ that we call allowable chains of axes. These chains limit at points of the non-conical limit set NC(G).

Although our constructions are carried out for the special system of curves based on the middle thirds Cantor set C, they apply immediately to any quasiconformal image of C. This follows because the crucial bounds on the lengths of the waist curves in our proof are quasiconformally quasi-invariant. Thus our main result is that for any surface representing a point in the Teichmüller space of X and a Fuchsian group representing it, there is a projection from the non-conical limit set to the boundary of the surface.

With more work, but using similar techniques, the same is true when the Riemann surface is the complement in the sphere of any Cantor set of bounded geometry.

The organization of the paper is as follows. In section 2 we set up notation for coding the middle thirds Cantor set C and the conjugacy from the action of E on C to the action of the one-sided shift on the space of one-sided sequences on two symbols. In section 3 we discuss the dynamics on the Riemann surface X of the conformal system E and obtain the upper bound on the lengths of the hyperbolic waist curves. The action of the system E(z), mapping subsurfaces of X onto X implies that the lengths of the hyperbolic waist curves are also bounded below. Although this bound is not needed in the proof of the main theorem, the proof that it exists involves the interaction of the conformal and hyperbolic structures on X in an interesting way. We therefore include it in an appendix. In section 4, we discuss the boundary of X in terms of sequences of hyperbolic waist curves on X and in section 5 we define the fundamental domain ω_{0} and the allowable chains of axes.

In section 6, which is the heart of the paper, we show that the set of limit points of allowable chains of axes L_{X}(G) coincides with the non-conical limit set NC(G) of the Fuchsian covering group G of X. In section 7 we show that our results hold true under quasiconformal deformation and in section 8 we describe Cantor sets of bounded geometry and their complements in the Riemann sphere and indicate how our theorem generalizes to them. Section 9 is the appendix.

2. The middle thirds Cantor set

Let C be the standard middle thirds Cantor set in the unit interval. It is obtained inductively: first remove (1/3, 2/3) from [0, 1] and at the n^{th} step remove the middle thirds of the remaining intervals. The points x ∈ C have unique representations of the form x = s_{1}s_{2} … where s_{i} ∈ {0, 1} and every sequence in {0, 1}^{∞} occurs. This representation arises as follows: represent all points in the interval [0, 1] by a triadic expansion

x = ∑_{j=1}^{∞} \frac{s_{j}}{3^{j}} ,

where s_{j} equals 0, 1 or 2. Now remove from the unit interval [0, 1] all of those points for which the the “digit” s_{j} is equal to 1 anywhere in the expansion. The remaining set of points is the middle thirds Cantor set C. Note that any point which admits two distinct expansions, s_{1} … s_{n}111 … and s_{1} … s_{n}2000 … and s_{i} ≠ 1 for i = 1 … n belongs to C.

Now replace 2 by 1 in every expansion and drop the “decimal” point. Then the correspondence i between codes x = s_{1}s_{2} …, where each s_{j} is either zero or one, and the points i(x) = ∑_{j=1}^{∞} \frac{2s_{j}}{3^{j}} in the Cantor set C contained in the complex plane ℂ is bijective. If we use the metric d_{C}(x, y) = ∑_{k} |s_{j} − t_{j}|/2^{j} on the space of codes, then i is a homeomorphism from the space of codes onto C. Moreover, the shift map Σ(s_{1}s_{2}s_{3} …) = s_{2}s_{3} … on the space of codes is conjugate to the expanding map E on C in the sense that

i(Σ(x)) = E(i(x)).

For readability, often we simply write x instead of i(x); in other words we identify the code x with the corresponding point i(x) in C.

3. The Riemann surface X

Let X be the Riemann surface X = ℂ̂ \ C.

3.1. The dynamics and the conformal structure. The map E defined in (1) has two inverse branches ϕ_{0}(z) = E_{0}^{−1}(z) and ϕ_{1}(z) = E_{1}^{−1}(z). As maps of ℂ they are conformal and contracting homeomorphisms. Let α_{−1} be the annulus {z|13/18 < |z − 1/2| < 15/18}. We label images of this annulus under successive applications of the mappings ϕ_{0} and ϕ_{1} in the following way: α_{0} = ϕ_{0}(α_{−1}) and α_{1} = ϕ_{1}(α_{−1}), α_{00} = ϕ_{0}(α_{0}), α_{01} = ϕ_{0}(α_{1}), α_{10} = ϕ_{1}(α_{0}) and α_{11} = ϕ_{1}(α_{1}). For x_{n} = s_{1}s_{2} … s_{n}, inductively define

α_{0x_{n}} = ϕ_{0}(α_{x_{n}}), α_{1x_{n}} = ϕ_{1}(α_{x_{n}}).

These annuli are contained in X and are mutually disjoint.

Since ϕ_{0}, ϕ_{1} are conformal, the modulus of α_{x_{n}} is the same for all truncations x_{n}. Moreover, the same is true of the extremal length m(Σ′_{x_{n}}) of the family Σ′_{x_{n}} of curves separating the boundary curves of α_{x_{n}}; we easily compute m(Σ′_{x_{n}}) = 2π/ log(15/13).

Take the circle |z − 1/2| = 14/18 as the core curve of α_{−1}; its images in the annuli α_{x_{n}} are their core curves and we denote them by Γ_{x_{n}}. The curves Γ_{0} and Γ_{1} both separate (in ℂ̂) the points with expansions 0s_{2}s_{3} … from the points with expansions 1t_{2}t_{3} …, where s_{i}, t_{j} ∈ {0, 1}.

Using the map ϕ_{0} it follows that the curve Γ_{00}, separates (in ℂ̂) the point with code 00s_{3}s_{4} … from all points with codes whose first or second entry is not 0, that is, points with codes of the form 1s_{2}s_{3} … or of the form 01s_{3}s_{4} . . .. It also follows that the curve Γ_{01} separates the points with codes 01s_{3}s_{4} … from all points with codes whose first entry is not 0 and whose second entry is not 1. Analogously, using the map ϕ_{1}, the curves Γ_{10} and Γ_{11} separate, respectively, points with codes 10s_{3}s_{4} … and 11s_{3}s_{4} … from points whose codes disagree in one of their first two entries.

Applying the maps ϕ_{0} and ϕ_{1}, according as s_{i} = 0 or 1 we see inductively that the curve Γ_{s_{1}...s_{n}} = ϕ_{s_{n}}(Γ_{s_{1}...s_{n−1}}) separates points with codes s_{1}s_{2} … s_{n}s_{n+1}s_{n+2} … from all points whose codes disagree in any one of their first n entries.

It is clear that the curves Γ_{x_{n}}, for all truncations x_{n} of all possible codes x, divide X into pairs of pants (three holed spheres). These curves are the Euclidean waist curves. These pairs of pants are glued together with the structure of a binary tree to form X. For a given x ∈ C the curves Γ_{x_{n}} are nested and their Euclidean diameters go to zero; therefore the intersection of their interiors is the unique point x ∈ ∂X.

3.2. The hyperbolic structure. By the Riemann mapping theorem, the universal cover of X is conformally equivalent to the upper half plane ℍ and X inherits a hyperbolic structure under projection. We denote the distance in this metric by d_{X} to distinguish it from the Euclidean distance d on X inherited from X as a subset of ℂ.

Each free homotopy class of simple closed curves on X has a unique geodesic representative with respect to the hyperbolic structure. Except for Γ_{0} and Γ_{1}, the curves Γ_{x_{n}} of the previous section are all in distinct free homotopy classes. Denote by γ_{x_{n}} the geodesic representative in the free homotopy class of Γ_{x_{n}}. We call γ_{x_{n}} a hyperbolic waist curve of level n. For readability, if the code x_{n} is not relevant we write γ_{k} for an arbitrary waist curve. Also, we omit the modifier Euclidean or hyperbolic for a waist curve if it is clear from the context.

Since Γ_{0} and Γ_{1} are freely homotopic, γ_{0} = γ_{1}. Note that by symmetry γ_{0} = 1/2 + iy, y ∈ ℝ. In general, Γ_{x_{n}} and γ_{x_{n}} are not the same curve.

Let l(γ_{k}) denote the hyperbolic length of γ_{k}. Then we have the following easy upper bound.

Proposition 1. There is a constant B > 1 such that, for all waist curves of every level, l(γ_{k}) < B.

Proof: Fix γ_{k}. Let Σ_{k} be the set of curves on X freely homotopic to γ_{k} and let m(Σ_{k}) be the extremal length of this family. The family Σ′_{k} of closed curves in the annulus α_{k} is a subset of Σ_{k} so that m(Σ_{k}) ≤ m(Σ′_{k}) = 2π/ log(15/13). Using hyperbolic geometry it is easy to prove (see e.g. [9]) that l(γ_{k}) ≤ πm(Σ_{k}).

Remark. There is also a lower bound on the lengths of the waist curves. Since its proof involves an interesting interaction between the conformal and hyperbolic structures of X, but is not necessary for our main theorem we defer it to an appendix.

On each pair of pants P_{k} ⊂ X there are three unique geodesic arcs η_{i}^{k}, i = 1, 2, 3 orthogonal to each distinct pair of hyperbolic waist curves. Using the hyperbolic cosine laws and properties of right-angled hyperbolic hexagons, [3], we can express the length of each η_{i}^{k} as a function of the lengths of the waist curves of P_{k}. Thus the following corollary follows from Proposition 1 and also from Proposition 13 in the appendix.

Corollary 2. There are uniform upper and lower bounds on the lengths of waist curves γ_{j} and the orthogonal arcs η_{i}^{j}, i = 1, 2, 3 on all pair of pants in X.

Remark. The bound we really use is the lower bound on the orthogonals to the waist curves. Thus for the remainder of the paper we let ϵ_{0} be a positive number less than the lengths of all of these orthogonals.

4. Characterizing the boundary

In this section we describe non-recurrent and adjacent sequences in general. Then we use the waist curves of X to specify a marking for the fundamental group π_{1}(X).

4.1. Allowable chains of waist curves. In section 3.2 we considered the special sequences of waist curves γ_{x_{n}} that characterize points in C. Here we discuss more general sequences of waist curves that determine points of C. Definition: A sequence of waist curves γ_{k} has the adjacency property if each consecutive pair γ_{k} and γ_{k+1} are boundary curves of a common pair of pants P_{k}.

Definition: The sequence γ_{k} is non-recurrent if for every level L there is k_{0} such that for every k > k_{0}, the level of γ_{k} is greater than L. Obviously, any subsequence of a non-recurrent sequence is non-recurrent.

Definition: A non-recurrent sequence satisfying the adjacency property is called an allowable chain of waist curves or simply an allowable chain.

It is clear that given x ∈ C the sequence γ_{x_{n}} constructed in section 3.2 is an allowable chain and that, in particular, allowable chains exist. In the next proposition we prove a converse.

Proposition 3. An allowable chain γ_{k} on X uniquely determines a point x ∈ C.

Proof: Without loss of generality, we may assume γ_{1} is at level L_{1} and, for k > 1, γ_{k} is at level L_{1} or higher. Because the sequence γ_{k} is non-recurrent, there is a largest value of k such that γ_{k} is at level L_{1}. Let k_{1} be this largest value. Because of the adjacency property, γ_{k_{1}+1} is at level L_{1} + 1. We let k_{2} be the largest value of k for which γ_{k} is at level L_{1} + 1 and, inductively, we let k_{j} be the largest value of k for which γ_{k} is at level L_{1} + j − 1.

Then γ_{k_{j}} is an allowable chain of waist curves which is minimal with respect to the adjacency property in the following sense. It has the adjacency property but none of its subchains do. Moreover, any subchain with the adjacency property eventually visits the same sequence of pairs of pants.

The pairs of pants P_{k_{j}} are adjacent and form a nested sequence in the Euclidean representation of X ⊂ ℂ and therefore uniquely determine a point x ∈ C. Because any other subchain must visit the same sequence of pairs of pants, no other subchain can nest on a different point in C.

4.2. The marking for X. In order to define a presentation for π_{1}(X, q) we first canonically choose a point on each pair of pants.

Denote by P_{x_{n}} the pair of pants in X bounded by the three waist curves γ_{x_{n}}, γ_{0x_{n}} and γ_{1x_{n}}. Below we will need to distinguish among these curves so we will call γ_{x_{n}} the waist of P_{x_{n}} and γ_{0x_{n}} and γ_{1x_{n}} its cuffs. On each pair of pants P_{x_{n}} we orient the cuffs so that for any choice of base point, the geodesic γ_{0x_{n}}γ_{1x_{n}} in the free homotopy class of a curve formed by going around γ_{0x_{n}} and then going around γ_{1x_{n}} is simple. The geodesic in the free homotopy class of γ_{0x_{n}}γ_{1x_{n}}^{−1} then has a single self-intersection point on P_{x_{n}} which we denote by p_{n}.

For each x_{n+1} equal to 0x_{n} or to 1x_{n}, the curve γ_{x_{n+1}} is a cuff of P_{x_{n}} and the waist of P_{x_{n+1}}. There is a family of geodesic arcs on the four holed sphere Y = P_{x_{n}} ∪ P_{x_{n+1}} joining p_{x_{n}} and p_{x_{n+1}}. All arcs in this family intersect γ_{x_{n+1}} but do not intersect any of the other waists or cuffs of Y. Among all arcs in this family there are at most two whose length is equal to d_{X}(p_{x_{n}}, p_{x_{n+1}}). There may be two if each of the points p_{x_{n}} and p_{x_{n+1}} lies on the common orthogonal to the cuffs of its pair of pants. Let δ_{x_{n}} be the shortest among these arcs and if there are two such arcs, choose one. Parameterize δ_{x_{n}} by t ∈ [n, n + 1].

Analogously, let σ be the shortest geodesic arc joining p_{0} ∈ P_{0} to p_{1} ∈ P_{1}. Let q be the intersection point of σ and γ_{0} = γ_{1} and set σ_{0} = σ ∩ P_{0}, σ_{1} = σ ∩ P_{1}. Parameterize σ_{i} by t ∈ [0, 1], i = 0, 1, σ_{i}(0) = q.

To each x_{n} = s_{1} … s_{n} let σ_{x_{n}} be the arc σ_{x_{n}} = σ_{s_{1}} ∪ δ_{x_{1}} ∪ δ_{x_{2}} … δ_{x_{n−1}} ∪ (δ_{x_{n}} ∩ P_{x_{n}}) and set

γ̂_{x_{n}} = σ_{x_{n}}γ_{x_{n}}σ_{x_{n}}^{−1}.

We have

Proposition 4. The fundamental group of X with basepoint q is isomorphic to the free group on the generators

⟨γ_{0}, γ̂_{x_{n}}; x_{n} = s_{1} … s_{n}, s_{i} ∈ {0, 1}, n = 1, … ∞⟩

factored by the relations that the words γ̂_{x_{n}}^{−1}γ̂_{0x_{n}} γ̂_{1x_{n}} are equal to the identity.

Proof: On each pair of pants P_{x_{n}}, for t = x_{n}, 0x_{n}, 1x_{n}, let γ̄_{t} = γ̂_{t} ∩ P_{x_{n}}. Then π_{1}(P_{x_{n}}, p_{x_{n}}) = ⟨γ̄_{x_{n}}, γ̄_{0x_{n}}, γ̄_{1x_{n}}; γ̄_{x_{n}}^{−1}γ̄_{0x_{n}} γ̄_{1x_{n}}⟩. By Van Kampen’s theorem π_{1}(X) is formed by taking free products with amalgamation along the common boundary of the fundamental groups of adjacent pairs of pants. The curves γ̂_{x_{n}} and γ̄_{x_{n}} are pointed curves freely homotopic to the geodesics γ_{x_{n}} by construction and thus generate the amalgamating subgroups.

5. Lifting the boundary

Let ρ : π_{1}(X, q) → P SL(2, ℝ) be a representation with discrete torsion free image G such that ℍ/G with conformal structure inherited from ℍ is holomorphically equivalent to X. As usual, we identify an element in P SL(2, ℝ) with its corresponding Möbius transformation.

We use the marking defined above to obtain a specific set of generators for G. Let g_{x_{n}} = ρ(γ̂_{n}) and let A_{x_{n}} denote the axis (invariant geodesic) of g_{x_{n}}. Then π(A_{x_{n}}) = γ_{x_{n}}. To specify the representation uniquely we normalize as follows. We assume that the axis A_{0} is the imaginary axis and that zero is the repelling fixed point of g_{0}. We also assume that the attracting fixed point of g_{01} is −1. Because of the symmetry of the surface X, the attracting fixed point of g_{10} is +1.

5.1. A canonical fundamental polygon. We can canonically choose lifts P̃_{x_{n}} of pairs of pants so that as x_{n} varies through all truncations of points x ∈ C we obtain a canonically determined fundamental domain ω_{0} for G. More specifically, consider the three axes A_{x_{n}}, A_{0x_{n}}, A_{1x_{n}} together with the three common perpendiculars joining A_{x_{n}} to A_{0x_{n}}, A_{0x_{n}} to A_{1x_{n}}, and A_{1x_{n}} to A_{x_{n}}. This configuration forms a hyperbolic hexagon Q_{x_{n}} in which each interior angle is 90^{o} and the side lengths are all bounded above and below by 1/ϵ_{0} and ϵ_{0}. Let Q̄_{x_{n}} be the reflection of Q_{x_{n}} in the orthogonal joining A_{0x_{n}} and A_{x_{n1}}. Then P̃_{x_{n}} = Q_{x_{n}} ∪ Q̄_{x_{n}} is a canonical octagon which covers the pair of pants P_{x_{n}}. The union of these octagons for all truncations x_{n} and all x ∈ C is the canonical fundamental domain ω_{0} for G.

We claim that ω_{0} is connected. More precisely, P̃_{x_{n}} and P̃_{x_{n}0} both contain boundary arcs on the axis A_{x_{n}0} and these arcs intersect; the same is true of course, for P̃_{x_{n}} and P̃_{1x_{n}}. We carry out the argument only in the first case. Note that the axes of the products g_{0x_{n}}g_{1x_{n}}^{−1} and g_{0x_{n}}^{−1}g_{1x_{n}} intersect in a point p̃_{x_{n}} that projects to p_{x_{n}}. Moreover, by consideration of the separation properties of these axes and the axes A_{x_{n}}, A_{0x_{n}}, A_{1x_{n}}, one sees that p̃_{x_{n}} lies inside the closure of P̃_{x_{n}}. There is a lift δ̃_{0x_{n}} at p̃_{x_{n}} which determines the marked generators g_{00x_{n}} and g_{10x_{n}} and therefore ends at p̃_{0x_{n}}. Further, for any integer k ≠ 0, the length of δ̃_{0x_{n}} is shorter than the length of any geodesic joining p̃_{x_{n}} to g_{0x_{n}}^{k}(p̃_{0x_{n}}). This implies either the boundary arcs of P̃_{x_{n}} and P̃_{0x_{n}} on A_{1x_{n}} overlap or δ̃_{0x_{n}} is orthogonal to A_{0x_{n}} and forms part of the boundary of P̃_{x_{n}} ∪ P̃_{0x_{n}}.

5.2. Allowable Chains of Axes. Let W be the totality of all hyperbolic waist curves on X and set W = π^{−1}(W). Note that any lift A_{k} of γ_{k} is an element of W and is an axis of an element of G conjugate to some generator. Because the waist curves are pairwise disjoint on X it follows that the axes in W are also disjoint.

Each sequence A_{k} in W determines a sequence of intervals I_{k} on the boundary of ℍ. For any A_{k} except the generator A_{0}, let I_{k} be the interval on ∂ℍ between the fixed points of A_{k}. For that special generator we take I_{0} to be either [−∞, 0] or [0, ∞].

We make the following definitions for chains of axes which are parallel to the definitions given at the beginning of the previous section for chains of waist curves.

Definition: A sequence of axes is called nested if the corresponding sequence of intervals I_{k} is nested. A sequence of axes has the adjacency property if the sequence of waist curves γ_{k} = π(A_{k}) on X has the adjacency property. A sequence of axes is non-recurrent if the corresponding sequence of waist curves γ_{k} = π(A_{k}) on X is non-recurrent.

Definition: A nested non-recurrent sequence of axes satisfying the adjacency property is called an allowable chain of axes.

Proposition 5. Any nested sequence of axes can be extended to have the adjacency property.

Proof: Let A_{k} and A_{k+1} be elements of the nested sequence of axes. Denote the endpoints of the intervals I_{k} by a_{k}, b_{k}. Since the sequence is nested, we assume a_{k} < a_{k+1} < b_{k+1} < b_{k}. Set L = a_{k+1} − a_{k}, M = b_{k+1} − a_{k+1}, R = b_{k} − b_{k+1} and T = b_{k} − a_{k}. Let d be the hyperbolic distance between A_{k} and A_{k+1}; it satisfies

cosh d − 1 = \frac{2RL}{MT}. (2)

Since there is a lower bound ϵ_{0} on the lengths of the orthogonals to the waist curves, there is an ϵ_{1} such that min{R/T, L/T } ≥ ϵ_{1} > 0. It follows that there are only finitely many axes in W between A_{k} and A_{k+1}; that is, axes with one endpoint in (a_{k}, a_{k+1}) and one endpoint in (b_{k+1}, b_{k}). Thus we can extend any nested chain by adding in finitely many axes between every pair until the adjacency property is satisfied.

Remark. Another way to obtain a bound on the number axes of hyperbolic elements of a group joining two disjoint intervals is given in [6]. The bound applies even to hyperbolic axes which may correspond to curves with self-intersection on X.

Lemma 6. If A_{k} is an allowable chain of axes, then the intervals I_{k} intersect in a single point x ∈ ∂ℍ.

Proof: We use the notation in the proof of proposition 5 above for the endpoints of the intervals and their lengths. The lemma will follow from the claim that if ϵ_{0} is the lower bound on the lengths of the orthogonals to the waist curves, then for all k, |I_{k+1}/I_{k}| < 1 − ϵ_{0}/2, which implies the lengths of I_{k} decrease geometrically to 0.

In the above notation |I_{k+1}/I_{k}| = M/T and \frac{L}{T} + \frac{M}{T} + \frac{R}{T} = 1. Then equation (2) yields \frac{d^{2}}{4} ≤ \frac{RL}{MT} . This implies either \frac{R}{T} or \frac{L}{T} is greater than or equal to d/2 and hence greater than ϵ_{0}/2. This, in turn, implies the upper bound \frac{M}{T} ≤ 1 − ϵ_{0}/2.

We now show that allowable chains of axes exist.

Proposition 7. Let x ∈ C and let A_{x_{n}} be the chain of axes of the generators g_{x_{n}}. Then A_{x_{n}} is an allowable chain.

Proof: We use the notation of sections 4.2 and 5.1. We use the points p̃_{x_{n}} in the lifts P̃_{x_{n}} forming the canonical polygon ω_{0}.

There is a unique geodesic σ̃ that joins the points p̃_{0} and p̃_{1} in ℍ. Its intersection q̃ = it with A_{0} projects to q and σ̃ is a uniquely defined lift of σ at q̃.

To show the A_{x_{n}} are nested we proceed by induction on n. The geodesic A_{0} = A_{x_{1}} divides ℍ into two hyperbolic half planes, one containing p̃_{0} and the other containing p̃_{1}. Denote by H_{1} the half plane containing p̃_{x_{1}}; it contains the connected lift P̃_{x_{1}} containing p̃_{x_{1}} and the boundary of this lift contains an arc of each of the axes A_{0} = A_{x_{1}}, A_{0x_{1}} and A_{1x_{1}}. Thus A_{0} does not separate A_{0x_{1}} and A_{1x_{1}} and they both lie in H_{1}.

We saw above that the lifted arc δ̃_{x_{n−1}} is a geodesic arc that joins p̃_{x_{n−1}} and p̃_{x_{n}} and crosses the geodesic A_{x_{n}}. Thus A_{x_{n}} divides ℍ into two hyperbolic half planes, one containing p̃_{x_{n−1}} and the other, which we denote by H_{n}, containing p̃_{x_{n}}.

We take as our induction hypothesis that H_{k} ⊂ H_{k−1} for all k ≤ n. Suppose now that H_{n+1} ⊄ H_{n}. Then δ̃_{x_{n}} joining p̃_{x_{n}} and p̃_{x_{n+1}} has to intersect A_{x_{n+1}} an even number of times. This is impossible because it must intersect it once by definition, and geodesics in ℍ intersect exactly once or not at all. Thus H_{n+1} ⊂ H_{n} and the induction is complete.

The adjacency property follows directly from the construction. Moreover since the level of A_{x_{n}} is precisely n the chain is non-recurrent.

Definition: The point ∩_{k}I_{k} for an allowable chain A_{k} is its limiting point.

We remark that there are infinitely many different allowable chains of axes with the same limiting point. Using the special chains A_{x_{n}}, however, we obtain the following result.

Proposition 8. There is a well-defined map π̃ from the limiting point x̃ of the allowable chain of axes A_{x_{n}} determined by the code of x = s_{1}s_{2} … to the corresponding point x = π̃(x) of C. Restricted to the boundary of the canonical polygon ω_{0}, this map is Hölder continuous.

Proof: Consider two points x̃ and ỹ on the x−axis which are on the boundary of the canonical fundamental polygon ω_{0}. Without loss of generality, assume x̃ and ỹ are both positive. We let A_{x_{n}} and A_{y_{n}} be the allowable chains of axes described at the beginning of section 5 associated to the points x̃ and ỹ. Also, let I_{x_{n}} be the interval on the real axis whose endpoints are the endpoints of the axis A_{x_{n}}. I_{x_{n+1}} is a subinterval of I_{x_{n}} and the four endpoints of these intervals divide I_{x_{n}} into three intervals, L, M = I_{x_{n+1}} and R. We let T = I_{x_{n}}. Using equation (2), we see that the lower bound ϵ_{0}, on the distances between A_{x_{n}} and A_{x_{n}+1} guarantees the existence of a number ϵ_{1} > 0 such that

min{L/T, M/T, R/T } > ϵ_{1},

for any x_{n} and any x̃ on the boundary of ω_{0}. Therefore, there is a positive number C such that the distance from x̃ to either of the endpoints of I_{x_{n}} is greater than or equal to Cϵ_{1}^{n}. Now suppose ỹ is in the boundary of ω_{0} and that |ỹ − x̃| < Cϵ_{1}^{n}. Then I_{x_{k}} = I_{y_{k}} for 0 ≤ k ≤ n. This implies the waist curves γ_{x_{k}} and γ_{y_{k}} are equal for 0 ≤ k ≤ n. In particular, the points x = π̃(x̃) and y = π̃(ỹ) in the Cantor set C corresponding to x̃ and ỹ satisfy |x − y| < (\frac{1}{3})^{n}. A simple computation shows that if we let α = (log(1/3)/ log(ϵ_{1}) the above inequalities imply

|x − y| ≤ C|x̃ − ỹ|^{α}

.

Note that any element g ∈ G determines an image g(ω_{0}) and allowable chains g(A_{x_{n}}) and that the same result holds for these chains and the boundary of g(ω_{0}).

Definition: The lifted boundary L_{X}(G) is the set of limiting points of all allowable chains of axes.

It is clear from the definitions that the image of an allowable chain of axes under an element of the group G is another allowable chain and hence that L_{X}(G) is G-invariant.

6. Conical and non-conical limit points

Suppose G is a Fuchsian group acting on ℍ with limit set Λ(G). Let c be a point in Λ(G) and let r be a geodesic ending at c. Given d > 0, a cone C_{d} based at c is the set of points in ℍ whose hyperbolic distance from the ray r is at most d.

The conical limit points of a Fuchsian group are defined as follows, see [13], where they are called radial limit points, or [11] or [10], where they are called points of approximation:

Definition: A point x ∈ Λ(G) is called a conical limit point if there is point p in ℍ and a cone C_{d} ⊂ ℍ based at x such that x is an accumulation point of the intersection of the G-orbit of p and the cone C_{d}. The conical limit set is the set of conical limit points and the non-conical limit set NC(G) is its complement in Λ(G).

Note that if x is a conical limit point then given any p′ ∈ ℍ there is some d′ such that a cone C_{d′} contains infinitely many points of the G-orbit of p′. It is obvious that the attracting fixed point x of a hyperbolic element g ∈ G is in the conical limit set because any cone C based at x and centered on A_{g} contains infinitely many points of {g^{n}(z)} for z ∈ A_{g} as n → ∞.

On the other hand if g is parabolic with fixed point x then x is in the nonconical limit set. To see this let D_{x} be a disk tangent to the real axis at x and precisely invariant under the cyclic subgroup generated by g. By precisely invariant, we mean g(D_{x}) = D_{x} and, if g′ is any element of G not equal to g^{n} for some integer n, then g′(D_{x}) ∩ D_{x} = ∅. Such a disk always exists for any primitive parabolic element of a Fuchsian group. In fact, such a disk can be chosen so that its image on the Riemann surface ℍ/G has hyperbolic area equal to one, [15]. Let C_{d} be any cone at x and let p ∈ D_{x} ∩ C_{d}. By the precise invariance, if g is not a power of g, then g′(p) ∉ D_{x}. By the parabolic action g^{n}(p) remains in D_{x} but leaves C_{d} as n → ∞. It follows that x is in the non-conical limit set.

It is clear from the definitions that the conical and non-conical limit sets are G-invariant.

We are now ready to prove our main result.

Theorem 9. If X is the Riemann sphere from which the Cantor set C is removed and G is a Fuchsian group representing X, then the set of non-conical limit points coincides with the set of limiting points of all allowable chains of axes, that is,

NC(G) = L_{X}(G).

Proof: Suppose that x̃ ∈ L_{X}(G) and let A_{k} be an allowable chain of axes limiting to x̃. Let C_{d} be any cone based at x̃. Denote by R_{k} the intersection of C_{d} and the region between A_{k} and A_{k+1}. By the adjacency property of the chain, the uniform lower bound on the distances between axes projecting to waist curves and the fact that the width of the cone is finite, there is a uniform upper bound on the areas of the R_{k}. The projection π(R_{k}) onto X is contained in a pair of pants P_{k}. By the uniform upper bound on the lengths of the waist curves there is a uniform upper bound on the the areas of the P_{k}. Since R_{k} is compactly contained in ℍ, the projection π, restricted to R_{k}, has image contained in P_{k} and is finite to one. Since the chain is non-recurrent, for any k, P_{k′} = P_{k} for only finitely many k′. It follows that any point in the projection of C_{d} onto X can have only finitely many preimages in C_{d} and therefore that x̃ ∈ NC(G).

Now suppose x̃ ∈ NC(G) and let C_{d} be a cone based at x̃. Let C_{d} consist of the set of points in the upper half plane with hyperbolic distance less than d from a geodesic ray r starting at a point z_{0} in ℍ and ending at x̃ on ℝ. Assume 2d is less than ϵ_{0}, the uniform lower bound on the lengths of the orthogonals to the waist curves.

Consider the arc π(r) on X and let P_{k} be the sequence of pairs of pants in X traversed, in order, by the arc π(r). If γ_{k} of level m_{k} is the waist of P_{k}, we claim that m_{k} goes to infinity with k. If not, for some k_{0} there is a sequence k_{j} → ∞ such that P_{k_{j}} = P_{k_{0}}. The lower bound ϵ_{0} on the lengths of the orthogonals implies the areas of the pairs of pants is uniformly bounded above. Thus, there must be a neighborhood N_{ϵ} of diameter ϵ in P_{k_{0}} with infinitely many disjoint preimages in C_{d}, which contradicts the assumption that x̃ ∈ NC(G).

Each waist curve γ_{k} lifts to a unique axis A_{k} that intersects C_{d}; these axes determine a nested chain which, by the above argument, is non-recurrent. The cone C_{d} is simply connected, and the order of the axes is determined by the ray r. Since the ray projects onto adjacent pairs of pants, the chain is allowable. The point x̃ is therefore in L_{X}(G).

7. Quasiconformal deformations

So far we have dealt only with the middle thirds Cantor set, the Riemann surface obtained by removing this Cantor set from the Riemann sphere and a normalized Fuchsian group representing this surface. In this section we vary all of these objects. To fix notation we now denote the middle thirds Cantor set by C_{0}, the corresponding Riemann surface by X_{0} = ℂ \ C_{0} and the normalized Fuchsian group by G_{0}.

The Teichmüller space T (X_{0}) is defined as follows. Consider the set of pairs (f, X) where f : X_{0} → X is a normalized quasiconformal homeomorphism to a variable Riemann surface X. Define an equivalence relation on this set by stipulating that (f, X) is equivalent to (g, Y) if there exists a conformal homeomorphism h : X → Y such that g^{−1}hf : X_{0} → X_{0} is homotopic to the identity through a homotopy which fixes pointwise the points of C_{0}. By definition the Teichmüller space T (X_{0}) consists of the set of these pairs factored by this equivalence relation. If (f, X) represents a point in T (X_{0}), then the map f induces an isomorphism Θ_{f} : G_{0} → G = fG_{0}f^{−1}. The group G is Fuchsian and ℍ/G is holomorphically equivalent to X.

We have the following facts:

• Let (f, X) be any representative of a point in T (X_{0}) and suppose f is a K-quasiconformal homeomorphism. Then f extends uniquely to a Kquasiconformal homeomorphism of the full Riemann sphere. This follows because for each annulus α_{x_{n}}, (see [2])

1/K ≤ \frac{mod(f(α_{x_{n}}))}{mod(α_{x_{n}})} ≤ K.

One uses this inequality to show f extends to a homeomorphism defined everywhere and then one shows that f is absolutely continuous on almost every horizontal and vertical line. Denote the extended map again by f.

• The symbolic dynamics carries over as well. If x = s_{1}s_{2} … ∈ C_{0} we assign f(x) = s_{1}s_{2} . . .. Let C = f(C_{0}). The expanding map fEf^{−1} commutes with the shift map on C. Moreover, since it is defined, it generates a (not necessarily conformal) dynamical system fEf^{−1} acting on X conjugate to the original system E acting on X_{0}.

• The construction of the set L_{X_{0}}(G_{0}) was based only on the length bounds. If a waist curve γ on X_{0} has length l(γ) and the waist curve in the free homotopy class of f(γ) has length l(f(γ)) on X, then the following inequality is well-known:

l(γ)/K ≤ l(f(γ)) ≤ Kl(γ),

(see for example [5], page 159). It follows that we can define the set L_{X}(G). The proof of the theorem NC(G) = L_{X}(G) holds mutatis mutandi. This is summarized by the following theorem.

Theorem 10. For any point [(f, X)] ∈ T (X_{0}) represented by a Fuchsian group G, the non-conical limit set coincides with the set of limiting points of allowable chains of axes. Moreover, the natural mapping from the boundary of the canonical fundamental polygon ω for G maps surjectively onto C and is Hölder continuous.

8. A Further Generalization

In this section we define a conformal, expanding dynamical system that generalizes the construction of C. Take the interiors of a disk D and m subdisks D_{1}, ⋯, D_{m} of D. We assume the subdisks are mutually disjoint and that the distances between their boundaries and between the boundary of any subdisk D_{j} and the the boundary of D is bounded below by a positive number. We let E_{j} be any affine transformation mapping D_{j} onto D and E be the join of the mappings E_{j}. Thus, E is an m to 1 mapping with domain ⋃_{j} D_{j} and range D and E restricted to D_{j} is equal to E_{j}. On D_{j}, E′ is constant and |E′| > 1.

We say the Ẽ is a locally homeomorphic, expanding conformal dynamical system if there is a a plane quasiconformal mapping f with the property that Ẽ = f ∘E∘f^{−1} is conformal on f( ⋃_{j} D_{j}) = ⋃_{j} D̃_{j}.

Now let E be a locally homeomorphic, expanding conformal dynamical system with range D and let Λ = ⋂_{k} E^{−k}(D). Written in this form, the Cantor set Λ is seen as the decreasing intersection the sets E^{−k}(D) each of which consists of m^{k} components.

If Y = ℂ̂ \ Λ, then, like X, it is a (no longer binary) tree of pairs of pants. The boundaries of the disks D_{j} and their inverse images under E are conformal waist curves. By construction, there is a uniform upper bound on the extremal length of the free homotopy class of any such waist curve. It follows that the hyperbolic lengths of the hyperbolic waist curves is also bounded above.

The definitions of allowable chains of waist curves and allowable chains of axes for a Fuchsian uniformization G for Y clearly extend to this case. Therefore the limit set L_{Y} (G) is well defined. Moreover, the construction of the fundamental polygon ω generalizes and from the the proof above we have

Theorem 11. Fix some locally homeomorphic expanding conformal dynamical system E_{0} with invariant Cantor set Λ_{0} and let Y_{0} = ℂ̂\Λ_{0}. For any point [(f, Y)] ∈ T (Y_{0}) represented by a Fuchsian group G, the non-conical limit set coincides with the set of limiting points of allowable chains of axes. Moreover, the natural mapping from the boundary of the canonical fundamental polygon ω for G maps surjectively onto C and is Hölder continuous.

9. Appendix

The existence of a lower bound on the hyperbolic lengths of the waist curves is a corollary to a stronger result valid for complements of Cantor sets generated by conformal dynamical systems. That result is called the hyperbolic distortion lemma; it measures the extent to which long compositions E^{n} of E deviate from being isometries in the hyperbolic metric on the Riemann surface close to its Cantor set boundary. For consistency of presentation, we prove this distortion lemma in the special case where the surface X is the complement in the sphere of the middle thirds cantor set C. We keep the notation set above.

Let ds = λ_{X}(z)|dz| denote the hyperbolic metric on X and for x_{n} = s_{1} … s_{n} set ϕ^{x_{n}}(z) = ϕ_{s_{n}} ∘ ϕ_{s_{n−1}} ∘ … ϕ_{s_{1}}(z). Then we have

Lemma 12 (Hyperbolic Distortion lemma). For every ϵ > 0 there exists a neighborhood U = U(ϵ) ⊃ C such that for every z ∈ U \ C and every x_{n},

\frac{1}{1 + ϵ} ≤ \frac{λ_{X}(ϕ^{x_{n}}(z))|(ϕ^{x_{n}})′(z)|}{λ_{X}(z)} ≤ 1 + ϵ.

Proof: A similar result is proved in lemma 3 of [7].

We begin the proof with three observations. Suppose ρ and λ are infinitesimal forms of metrics on domains which contain C. We say λ and ρ are asymptotic at C if, for every ϵ > 0, there is a neighborhood U of C, such that if z is in U \ C then

\frac{1}{1 + ϵ} ≤ \frac{λ(z)}{ρ(z)} ≤ 1 + ϵ.

The first observation is that this lemma is true for a metric λ if and only if it is true for any metric ρ asymptotic to λ at C.

For the second observation, suppose Ω_{1} ⊃ Ω_{2} are plane domains containing C and ρ_{1} and ρ_{2} are the Poincaré metrics for the domains Ω_{1} \ C and Ω_{2} \ C. We claim ρ_{1} and ρ_{2} are asymptotic at C. Moreover,

ρ_{1}(z) < ρ_{2}(z) < \frac{1 + e^{−2d}}{1 − e^{−2d}} ρ_{1}(z),

where d is the Poincaré distance from z to the boundary of Ω_{2} measured in the ρ_{1}−metric. The first inequality follows from Schwarz’s lemma applied to the inclusion map from Ω_{2} \ C into Ω_{1} \ C. To see the second inequality, represent the fundamental group for Ω_{1} \ C as a Fuchsian group G acting on the unit disk, 𝔻 = {w : |w| < 1}, in such a way that the universal covering π projects 0 to z. Among all holomorphic functions f defined on the unit disk and mapping 0 to z, the minimum of the ratio \frac{1}{|f′(0)|} and \frac{1}{|π′(0)|} is 1. It is therefore realized by the function π and ρ_{1}(z) = \frac{1}{|π′(0)|}. From the definition of d, π^{∗}ρ_{2}(z) = ρ_{2}(π(z))|π′(z)| is a complete, Riemannian metric of constant negative curvature equal to −1 on a domain contained in the unit disk and containing the disk of radius r, where r is related to d by the formula

\frac{1}{2} log \frac{1 + r}{1 − r} = d.

Since the formula for the infinitesimal form of Poincaré’s metric on a disk of radius r is

\frac{r|dz|}{r^{2} − |z|^{2}} ,

we conclude ρ_{2}(π(0))∥π′(0)| ≤ \frac{1}{r}, which implies

ρ_{2}(z) ≤ \frac{1}{r} \frac{1}{|π′(0)|}.

Therefore,

ρ_{2}(z) ≤ \frac{e^{2d} + 1}{e^{2d} − 1}ρ_{1}(z). (3)

To see that ρ_{1} and ρ_{2} are asymptotic at C we only have to observe that the set of z such that the distance from z to the boundary of Ω_{2} measured in the Poincaré metric ρ_{1} is larger than a constant M determines a neighborhood U of the set C.

For the third observation, let Δ = {z : |z − 1/2| < 11/18} be the interior of Γ_{−1} and let Y_{0} = Δ \ C. Next let Y_{1} = (ϕ_{s_{1}}(Δ \ C), Y_{k} = ϕ_{s_{k}} ∘ ⋯ ∘ ϕ_{s_{1}}(Δ \ C) and let λ, λ_{0}, λ_{1}, λ_{2}, … be the infinitesimal forms of the Poincaré metrics on X, Y_{0}, Y_{1}, Y_{2}, …, respectively. Let δ_{1} be the minimum λ−distance from ∂Y_{0} to ∂Y_{1} and, in general, δ_{k} be the minimum λ−distance from ∂Y_{k−1} to ∂Y_{k}. We claim

δ_{1} < δ_{2} < ⋯ < δ_{n}.

First we show δ_{1} < δ_{2}. We may assume Y_{1} = ϕ_{0}(Δ \ C). Let λ̃ be the infinitesimal form of the Poincaré metric on the Ỹ = \overline{ℂ} \ ϕ_{0}(C). Then the map z ↦ 3z from Ỹ to X is an isometry and it maps Y_{1} to Y_{0}. Therefore, the distance δ_{1} from the boundary of Y_{0} to the boundary of Y_{1} in the λ−metric is equal to the distance δ̃_{2} from the boundary of Y_{1} to the boundary of Y_{2} in the λ̃−metric. But since λ̃(z) < λ(z), δ̃_{2} < δ_{2}. To show δ_{k} < δ_{k+1}, note that Y_{k+1} = ϕ(Y_{k}), where ϕ(z) = \frac{1}{3}z + (const). Now let λ̃ be the Poincaré metric for the extended complex plane with ϕ(C) removed and apply the same argument as before.

To begin the proof of the lemma, first note that

λ < λ_{0} < λ_{1} < ⋯ < λ_{n}.

Now assume z is in a small neighborhood of C and contained in Y_{n} and let d_{k} be the λ_{k−1}−distance from z to the boundary of Y_{k}. Then from the second observation above

λ_{k}(z) < λ_{k−1}(z)\frac{1 + e^{−2d_{k}}}{1 − e^{−2d_{k}}} .

Since λ_{k}(ϕ_{k−1}(z))|ϕ′_{k−1}(z)| = λ_{k}(z) and λ_{n}(ϕ^{x_{n}}(z))|(ϕ^{x_{n}})′(z)| = λ_{0}(z), we find that

λ_{0}(z) < λ_{n}(z) = λ_{0}(ϕ^{x_{n}}(z))|(ϕ^{x_{n}})′(z)| =

(λ_{n})/λ_{n−1}) ⋯ (λ_{1}/λ_{0})(λ_{0}) < λ_{0}(z) ∏_{k=1}^{n} \frac{1 + e^{−2d_{k}}}{1 − e^{−2d_{k}}} . (4)

However, each d_{k} is bigger than or equal to ∑_{j=k}^{n} δ_{j} and each δ_{j} ≥ δ_{1}. Therefore the product in (4) is dominated by

(\frac{1 + e^{−2d_{1}}}{1 − e^{−2d_{1}}} ) ∏_{k=2}^{n} \frac{1 + α^{2k}}{1 − α^{2k}} , (5)

where α = e^{−δ_{1}} < 1. Since the second factor in (5) is bounded independently of n and the first factor can be made less than 1 + ϵ by choosing the neighborhood U sufficiently small, the lemma is proved.

Using this proposition we can prove

Proposition 13. There is a constant M > 0 such that for all waist curves γ_{k}, l(γ_{k}) > M.

Proof: Fix ϵ > 0. Since the lengths of the hyperbolic waist curves are bounded above we can fix k = x_{m} such that γ_{x_{m}} ⊂ U(ϵ). By the hyperbolic distortion lemma, for every truncation x_{n} we have

l(γ_{x_{m}x_{n}}) = ∫_{γ_{xmxn}} λ(z)|dz|
≥ \frac{1}{1 + ϵ} ∫_{γ_{xmxn}} λ(ϕ^{x_{n}}(z))|(ϕ^{x_{n}})′(z)||dz|
= \frac{1}{1 + ϵ} ∫_{E^{xn} (γ_{xmxn})} λ(z)|dz|
≥ \frac{1}{1 + ϵ} ∫_{γ_{xm}} λ(z)|dz|
= \frac{1}{1 + ϵ}l(γ_{x_{m}}).

To find the constant M pick a neighborhood U corresponding to the choice of ϵ = 1/2 in Lemma 12. All of the waist curves at level m for m bigger than or equal to some m_{0} are contained in U. Let M be less than 2/3 multiplied by the minimum of the hyperbolic lengths of all the waist curves γ_{x_{k}} for k ≤ m_{0}. Then from the above inequality the hyperbolic lengths of all the waist curves γ_{x_{n}} for n ≥ m_{0} is greater than or equal to M. Moreover, each of the hyperbolic lengths of waist curves γ_{x_{k}} for k ≤ m_{0} is also greater than or equal to M and so this inequality holds for all waist curves.

Remark. There is more direct way to obtain a uniform lower bound on the lengths of the hyperbolic waist curves. This also generalizes to complements of Cantor sets of bounded geometry. One explicitly constructs a planar Riemann surface Ω which is a binary tree of pairs of pants. One arranges for Ω to have one hole “at the top” and to be such that all of its waist curves have hyperbolic length equal to one. Also, one constructs Ω so that there are conformal maps G_{0} and G_{1} from the right and left subtrees of Ω onto Ω mapping pants to pants. By using compositions of the mappings G_{0}, G_{1}, E_{0} and E_{1}, one can construct a quasiconformal mapping from Ω to a subdomain of X. Combining this result with inequality (3) and the Teichmüller inequality bounding the distortion of the length of a closed hyperbolic geodesic in a given homotopy class by a quasiconformal mapping (Lemma 2, page 159, of [5]), one obtains the uniform bound.

References

[1] L. V. Ahlfors, Lectures on Quasiconformal Mappings, Van Nostrand Studies 10, 1966.

[2] , Conformal Invariants; Topics in Geometric Function Theory, McGraw-Hill, 1973.

[3] A. F. Beardon, The Geometry of Discrete Groups, Graduate Texts in Mathematics, Springer-Verlag, 1983.

[4] A. F. Beardon and B. Maskit, Limit points of Kleinian groups and finite-sided polyhedra, Acta Math., 132 (1974) 1-12.

[5] F. Gardiner, Teichmüller Theory and Quadratic Differentials, Wiley-Interscience, John Wiley & Sons, 1987.

[6] , An inequality for conjugacy classes in Fuchsian groups, London Math. Soc. Lecture Note Series 111, (1984) 275-280.

[7] F. Gardiner and D. Sullivan, Lacunary series as quadratic differentials in conformal dynamics, Contemporary Mathematics, AMS 169 (1994) 307-330.

[8] E. Hopf, Ergodic theory and the geodesic flow on surfaces of constant negative curvature, Bul. Amer. Math. Soc., 77(1971) 863-877.

[9] B. Maskit, Comparison of hyperbolic and extremal lengths, Ann. Acad. Sci. Fenn., 10(1985) 381-386.

[10] Kleinian Groups Springer-Verlag, Series of Comprehensive Studies in Mathematics, 287 1988.

[11] P. Nicholls, The Ergodic Theory of Discrete Groups, Cambridge Univ. Press, (1989).

[12] S. Patterson, The limit set of a Fuchsian group, Acta Math., 136(1976) 241-273.

[13] Lectures on measures on limit sets of Kleinian groups, London Math. Soc. Lecture Series, 111 (1987) 281-323.

[14] I. Petrovic, A quasiconformal model for period doubling, Thesis, Graduate School and University Center of CUNY, (1998) preprint.

[15] H. Shimizu, On discontinuous groups in higher-dimensional spaces, Annals of Math., 77 (1959) 33-71.

[16] D. Sullivan, The density at infinity of a discrete group of hyperbolic motions, Publ. Math. IHES, 50(1979) 171-202.

Frederick P. Gardiner, Dept. of Math., Brooklyn College, CUNY, 11210

E-mail address: FGardiner@gc.cuny.edu

Linda Keen, Dept. of Math., Lehman College, CUNY, 10468

E-mail address: linda@alpha.lehman.cuny.edu