AN EXPLICIT BOUND FOR UNIFORM PERFECTNESS OF THE JULIA SETS OF RATIONAL MAPS

TOSHIYUKI SUGAWA

Abstract. A compact set C in the Riemann sphere is called uniformly perfect if the moduli of annuli separating C are bounded. Mañé-da Rocha and Hinkkanen showed independently uniform perfectness of the Julia sets of rational maps of degree ≥ 2, but they presented no explicit bounds for uniform perfectness. In this note, we shall provide such an explicit bound and, as a result, we give another proof of uniform perfectness of the Julia sets. As an application, we refer to a lower estimate of the Hausdorff dimension of the Julia sets. We also give a concrete bound for the family of quadratic polynomials f_{c}(z) = z^{2} + c in terms of the parameter c.

1. Introduction

Let C be a closed set in the Riemann sphere ℂ̂ with #C ≥ 3 and Ω its complement. We say C is uniformly perfect if there exists a constant 0 < c < 1 such that C ∩ {z ∈ ℂ; cr < |z − a| < r} ≠ ∅ for any a ∈ C and 0 < r < diam(C), where diam denotes the Euclidean diameter.

The notion of uniform perfectness first appeared in Beardon-Pommerenke [3], and was investigated more deeply by Pommerenke [12] and [13], and afterwards by many authors (see [15] and its references). By definition, the sets with some kind of self-similarities are expected to have uniform perfectness. In fact, the limit sets are known to be uniformly perfect for a wide class of Kleinian groups (cf. Sugawa [16]). On the other hand, Pommerenke [13] first showed the uniform perfectness of the Julia sets of hyperbolic rational maps. Later, Mañé-da Rocha [11] and Hinkkanen [7] proved in the case of general rational maps of degree ≥ 2, independently. For a simpler proof, see the textbook [4] by Carleson and Gamelin. But, their proofs are done by contradiction, thus no explicit bounds for uniform perfectness are given. In this note, we shall present such an explicit bound and also exhibit some applications of this result. Our proof employs the hyperbolic geometry, and hence is different from ones of the above authors. As Hinkkanen remarked in [7], we should note that there exists an entire function whose Julia set is not uniformly perfect. We also note that Hinkkanen and Martin [8] recently proved uniform perfectness for the Julia sets of finitely generated rational semigroups.

We will state the main result in the next section as well as fundamental definitions and notation. In Section 3, we shall discuss the connection between branched coverings and uniform perfectness, which will be a key to prove our main theorem. Section 4 is devoted to prove our main theorem, and we make an essential use of Sullivan’s No Wandering Domains Theorem. We shall give applications of the main result to estimations of Hausdorff dimension, the (logarithmic) capacity density of the Julia set and the Poincaré metric of the Fatou set in Section 5. In the last section, as a special case, we investigate the quadratic family of polynomials f_{c}(z) = z^{2} + c with c ∈ ℂ outside the Mandelbrot set. We have rather satisfactory result in the case c < −2. For a general c, we shall explain a method using a holomorphic motion of the Julia set.

Date: April 6, 1998.

1991 Mathematics Subject Classification. Primary 30D05, Secondary 30F45.

Key words and phrases. uniformly perfect, Julia set.

Finally, the author would like to express his hearty gratitude to Professors M. Taniguchi and K. Matsuzaki for helpful comments.

2. Main result

Let C be a closed set in the Riemann sphere containing at least three points and Ω its complement. We denote by A_{Ω} and A_{Ω}^{∘} the sets of annuli and round annuli, respectively, in Ω separating C, where annuli mean doubly connected domains and round annuli do special annuli of the form {z ∈ ℂ; r_{1} < |z −a| < r_{2}} for some a ∈ ℂ and 0 ≤ r_{1} < r_{2} ≤ ∞, and we say that an annulus A separates C if A ∩ C = ∅ and if each component of ℂ̂ \ A intersects C. The modulus m(A) of an annulus A is the number m such that A is conformally equivalent to the round annulus {z; 1 < |z| < e^{m}}. We set

M_{Ω} = sup_{A∈A_{Ω}} m(A), M_{Ω}^{∘} = sup_{A∈A_{Ω}^{∘}} m(A),

and call them the modulus and the round modulus of Ω. (If A_{Ω} and/or A_{Ω}^{∘} is empty, then we define M_{Ω} = 0 and/or M_{Ω}^{∘} = 0, respectively.) For these constants, it is known that \frac{1}{2}M_{Ω} − 1.7332 ⋯ ≤ M_{Ω}^{∘} ≤ M_{Ω}, and that if Ω ⊂ ℂ, M_{Ω} − 2.8911 ⋯ ≤ M_{Ω}^{∘} ≤ M_{Ω} (cf. [15]).

It is easily verified that C is uniformly perfect if and only if M_{Ω}^{∘} < ∞, equivalently M_{Ω} < ∞.

Next, for the later use, we consider quantities determined by the hyperbolic geometry of Ω. Let C be a closed set in the Riemann sphere with #C ≥ 3 and Ω = ℂ̂\C. Then each component D of Ω is hyperbolic, i.e., there exists a holomorphic universal covering map p : ℍ → D from the upper half plane onto D. Thus D can be regarded as the quotient space ℍ/Γ of ℍ over the covering transformation group Γ = {γ ∈ PSL(2, ℝ); p ∘ γ = p}. Since the hyperbolic (or Poincaré) metric ρ_{ℍ}(z)|dz| = \frac{|dz|}{2Imz} is invariant under the action of PSL(2, ℝ), D inherits the hyperbolic metric ρ_{D}(z)|dz| so that p : ℍ → D is a local isometry with respect to the hyperbolic metric, i.e., ρ_{ℍ} = p^{∗}ρ_{D}. Therefore, we can define the hyperbolic metric ρ_{Ω} of Ω componentwise.

The hyperbolic distance d_{Ω}(z_{0}, z_{1}) of a pair of points z_{0}, z_{1} in the same component of Ω can be defined by inf_{α} ∫_{α} ρ_{Ω}(z)|dz|, where the infimum is taken over all paths joining z_{0} to z_{1} in Ω. We also difine d_{Ω}(z_{0}, z_{1}) = +∞ if z_{0} and z_{1} do not belong to the same component of Ω. For z ∈ Ω we denote by ι_{Ω}(z) the injectivity radius of Ω at z, that is, ι_{Ω}(z) is the maximal radius r so that the hyperbolic disk {w ∈ Ω; d_{Ω}(z, w) < r} is simply connected.

Let C_{Ω} denote the set of free homotopy classes of non-trivial loops in Ω, where a loop (=closed curve) is called non-trivial if this is not null-homotopic (=contractible) in Ω. For a loop α in Ω, we define the length of it by

ℓ_{Ω}(α) = ∫_{α} ρ_{Ω}(z)|dz|,

and for the free homotopy class [α] represented by α, we define

ℓ_{Ω}[α] = inf_{α′∈[α]} ℓ_{Ω}(α′).

Finally, we set

L_{Ω} = inf_{[α]∈C_{Ω}} ℓ_{Ω}[α].

(If C_{Ω} is an empty set, we set L_{Ω} = +∞.) We remark that the injectivity radius ι_{Ω}(z) is equal to half the infimum of lengths of non-trivial loops in Ω passing through z. In particular, L_{Ω} is nothing other than twice the (global) injectivity radius inf_{z∈Ω} ι_{Ω}(z) of Ω.

Concerning the constant L_{Ω}, the following estimate is fundamental.

Proposition 2.1 ([15]).

L_{Ω} ≤ \frac{π^{2}}{M_{Ω}} ≤ min\{L_{Ω}e^{L_{Ω}}, \frac{L_{Ω}^{2}}{2} coth^{2}(L_{Ω}/2)\}.

In particular, M_{Ω} < ∞ if and only if L_{Ω} > 0.

In order to estimate M_{Ω} from above, by this proposition, we have only to do L_{Ω} from below. Now let us state the main theorem. For basic definitions and results about the complex dyamics of the rational maps, we refer to the textbook [2] by Beardon as a general reference.

Let f : ℂ̂ → ℂ̂ be a rational map of degree d ≥ 2. We denote by J = J_{f} and Ω = Ω_{f} the Julia set and the Fatou set of f, respectively. (In other words, Ω_{f} is the domain of normality of the iteration family {f^{n}}_{n=1,2,···} of f and J_{f} = ℂ̂ \ Ω_{f}.) Note that Ω_{f} is completely invariant under f, precisely, f(Ω_{f}) = Ω_{f} = f^{−1}(Ω_{f}).

We denote by Crit(f) the set of critical points of f in the Fatou set Ω_{f} and let U_{1}, ⋯, U_{s} be the complete list of the components of Ω_{f} which contains at least one critical point of f and is not simply connected. And we set W_{j} = f(U_{j}) and C_{j} = Crit(f) ∩ U_{j} for j = 1, ⋯, s. Note here that #Crit(f) ≤ 2d−2, so s ≤ 2d−2. Now we introduce two kinds of curve family: S(v_{1}, v_{2}) and T (v), for v_{1}, v_{2}, v ∈ f(C_{j}) with v_{1} ≠ v_{2}. Let S(v_{1}, v_{2}) and T (v) consist of the loops β : S^{1} → W_{j}, where S^{1} denotes the unit circle {z ∈ ℂ; |z| = 1}, satisfying the conditions (a), (b), (c) and (a), (b’), (c), respectively, in the following:

(a) β is contractible in W_{j}.

(b) β passes through v_{1} and v_{2}. (b’) β passes through v essentially two times, at least.

(c) There exists a non-trivial loop α in U_{j} such that f_{∗}(α) = β.

More precisely, the condition (b’) says that there exist distinct points ζ_{0} and ζ_{1} in S^{1} with β(ζ_{0}) = β(ζ_{1}) = v such that the restrictions β|\overline{I_{1}} and β|\overline{I_{2}} of the loop β are both nontrivial closed curves in W_{j}, where I_{1} and I_{2} are the connected component of S^{1} \ {ζ_{0}, ζ_{1}}. In particular, T (v) is empty if W_{j} is simply connected. And we set

a_{j}(v_{1}, v_{2}) = inf_{β∈S(v_{1},v_{2})} ℓ_{Ω}(β), b_{j}(v) = inf_{β∈T (v)} ℓ_{Ω}(β) and

a_{j} = min_{v_{1},v_{2}∈f(C_{j}),v_{1}≠v_{2}} a_{j}(v_{1}, v_{2}), b_{j} = min_{v∈f(C_{j})} b_{j}(v),

where we set a_{j} = +∞ if #f(C_{j}) = 1.

Finally, let A_{1}, ⋯, A_{t} be all of the cycles of Herman rings of f. We note here that, by Shishikura’s theorem, 0 ≤ t ≤ d − 2, in particular, if d = 2 there are no Herman rings. And, since the Julia set has no isolated points, the Herman rings have finite moduli, so L_{A_{k}} > 0 for all k.

Now we are ready to state our main theorem.

Theorem 2.2 (Main Thoerem). For an arbitrary rational map f : ℂ̂ → ℂ̂ of degree d ≥ 2, the following holds.

L_{Ω_{f}} ≥ min{a_{1}, ⋯, a_{s}, b_{1}, ⋯, b_{s}, L_{A_{1}}, ⋯, L_{A_{t}}}.

The proof of this theorem will be given in Section 4.

For any β ∈ S(v_{1}, v_{2}), it is clear by definition that ℓ_{Ω}(β) ≥ 2d_{Ω}(v_{1}, v_{2}). Similarly, for β ∈ T (v), we have ℓ_{Ω}(β) ≥ 4ι_{Ω}(v). Thus, we conclude that a_{j}(v_{1}, v_{2}) ≥ 2d_{Ω}(v_{1}, v_{2}) and b_{j}(v) ≥ 4ι_{Ω}(v) and hence have the following

Corollary 2.3. Under the same situation as the Main Theorem, it follows that

L_{Ω_{f}} ≥ min{C_{1}, C_{2}, C_{3}}(> 0),

where

C_{1} = min_{v_{1}≠v_{2}∈f(Crit(f))} 2d_{Ω_{f}} (v_{1}, v_{2}),
C_{2} = min_{v∈f(Crit(f))} 4ι_{Ω_{f}} (v), and
C_{3} = min_{k=1,··· ,t} L_{A_{k}}.

In particular, the Julia set J_{f} is uniformly perfect.

Remark 1. As is well-known, any polynomial has no Herman rings. In general, if there exsits a Herman ring A, it is known that the boundary of A is contained in the closure of forward orbits of the critical points of f. Therefore, if each critical point of f is (pre)periodic or contained in a (super)attracting or parabolic basin, then we can conclude that f has no Herman rings. We also note that a cycle of (super)attracting or parabolic components always contains a critical point, thus a component of it appears as a member of the list U_{1}, ⋯, U_{s}.

Remark 2. Let B_{1} and B_{2} be connected components of a cycle of Herman rings A_{j}. Then B_{2} = f^{l}(B_{1}) for some l ∈ ℕ. Since f^{m} : B_{1} → B_{1} is known to be analytically conjugate to an irrational rotation of a round annulus, where m is the period of A_{j}, we can see that f^{l} : B_{1} → B_{2} is biholomorphic. Hence, L_{A_{j}} is equal to the hyperbolic length of the core curve of any component of A_{j}.

Remark 3. A pair of critical values v_{1}, v_{2} can accidentally be very close to each other in Ω, i.e., d_{Ω}(v_{1}, v_{2}) is very small, while a_{j}(v_{1}, v_{2}) is not so small. (The phenomenon T (v) ≠ ∅ can be considered as a limiting case of the above situation.) So, the formulation in the above corollary does not always provide a good estimate for uniform perfectness.

3. Branched coverings and uniform perfectness

In this section, we shall investigate the connection between branched coverings and uniform perfectness. Let f : U → W be a holomorphic (possibly branched) covering map from a (connected) hyperbolic Riemann surface U onto another W. Precisely speaking, for each point w ∈ W there exists an open neighborhood V of w satisfying the condtion: For each component Ṽ of f^{−1}(V) there exist a natural number n ≥ 1 and conformal homeomorphisms φ : Ṽ → Δ_{r} and ψ : V → Δ_{r^{n}} with ψ(w) = 0 such that ψ ∘ f ∘ φ^{−1}(ζ) = ζ^{n}, where Δ_{r} denotes the disk {|ζ| < r}.

When a loop α is freely homotopic to another α′ in U, f_{∗}α := f ∘α is freely homotopic to f_{∗}α′. Therefore, the natural homomorphism f_{∗} : C_{U} → C_{W} can be defined by f_{∗}[α] = [f_{∗}α].

First suppose that f is unbranched, then by the homotopy lifting property we can see that the induced map f_{∗} is injective. And moreover ℓ_{U}[α] = ℓ_{W} [f_{∗}α] because f is a local isometry, therefore we have the next

Proposition 3.1. If f : U → W is an unbranched holomorphic covering map, then L_{U} ≥ L_{W} .

In the case when f is branched, we need more efforts to estimate L_{U} from below. In fact, for any finitely connected planar Jordan domain U, it is known that there exists a branched holomorphic covering map from U onto the unit disk (so-called the Ahlfors map), thus L_{U} cannot be estimated from below by only the data of W (in this case, L_{W} = +∞).

Let Crit(f) be the set of critical points of f and for v_{1}, v_{2}, v ∈ f(Crit(f)) with v_{1} ≠ v_{2} define the curve families S(v_{1}, v_{2}) and T (v) by the same way as in the previous section. And we set

a(v_{1}, v_{2}) = inf_{β∈S(v_{1},v_{2})} ℓ_{W} (β), b(v) = inf_{β∈T (v)} ℓ_{W} (β), and

a = inf_{v_{1}≠v_{2}∈f(Crit(f))} a(v_{1}, v_{2}), b = inf_{v∈f(Crit(f))} b(v).

Then, the following lemma is a key step to our proof of the Main Theorem.

Lemma 3.2. For a non-trivial loop α in U such that β = f_{∗}α is contractible in W, it follows that

ℓ_{U}[α] ≥ min{a, b}.

Proof. First we show that ℓ_{U}[α] > 0. In fact, if ℓ_{U}[α] = 0 then α surrounds a puncture of U, in other words, there exists a holomorphic injection g : Δ^{∗} = Δ \ {0} → U such that α is freely homotopic to ε^{n} for some integer n ≠ 0, where ε = g({|ζ| = 1/2}). As is easily seen, f(g(Δ^{∗})) is a neighborhood of a puncture of W and thus β is freely homotopic to non-zero multiple of a simple loop around the puncture in W. On the other hand, β is contractible in W, therefore W must be conformally equivalent to the complex plane ℂ, but this is impossible because W is hyperbolic. Hence we have ℓ_{U}[α] > 0. In particulur, we see that ℓ_{U}[α] = ℓ_{U}(α_{0}) for the closed geodesic α_{0} freely homotopic to α in U. So, for the proof, it suffices to that ℓ_{U}(α) ≤ min{a, b} in the case α is a smooth curve. Approximating α by another smooth curve if necessary, we may further assume that α does not pass any critical point. Here, we should observe ℓ_{U}(α) ≥ ℓ_{W} (β) by the Schwarz-Pick lemma: f^{∗}ρ_{W} ≤ ρ_{U}.

Let p : Δ → W be a holomorphic universal covering map of W from the unit disk Δ and set C = p^{−1}(f(Crit(f))). Since β is contractible, a lift β̃ : S^{1} → Δ of β via p is closed. Let K be the holomorphically convex hull of β̃(S^{1}) in Δ. In other words, K = Δ \ D_{0}, where D_{0} is the relatively non-compact component of Δ \ β̃(S^{1}) in Δ.

Now we show that #(K ∩ C) ≥ 2. If K ∩ C is an empty set, then it is clear that β is homotopic to a point with a homotopy in W \f(Crit(f)). Since f : U \f^{−1}(f(Crit(f))) → W \ f(Crit(f)) is an unbranched covering map, this homotopy can be lifted via f to a homotopy from α to a point, but this contradicts the assumption that α is non-trivial. Next, suppose that K ∩ C consists of one point ζ_{0}. By assumption, we note that ζ_{0} ∈ K \ β̃(S^{1}). Then it is not difficult to see that the loop β̃ is freely homotopic to ε^{n} in Δ \ C, where ε is a sufficiently small simple loop around ζ_{0} in Δ \ C and n is the winding number of β̃ around ζ_{0}. This implies that β is freely homotopic to p_{∗}(ε^{n}) in W \f(Crit(f)), therefore α is freely homotopic to a loop δ with f_{∗}δ = p_{∗}(ε^{n}). In particular, ℓ_{U}[α] ≤ ℓ_{U}(δ) and the length of δ can be arbitrarily small, therefore ℓ_{U}[α] = 0, this is not the case. Now we have proved that #(K ∩ C) ≥ 2.

Here we recall that β̃ is parametrized by S^{1} = {z ∈ ℂ : |z| = 1}. For each θ ∈ ℝ, we denote by S_{θ} the hyperbolic segment joining β̃(1) and β̃(e^{iθ}) in Δ. Now we define positive numbers θ_{+} and θ_{−} by

θ_{±} = max{θ ≥ 0; S_{±u} ∩ C = ∅ for all u ∈ [0, θ)}.

Then we see that θ_{+} + θ_{−} ≤ 2π and if the equality occurs we have #(S_{θ_{+}} ∩ C) ≥ 2 since K ⊂ ⋃_{t∈ℝ} S_{t} and #(K ∩ C) ≥ 2. In any case, there exist distinct two points ṽ_{+} and ṽ_{−} such that ṽ_{±} ∈ S_{±θ_{±}} ∩ C. We put v_{±} = p(ṽ_{±}).

Let u_{n}^{±} (n = 1, 2, ⋯) be an increasing sequence of positive numbers which converges to θ_{±} for each signature, and β̃_{n} the curve obtained from β̃ by replacing its subarcs β̃|_{I_{n}^{+}}, β̃|_{I_{n}^{−}} by the hyperboic segments S_{u_{n}^{+}}, S_{−u_{n}^{−}}, respectively, where I_{n}^{±} denotes the subinterval {e^{iθ}; ±θ ∈ [0, u_{n}^{±}]} of S^{1}. And set β_{n} = p_{∗}β̃_{n} for each n = 1, 2, · · · . By construction, β_{n} is freely homotopic to β in W \ f(Crit(f)), and it holds that ℓ_{W} (β_{n}) = ℓ_{Δ}(β̃_{n}) ≤ ℓ_{Δ}(β̃) = ℓ_{W} (β). Let α_{n} be the lift of β_{n} via f determined by α_{n}(1) = α(1), then α_{n} is closed and homotopic to α.

Let α′ = lim α_{n} and β′ = f_{∗}α′. Then, we note that ℓ_{W} (β′) = lim ℓ_{W} (β_{n}) ≤ ℓ_{W} (β). Further, we can see that β′ ∈ S(v_{+}, v_{−}) or β′ ∈ T (v_{+}) according to that v_{+} ≠ v_{−} or not. Therefore, we can compute as follows.

ℓ_{U}(α) ≥ ℓ_{W} (β) ≥ ℓ_{W} (β′) ≥ min{a(v_{+}, v_{−}), b(v_{+})} ≥ min{a, b}.

If β = f_{∗}α is not contractible in W, then ℓ_{U}(α) ≥ ℓ_{W} (β) ≥ L_{W} . Whence we have the following

Corollary 3.3. Let f : U → W be a holomorphic branched covering between hyperbolic Riemann surfaces U and W. Then it follows that

L_{U} ≥ min{L_{W} , a, b},

where the constants a and b are as in the above.

4. Proof of the Main Theorem

Let α be a non-trivial closed curve in Ω = Ω_{f}. In order to prove our main theorem, we should show that ℓ_{Ω}(α) ≥ C, where C = min{a_{1}, ⋯, a_{s}, b_{1}, ⋯, b_{s}, L_{A_{1}}, ⋯, L_{A_{t}}}. We denote by α_{n} the image f^{n} ∘ α = (f^{n})_{∗}(α) of α under the n-th iterate of f. We note here that ℓ_{Ω}(α) ≥ ℓ_{Ω}(α_{1}) ≥ ℓ_{Ω}(α_{2}) ⋯ by the Schwarz-Pick lemma. Let U be the component of Ω containing α. Then, by Sullivan’s No Wandering Domains Theorem, U is eventually periodic, i.e., D = f^{k}(U) is a periodic component for some integer k. As is well-known, a periodic component D is one of the following:

1. a (super)attracting immediate basin. In this case, the sequence of curves α_{n} is attracted to a (super)attracting cycle (in Ω), in particular, α_{n} is contractible in Ω for sufficiently large n.

2. a parabolic immediate basin. In this case, a subsequence of α_{n} is absorbed by a simply connected attracting petal (in Ω), therefore α_{n} is contractible in Ω, too, for sufficiently large n.

3. a Siegel disk. In this time, D is simply connected itself, thus α_{k} is of course contractible in D.

4. a Herman ring.

Hence, we can conclude that if α_{n} is non-trivial for any n, then α_{n} is contained in a cycle of Herman rings A_{j} for sufficiently large n. In this case, α_{n} is freely homotopic to a non-zero multiple of the core curve β of a component of A_{j}, thus ℓ_{Ω}(α_{n}) ≥ ℓ_{A_{j}}(β) = L_{A_{j}}.

In particular, we have ℓ_{Ω}(α) ≥ L_{A_{j}} ≥ C.

Otherwise, there exists an integer n ≥ 0 such that α_{n} is non-trivial while α_{n+1} is trivial in Ω. Since ℓ_{Ω}(α) ≥ ℓ_{Ω}(α_{n}), we may assume that n = 0, in other words, α is non-trivial but β = f_{∗}(α) is contractible in Ω. If f : U → W := f(U) is unbranched covering, a homotopy connecting β with a constant curve in W can be lifted to a homotopy connecting α with a constant curve in U via f, thus α is contractible in U, this is a contradiction. Therefore, f : U → W must be branched, i.e., U = U_{j} for some j = 1, ⋯, s. Now we can apply the key lemma in the previous section! By Lemma 3.2, we have ℓ_{Ω}(α) ≥ min{a_{j}, b_{j}} ≥ C, thus the proof is now completed.

5. Applications

We now present some applications of the main result. For a rational map f : ℂ̂ → ℂ̂ of degree d ≥ 2, we have seen that L_{Ω_{f}} ≥ C, where Ω_{f} denotes the Fatou set ℂ̂ \ J_{f} and C > 0 is the constant which appears in Theorem 2.2 or Corollary 2.3.

First of all, we state a result concerning Hausdorff dimension. The following theorem is essentially due to Järvi-Vuorinen [9], while a quantitative version as in the following can be found in [15].

Theorem 5.1. The Hausdorff dimension of the Julia set J_{f} of a rational map f can be estimated as

H-dim(J_{f}) ≥ \frac{log 2}{log(2e^{M_{Ω_{f}}^{∘}} + 1)} ≥ \frac{log 2}{M_{Ω_{f}}^{∘} + log 3} ≥ \frac{log 2}{π^{2}/L_{Ω_{f}} + log 3}.

In particular, any rational map of degree≥ 2 has always the Julia set of positive Hausdorff dimension. This is a well-known fact and is also shown in [4] by uniform perfectness of the Julia set in another context.

The next theorem ensures the regularity of the Julia set in the sense of Dirichlet (cf. [17]) by Wiener’s criterion.

Theorem 5.2 (Pommerenke [12]. See also [15]). Let f be a rational map of degree at least two. Then, for each point a ∈ J_{f} and 0 < r < diamJ_{f}, it holds that Cap(J_{f} ∩ B(a, r)) ≥ cr, where c ≤ 1 is a constant satisfying log 1/c ≤ M_{Ω_{f}}^{∘} + 7 log 2, diam stands for the Euclidean diameter, Cap the logarithmic capacity and B(a, r) is the closed disk centered at a with radius r.

In fact, the above property characterizes uniform perfectness of J_{f} (see [12]). Similarly, we can state a characterization of uniform perfectness of the closed set in terms of Hausdorff contents [9] (see also [15]).

Finally, we mention the estimate of the hyperbolic (or Poincaré) metric ρ(z)|dz| = ρ_{Ω}(z)|dz| of Ω = Ω_{f} in terms of the distance function δ(z) = δ_{Ω}(z) = dist(z, J_{f}) = inf_{a∈J_{f}} |z − a|, provided that ∞ ∈ J_{f}. It is always true that ρ(z) ≤ 1/δ(z). On the other hand, if Ω is simply connected, it is well-known that ρ(z) ≥ 1/4δ(z), while this kind of inequality need not hold in general, even in the case ∂Ω is a perfect set. But this is true in our situation, indeed the validity of this inequality characterizes uniform perfectness.

Theorem 5.3 (cf. [15]). For a rational map f of degree at least two, we set L = L_{Ω_{f}} . If ∞ ∈ J_{f} then we have

\frac{1}{4} tanh L/2 ≤ inf_{z∈Ω_{f}} ρ_{Ω_{f}} (z)δ_{Ω_{f}} (z) ≤ \frac{\sqrt{3}L}{\sqrt{π^{2} + 4L^{2}}} .

For other applications and characterizations of uniform perfectness, see [15] and its references.

6. Quadratic polynomials

In this section, as the simplest example, we shall consider the quadratic polynomials f(z) = f_{c}(z) = z^{2} + c and attempt to give concrete lower and upper bounds for the uniform perfectness constant L_{Ω_{f}} (abbreviated by L_{c}) of the Jula set J_{c} of f_{c} in terms of the parameter c. (For a general rational map f, we may estimate L_{Ω_{f}} in the similar way as below, in principle.) For general results of the dynamics of quadratic polynomials, the reader will find a good account in the book [4] by Carleson and Gamelin.

Since f_{c} is a polynomial, the point at infinity is a superattracting fixed point of f_{c}. And 0 is a unique finite critical point of f_{c} and c is the corresponding critical value. Let M denote the Mandelbrot set {c ∈ ℂ; (f_{c}^{n}(0))_{n=1,2,···} is a bounded sequence}. As is wellknown, c ∈ M if and only if the Julia set J_{c} is connected, in which case L_{c} = +∞ since Ω = Ω_{c} := ℂ̂ \ J_{c} is simply connected, thus we have nothing to do. So we assume that c ∉ M in the sequel. In this case, the Julia set J_{c} is a Cantor set, therefore the Fatou set Ω_{c} is connected. In order to estimate L_{c} from below, by Corollary 2.3, it is sufficient to estimate d_{Ω}(c, ∞), ι_{Ω}(c) and ι_{Ω}(∞) from below. To accomplish it, we may utilize the monotonicity property of the hyperbolic metrics. If we find a hyperbolic domain Ω̃ containing Ω which is easier to estimate its hyperbolic metric, then ρ_{Ω} ≥ ρΩ̃ by the Schwarz-Pick lemma. Therefore, it holds that d_{Ω}(a, b) ≥ dΩ̃(a, b) for any a, b ∈ Ω. On the other hand, it is not always true that ι_{Ω}(a) ≥ ιΩ̃(a), but we can avoid this difficulty as follows. Fix a ∈ Ω. Let D be an arbitrary simply connected subdomain of Ω containing a, then we have

ι_{Ω}(a) ≥ inf_{w∈∂D} d_{Ω}(a, w) ≥ inf_{w∈∂D} dΩ̃(a, w).

The most useful (but not necessarily sufficient) domain D is thought to be a thrice punctured sphere, since it has been studied for a long time and its hyperbolic metric can be expressed almost explicitly (see, for example, [1] and [3]). Any thrice punctured sphere is conformally (indeed, Möbius) equivalent to the canonical one: D_{0} = ℂ̂ \ {0, 1, ∞}. The following is the precise version of Landau’s theorem due to Hempel [6]: The hyperbolic metric ρ_{0}(z)|dz| of D_{0} satisfies

ρ_{0}(z) ≥ \frac{1}{2|z|(| log |z|| + K)}, (6.1)

where K = Γ(\frac{1}{4})^{4}/4π^{2} = 4.3768796 ⋯, and the equality occurs if z = −1. Note that this estimate is efficient only on the half plane Rez ≤ \frac{1}{2}, otherwise we have only to use the functional equation ρ_{0}(1 − z) = ρ_{0}(z).

In order to find out a thrice punctured sphere containing Ω, we have only to specify three points a_{1}, a_{2}, a_{3} in the Julia set, for example, repelling periodic points or their inverse images. In our present case, any periodic point is repelling since c ∉ M. For example, fixed points of f_{c} are solutions of the equation: z^{2} + c = z, thus (1 ± \sqrt{1 − 4c})/2. We note that if α is in J_{c}, so is −α. If we selected the three points a_{1}, a_{2}, a_{3} in the Julia set, let T be the Möbius transformation mapping a_{1}, a_{2} and a_{3} to 0, 1 and ∞, respectively. Then, d_{Ω_{c}}(c, ∞) ≥ d_{ℂ̂\{a_{1},a_{2},a_{3}}}(c, ∞) = d_{0}(T (c), T (∞)), where d_{0} denotes the hyperbolic distance in D_{0}, however it seems impossible to estimate ι_{Ω_{c}}(c) and ι_{Ω_{c}}(∞) by only the data a_{j}.

For simplicity, we further assume that c < −2 for a moment. Set α = (1 + \sqrt{1 − 4c})/2 and β = 1 − α, then these are fixed points of f = f_{c}. Then we see that S = ∪_{n=1}^{∞}f^{−n}(α) ⊂ [−α, α], hence J_{c} ⊂ [−α, α] since \overline{S} = J_{c}.

For the later convenience, we set t = \sqrt{1 − 4c} − 3 > 0. Let T (z) = \frac{(β−α)(z+α)}{(β+α)(z−α)} =

(3 + t)\frac{α+z}{α−z}. Then T (∞) = −(3 + t) and T (c) = −\frac{t(3+t)}{4+t}. We also note that T (J_{c}) ⊂ [0, ∞]. Using (6.1), we can calculate as

d_{Ω_{c}}(c, ∞) ≥ d_{0}(T (c), T (∞)) = ∫_{T (∞)}^{T (c)} ρ_{0}(x)dx ≥ \frac{1}{2}Q(t),

where

Q(t) = \{ \begin{matrix}log \frac{log(3 + t) + K}{K} + log \frac{log \frac{4+t}{t(3+t)} + K}{K} & if t ≤ t_{0} \\ log \frac{log(3 + t) + K}{log \frac{4+t}{t(3+t)} + K} & if t > t_{0} \end{matrix}

and t_{0} = 0.38297 ⋯ is the positive root of the equation 3 + t = (4 + t)/t(3 + t).

Next, we shall estimate the injectivity radii of Ω_{c} at c and ∞. As a preparation, we consider the quantity h(a) = ι_{D_{0}}(−a) for a > 0. First we assume that 0 < a ≤ 1. Let Δ be the domain defined by \{τ ∈ ℍ; 0 < Reτ < 1, |τ − \frac{1}{2}| > \frac{1}{2}\}, and λ : Δ → ℍ the conformal homeomorphism from Δ onto the upper half plane ℍ which maps 0, 1, ∞ to 1, ∞, 0, respectively. We denote by g : ℍ → Δ the inverse map of λ. Then, as is well-known, λ is analytically continued to the universal covering map of D_{0} from ℍ by the reflection principle, in particular, 1/2Imτ = ρ_{0}(λ(τ))|λ′(τ)|. The map λ is nothing but the classical elliptic modular function. For the point τ_{0} = (e^{iθ} + 1)/2 = g(1 + a) ∈ g((1, 2]) (π/2 ≤ θ < π) we can see that d_{ℍ}(τ_{0}, g((0, 1))) ≤ d_{ℍ}(τ_{0}, g((−∞, 0))) and that the shortest hyperbolic segment γ connecting τ_{0} and g((0, 1)) = {ti; t > 0} is contained in \{τ ∈ Δ; Reτ ≤ \frac{1}{2}\}. Noting that λ(\{τ ∈ Δ; Reτ = \frac{1}{2}\}) = \{z ∈ ℍ; |z − 1| = 1\}, we have h(a) = ι_{D_{0}}(−a) = ι_{D_{0}}(1 + a) = ∫_{λ_{∗}γ} ρ_{0}(z)|dz| and λ_{∗}γ is contained in \{z ∈ \overline{ℍ}; |z − 1| ≤ 1\}. Denote by β the closed curve obtained as the union of 1 − λ_{∗}γ and its complex conjugate, then |β| ≤ 1 and 2h(a) = ∫_{β} ρ_{0}(z)|dz|. Note that |dz| ≥ (|dr| + r|dθ|)/\sqrt{2}, where z = re^{iθ}. Put a_{0} = min |β|, then by (6.1) we have

2h(a) ≥ ∫_{β} \frac{|dz|}{2|z|(− log |z| + K)} ≥ ∫_{β} \frac{|dr| + r|dθ|}{2r\sqrt{2}(− log r + K)}

≥ \frac{2}{2\sqrt{2}} log (\frac{− log a_{0} + K}{− log a + K} ) + \frac{1}{2\sqrt{2}} \frac{2π}{− log a_{0} + K}

≥ \frac{π/\sqrt{2}}{− log a + K} ,

because K > π.

Next, we consider the case a > 1. Since the Möbius transformation z ↦ 1/z preserves D_{0}, we have h(a) = h(1/a), thus h(a) ≥ π/2\sqrt{2}(log a + K). Therefore, for any a > 0, we have

ι_{D_{0}}(−a) ≥ \frac{π}{2\sqrt{2}(| log a| + K)}.

Letting D = ℂ\[0, ∞), we can estimate the injectivity radius of Ω_{c} at T^{−1}(−a) as follows:

ι_{Ω_{c}}(T^{−1}(−a)) = ι_{T (Ω_{c})}(−a) ≥ inf_{w∈∂D} d_{0}(−a, w) = ι_{D_{0}}(−a) ≥ \frac{π}{2\sqrt{2}(| log a| + K)}.

Hence,

min{2d_{Ω_{c}}(c, ∞), 4ι_{Ω_{c}}(c), 4ι_{Ω_{c}}(∞)}

(6.2) ≥ R(t) := min \{Q(t), \frac{\sqrt{2}π}{log(3 + t) + K} , \frac{\sqrt{2}π}{log \frac{4+t}{t(3+t)} + K} \}

= \{ \begin{matrix}\frac{\sqrt{2}π}{log \frac{4+t}{t(3+t)} + K} & if 0 < t < 0.12626 ⋯, \\ log \frac{log(3 + t) + K}{K} + log \frac{log \frac{4+t}{t(3+t)} + K}{K} & if 0.12626 ⋯ < t ≤ t_{0}, \\ log \frac{log(3 + t) + K}{log \frac{4+t}{t(3+t)} + K} & if t_{0} = 0.38297 ⋯ < t, \end{matrix}

where K = Γ(\frac{1}{4})^{4}/4π^{2} = 4.3768796 · · · .

In contrast, the estimation of L_{c} from above is rather easy. In the same assumption as the above, we set γ = \sqrt{−c − α} > 0. Note that γ ∈ J_{c} ⊂ [−α, α] since f_{c}(γ) = −α ∈ J_{c}. Then, for x ∈ (−γ, γ), we see that f_{c}(x) = x^{2} + c < γ^{2} + c = −α, hence (−γ, γ) ⊂ Ω_{c}. This implies that the annulus A = ℂ̂ \ ([−α, −γ] ∪ [γ, α]) separates the Julia set J_{c}. The Möbius transformation T (z) = \frac{γ+z}{α−z} maps A onto Teichmüller’s extremal domain ℂ̂ \ ([−r_{1}, 0] ∪ [r_{2}, +∞]), where r_{1} = (α − γ)/2α and r_{2} = 2γ/(α − γ). Thus we have m(A) = 2μ(\sqrt{r_{1}/(r_{1} + r_{2})}), where μ(r) denotes the modulus of Grötzsch’s extremal domain 𝔻 \ [0, r] for 0 < r < 1, where 𝔻 denotes the unit disk (see [10]). The behaviour of the function μ(r) is well understood. Amongst them, it will be useful to record the following (cf. [10]):

log \frac{(1 + \sqrt{1 − r^{2}})^{2}}{r} < μ(r) < log \frac{2(1 + \sqrt{1 − r^{2}})}{r} < log \frac{4}{r} and

μ(r)μ (\frac{1 − r}{1 + r}) = \frac{π^{2}}{2} .

In particular, we can see that

m(A) = 2μ (\frac{α − γ}{α + γ} ) = \frac{π^{2}}{μ(γ/α)}.

Therefore, we have L_{c} ≤ π^{2}/M_{Ω_{c}} ≤ π^{2}/2μ((α − γ)/(α + γ)) = μ(γ/α). Noting Corollary 2.3, we summarize the results obtained above.

Theorem 6.1. For c < −2, the Fatou set Ω_{c} of f_{c}(z) = z^{2} + c satisfies

R(t) ≤ L_{Ω_{c}} ≤ \frac{π^{2}}{2μ(\frac{α−γ}{α+γ})} = μ (\frac{γ}{α}) , (6.3)

where t = \sqrt{1 − 4c} − 3 > 0, α = (1 + \sqrt{1 − 4c})/2 > 2, γ = \sqrt{−c − α} > 0, R(t) is the function defined by (6.2) and μ(r) denotes the modulus of Grötzsch’s extremal domain 𝔻 \ [0, r].

Remark . Since (α − γ)/(α + γ) = 2α/(α + γ)^{2} ∼ 1/t, the upper bound in (6.3) behaves like π^{2}/2 log t as t → ∞, while R(t) ∼ 4/t log t.

When t → +0, the upper bound in (6.3) is \frac{1}{2} log 1/t+O(1) since γ/α = \sqrt{t/8}(1+O(t)), however R(t) ∼ \sqrt{2}π/ log 1/t.

The badness of the lower bound R(t) is mainly caused by having replaced the Julia set with only three points in the estimation when the critical values are very close to the Julia set.

In order to get a result for any c ∈ ℂ \ M, we can use the following fundamental property of the uniform perfectness constants (cf. [15]).

Proposition 6.2. The constants M_{Ω} and L_{Ω} are quasi-invariant, in other words, if f : Ω → Ω′ is a K-quasiconformal homeomorphism (K ≥ 1) then

M_{Ω}/K ≤ M_{Ω′} ≤ KM_{Ω} and L_{Ω}/K ≤ L_{Ω′} ≤ KL_{Ω}.

Next we consider a holomorphic motion of the Julia set, which is an important tool introduced by Mañé-Sad-Sullivan. For a subset E of ℂ̂ and a pointed hyperbolic Riemann surface (or, more generally, complex hyperbolic manifold) (X, x_{0}) with hyperbolic distance d_{X}, a map F : X × E → ℂ̂ is called a holomorphic motion of E parametrized by (X, x_{0}) if the following holds:

1. F (·, a) : X → ℂ̂ is holomorphic for each a ∈ E,

2. F_{x} := F (x, ·) : E → ℂ̂ is injective for each x ∈ X, and

3. F_{x_{0}} is the identity map of E.

We can state the optimal λ-lemma proved by Słodkowsky [14] (see also [5]) as in the following form.

Theorem 6.3 (Optimal λ-lemma). Let F be a holomorphic motion of a subset E of the Riemann sphere parametrized by a simply connected hyperbolic Riemann surface X with basepoint x_{0}. Then, F can be extended to a holomorphic motion F̃ of the whole sphere ℂ̂ parametrized by (X, x_{0}) with the following properties.

1. F̃ : X × ℂ̂ → ℂ̂ is (jointly) continuous,

2. for each x ∈ X, the map F̃_{x} : ℂ̂ → ℂ̂ is a quasiconformal homeomorphism with Beltrami coefficient μ_{x} = ∂z̄F̃_{x}/∂_{z}F̃_{x} satisfying d_{T} (μ_{x}, 0) ≤ d_{X}(x, x_{0}), where d_{T} denotes the Teichmüller distance

d_{T} (μ, ν) = ess. sup_{z∈ℂ}d_{𝔻}(μ(z), ν(z)) = arctanh (∥\frac{μ − ν}{1 − ν̄μ}∥_{∞}) .

Now we construct a holomorphic motion of the Julia set by the standard method (see, for instance, [5]). As is well-known (cf. [4]), the functions Φ_{n}(c) := (f_{c}^{n}(c))^{2^{−n}} converges locally uniformly in ℂ \ M to a holomorphic function Φ(c), which is, in turn, a conformal mapping of ℂ \ M onto D := {z ∈ ℂ; |z| > 1}, where we take the branch Φ_{n} so as to Φ_{n}(c) = c + O(1) as c → ∞. By the symmetry of the Mandelbrot set M, we note that \overline{Φ(z̄)} = Φ(z), in particular, Φ((−∞, −2)) = (−∞, −1).

Now we define the function p : H = {ζ ∈ ℂ; Reζ > 0} → D by p(ζ) = −e^{ζ}, then q := Φ^{−1} ∘ p is a universal covering map of ℂ \ M from the right half plane H. Fix an arbitrary point c_{1} in ℂ \ M. Then there exists a ζ_{1} in H such that q(ζ_{1}) = c_{1} and −π < Imζ_{1} ≤ π. Set ζ_{0} = |ζ_{1}|, c_{0} = q(ζ_{0}) and let E_{0} be the set of repelling periodic points of f_{c_{0}}. Then, considering the roots of the equation f_{c}^{n}(z) = z, where c = q(ζ) and n is taken over all positive integers, we can obtain a holomorphic motion of the set E_{0} parametrizedby the right half plane H with basepoint ζ_{0} because f_{c} has no parabolic periodic points for c ∈ ℂ \ M, thus the roots do not collide. By the optimal λ-lemma, we have a holomorphic motion F : H × ℂ̂ → ℂ̂ such that F_{ζ_{0}} = id and that F_{ζ}(E_{0}) is the set of repelling periodic points of f_{q(ζ)}. (We can take an F compatible with the dynamics, thus F_{ζ} is a quasiconformal conjugate of f_{c_{0}} to f_{q(ζ)}, but we do not this property here.) Since the set of repelling periodic points is dense in the Julia set, F_{ζ}(J_{c_{0}}) = J_{q(ζ)} holds. Hence, by Proposition 6.2 and the second property in Theorem 6.3, we have

1/K ≤ L_{Ω_{c_{1}}}/L_{Ω_{c_{0}}}, M_{Ω_{c_{1}}}/M_{Ω_{c_{0}}} ≤ K,

where \frac{1}{2} log K = d_{H}(ζ_{1}, ζ_{0}).

Now we compute K. We write ζ_{1} = |ζ_{1}|e^{iθ} = z_{0}e^{iθ} with θ ∈ (−π/2, π/2). Then we can calculate as

d_{H}(ζ_{1}, ζ_{0}) = arctanh (|\frac{ζ_{1} − ζ_{0}}{ζ_{1} + ζ_{0}} |) = arctanh (tan \frac{θ}{2}) = \frac{1}{2} log (\frac{1 + sin θ}{cos θ}) .

Now we have shown the following.

Theorem 6.4. For an arbitray c ∈ ℂ \ M, take a point ζ = re^{iθ} with θ ∈ (−π/2, π/2) such that −e^{ζ} = Φ(c) and that −π < Imζ = r sin θ ≤ π. Then, we have the following estimates:

L_{Ω_{c_{0}}}/K ≤ L_{Ω_{c}} ≤ KL_{Ω_{c_{0}}} and M_{Ω_{c_{0}}}/K ≤ M_{Ω_{c}} ≤ KM_{Ω_{c_{0}}},

where c_{0} < −2 is the number determined by Φ(c_{0}) = −e^{r}, and

K = \frac{1 + sin θ}{cos θ}.

We remark that log |Φ(c)| = r cos θ is Green’s function of the domain ℂ̂ \ M with pole at the infinity.

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Department of Mathematics, Kyoto University, 606-8502 Kyoto, Japan

E-mail address: sugawa@kusm.kyoto-u.ac.jp