UNIFORM PERFECTNESS OF THE JULIA SETS OF QUADRATIC POLYNOMIALS

TOSHIYUKI SUGAWA

Abstract. The author gave in [12] an explicit estimate of uniform perfectness of the Julia sets of general rational maps of degree d ≥ 2. In this note, we will present some exapmles of such an estimate for quadratic polynomials.

1. Uniform perfectness of Julia sets

A compact set C with #C > 1 in the Riemann sphere ℂ̂ is said to be uniformly perfect if C ∩ {z ∈ ℂ; cr < |z − a| < r} ≠ ∅ for any a ∈ C \ {∞} and 0 < r < diamC, where 0 < c < 1 is a constant and diam stands for the diameter with respect to the Euclidean metric. It is easy to see that C = ℂ̂ \ D is uniformly perfect if and only if finite is the supremum M_{D}^{∘} of the moduli of essential round annuli in D separating C, where the modulus of the round annulus {z; r_{1} < |z − a| < r_{2}} is defined by log r_{2}/r_{1} and an annulus in D is said to separate C if both of two components of the complement of the annulus intersect C. Teichmüller’s theorem tells us that the condition M_{D}^{∘} < ∞ is equivalent to finiteness of the supremum M_{D} of the moduli of essential (not necessarily round) annuli in D separating C. We remark that the above equivalences are quantitative (for more information, see [13]).

The notion of uniform perfectness first appeared in [1] and then was intensively investigated by Pommerenke in [8] and [9]. In [9] Pommerenke proved that the Julia set of a hyperbolic rational map of degree≥ 2 is uniformly perfect. Afterwards, Mañé-da Rocha [7] and Hinkkanen [4] independently proved the uniform perfectness for general rational maps of degree≥ 2. But their proofs are made by contradiction argument, thus no explicit bounds for uniform perfectness were given so far. Recently, the author gave in [12] another proof for uniform perfectness of the Julia sets of general rational maps with explicit bounds by using the hyperbolic geometry. We shall present this estimate in the following.

We begin with several definitions and notation needed later. Let C be a compact set in ℂ̂ containing at least three points and D its complement. Then each component D_{0} of D has a holomorphic universal covering map p : 𝔻 → D_{0} from the unit disk 𝔻. Since the hyperbolic metric ρ_{𝔻} = (1 − |z|^{2})^{−1}|dz| is invariant under the action of the cover transformation group Γ < Möb, this metric induces a Riemannian metric ρ_{D_{0}}(z)|dz|, which is also called the hyperbolic metric, that is, ρ_{D_{0}}(p(z))|p′(z)| = (1 − |z|^{2})^{−1}|dz|. We define ρ_{D} by ρ_{D} = ρ_{D_{0}} on each component D_{0}. For a piecewise smooth curve α in D the hyperbolic length ℓ_{D}(α) of α is defined as ∫_{α} ρ_{D}(z)|dz|. We set

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

Key words and phrases. uniformly perfect, Julia set, quadratic polynomial.

d_{D}(z, w) = inf_{α} ℓ_{D}(α), ι_{D}(z) = \frac{1}{2} inf_{β} ℓ_{D}(β),

where α runs over all curves joining z and w in D and β runs over all nontrivial loops passing through z in D, which are called the hyperbolic distance and the injectivity radius of D, respectively. Let L_{D} be the infimum of hyperbolic lengths of nontrivial loops in D. (If D is simply connected we define L_{D} = +∞.) In other words, L_{D} = 2 inf_{z∈D} ι_{D}(z). The following result is essential for our argument.

Theorem 1.1 ([13]). The following inequalities hold for any hyperbolic open set D of ℂ̂ with the hyperbolic metric.

\frac{1}{2}M_{D} − 1.7332 ⋯ ≤ M_{D}^{∘} ≤ M_{D}, and

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

In particular, the complement ℂ̂ \ D is uniformly perfect if and only if L_{D} > 0.

Let f : ℂ̂ → ℂ̂ be a rational map of degree d ≥ 2 and denote by J_{f} and Ω_{f} the Julia set and the Fatou set of f. In other words, Ω_{f} is the maximal open set where the iterates f^{n} (n = 1, 2, ⋯) form a normal family and J_{f} is its complement. Since d ≥ 2 the Julia set is non-empty and perfect, hence J_{f} is uncountable. In particular, Ω_{f} is hyperbolic. Let U_{1}, ⋯, U_{s} be a complete list of the components of Ω_{f} containing a critical point of f. The number of critical points of f with counting multiplicities is known to be 2d − 2, thus s ≤ 2d − 2. We denote by C_{j} the set of critical points of f contained in U_{j} for j = 1, ⋯, s. Then we consider the following two families of loops in W_{j} = f(U_{j}). For v_{1}, v_{2}, v ∈ f(C_{j}) with v_{1} ≠ v_{2} we set

S(v_{1}, v_{2}) ={β; β is a trivial loop in W_{j} through v_{1}, v_{2} with a nontrivial lift in U_{j}},
T (v) ={β; β is a trivial loop in W_{j} through v essentially

at least two times with a nontrivial lift in U_{j}}.

More precisely, the statement that a loop β : S^{1} → W_{j} passes through v at least two times means 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}}.

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.

Next, let A_{1}, ⋯, A_{t} be the complete system of representatives of the cycles of Herman rings of f. Remark that Shishikura’s theorem says that 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 in a position to state the above-mentioned theorem.

Theorem 1.2 ([12]).

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

Since β ∈ S(v_{1}, v_{2}) satisfies ℓ_{W_{j}}(β) ≥ 2d_{W_{j}}(v_{1}, v_{2}), we have a_{j}(v_{1}, v_{2}) ≥ 2d_{W_{j}}(v_{1}, v_{2}). Similarly we obtain b_{j}(v) ≥ 4ι_{W_{j}}(v). Hence, we have the following

Corollary 1.3. Let V be the set of critical values of f contained in the Fatou set Ω_{f}. Then we have

L_{Ω_{f}} ≥ min{K_{1}, K_{2}, K_{3}},

where

K_{1} = min_{v_{1}≠v_{2}∈V} 2d_{Ω_{f}} (v_{1}, v_{2}), K_{2} = min_{v∈V} 4ι_{Ω_{f}} (v)

and K_{3} = min{L_{A_{1}}, ⋯, L_{A_{t}}}. In particular, L_{Ω_{f}} > 0, equivalently, J_{f} is uniformly perfect.

In order to prove the above theorem, first we remark that the Schwarz-Pick lemma implies ℓ_{Ω_{f}} (α) ≥ ℓ_{Ω_{f}} (f_{∗}α) for any loop α in Ω_{f}. For any nontrivial loop α in Ω_{f}, by virtue of Sullivan’s No Wandering Domains Theorem, the image curve α_{n} = (f^{n})_{∗}α becomes contractible for sufficiently large n unless α_{n} lands on some Herman ring (in this case, ℓ_{Ω_{f}} (α) ≥ ℓ_{Ω_{f}} (α_{n}) ≥ K_{3}), thus the following lemma completes the proof.

Lemma 1.4. Let f : U → W be a branched holomorphic covering map between hyperbolic Riemann surfaces. We define two families of curves S(v_{1}, v_{2}) and T (v) for critical values v_{1}, v_{2}, v with v_{1} ≠ v_{2} of f as in the same way as above. And define a(v_{1}, v_{2}), b(v) as in the above and set a = inf_{v_{1}≠v_{2}} a(v_{1}, v_{2}) and b = inf_{v} b(v). Then, for any nontrivial loop α such that f_{∗}α is trivial, it holds that

ℓ_{U}(α) ≥ ℓ_{W} (f_{∗}α) ≥ min{a, b}.

For a rigorous proof of this lemma, see [12]. In this note, let us be content to exhibit only a simplified example illustrating an essence of the proof.

For an r > 1 we set U = U(r) = {z ∈ ℂ; 1/r < |z| < r}. And, we consider the Jeukowsky transformation f(z) = z + 1/z and put

f(U(r)) = W (r) = \{w = u + iv ∈ ℂ; (\frac{u}{r + 1/r} )^{2} + (\frac{v}{r − 1/r} )^{2} < 1\} .

Then f : U → W is a two-sheeted branched analytic covering with critical points ±1. Let α be a nontrivial loop in U, then one can observe that the image loop β = f_{∗}α “surrounds” both critical values ±2. Thus it is not difficult to see that ℓ_{W} (β) ≥ a(2, −2) = 2d_{W} (2, −2). Since W is simply connected, it holds that 1/4 ≤ δ_{W} (z)ρ_{W} (z) ≤ 1, where δ_{W} (z) = inf_{a∈∂W} |z − a|. Letting A = r + 1/r, then we have

d_{W} (2, −2) ≍ 2 ∫_{0}^{2} \frac{dx}{δ_{W} (x)} = 2 ∫_{0}^{4/A} \frac{dx}{(r − 1/r)\sqrt{1 − x^{2}/4}} + 2 ∫_{4/A}^{2} \frac{dx}{A − x}

= \frac{2}{r − 1/r} arcsin (\frac{2}{r + 1/r} ) + 2 log (1 + \frac{2}{r + 1/r} )

In particular, we note that d_{W (r)}(2, −2) ∼ 1/(r−1) as r → 1+ and d_{W (r)}(2, −2) ∼ 1/r as r → +∞. In this case, in fact, we can calculate L_{U(r)} explicitly as π^{2}/ log r. Thus, the estimate L_{U} ≥ 2d_{W} (2, −2) is not so good when r tends to +∞.

Next, let φ be a conformal map from W onto the unit disk 𝔻 so that ξ := φ(2) > 0 and η := φ(−2) is very close but not equal to −ξ. Now we consider the branched covering map g : U → 𝔻 of degree 4 defined by g(z) = φ(f(z))^{2}. Then the critical values of g is ξ^{2}, η^{2} and 0. In this case, d_{𝔻}(ξ^{2}, η^{2}) is very close to zero but a(ξ^{2}, η^{2}) is not so small because any element of S(ξ^{2}, η^{2}) goes a long way round another critical value 0. Hence we cannot expect that the estimate in Corollary 1.3 would be always sufficiently good.

We conclude this section by giving some applications of the estimate of uniform perfectness. The first is due to Pommerenke.

Theorem 1.5 (Pommerenke [9]). A compact set C with #C > 1 in ℂ̂ is uniformly perfect if and only if there exists a positive constant c such that

Cap(C ∩ B(a, r)) ≥ cr

for any a ∈ C and 0 < r < diamC, where Cap denotes the logarithmic capacity and B(a, r) = {z; |z −a| ≤ r}. In particular, a uniformly perfect set is regular in the sense of Dirichlet.

In fact, the above constant c is explicitly estimated by the uniform perfectness constant and vice versa (see [8]). The next result is essentially due to Järvi-Vuorinen [5], however the following quantitative form appeared in [13].

Theorem 1.6. The Hausdorff dimension of a uniformly perfect set C = ℂ̂ \ D can be estimated as

H-dimC ≥ \frac{log 2}{log(2e^{M_{D}^{∘}} + 1)} ≥ \frac{log 2}{M_{D}^{∘} + log 3} ≥ \frac{log 2}{π^{2}/L_{D} + log 3}.

2. Explicit estimate in quadratic polynomial case

In this section, we shall present a typical example of estimating the constant L_{Ω_{f}} . We consider here the quadratic polynomial f_{c}(z) = z^{2}+c where c is a parameter in ℂ. For brevity, we set J_{c} = J_{f_{c}} and Ω_{c} = Ω_{f_{c}} etc. Our aim here is to estimate L_{c} = L_{Ω_{c}} from below. If c is in the Mandelbrot set M = {c ∈ ℂ; {f^{n}(0); n ∈ ℕ} is bounded }, then it is known that J_{c} is connected, so L_{c} = +∞. Therefore we have nothing to do in this case. In the case c ∉ M the Julia set J_{c} is known to be a totally disconnected (Cantor type) set and Ω_{c} = {z ∈ ℂ̂; f^{n}(z) → ∞ as n → ∞} is connected. The critical values of f are c and ∞, so what we should do is estimate the hyperbolic distance d_{c}(c, ∞) and the injectivity radii ι_{c}(c) and ι_{c}(∞) by Corollary 1.3.

For simplicity, we shall make an additional assumption that c < −2. Then J_{c} is contained in the interval [−α, α], where α = (1 + \sqrt{1 − 4c})/2 is a fixed point of f_{c} with α > 2. We denote another fixed point of f_{c} by β. Note here that α + β = 1, and ±α, β ∈ J_{c}. In particular, the hyperbolic domain D_{0} = ℂ̂ \ {α, −α, β} contains Ω_{c}, thus the Schwarz-Pick lemma yields that ρ_{D_{0}} ≤ ρ_{c} = ρ_{Ω_{c}} on Ω_{c}. Let T be the Möbius transformation taking α, −α and β to ∞, 0 and 1, respectively. Then T can be expressed by T (z) = (3 + t)(α + z)/(α − z), thus T (∞) = −(3 + t) and T (c) = −t(3 + t)/(4 + t), where t = \sqrt{1 − 4c} − 3 > 0. We notice that T (J_{c}) ⊂ [0, +∞]. The bihaviour of the hyperbolic metric σ(z)|dz| of the canonical domain D = ℂ̂ \ {∞, 0, 1} is well understood. For instance, the precise version of Landau’s theorem due to Hempel [3] says that

σ(z) ≥ \frac{1}{2|z|(| log |z|| + H)}, (2.1)

where H = Γ(\frac{1}{4})^{4}/4π^{2} = 4.3768796 ⋯ and the equality occurs when z = −1. Note that this inequality is actually efficient only on the half plane Rez ≤ \frac{1}{2}, otherwise we have only to note that σ(1 − z) = σ(z). We also note that Solynin and Vuorinen [11] gave a recursive procedure for computing σ(z).

In order to estimate the metric σ, we shall analyze the universal covering map of D. Let Δ be the domain \{τ ∈ ℍ; 0 < Imτ < 1, |τ − \frac{1}{2}| > \frac{1}{2}\}, and λ : Δ → ℍ the conformal homeomorphism from Δ onto the upper half plane ℍ taking 0, 1, ∞ to 1, ∞, 0, respectively. By the reflection principle, λ is analytically continued to the holomorphic universal covering map of D from the upper half plane, which will be also denoted by λ, in particular, we see 1/2Imτ = σ(λ(τ))|λ′(τ)|. The map λ : ℍ → D is nothing more than the classical elliptic modular function. Noting that the hyperbolic line {z ∈ ℍ; Rez = 1} is mapped onto (−∞, 0) by λ, one can show that the segment γ = [T (∞), T (c)] is the shortest hyperbolic geodesic joining T (∞) and T (c) in D. Thus we can estimate as

d_{W} (∞, c) ≥ d_{_{ℂ}\{α,−α,β}}(∞, c) = d_{D}(T (∞), T (c)) = ∫_{γ} σ(z)|dz|

≥ ∫_{γ} \frac{|dz|}{2|z|(| log |z|| + H)} = \frac{1}{2} log \frac{log(3 + t) + H}{H} + \frac{1}{2} |log \frac{log \frac{4+t}{t(3+t)} + H}{H} | .

Next, we shall explain the estimation of the injectivity radii. We remark that Ω_{f} ⊂ Ω does not necessarily imply ι_{Ω_{f}} (z) ≥ ι_{Ω}(z), however ρ_{Ω_{f}} ≥ ρ_{Ω} implies that ι_{Ω_{f}} (z) ≥ inf_{w∈∂V} d_{Ω_{f}} (z, w) ≥ inf_{w∈∂V} d_{Ω}(z, w) for any simply connected subdomain V of Ω_{f} containing z. We shall take ℂ̂ \ [−α, α] as V. In this case, for any x ∈ [−∞, −α) ∪ (α, +∞], we have

ι_{Ω_{f}} (x) ≥ inf_{y∈[−α,α]} d_{Ω_{f}} (x, y) ≥ inf_{y∈[−α,α]} d_{_{ℂ}\{α,−α,β}}(x, y) = inf_{s>0} d_{D}(T (x), s) = ι_{D}(T (x)). (2.2)

Now we estimate I(a) = ι_{D}(−a) = ι_{D}(1 + a) for a > 0. Let φ : ℍ → Δ be the inverse map of λ : Δ → ℍ. Since Δ is a Jordan domain, Carathéodory’s theorem implies that φ extends to a homeomorphism from \overline{Δ} onto \overline{ℍ}. First we assume that 0 < a ≤ 1. Then τ_{0} = φ(1 + a) can be expressed by (e^{iθ} + 1)/2 with π/2 ≤ θ < π. Thus d_{ℍ}(τ_{0}, φ((0, 1))) ≤ d_{ℍ}(τ_{0}, φ((−∞, 0))) and the shortest hyperbolic segment γ joining τ_{0} and φ((0, 1)) = \{yi; y > 0\} is contained in \{τ ∈ \overline{Δ}; Reτ ≤ \frac{1}{2}\}. Because λ(\{τ ∈ Δ; Reτ = \frac{1}{2}\}) = \{z ∈ ℍ; |z − 1| = 1\}, it follows that ι_{D}(1 + a) = ∫_{λ_{∗}γ} σ(z)|dz| and λ_{∗}γ is contained in {|z − 1| ≤ 1}. Let ν the loop in D obtained as the union of 1 − λ_{∗}γ and its complex conjugate. Then |ν| ≤ 1 and 2I(a) = ∫_{ν} σ(z)|dz|. Set a_{0} = min |ν|. Noting that |dz| ≥ (|dr| + r|dθ|)/\sqrt{2}, where z = re^{iθ}, we have

2I(a) ≥ ∫_{ν} \frac{|dz|}{2|z|(− log |z| + H)} ≥ ∫_{ν} \frac{|dr| + r|dθ|}{2r\sqrt{2}(− log r + H)}

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

≥ \frac{π/\sqrt{2}}{− log a + H} ,

since the function h(x) = log x + π/x is increasing in x > 0 thus h(x) > h(H) > 0 for x > H. Hence I(a) ≥ π/2\sqrt{2}(− log a + H) for 0 < a ≤ 1.

In the case 1 < a, by I(a) = I(1/a), we have I(a) ≥ π/2\sqrt{2}(log a + H). In any case, I(a) ≥ π/2\sqrt{2}(| log a| + H). Combining this with (2.2), we obtain

K := min{2d_{c}(c, ∞), 4ι_{c}(c), 4ι_{c}(∞)}

≥ min \{log \frac{log(3 + t) + K}{K} + |log \frac{log \frac{4+t}{t(3+t)} + K}{K} | , \frac{\sqrt{2}π}{log(3 + t) + K} , \frac{\sqrt{2}π}{log \frac{4+t}{t(3+t)} + K} \}

= \frac{\sqrt{2}π}{log m + K} ,

where m = max\{3 + t, \frac{4+t}{t(3+t)}\}. By Corollary 1.3 we have the next

Theorem 2.1. For c < −2, the Fatou set Ω_{c} of f_{c}(z) = z^{2} + c satisfies

L_{Ω_{c}} ≥ \frac{\sqrt{2}π}{log m + H} ,

where m = max\{\sqrt{1 − 4c}, \frac{\sqrt{1−4c}+1}{\sqrt{1−4c}(\sqrt{1−4c}−3)}\} and H = Γ(\frac{1}{4})^{4}/4π^{2} = 4.3768796 · · · .

On the other hand, it is relatively easy to obtain an upper bound for L_{c}. For simplicity, we assume that c < −2 again and use the same notation as above. Let γ = \sqrt{−c − α} > 0, then one can observe that

f_{c}(x) = x^{2} + c < γ^{2} + c = −α

for x ∈ (−γ, γ), thus (−γ, γ) ⊂ Ω_{c}. This says that the annulus A = ℂ̂ \ ([γ, α] ∪ [−α, −γ]) separates the Julia set J_{c}. By definition, we have M_{Ω_{c}} ≥ m(A). Here we note that A is conformally mapped to Teichmüller’s extremal domain ℂ̂ \ ([−r_{1}, 0] ∪ [r_{2}, +∞]) by the Möbius transformation T (z) = \frac{γ+z}{α−z}, where r_{1} = (α − γ)/2α and

r_{2} = 2γ/(α − γ). It is known that 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 and this quantity satisfies the following (cf. [6]):

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 obtain

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

Combining these with Thoerem 1.1, we get Theorem 2.2. For c < −2, the Fatou set Ω_{c} of f(z) = z^{2} + c satisfies

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

where α = (1 + \sqrt{1 − 4c})/2 > 2 and γ = \sqrt{−c − α} > 0.

Since (α−γ)/(α+γ) = 1/2\sqrt{−c}+O(|c|^{−1}) as c → −∞ and γ/α = \sqrt{ε/6}(1+O(ε)) as ε := −c − 2 → +0, we can see that L_{Ω_{c}} ≤ \frac{π^{2}}{log |c|}(1 + o(1)) as c → −∞ and L_{Ω_{c}} ≤ log \frac{1}{ε}(1 + o(1)) as ε = −c − 2 → +0.

3. Estimate by Green’s function

In this section, we explain an estimate of the uniform perfectness constant by Green’s function, which is weaker than one in the previous section but easier to make in most cases. Let f(z) = c_{0}z^{d} + ⋯ + c_{d−1}z + c_{d} be a polynomial of degree d ≥ 2 and Ω the immediate basin of ∞. (More generally, f may be a rational map with a super attractive fixed point z_{0} such that f^{−1}(z_{0}) ∩ Ω = {z_{0}}, where Ω denotes the immediate basin of z_{0}.)

As is well-known (cf. [2]), Green’s function g(z) = G(z, ∞) of Ω with pole at ∞ can be expressed by

g(z) = lim_{n→∞} d^{−n} log |f^{n}(z)|.

Since g(z) = log |z| + (d − 1)^{−1} log |c_{0}| + o(1) as z → ∞, we know that Cap(J_{f}) = 1/ \sqrt[d−1]{|c_{0}|}. We also note that the following functional equation:

g(f(z)) = d · g(z). (3.1)

On the other hand, by Myrberg’s theorem (cf. [14]), Green’s function G(z, a) of the domain Ω with pole at a can be written by

G(p(z), a) = ∑_{γ∈Γ} log |\frac{1 − \overline{γ(0)}z}{z − γ(0)} | ,

where p : 𝔻 → Ω is a holomorphic universal covering of Ω with p(0) = a and Γ its cover transformation group. For a w = p(z) ∈ Ω with z ∈ 𝔻 there exists a γ ∈ Γ

such that d_{Ω}(w, a) = d_{𝔻}(z, γ(0)) = arctanh |\frac{z−γ(0)}{1−\overline{γ(0)}z} | , therefore

G(w, a) ≥ log |\frac{1 − \overline{γ(0)}z}{z − γ(0)} | = − log tanh(d_{Ω}(w, a)),

equivalently,

d_{Ω}(w, a) ≥ −\frac{1}{2} log tanh(G(w, a)/2). (3.2)

For simplicity, we assume that f(z) = f_{c}(z) = z^{2} +c with c outside the Mandelbrot set in the sequel. In this case, 0 and its backward orbit under f form the set of critical points for Green’s function g_{c}(z) = G_{c}(z, ∞), thus we can see that the subdomain Ω′ = {z ∈ Ω; g_{c}(z) > g_{c}(0)} is simply connected.

Now we consider the Böttcher coordinate ψ of f at ∞, i.e., ψ(z) = lim_{n→∞}(f^{n}(z))^{−d^{−n}} satisfying the functional equation ψ(f(z)) = ψ(z)^{2}, by which ψ can be analytically continued near any point z_{0} so far as ψ is already defined near f(z_{0}). Therefore, φ = ψ ∘ p can be analytically continued to a holomorphic map from 𝔻 to itself, where p : 𝔻 → Ω is a universal covering map of Ω with p(0) = ∞. Let V be the connected component of p^{−1}(Ω′) containing 0 and s : Ω′ → V the inverse map of p|_{V} . We note that φ(V) = {|z| < e^{−g_{c}(0)}}.

Since Ω′ is simply connected, any nontrivial loop γ passing through c must escape from Ω′, thus it contains two parts γ_{1} and γ_{2} both of which start from c and end at some points in ∂Ω′ and entirely contained in \overline{Ω}′. Since |φ(s(c))| = e^{−g_{c}(c)} and g_{c}(c) = 2g_{c}(0) by (3.1), we can estimate as

ℓ_{c}(γ_{j}) = ℓ_{𝔻}(s_{∗}γ_{j}) ≥ ℓ_{𝔻}(φ_{∗}(s_{∗}γ_{j})) ≥ d_{𝔻}(e^{−g_{c}(c)}, e^{−g_{c}(0)})

= \frac{1}{2} log \frac{tanh g_{c}(c)/2}{tanh g_{c}(0)/2} = \frac{1}{2} log \frac{tanh g_{c}(0)}{tanh g_{c}(0)/2}.

Thus we have ι_{c}(c) ≥ (ℓ_{c}(γ_{1}) + ℓ_{c}(γ_{2}))/2 ≥ \frac{1}{2} log \frac{tanh g_{c}(0)}{tanh g_{c}(0)/2}. In the same fashon, we obtain ι_{c}(∞) ≥ −\frac{1}{2} log tanh g_{c}(0)/2. Summing up these esitimates, we obtain the following result by Corollary 1.3.

Theorem 3.1. Let Ω_{c} be the Fatou set of a quadratic polynomial f_{c}(z) = z^{2} + c with c outside the Mandelbrot set. If we denote Green’s function of Ω_{c} with pole at ∞ by g_{c}(z), we have the following estimate:

L_{Ω_{c}} ≥ \frac{1}{2} log \frac{tanh g_{c}(0)}{tanh g_{c}(0)/2} = \frac{1}{2} log \frac{2}{1 + tanh^{2}(g_{c}(0)/2)}.

Finally, we make some comparison between the above estimates and known results. By Theorem 1.6, Theorems 2.1 and 3.1 produces the following inequalities.

(3.3) H-dimJ_{c} ≥ \frac{\sqrt{2} log 2}{π(log m + H) + \sqrt{2} log 3}, and

(3.4) H-dimJ_{c} ≥ \frac{log 2}{2π^{2}/ log \frac{tanh g_{c}(0)}{tanh g_{c}(0)/2} + log 3},

where m = max\{\sqrt{1 − 4c}, \frac{\sqrt{1−4c}+1}{\sqrt{1−4c}(\sqrt{1−4c}−3)}\} and c < −2 in (3.3) and c ∉ M in (3.4).

On the other hand, Ransford [10] proved the following estimate for any c ∈ ℂ \ M:

\frac{log 2}{g_{c}(0) + log 2} ≤ H-dimJ_{c} ≤ \frac{log 2}{g_{c}(0) + (e^{−g_{c}(0)} + 1)^{−1} log 2}.

Note that G_{M}(c, ∞) = g_{c}(c) = 2g_{c}(0), where G_{M} is Green’s function of the exterior of the Mandelbrot set with pole at ∞. Since g_{c}(0) = \frac{1}{2}G_{M}(c, ∞) = \frac{1}{2} log |c| + o(1) as c → ∞ (cf. [2]), we see that H-dimJ_{c} = \frac{2 log 2}{log |c|}(1 + o(1)) at ∞. On the other hand,

(3.3) yields that H-dimJ_{c} ≥ \frac{2\sqrt{2}π}{log |c|} as c → −∞, which is rather good estimate. But

(3.4) yields only that H-dimJ_{c} ≥ log 2/π^{2}\sqrt{|c|} as c → ∞.

Acknowledgement. It is pleasure to express the author’s gratitude to Professor Masashi Kisaka for informing him references concerning the Hausdorff dimension of the Julia sets of quadratic polynomials.

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Department of Mathematics, Kyoto University, 606-01 Sakyo-ku, Kyoto, Japan E-mail address: sugawa@kusm.kyoto-u.ac.jp