CIRCLE PACKINGS IN THE UNIT DISC

Tomasz Dubejko

Abstract. A Bl-packing is a (branched) circle packing that “properly covers” the unit disc. We establish some fundamental properties of such packings. We give necessary and sufficient conditions for their existence, prove their uniqueness, and show that their underlying surfaces, known as carriers, are quasiconformally equivalent to surfaces of classical Blaschke products. We also extend the approximation results of [D1] to general combinatorial patterns of tangencies in Bl-packings. Finally, a branched version of the Discrete Uniformization Theorem of [BSt1] is given.

Introduction

This is a continuation of the work initiated in [D1]. We are interested in (infinite) circle packings of the type considered in [D1], which we call Bl-packings. These are (branched) circle packings that are contained in the closure of the unit disc D, their circles do not accumulate in D, and their boundary circles, if any, are internally tangent to ∂D. The existence of finite Bl-packings was proved in [D1]; the existence of infinite ones will be proved in this paper.

Our overall aim is to describe such packings, show similarities that they share with branched surfaces of classical Blaschke products, and establish their distortion properties. We first prove that large collections of finite Bl-packings possess, under some rather natural restrictions, uniform bounds on their functions of local distortion which are defined in terms of ratios of radii of neighboring circles (Lemma 3.1). This result turns out to be very useful tool for our goals. It implies that large families of finite discrete Blaschke products of [D1] are K-quasiregular (Corollary 3.2). In particular, this allows to extend approximation schemes of [D1] to general combinatorics and to show that every classical Blaschke product can be approximated uniformly on compact subsets of D by discrete Blaschke products in combinatorially less restrictive setup then that of [D1] (Theorem 3.3).

Lemma 3.1 is also used in proving the existence of infinite Bl-packings (Theorem 4.1). Combining this with results of [DSt] and [D3] we obtain a branched version of the Discrete Uniformization Theorem of [BSt1], which shows the consistency in circle packing theory and its analogy to the classical theory of Riemann surfaces.

Typeset by AMS-TEX

1991 Mathematics Subject Classification. 52C15, 30C62, 30G25.

Key words and phrases. circle packing, discrete analytic maps, quasiconformal mappings.

Research at MSRI is supported in part by NSF grant DMS-9022140.

We finish our investigation of properties of Bl-packings with Theorem 4.4 which generalizes Lemma 3.1 and quasiregularity result for finite discrete Blaschke products to infinite Bl-packings and infinite discrete Blaschke products. The quantitative statement in Theorem 4.4 contains the inverse result to the Discrete Schwarz Lemma of [BSt2].

1. Preliminaries

All simplicial 2-complexes considered in this paper are assumed to be oriented and simplicially equivalent to triangulations of a disc in the complex plane C. They could be infinite, possibly with boundary. If 𝕂 is such a complex then 𝕂^{0}, int 𝕂^{0}, bd 𝕂^{0}, 𝕂^{1}, and 𝕂^{2}denote the sets of vertices, interior vertices, boundary vertices, edges, and faces of 𝕂, respectively. If u and v are neighboring vertices in a complex, shortly u ∼ v, then uv stands for the edge between them. The degree of a complex 𝕂 is defined by deg(𝕂) := sup_{v∈𝕂^{0}}{deg(v)}, where deg(v) is the number of vertices adjacent to v. A set {(v_{1}, k_{1}), …, (v_{m}, k_{m})} of pairs of interior vertices of 𝕂 and positive integers is called a branch structure for 𝕂 if for every simple closed edge path Γ in 𝕂 the number of its edges is at least 3 + ∑_{i=1}^{m} ind_{Γ}(v_{i})k_{i}, where ind_{Γ}(v_{i}) is 1 when v_{i} is enclosed by Γ and 0 otherwise.

A collection {C(v)}_{v∈𝕂^{0}} of circles in C is said to be a circle packing for a complex 𝕂 if ⟨C(u), C(v), C(w)⟩ is a positively oriented triple of mutually and externally tangent circles in C whenever ⟨u, v, w⟩ is an oriented faces of 𝕂. A circle packing P = {C_{P}(v)} for 𝕂 determines the simplicial map S_{P} : 𝕂 → C by the condition that S_{P}(v) is equal to the center of C_{P}(v); the geometric simplicial complex S_{P}(𝕂) is called the carrier of P, carr(P). For each △ = ⟨u, v, w⟩ ∈ 𝕂^{2} let α_{P}(v, △) denote the angle sum at S_{P}(v) in the Euclidean triangle S_{P}(△). If v ∈ int 𝕂^{0} then Θ_{P}(v) := ∑_{△∈𝕂^{2}} α_{P}(v, △) turns out to be a positive integer multiple of 2π and is called the (interior) angle sum at v induced by P; if v ∈ bd 𝕂^{0} then γ_{P}(v) := ∑_{△∈𝕂^{2}} α_{P}(v, △) is called the boundary angle sum at v induced by P.

We define the set of branch vertices of P as br_{V} (P) := {v ∈ int 𝕂^{0} : Θ_{P}(v) > 2π} and the branch set of P as br(P) := {(v, ord_{P}(v))}_{v∈br_{V} (P)}, where ord_{P}(v) := \frac{Θ_{P} (v)}{2π} − 1. A circle packing is said to be univalent if all its circles have mutually disjoint interiors, in which case its set of branch vertices must be empty.

Suppose P and Q are circle packings for a complex 𝕂. Then the simplicial map f : carr(P) → carr(Q), f(S_{P}(v)) = S_{Q}(v) for v ∈ 𝕂^{0}, is called the cp-map (read, circle packing map) from P to Q; in this case P and Q are called the domain and range packings of f, respectively. The associated affine map f^{#} : carr(P) → (0, ∞) determined by f^{#}(S_{P}(v)) = \frac{r_{Q}(v)}{r_{P} (v)}, v ∈ 𝕂^{0}, is called the ration map from P to Q, where r_{∗}(v) is the radius of the circle C_{∗}(v). Notice that if P is univalent then f and f^{#} are functions defined in a domain of C (=carr(P)).

Recall that the valence val(f) of a function f is the least upper bound on the number of elements in the preimage under f of any given point from the range. The valence of a packing P = {C_{P}(v)} is defined by val(P) := sup{n : ⋂_{i=1}^{n} D_{P}(v_{i}) ≠ ∅}, where D_{P}(v) is the closed disc bounded by C_{P}(v). It is easy to see that val(f) ≤ val(P) for every cp-map f whose domain packing is univalent and whose range packing is P.

Finally, we will say that a map f : Ω → Ω, Ω ⊂ C, is an automorphism of Ω if it is 1-to-1, onto, and conformal. We note that all automorphisms of C are similarities of C, and all automorphisms of D are Möbius transformations fixing D.

2 Bl-packings and discrete Blaschke products

We begin with the following definition.

Definition 2.1. Let 𝕂 be a (finite or infinite) triangulation of a disc. A circle packing P for 𝕂 is said to be a Bl-packing if P is contained in \overline{D}, its circles have no accumulation point in D, and its boundary circles, if any, are internally tangent to ∂D.

(a)

(b)

Figure 1*. Bl-packings for the same complex: (a) univalent and

(b) 3-fold branched (with black circles corresponding to branch vertices of order 1) and its decomposition into locally univalent sheets.

*This figure was created with the help of CirclePack by the kind permission of Ken Stephenson

The existence of finite Bl-packings (i.e., Bl-packings for finite complexes) was proved in [D1]. The existence of infinite ones is more subtle and will be shown in Section 4. Leaving aside the issue of existence we now establish some fundamental properties of Bl-packings.

Proposition 2.2. Suppose P is a Bl-packing for 𝕂. Then the following hold. (1) There exists a univalent Bl-packing for 𝕂.

(2) If P̃ is a univalent Bl-packing for 𝕂 then there exists an extension f̄: D → D of the cp-map f from P̃ to P such that f̄ = ϕ ∘ h, where ϕ : D → D is a classical, finite Blaschke product and h : D → D is a homeomorphism. Moreover,

val(P) = val(f̄) = 1 + ∑_{v∈br_{V} (P)} ord_{P}(v),

and the branch set of ϕ is equal to {(h(SP̃(v)), ord_{P}(v))}_{v∈br_{V} (P)}.

(3) If Q is Bl-packing such that br(Q) = br(P) then there exists a Möbius transformation M preserving D such that Q = M(P), i.e. Bl-packings are uniquely determined, up to automorphisms D, by their branch sets.

Proof. (1) This part follows easily, by switching to hyperbolic metric in D, from the Perron-type arguments of [BSt2] (see [BSt2, Thm. 4]).

(2) We first define a map f̄. If 𝕂 has no boundary then we set f̄ to be f. For 𝕂 with boundary the construction of f̄ is as follows.

Suppose u, v ∈ bd 𝕂^{0}, u ∼ v. Define λ_{∗}(v) := C_{∗}(v) ∩ ∂D and μ_{∗}(u, v) := C_{∗}(u) ∩ C_{∗}(v), where ∗ denotes P or P̃. Also, if there exists a component of D\(C_{∗}(u)∪C_{∗}(v)) that has empty intersection with ⋃_{△∈𝕂^{2}:uv⊂△} S_{∗}(△), we denote it by A_{∗}(u, v) (there could be at most one such a component). Let M(u, v) be the Möbius transformation mapping \overline{AP̃(u, v)} onto \overline{A_{P}(u, v)} with points λP̃(u), λP̃(v), and μP̃(u, v) mapped to λ_{P}(u), λ_{P}(v), and μ_{P}(u, v), respectively (Fig. 2). Thus f and the maps M(u, v), u, v ∈ bd 𝕂^{0}, u ∼ v, induce a well-defined and continuous function f̂: carr(P̃) ∪ (⋃_{v∈bd 𝕂^{0}} CP̃(v)) ∪ (⋃\begin{matrix}u,v∈bd 𝕂^{0} \\ u∼v \end{matrix} AP̃(u, v)) → D. To obtain f̄

we extend each f̂|_{CP̃ (v)}, v ∈ bd 𝕂^{0}, radially to the interior of DP̃(v).

CP̃(v) C_{P}(u)

f̄
C_{P}(u)

CP̃(u)

AP̃(u, v) A_{P}(u, v)

P̃ P

Figure 2. The extension.

Now recall that a continuous function g : X → Y, X, Y ⊂ C, is proper if preimage of every compact set in Y is compact. Definitions of Bl-packings and cpmaps together with the above construction imply that f̄ is a proper mapping. Thus, Stoilow’s theorem [LV] and the fact that an analytic proper function Φ : D → D is a finite Blaschke product show that f̄ = ϕ ∘ h : D → D, where h : D → D is a homeomorphism and ϕ : D → D is a finite Blaschke product. In particular, val(P) = val(ϕ) = 1 + ∑_{v∈br_{V} (P)} ord_{P}(v) < ∞.

(3) The assertion of this part follows from the Discrete Schwarz and Distortion Lemmas of [DSt]. □

In [D1] the notion of finite, discrete Blaschke products was introduced. In the light of the last proposition it is natural to extend this notion to the infinite case.

Definition 2.3. A discrete Blaschke product is a cp-map between two Bl-packings with the domain packing being univalent.

If f is a discrete Blaschke product then the extension of f constructed in the proof of Proposition 2.3(3) will be denoted by f̄.

3. Finite Bl-packings and discrete Blaschke products: quasiregularity and approximation

In this section we are primarily interested in a local behavior of finite Bl-packings and discrete Blaschke products. We establish here our key result, Lemma 3.1, which is followed by two applications. The first application implies that discrete Blaschke products are quasiregular mappings. The second one shows that approximation schemes of [D1] extend to general combinatorics.

For the purpose of studying local behavior of Bl-packings we introduce a function that measures local distortion of packings. If P is a circle packing for 𝕂 then the function μ_{P} : 𝕂^{0} → (0, ∞) defined by

μ_{P}(v) := max_{\begin{matrix}u∈𝕂^{0} \\ u∼v \end{matrix}} \frac{1}{2} (\frac{r_{P}(u)}{r_{P}(v)} + \frac{r_{P}(v)}{r_{P}(u)})

will be called the function of local distortion of P.

We can now state

Lemma 3.1. Let d and ℓ be positive integers, and ρ ∈ (0, 1). There exists a constant κ = κ(d, ℓ, ρ) ≥ 1, depending only on d, ℓ, and ρ, such that if P is a Bl-packing for a complex 𝕂 satisfying 1) deg(𝕂) ≤ d, 2) val(P) ≤ ℓ, and 3) S_{P}(int 𝕂^{0}) ∩ {|z| ≤ ρ} ≠ ∅ and {S_{P}(v)}_{v∈br_{V} (P)} ⊂ {|z| ≤ ρ} then

(*)  μ_{P}(v) ≤ κ for all v ∈ 𝕂^{0}.

The importance of this lemma lies in the fact that (*) holds for all vertices in the complex (cf. [D1, Lemma 6.1] and [D2, Corollary 4.9]). Its proof will be given in the last section. Now couple remarks regarding its hypotheses. It is immediate that for univalent Bl-packings only the conditions 1) and the first part of 3) are necessary; the remaining conditions are satisfied anyway in this case. For Bl-packings which are not univalent it is not hard to see that all conditions 1) – 3) are necessary, though the first part of the condition 3) can be removed because it follows from the second part.

The following is almost an immediate consequence of Lemma 3.1.

Corollary 3.2. Let d, ℓ, and ρ be as in Lemma 3.1. Then there exists a constant K = K(d, ℓ, ρ) ≥ 1, depending only on d, ℓ, and ρ, such that if f is a discrete Blaschke product whose domain and range packings satisfy 1)–3) of Lemma 3.1 then the corresponding extension map f̄: D → D is K-quasiregular.

Proof. From (*) it follows that there is a constant κ′ = κ′(d, ℓ, ρ) ≥ 1 such that for every f satisfying hypotheses of the corollary \frac{1}{κ′} α_{R_{f}} (v, △) ≤ α_{D_{f}} (v, △) ≤ κ′α_{R_{f}} (v, △) for all v ∈ 𝕂^{0} and △ = ⟨u, v, w⟩ ∈ 𝕂^{2}, where D_{f} and R_{f} are the domain and range packings of f, respectively, and 𝕂 is their complex. Hence, the definition of discrete Blaschke products implies that there is K_{1} = K_{1}(d, ℓ, ρ) such that f̄|_{carr(D_{f})} = f is K_{1}-quasiregular. It is also easy to see that (*) and the definition of extensions of discrete Blaschke products imply that there exists K_{2} = K_{2}(d, ℓ, ρ) such that f̄|_{D\carr(D_{f})} is K_{2}–quasiregular. By taking K := max_{i=1,2}{K_{i}} we obtain the assertion of the corollary. □

In [D1] a constructive method using hexagonal complexes was developed to show that every classical Blaschke product can be approximated uniformly on compact subsets of D by discrete Blaschke products. Corollary 3.2 allows for a generalization of this approximation scheme to other then hexagonal combinatorics.

Theorem 3.3. Let ϕ be a finite Blaschke product. Write {(x_{1}, k_{1}), …, (x_{m}, k_{m})} for the branch set of ϕ. Suppose {D_{n}} and {R_{n}} are sequences of finite Bl-packings such that

(1) D_{n} and R_{n} are packings for the same complex 𝕂_{n}, and D_{n} is univalent,

(2) there exists d such that deg(𝕂_{n}) ≤ d for all n,

(3) lim_{n→∞} σ_{n} = 0, where σ_{n} := sup_{v∈𝕂_{n}^{0}}{r_{D_{n}}(v)},

(4) for each n there are vertices v̇_{n}, v̈_{n} ∈ 𝕂_{n}^{0} such that S_{D_{n}}(v̇_{n}) = 0, S_{R_{n}}(v̇_{n}) = ϕ(0), lim_{n→∞} S_{D_{n}}(v̈_{n}) = 1/2, and Arg(S_{R_{n}}(v̈_{n})) = Arg(ϕ(1/2)), where Arg(z) denotes the argument of the complex number z,

(5) each br_{V} (R_{n}) can be written as a disjoint union of sets V_{1}(n), …, V_{m}(n) such that for each n one has ∑_{v∈V_{i}(n)} ord_{R_{n}}(v) = k_{i}, i = 1, …, m, and lim_{n→∞} S_{D_{n}}(V_{n}(i)) = {x_{i}}, i = 1, …, m, as the set limit.

Denote by f_{n} the discrete Blaschke product from D_{n} to R_{n}. Then the functions f_{n} and f_{n}^{#} converge uniformly on compact subsets of D to ϕ and |ϕ′|, respectively.

Proof. The proof will be the same as the proof of Theorem 7.2 in [D1] except for two modifications. The first modification is required in order to show that lim_{n→∞} δ_{n} = 0, where δ_{n} := sup_{v∈𝕂_{n}^{0}}{r_{R_{n}}(v)}. The second one is needed in proving that lim_{n→∞} f_{n} is a holomorphic function.

By applying if necessary the automorphism Φ of D such that Φ(ϕ(0)) = 0 and Arg(Φ(ϕ(1/2))) = Arg(ϕ(1/2)) to packings R_{n}, we can assume that ϕ(0) = 0.

Recall [LV] that a K-quasiconformal homeomorphism g : D → D, g(0) = 0, satisfies

(3.1)  |g(z_{1}) − g(z_{2})| ≤ 16|z_{1} − z_{2}|^{1/K} for all z_{1}, z_{2} ∈ \overline{D}.

Recall also that for each n the extension map f̄_{n} of f_{n} has a decomposition f̄_{n} = ϕ_{n} ∘ h_{n}, where ϕ_{n} is a Blaschke product and h_{n} is a self-homeomorphism of D, h_{n}(0) = 0, h_{n}(1/2) ∈ (0, 1), and ϕ_{n}(0) = 0 (because ϕ(0) = 0).

The condition (5) implies that each ϕ_{n} is a Blaschke product of degree ℓ = 1 + ∑_{i=1}^{m} k_{i}; in particular val(R_{n}) ≤ ℓ for every n. Moreover, (5) and the Discrete Schwarz Lemma of [BSt2] imply that there is ρ such that S_{D_{n}}(br_{V} (R_{n})) and S_{R_{n}}(br_{V} (R_{n})) are contained in {|z| ≤ ρ} for all n. Therefore, by Lemma 3.1, the functions h_{n} are K-quasiconformal and form a normal family whose every convergent subsequence has a K-quasiconformal self-homeomorphism of D as its limit [LV, Ch. II, Thm. 5.5]. This gives that {ϕ_{n}} is not only a normal family but it is also compact in the class of Blaschke products, and it follows from (3.1) and the equicontinuity of normal families that lim_{n→∞} δ_{n} = 0.

Thus we are now in a position of copying all quasiregular arguments (except for the argument involving conformality) used in the proof of Theorem 7.2 in [D1] to our current setting. By replacing the hexagonal complexes H(n) of [D1] by the complexes 𝕂_{n} and by repeating the arguments of [D1] we arrive to the following conclusion: from every subsequence of {f_{n}} we can choose further subsequence {f_{n_{l}}} such that: 1) h_{n_{l}} → h uniformly on compacta of D as l → ∞, where h : D → D is a K-quasiconformal homeomorphism, h(0) = 0, and h(1/2) ∈ (0, 1), and 2) ϕ_{n_{l}} → ψ uniformly on compacta of D as l → ∞, where ψ is a Blaschke product with branch set {(h(x_{1}), k_{1}), …, (h(x_{m}), k_{m})}.

We will now show that h is conformal; this involves the second modification. Let z ∈ D \ {x_{1}, …, x_{m}}. The uniform convergence of {h_{n_{l}}} and {ϕ_{n_{l}}} implies that there is ϵ > 0 and L such that f_{n_{l}}|_{B_{ϵ}(z)} is 1-to-1 for every l > L, where B_{ϵ}(z) = {ζ : |ζ − z| < ϵ}. In particular, the portion of R_{h_{l}} corresponding to the portion of D_{h_{l}} contained in B_{ϵ}(z) is univalent. Since lim_{n→∞} δ_{n} = 0, Theorem 2.2 of [HR] shows that the quasiconformal distortion of f_{n_{l}} at z goes to 0 as l → ∞. Hence we obtain that h is 1-quasiconformal at z. Thus h is 1-quasiconformal in D \ {x_{1}, …, x_{m}}, and it follows that h is conformal in D.

By Riemann mapping theorem, h is the identity function and ψ = ϕ. Now standard arguments imply that the full sequence {f_{n}} converges uniformly on compact subsets of D to ϕ.

The convergence of {f_{n}^{#}} to |ϕ′| is a consequence of Theorem 1 of [DSt]. □

4. Existence and properties of (infinite) Bl-packings

As we have mentioned earlier, the author gave in [D1] the necessary and sufficient conditions, Theorem 2.5, for the existence of finite Bl-packings. With the help of Lemma 3.1 we are now able to extend this result to the infinite case, and show the existence of infinite Bl-packings.

Theorem 4.1. Let 𝕂 be a triangulation of a disc with deg(𝕂) < ∞. Then there exists a Bl-packing for 𝕂 with branch set 𝔅 if and only if there exists a univalent Bl-packing for 𝕂 and 𝔅 is a branch structure for 𝕂.

Proof. Since the necessity is a straightforward consequence of Proposition 2.2 and Theorems 2.2 of [D1], we only need to be concerned with the sufficient condition in our assertion and, due to [D1, Thm. 2.5], only when 𝕂 is infinite.

Let 𝕂_{1} ⊂ 𝕂_{2} ⊂ … be a sequence of subcomplexes of 𝕂 such that {b_{1}, …, b_{m}} ⊂ 𝕂_{1}, each 𝕂_{j} is a finite triangulation of a disc, and ⋃_{j} 𝕂_{j} = 𝕂. Let u_{0} and u_{1} be two designated vertices in 𝕂_{1}. Denote by P_{j} the univalent Bl-packing for 𝕂_{j} such that S_{P_{j}}(u_{0}) = 0 and S_{P_{j}}(u_{1}) ∈ (0, 1). It follows from [BSt1, Lemma 5] and the hypothesis that the sequence of packings {P_{j}} has the geometric limit which is a Bl-packing P for 𝕂 such that S_{P}(u_{0}) = 0 and S_{P}(u_{1}) ∈ (0, 1). In particular, there is ρ, 0 < ρ < 1, such that S_{P_{j}}({b_{1}, …, b_{m}}) ⊂ {|z| < ρ} for every j, and for each v ∈ 𝕂^{0}, lim_{n→∞} r_{P_{j}}(v) = r_{P}(v). By the hypothesis and [D1, Thm. 2.5], for each j there exists a Bl-packing Q_{j} for 𝕂_{j} such that br(Q_{j}) = 𝔅, S_{Q_{j}}(u_{0}) = 0 and S_{Q_{j}}(u_{1}) ∈ (0, 1). We will show that the sequence {Q_{j}} has the geometric limit.

From Theorem 4 of BSt1 it follows that S_{Q_{j}}({b_{1}, …, b_{m}}) ⊂ {|z| < ρ} for every j. Write f_{j} for the discrete Blaschke product from P_{j} to Q_{j}. Let f̄_{j} be the corresponding extension. Then, by Corollary 3.2, {f̄_{j}} is a family of K-quasiregular mappings. Moreover, for each j, f̄_{j} = ϕ_{j} ∘ h_{j}, where h_{j} : D → D is K-quasiconformal homeomorphism, h_{j}(S_{P_{j}}(u_{0})) = h_{j}(0) = 0, h_{j}(S_{P_{j}}(u_{1})) ∈ (0, 1), and ϕ_{j} is a Blaschke product, ϕ_{j}(0) = 0. Since h_{j}’s are uniformly bounded, {h_{j}} is a normal family. Thus, by taking a subsequence if necessary, we may assume that {h_{j}} converges uniformly on compacta of D to a K-quasiconformal homeomorphism h of D onto D (see [LV, Ch. II, Thm. 5.5]). In particular, there exists ρ̄, 0 < ρ̄ < 1, such that h_{j}(S_{P_{j}}({b_{1}, …, b_{m}})) ⊂ {|z| < ρ̄} for all j, i.e. the critical points of each ϕ_{j} are in {|z| < ρ̄}. Similarly, since {ϕ_{j}} is a normal family of Blaschke products of degree 1 + ∑_{i=1}^{m} k_{i} and 0 is a fixed point for all ϕ_{j}’s, we can assume that {ϕ_{j}} converges uniformly in \overline{D} to a Blaschke product ϕ : \overline{D} → \overline{D} whose critical set is equal to {(h(S_{P}(b_{j})), k_{i})}_{i=1}^{m}. Hence the sequence {Q_{j}} has the geometric limit Q which is a circle packing for 𝕂 such that ϕ ∘ h is the cp-map from P to Q. Thus Q is the sought after Bl-packing. □

We will now combine the above result with a similar one proved in [D3] for circle packings in C to show the consistency in circle packing theory and its analogy to the classical theory of (branched) surfaces. For this purpose we need to introduce a new term.

Recall that a cp-map f is said to be a discrete complex polynomial (see [D2]) if the domain packing of f is univalent and there exists a decomposition f = ψ ∘ h, where ψ is a complex polynomial and h is self-homeomorphism of C. A circle packing which is the range packing of a discrete complex polynomial will be called a Pl-packing.

By putting together Corollary 4.3 of [D3], Proposition 2.2, and Theorem 3.3 we obtain the following branched version of the Discrete Uniformization Theorem of [BSt1]:

Uniformization. If 𝕂 is a triangulation of a disc with deg(𝕂) < ∞ then one and only one of the following holds:

(1) for every branch structure 𝔅 for 𝕂 there exists a Bl-packing for 𝕂 with branch set 𝔅;

(2) for every branch structure 𝔅 for 𝕂 there exists a Pl-packing for 𝕂 with

branch set 𝔅. Moreover, packings in (1) and (2) are determined uniquely up to automorphisms of D and C, respectively.

Remark 4.2. The reader can notice that the above statement is not the complete branched analog of the Discrete Uniformization Theorem of [BSt1]; the case which we have left out is that of triangulations of the 2-sphere. We do not know of any satisfactory answer to a question of the existence and uniqueness of branched circle packings for triangulations of the 2-sphere. The best result so far is due to [BoSt] which gives the existence and uniqueness of such packings assuming that a half of the total branching occurs at one of vertices of a triangulation.

Remark 4.3. We expect the above result to be true for every triangulation of a disc, not necessarily with bounded degree. In the univalent case, it was shown in [HSc] that if 𝕂 is a triangulation of a disc (deg(𝕂) ≤ ∞) then there exits either a univalent Pl-packing for 𝕂 or a univalent Bl-packing for 𝕂, but not both. Moreover, in [D2] it was shown that if 𝕂 has boundary and 𝔅 is a branch structure for 𝕂 then there exists a circle packing for 𝕂 contained in D with branch set 𝔅, hence the Perron-type arguments of [BSt2] imply the existence of a Bl-packing for 𝕂 with branch set 𝔅.

We finish this section by extending the results from §3 to infinite Bl-packings.

Theorem 4.4. Suppose 𝕂 is a finite or infinite triangulation of a disc, P̃ and P are Bl-packings for 𝕂, and P̃ is univalent. Let d and ℓ be positive integers, and ρ ∈ (0, 1). In addition to the constants κ(d, ℓ, ρ) and K(d, ℓ, ρ) of Lemma 3.1 and Corollary 3.2, respectively, there exist constants λ = λ(d, ℓ, ρ) and η = η(d, ℓ, ρ), depending only on d, ℓ, and ρ, such that if deg(𝕂) ≤ d, val(P) ≤ ℓ, and S_{P}(br_{V} (P)) ⊂ {|z| ≤ ρ} then:

(1) μ_{P}(v) ≤ κ for all v ∈ 𝕂^{0};

(2) if SP̃(int 𝕂^{0}) ∩ {|z| ≤ ρ} ≠ ∅ and f is the discrete Blaschke product from P̃ to P then the corresponding extension map f̄: D → D is K-quasiregular;

(3) if SP̃(v_{0}) = S_{P}(v_{0}) = 0 ∈ D for some v_{0} ∈ int 𝕂^{0} then

λ(rP̃(v_{0}))^{η} ≤ r_{P}(v_{0}) ≤ rP̃(v_{0}).

Proof. If 𝕂 is finite then (1) and (2) are just restatements of Lemma 3.1 and Corollary 3.2. If 𝕂 is infinite then the assertions in (1) and (2) follow Lemma 3.1, Corollary 3.2, and limit-type arguments similar to those used in the proof of Theorem 4.1.

We will now prove (3). The inequality r_{P}(v_{0}) ≤ rP̃(v_{0}) is a consequence of the Discrete Schwarz Lemma of [DSt] (see also [BSt2, Thm. 4]). To show the other inequality in (3) we refer to quasiregular arguments. Recall that if g : Ω → C, Ω ⊂ C, is a K-quasiregular mapping, A ⊂ Ω is open, bounded, \overline{A} ⊂ Ω, and ∂g(A) = g(∂A), then for every compact subset B of A,

(4.1)  cap(A, B) ≤ K val(g) cap(g(A), g(B))

where cap(A, B) is the conformal capacity of the condenser (A, B) (see [D1],[V]). From (4.1) it follows (cf. [D1, Sec. 4.6]) that there exists a constant ρ′(ℓ, ρ), depending only on ℓ and ρ, such that if ϕ is a Blaschke product, ϕ(0) = 0, val(ϕ) ≤ ℓ, and the images of the branch points of ϕ are in {|z| ≤ ρ}, then the branch points of ϕ are in {|z| ≤ ρ′}. Recall that the function f̄ has a decomposition f̄ = ϕ ∘ h, where ϕ is a Blaschke product, val(ϕ) ≤ ℓ, and h is a self-homeomorphism of D. Since SP̃(v_{0}) = S_{P}(v_{0}) = 0, we can assume that ϕ(0) = 0 and h(0) = 0. Thus SP̃(br_{V} (P)) ⊂ {|z| ≤ ρ′} because S_{P}(br_{V} (P)) ⊂ {|z| ≤ ρ}, that is the images of the branch points of ϕ are in {|z| ≤ ρ}. Hence, by (2), f̄ is K-quasiregular, K = K(d, ℓ, ρ̄), where ρ̄ = max{ρ, ρ′(ℓ, ρ)}. From the definition of f̄, (4.1), and the fact that f̄(D) = D we have

cap(D, SP̃(st(v_{0}))) ≤ K ℓ cap(f̄(D), f̄(SP̃(st(v_{0})))) = K ℓ cap(D, S_{P}(st(v_{0}))),

where st(v) is the union of all faces in 𝕂 having v as the common vertex. The left-hand side inequality in (3) now follows from (1) and the fact that CP̃(v) ⊂ SP̃(st(v)). □

Remark 4.5. Notice that the left-hand side inequality in (3) is the inverse result to the Discrete Schwarz Lemma of [BSt2] and [DSt].

5. Proof of Lemma 3.1

We will now show Lemma 3.1.

Proof of Lemma 3.1. Suppose that the assertion of the lemma is not true. Then there exist a sequence of Bl-packings {P_{n}} satisfying 1) – 3) and a sequence of pairs {v_{n}, w_{n}} such that v_{n}w_{n} is an edge in 𝕂_{n} and r_{P_{n}}(v_{n})/r_{P_{n}}(w_{n}) ≥ n, where 𝕂_{n} is the complex of P_{n}. Let P̃_{n} be a univalent Bl-packing for 𝕂_{n}. Let f̄_{n} be the extension of the discrete Blaschke product f_{n} from P̃_{n} to P_{n}. For v ∈ bd 𝕂_{n}^{0} we define

A_{n}(v) := DP̃_{n}(v) ∪ ⋃_{\begin{matrix}△∈𝕂_{n}^{2} \\ v∈△ \end{matrix}} SP̃_{n}(△) ⋃_{\begin{matrix}u∈𝕂_{n}^{0} \\ u∼v \end{matrix}} AP̃_{n}(u, v),

where the notation is as in Section 2. From the definition of f̄_{n} it follows that f̄_{n}|_{A_{n}(v)} is 1-to-1. Since f̄_{n} : D → D is a branched covering, f̄_{n}(A_{n}(v)) cannot contain images of all branch points of f̄_{n}, i.e. f̄_{n}(br_{V} (P_{n})) \ f̄_{n}(A_{n}(v)) ≠ ∅. In particular, f̄_{n}(br_{V} (P_{n})) \ C_{P_{n}}(v) ≠ ∅ because C_{P_{n}}(v) ⊂ f̄_{n}(A_{n}(v)). This and the condition 3) imply

(5.1)  r_{P_{n}}(v) ≤ \frac{1}{2}(1 + ρ) for v ∈ bd 𝕂_{n}^{0}.

Note. In the remainder of the proof the existence and convergence of various quantities will be taken for granted; this is due to the fact that one can always apply diagonalization techniques or take a subsequence if necessary because deg(𝕂_{n}) ≤ d for all n.

We may assume without loss of generality (see Note) that there exists m such that for each n there are vertices w_{n}^{1}, …, w_{n}^{m}, u_{n} with the following properties:

P(1) w_{n}^{1} := w_{n},

P(2) w_{n}^{1}, …, w_{n}^{m}, u_{n} are consecutive neighbors of v_{n} in 𝕂_{n} listed in the clockwise

order, P(3) lim_{n→∞} \frac{r_{n}(v_{n})}{r_{n}(w_{n}^{i})} = ∞, i = 1, …, m, and lim_{n→∞} \frac{r_{n}(v_{n})}{r_{n}(u_{n})} < ∞, where r_{n}(·) :=

r_{P_{n}}(·).

The following two observations can be helpful in seeing that the property P(3) can be met.

(a) If v_{n} ∈ bd 𝕂_{n}^{0} for every n then, since deg(𝕂_{n}) ≤ d and (5.1),

\overline{lim}_{n→∞} max_{\begin{matrix}v∈𝕂_{n}^{0} \\ v∼v_{n} \end{matrix}}\{r_{n}(v)/r_{n}(v_{n})\} > 0,

otherwise the circles {C_{P_{n}}(v)}_{v: v∼v_{n}} would not chain around C_{P_{n}}(v_{n}) from ∂D to ∂D. (Figure 3(a).) Moreover, w_{n} cannot be the last of the consecutive neighbors of v_{n} in 𝕂_{n}^{0} listed in the counterclockwise order, otherwise there would exist m ≥ 1 such that w̃_{n}^{1} := w_{n}, w̃_{n}^{2}, …, w̃_{n}^{m}, ũ_{n} are consecutive neighbors of v_{n} in 𝕂_{n}^{0} listed in the counterclockwise order, lim_{n→∞} r_{n}(v_{n})/r_{n}(w̃_{n}^{i}) = ∞, i = 1, …, m, and lim_{n→∞} r_{n}(v_{n})/r_{n}(ũ_{n}) < ∞; this would imply, however, that C_{P_{n}}(ũ_{n}) must intersect {|z| > 1} for sufficiently large n. This is impossible. (Figure 3(b).)

(b) Similarly, if v_{n} ∈ int 𝕂_{n}^{0} for every n then, because deg(𝕂_{n}) ≤ d,

\overline{lim}_{n→∞} max_{\begin{matrix}v∈𝕂_{n}^{0} \\ v∼v_{n} \end{matrix}}\{r_{n}(v)/r_{n}(v_{n})\} > 0,

otherwise the circles {C_{P_{n}}(v)}_{v: v∼v_{n}} would not wrap around C_{P_{n}}(v_{n}).

C_{P_{n}}(w̃_{n}^{i})
C_{P_{n}}(ũ_{n})
C_{P_{n}}(v_{n})
C_{P_{n}}(v_{n})
C_{P_{n}}(v)

P_{n} P_{n}

(a) (b)

Figure 3. The impossible.

We will now describe an inductive process. We are going to define a doublesequence of sets of vertices {w_{n}^{1}(j), …, w_{n}^{m(j)}(j), u_{n}(j)} ⊂ 𝕂_{n}^{0}, n = 1, 2, …, and j = 1, 2, …, where m(j) depends only on j.

Set m(1) := m, w_{n}^{i}(1) := w_{n}^{i} for i = 1, …, m(1), and u_{n}(1) := u_{n}. For purely notational purposes we also set u_{n}(0) := v_{n}. Suppose the sequences

{w_{n}^{1}(j), …, w_{n}^{m(j)}(j), u_{n}(j)}_{n=1}^{∞}

have been defined for j = 1, …, N. We now define the sequence

{w_{n}^{1}(N + 1), …, w_{n}^{m(N+1)}(N + 1), u_{n}(N + 1)}_{n=1}^{∞}

to be any sequence such that

℘(1) w_{n}^{1}(N + 1) := w_{n}^{m(N)}(N),

℘(2) w_{n}^{1}(N + 1), …, w_{n}^{m(N+1)}(N + 1), u_{n}(N + 1) are consecutive neighbors of

u_{n}(N) in 𝕂_{n} listed in the clockwise order, ℘(3) lim_{n→∞} \frac{r_{n}(v_{n})}{r_{n}(w_{n}^{i}(N+1))} = ∞, i = 1, …, m(N + 1), and lim_{n→∞} \frac{r_{n}(v_{n})}{r_{n}(u_{n}(N+1))} <

∞. The existence of such a sequence easily follows from the fact that

lim_{n→∞} \frac{r_{n}(u_{n}(N))}{r_{n}(w_{n}^{m(N)}(N))} = 0, lim_{n→∞} \frac{r_{n}(u_{n}(N−1))}{r_{n}(w_{n}^{m(N)}(N))} = 0,

and the following slight modifications of the earlier observations (see also Note). (a′) If u_{n}(N) ∈ bd 𝕂_{n}^{0} for every n then, since deg(𝕂_{n}) ≤ d and (5.1),

(5.2)  \overline{lim}_{n→∞} max_{\begin{matrix}v∈𝕂_{n}^{0}\\{u_{n}(N−1)\} \\ v∼u_{n}(N) \end{matrix}}\{r_{n}(u_{n}(N))/r_{n}(v)\} > 0.

For if the above were not true and lim_{n→∞} \frac{r_{n}(u_{n}(N−1))}{r_{n}(u_{n}(N))} > 0 then, in order to preserve tangencies in P_{n} among the circles C_{P_{n}}(v), v ∼ u_{n}(N), and the circle C_{P_{n}}(u_{n}(N)), C_{P_{n}}(u_{n}(N − 1)) would have to intersect {|z| > 1} for sufficiently large n, which is impossible. When lim_{n→∞} \frac{r_{n}(u_{n}(N−1))}{r_{n}(u_{n}(N))} = 0 the arguments of (a) can be used to show (5.2).

(b′) If u_{n}(N) ∈ int 𝕂_{n}^{0} for every n then it follows from the arguments of (b) that

(5.2) holds in this case as well. It may a priori happen that after certain number of induction steps repetitions may start occuring. We will show that given L there exists M_{L} and a subsequence of indices {n_{k}} such that for each k the number of different vertices (of 𝕂_{n_{k}}) in {u_{n_{k}}(0), …, u_{n_{k}}(M_{L})} is at least L.

Suppose that this is not true. Write U_{n}(k, l) for the set {u_{n}(k), …, u_{n}(l)} and #U_{n}(k, l) for the number of different vertices in U_{n}(k, l). Then there exists L such that for every M there is n_{M} such that #U_{n}(k, l) ≤ L for all n ≥ n_{M}. In particular, U_{n}(0, M) ⊆ B_{n}(v_{n}, L) for all n ≥ n_{M}, where B_{n}(v, ϱ) denotes the ball in 𝕂_{n}^{0} with center v and radius ϱ for the combinatorial metric in 𝕂_{n}^{0}. Moreover, we can assume (see Note) that the double-sequence {u_{n}(j)}_{j,n=1}^{∞} has the following properties:

(1) there are indices j_{i}, l_{i}, and η_{i}, i = 1, 2, ..., satisfying

0 ≤ j_{1} < j_{2} < …, 1 ≤ l_{1}, l_{2}, ⋯ ≤ L − 1, j_{i} + l_{i} < j_{i+1} + l_{i+1},

η_{1} ≤ η_{2} ≤ …, and η_{i} ≥ n_{i} (n_{i} as above),

such that for every n ≥ η_{i}, #U_{n}(j_{i}, j_{i}+l_{i}) = l_{i}+1 and u_{n}(j_{i}) = u_{n}(j_{i}+l_{i}+1) (i.e., Γ_{n}(i) := u_{n}(j_{i})u_{n}(j_{i} + 1) ∪ ⋯ ∪ u_{n}(j_{i} + l_{i})u_{n}(j_{i} + l_{i} + 1) is a simple closed edge-path in 𝕂_{n} of length at most L), and #U_{n}(j_{i} + 1, j_{i+1} + l_{i+1}) = j_{i+1}+l_{i+1}−j_{i} (i.e., u_{n}(j_{i}+1)u_{n}(j_{i}+2)∪⋯∪u_{n}(j_{i+1}+l_{i+1}−1)u_{n}(j_{i+1}+l_{i+1}) is a simple edge-path in 𝕂_{n})

(2) for all n ≥ η_{i} the number of vertices in B_{n}(u_{n}(0), L) ∩ 𝕂_{n}^{0} enclosed by Γ_{n}(i)

is constant, say χ(i). Notice that (2) follows from the fact that the number of vertices of 𝕂_{n} contained in B_{n}(v_{n}, L) (recall, u_{n}(0) = v_{n}) is at most Ld^{L} because deg(𝕂_{n}) ≤ d for all n. Also, observe that (a′) and (b′) imply that each l_{i} must be at least 2.

Let ι := min{i : χ(i) = min_{j}{χ(j)}}. Write Γ_{n} for Γ_{n}(ι). Let 𝕆_{n} be the subcomplex of 𝕂_{n} bounded by Γ_{n}. (The path Γ_{n} is included in 𝕆_{n}.) From ℘(1), ℘(2), and the induction definition it follows that for each n one of the following two cases occurs. Case 1:

( ⋃_{j=j_{ι}+1}^{l_{ι}}{w_{n}^{i}(j)}_{i=1}^{m(j)}) ∩ 𝕆_{n} = ∅;

Case 2:

⋃_{j=j_{ι}+1}^{l_{ι}}{w_{n}^{i}(j)}_{i=1}^{m(j)} ⊆ 𝕆_{n}.

By taking a subsequence if necessary, we can assume that either Case 1 or Case 2, but not both, holds for all n (see Fig. 4).

We now recall (cf. [D1]) that if 𝕂 is a finite triangulation of a disc then the following equation, which is the Euclidean version of Gauss-Bonnet formula, describes the relationship between interior and boundary angle sums induced by a circle packing P for 𝕂:

(5.3)  ∑_{v∈int 𝕂^{0}}(2π − Θ_{P}(v)) = 2π − ∑_{v∈bd 𝕂^{0}}(π − γ_{P}(v)).

Let

β_{n}(j) := α_{P_{n}}(u_{n}(j), ⟨u_{n}(j), u_{n}(j − 1), w_{n}^{1}(j)⟩)+

∑_{i=1}^{m(j)−1} α_{P_{n}}(u_{n}(j), ⟨u_{n}(j), w_{n}^{i}(j), w_{n}^{i+1}(j)⟩)+

α_{P_{n}}(u_{n}(j), ⟨u_{n}(j), u_{n}(j + 1), w_{n}^{m(j)}(j)⟩),

and γ_{n}(j) := ∑_{△∈𝕆_{n}} α_{P_{n}}(u_{n}(j), △), i.e. γ_{n}(j) is the boundary angle sum at u_{n}(j) induced by P_{n}|_{𝕆_{n}}.

Suppose that Case 1 holds for all n. Then u_{n}(j_{ι} + 1), …, u_{n}(j_{ι} + l_{ι}) ∈ int 𝕂_{n}^{0}. Hence γ_{n}(j) ≥ 2π − β_{n}(j) for j = j_{ι} + 1, …, j_{ι} + l_{ι} and all n. By (5.3), (5.4)

0 ≥ ∑_{v∈int 𝕆_{n}^{0}}(2π − Θ_{P_{n}}(v)) = 2π − ∑_{j=j_{ι}}^{l_{ι}}(π − γ_{n}(j)) ≥ π + γ_{n}(j_{ι}) + ∑_{j=j_{ι}+1}^{l_{ι}}(π − β_{n}(j)).

w_{n}^{m(j)}(j)

u_{n}(j + 1)

w_{n}^{2}(j)

u_{n}(j) u_{n}(j_{ι} + l_{ι})

u_{n}(j − 1) Γ_{n}

u_{n}(j_{ι}) u_{n}(j_{ι} +l_{ι} +1)
w_{n}^{1}(j)

u_{n}(j_{ι} + 1)

Case 1

u_{n}(j − 1)

w_{n}^{1}(j)

u_{n}(j) w_{n}^{2}(j) u_{n}(j_{ι} + 1)

w_{n}^{m(j)}(j)
Γ_{n}

u_{n}(j_{ι}) u_{n}(j_{ι}+l_{ι}+1) u_{n}(j + 1)

u_{n}(j_{ι} + l_{ι})

Case 2

Figure 4. The cases.

From ℘(3) we have

(5.5)  lim_{n→∞} \frac{r_{n}(u_{n}(j))}{r_{n}(w_{n}^{i}(j))} = ∞ for i = 1, …, m(j) and any j.

Hence

(5.6)  lim_{n→∞} β_{n}(j) = 0 for every j.

This and (5.4) give a contradiction. Thus Case 1 cannot occur.

Suppose Case 2 holds. Then ℘(1) and ℘(2) imply that either u_{n}(j_{ι} + l_{ι} + 2) ∈ int 𝕆_{n}^{0} or u_{n}(j_{ι} + l_{ι} + 2) = u_{n}(j_{ι} + 1). By choosing a subsequence if necessary, we may assume that one of these two possibilities holds for all n. We will show that the first possibility must be ruled out.

If u_{n}(j_{ι} + l_{ι} + 2) ∈ int 𝕆_{n}^{0} for all n then it follows from ℘(1) – ℘(3) that the path u_{n}(j_{ι} + l_{ι} + 1)u_{n}(j_{ι} + l_{ι} + 2) ∪ ⋯ ∪ u_{n}(j_{ι+1} + l_{ι+1})u_{n}(j_{ι+1} + l_{ι+1} + 1) must stay in int 𝕆_{n}^{0} for every n ≥ η_{ι+1}. (Notice that for all sufficiently large n (see Note), say n ≥ η_{ι+1},

{u_{n}(j_{ι} + l_{ι} + 1), …, u_{n}(j_{ι+1} + l_{ι+1} + 1)} ∩ ( ⋃_{j=j_{ι}+1}^{j_{ι}+l_{ι}+1}{w_{n}^{i}(j)}_{i=1}^{m(j)}) = ∅

because of ℘(1) – ℘(3), and either u_{n}(j_{ι+1} + l_{ι+1} + 1) ∈ int 𝕆_{n}^{0}, in which case Γ_{n}(ι + 1) is contained in the interior of 𝕆_{n}, or u_{n}(j_{ι+1} + l_{ι+1} + 1) = u_{n}(j_{ι} + l_{ι} + 1), in which case j_{ι+1} = j_{ι} + l_{ι} + 1 and Γ_{n}(ι + 1) is the path of the type considered in Case 1, i.e. cannot exist.) Now, since Γ_{n}(ι + 1) ⊂ B_{n}(u_{n}(0), L) for every n, it is easy to see that χ(ι + 1) < χ(ι), and we get a contradiction with the definition of ι.

Thus we are left with the second possibility, i.e. u_{n}(j_{ι} + l_{ι} + 2) = u_{n}(j_{ι} + 1). In this case j_{ι+1} = j_{ι} + 1 and l_{ι+1} = l_{ι}, and (5.3) implies

(5.7)  ∑_{v∈int 𝕆_{n}^{0}}(2π − Θ_{P_{n}}(v)) = 2π − ∑_{j=j_{ι+1}+1}^{j_{ι+1}+l_{ι+1}+1}(π − β_{n}(j)).

Since P_{n}’s are packings, by Theorem 4.1,

l_{ι+1} + 1 ≥ 3 + 2 ( ∑_{v∈br_{V} (P_{n})∩int 𝕆_{n}^{0}} ord_{P_{n}}(v)) .

Hence, (5.7) and (5.5) imply the following contradiction

−2π ( ∑_{v∈br_{V} (P_{n})∩int 𝕆_{n}^{0}} ord_{P_{n}}(v)) = ∑_{j=j_{ι+1}+1}^{j_{ι+1}+l_{ι+1}+1}(π−β_{n}(j)) ⟶_{n→∞} 2π−(l_{ι+1}+1)π ≤

− π − 2π ( ∑_{v∈br_{V} (P_{n})∩int 𝕆_{n}^{0}} ord_{P_{n}}(v)) .

Thus we have shown that Case 2 cannot occur neither. Therefore for any given L there exists M_{L} and a subsequence of indices {n_{k}} such that for each k the number of different vertices (of 𝕂_{n_{k}}) in {u_{n_{k}}(0), …, u_{n_{k}}(M)} is at least L.

Let L := 2(ℓ + 1). We may assume that #U_{n}(0, M_{L}) ≥ L for every n. We will show that for each sufficiently large n there is a point x_{n} ∈ D which is covered by at least ℓ + 1 different discs from {D_{n}(0), …, D_{n}(M_{l})}, where D_{n}(j) := D_{P_{n}}(u_{n}(j)). This, however, will imply that packings P_{n} are of valence no less then ℓ + 1 in contradiction to our assumption 2), and the proof of the lemma will be complete.

We now show the existence of x_{n}’s. First we “multiply” D (i.e., scale and rotate) by the factor ζ_{n} ∈ C so that ζ_{n}C_{n}(0) has the radius equal to 1 and so that the tangency point of ζ_{n}C_{n}(0) and ζ_{n}C_{n}(1) is the point 1 ∈ C, where C_{n}(j) := C_{P_{n}}(u_{n}(j)). From ℘(3) it follows that there exists δ > 0 such that for each j, j = 1, …, M_{L}, the radius of ζ_{n}C_{n}(j) is at least δ for all sufficiently large n. Moreover, ℘(1) – ℘(3) imply that

{ζ_{n}C_{n}(w_{n}^{1}(1)), …, ζ_{n}C_{n}(w_{n}^{m(1)}(1)), ⋯, ζ_{n}C_{n}(w_{n}^{1}(M_{L})), …, ζ_{n}C_{n}(w_{n}^{m(M_{L})}(M_{L}))} is a chain of circles such that, as n → ∞, it collapses geometrically to a point in C which is 1. From the construction of the chain it follows that the tangency points of pairs {ζ_{n}C_{n}(j), ζ_{n}C_{n}(j +1)}, j = 0, …, M_{L} −1, have the common limit point in C, as n → ∞, which is 1. As each ζ_{n}C_{n}(j), j = 0, …, M_{L} −1, has its radius no smaller then δ, it is easy to see that for each sufficiently large n there is a point y_{n} ∈ C which is covered by at least a half of different discs from {ζ_{n}D_{n}(0), …, ζ_{n}D_{n}(M_{l})}, i.e. by at least L/2 = ℓ + 1 of different discs. We now take x_{n} := y_{n}/ζ_{n} and the proof is finished. □

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Mathematical Sciences Research Institute, Berkeley, CA 94720 E-mail address: tdubejko@msri.org