数理解析研究所講究録
1042 巻 1998 年 39-51

39

AN EXPLICIT BOUND FOR UNIFORM PERFECTNESS OF THE
JULIA SETS OF RATIONAL MAPSTOSHIYUKI SUGAWA
in the Riemann sphere is called uniformly perfect if the
ABSTRACT. A compact set
Rocha and Hinkkanen showed
are bounded.
moduli of annuli separating, but
independently uniform perfectness of the Julia sets of rational maps of degree
provide
shall
note,
we
this
In
perfectness.
uniformforboundsexplicitnothey presented
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 alower estimate of the Hausdorff dimensionof 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 .
$C$

$C$

$\mathrm{M}\mathrm{a}\tilde{\mathrm{n}}\acute{\mathrm{e}}- \mathrm{d}\mathrm{a}$$\geq 2$$c$

1. INTRODUCTION

its complement.
be a closed set in the Riemann sphere with $\# C\geq 3$ andLetthat $C\cap\{z\in$such
We say is uniformly perfect if there exists a constant $0<c<1$, where diam denotes the$\mathbb{C};cr<|z-a|<r\}\neq\emptyset$ for any $a\in C$ andEuclidean diameter.
The notion of uniform perfectness first appeared in Beardon-Pommerenke [3], and wasinvestigated 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
Rocha [11] and Hinkkanen [7] proved in the case of general rational
maps. Later,, independently. For a simpler proof, see the textbook [4] by Carleson
maps of degree
and Gamelin. But, their proofs are done by contradiction, thus no explicit bounds foruniform perfectness are given. In this note, we shall present such an explicit bound andalso 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], weshould 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 andnotation. In Section 3, we shall discuss the connection between branched coverings anduniform 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$\Omega$$\hat{\mathbb{C}}$

$C$

$C$

$0<r<\mathrm{d}\mathrm{i}\mathrm{a}\mathrm{m}(C)$$\mathrm{M}\mathrm{a}\tilde{\mathrm{n}}\acute{\mathrm{e}}-\mathrm{d}\mathrm{a}$$\geq 2$

Date: April 6, 1998.
1991 Mathematics Subject Classification. Primary
Key words and phrases. uniformly perfect, Julia set.

$30\mathrm{D}05$, Secondary$30\mathrm{F}45$.

40
dimension, the (logarithmic) capacity density of the Julia set and the Poincar\’e metricof the Fatou set in Section 5. In the last section, as a special case, we investigate theoutside the Mandelbrot set.
quadratic family of polynomials $f_{c}(z)=z^{2}+c$ withWe have rather satisfactory result in the case $c<-2$ . For a general , we shall explain amethod using a holomorphic motion of the Julia set.
M. Taniguchi
Finally, the author would like to express his hearty gratitude toand K. Matsuzaki for helpful comments.
$c\in \mathbb{C}$$c$P.rofessors

2.

$\mathrm{M}\mathrm{A}\mathrm{l}\mathrm{N}$RESULT

Let be a closed set in the Riemann sphere containing at least three points and its
andthe sets of annuli and round annuli, respectively,
complement. We denote byinseparating , where annuli mean doubly connected domains and round annuli do
and $0\leq r_{1}<r_{2}\leq\infty$ ,special annuli of the form $\{z\in \mathbb{C};r_{1}<|z-a|<r_{2}\}$ for some$A\cap
C=\emptyset$
separates
and we say that an annulus
if
and if each component ofof an annulus is the number $m$ such that isintersects . The
conformally equivalent to the round annulus $\{z;1<|z|<e^{m}\}$ . We set$C$$\Omega$$A_{\Omega}$$A_{[mathring]_{\Omega}}$$C$$\Omega$$a\in \mathbb{C}$

$A$

$C$

$\hat{\mathbb{C}}\backslash A$$C$$M_{\Omega}= \sup_{A_{\Omega}A\in}m(A)$

$A$

$A$

$\mathrm{m}\mathrm{o}\mathrm{d}\mathrm{u}\mathrm{l}\dot{\mathrm{u}}\mathrm{s}m(A)$,$M_{[mathring]_{\Omega}}= \sup_{A\in A_{\Omega}\mathrm{o}}m(A)$,and call them the modulus and the round modulus of . (Ifis empty,, respectively.) For these constants, it is known
then we define, and that if,
that(cf. [15]).
It is easily verified thatis uniformly perfect if and only if, equivalently$\Omega$$A_{\Omega}\mathrm{a}\mathrm{n}\mathrm{d}/\mathrm{o}\mathrm{r}A_{\Omega}^{\mathrm{o}}$$M_{\Omega}=0\mathrm{a}\mathrm{n}\mathrm{d}/\mathrm{o}\mathrm{r}M_{[mathring]_{\Omega}}=0$$\Omega\subset \mathbb{C}$$\frac{1}{2}M_{\Omega}-1.7332\cdots\leq M_{[mathring]_{\Omega}}\leq M_{\Omega}$$M_{\Omega}-2.8911\cdots\leq M_{[mathring]_{\Omega}}\leq M_{\Omega}$$C$$M_{\Omega}<\infty$$M_{[mathring]_{\Omega}}<\infty$.

Next, for the later use, we consider quantities determined by the hyperbolic geometryof . Let be a closed set in the Riemann sphere with $\# C\geq 3$ and. Then eachcomponent $D$ of is hyperbolic, i.e., there exists a holomorphic universal covering map$p:\mathbb{H}arrow D$ from the upper half plane onto $D$ . Thus $D$ can be regarded as the quotient
spaceof over the covering transformation group.
themetrichyperbolic (or Poincar\’e)
is invariant under the action
Since, $D$ inherits the hyperbolic metric $\rho_{D}(z)|dz|$ so that :ofis a local
isometry with respect to the hyperbolic metric, i.e.,. Therefore, we can define
the hyperbolic metricof componentwise.
The hyperbolic distanceof a pair of points
in the same component of, where the infimum is taken over all paths joining
can be defined bytoin . We also difine $d_{\Omega}(z_{0}, z1)=+\infty$ if
anddo not belong to the same
component of . For
we denote by
the injectivity radius of at , that is,is the maximal radius so that the hyperbolic disk $\{w\in\Omega;d_{\Omega}(z,w)<r\}$ is simply
connected.
Letdenote the set of free homotopy classes of non-trivial loops in , where a loop(
curve) is called non-trivial if this is not null-homotopic
in .
For a loop in , we define the length of it by$C$$\Omega$$\Omega=\hat{\mathbb{C}}\backslash C$$\Omega$$\mathbb{H}$$\mathbb{H}/\Gamma$$\Gamma=\{\gamma\in \mathrm{P}\mathrm{S}\mathrm{L}(2, \mathbb{R});p\mathrm{o}\gamma=p\}$$\rho_{\mathbb{H}}(z)|dz|--\frac{|dz|}{2{\rm Im} z}$$\mathrm{P}\mathrm{S}\mathrm{L}(2, \mathbb{R})$$p$$\mathbb{H}arrow D$$\rho_{\mathbb{H}}=p^{*}\rho_{D}$$\Omega$$\rho_{\Omega}$$d_{\Omega}(z0, z_{1})$$\Omega$$z_{0},$ $z_{1}$$\inf_{\alpha}\int_{\alpha}\rho_{\Omega}(z)|dz|$$\Omega$$z_{0}$$z_{1}$$z_{0}$$\Omega$$\iota_{\Omega}(z)$$z\in\Omega$$z_{1}$$\Omega$$\iota_{\Omega}(z)$

$z$

$r$

$\Omega$$C_{\Omega}$$=\mathrm{c}1_{\mathrm{o}\mathrm{s}}\mathrm{e}\mathrm{d}$$(=\mathrm{c}\mathrm{o}\mathrm{n}\mathrm{t}\mathrm{r}\mathrm{a}\mathrm{C}\mathrm{t}\mathrm{i}\mathrm{b}\mathrm{l}\mathrm{e})$$\alpha$$\Omega$$l_{\Omega}( \alpha)=\int_{\alpha}p_{\Omega}(z)|dZ|$,$\Omega$

41
and for the free homotopy class$[\alpha]$

represented by a, we define$\ell_{\Omega}[\alpha]=,\inf_{\mathrm{J}}\ell\Omega(\alpha)\alpha\in \mathrm{I}\alpha J$.

Finally, we set$L_{\Omega}=$

inf$\ell_{\Omega}[\alpha]$.

$[\alpha]\in C_{\Omega}$

is an empty set, we set $L_{\Omega}=+\infty.$ ) We remark that the injectivity radius(Ifpassing through . In
is equal to half the infimum of lengths of non-trivial loops in) ofradiusinjectivity(global)theis nothing other than twice
particular,$\iota_{\Omega}(z)$$C_{\Omega}$$\Omega$$z$$\inf_{z\in\Omega}\iota_{\Omega(}Z$$L_{\Omega}$$\Omega$.

Concerning the constant$L_{\Omega}$, the following estimate is fundamental.

Proposition 2.1 ([15]).
$L_{\Omega} \leq\frac{\pi^{2}}{M_{\Omega}}\leq\min\{L_{\Omega}e^{\iota_{\Omega}}, \frac{L_{\Omega}^{2}}{2}\coth^{2}(L\Omega/2)\}$

In particular,

if and only if$M_{\Omega}<\infty$$L_{\Omega}>0$

.

.

from
from above, by this proposition, we have only to do
In order to estimate
below. Now let us state the main theorem. For basic definitions and results about thecomplex dyamics of the rational maps, we refer to the textbook [2] by Beardon as a generalreference.
be a rational map of degree $d\geq 2$ . We denote by $J=J_{j}$ andLet :is the domain of
the Julia set and the Fatou set of , respectively. (In other words,
is) Note thatof andnormality of the iteration family$f(\Omega_{f})=\Omega_{f}=f^{-1}(\Omega_{j})$.
completely invariant under , precisely,
and let
$(f)$
set
Fatou
in
the
the set of critical points of
We denote by Crit
which contains at least one critical point
be the complete list of the components offorof and is not simply connected. And we set $W_{j}=f(U_{j})$ and, so $s\leq 2d-2$ . Now we introduce two kinds, . Note here that
$j=1,$$v_{2},v\inf(C_{j})$ with
. Let $S(v_{1,2}v)$ and, forof curve family: $S(v_{1,2}v)$ and, wheredenotes the unit circle $\{z\in \mathbb{C};|z|=1\}$ ,consist of the loops :, respectively, in the following:
satisfying the conditions (a), (b), (c) and (a),$L_{\Omega}$$M_{\Omega}$$f$$\Omega=\Omega_{f}$$\hat{\mathbb{C}}arrow\hat{\mathbb{C}}$$f$$\Omega_{f}$$J_{f}=\hat{\mathbb{C}}\backslash \Omega_{f}.$$f$$\{f^{n}\}_{n=1,2},\cdots$$\Omega_{f}$

$f$

$f$

$U_{1},$$\Omega_{f}$$\cdots,$$U_{s}$$\Omega_{f}$$C_{j}=\mathrm{C}\mathrm{r}\mathrm{i}\mathrm{t}(f)\cap U_{j}$$f$$\cdots$$\#^{\mathrm{c}\mathrm{r}\mathrm{i}\mathrm{t}}(f)\leq 2d-2$$s$$\mathcal{T}(v)$$\beta$$\mathcal{T}(v)$$v_{1}\neq v_{2}$$v_{1},$$\mathrm{S}^{1}$$\mathrm{S}^{1}arrow W_{j}$$(\mathrm{b}’),$$(\mathrm{c})$.
is contractible inpasses through and .
passes through essentially two times, at least.
.
such that(c) There exists a non-trivial loop in
and insays that there exist distinct points
More precisely, the conditionof the loop are both nonandsuch that the restrictions
with.
, where and are the connected component oftrivial closed curves in
is simply connected. And we set
is empty if
In particular,
and
(a)
(b)

$W_{j}$$\beta$$\beta$$v_{1}$$v_{2}$$v$$(\mathrm{b}’)\beta$$\alpha$$f_{*}(\alpha)=\beta$$U_{j}$$\mathrm{S}^{1}$$\zeta_{0}$$(\mathrm{b}’)$$\beta(\zeta_{0})=\beta(\zeta_{1})=v$$\beta|_{\overline{I_{1}}}$$I_{1}$$W_{j}$$\mathcal{T}(v)$$\beta$$\beta|_{\overline{I_{2}}}$$\mathrm{S}^{1}\backslash \{\zeta_{0}, (_{1}\}$$I_{2}$$W_{i}$$a_{j}(v1, v_{2})= \beta\in S(\inf_{v_{2}v1)},\ell\Omega(\beta),$$b_{j}(v)= \beta\in \mathcal{T}()\inf_{v}l_{\Omega}(\beta)$$a_{j}= \min_{v_{1},v2\in f\langle C_{j})v1\neq v2},a_{j}(v1, v2)$

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

$\zeta_{1}$,$b_{j}= \min_{jv\in j(C)}b_{j}(v)$,42Finally, letbe all of the cycles of Herman rings of . We note here that, by
Shishikura’s theorem, $0\leq t\leq d-2$ , 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.
Now we are ready to state our main theorem.
$A_{1},$$\cdots,$$f$$A_{t}$$k$

Theorem 2.2 (Main Thoerem). For an arbitrary rationd map$d\geq 2$ , the following holds.

:$f$$L_{\Omega_{f}} \geq\min\{a_{1}, \cdots,a_{S}, b1, \cdots, b_{s’ A_{1}}L, \cdots,L_{A_{t}}\}$$\hat{\mathbb{C}}arrow\hat{\mathbb{C}}$of degree.

The proof of this theorem will be given in Section 4.
. Similarly, for
For any $\beta\in S(v_{1},v2)$ , it is clear by definition that
, we have. Thus, we conclude that $a_{j}(v_{1},v_{2})\geq 2d_{\Omega}(v1,v2)$ andand hence have the following$l_{\Omega}(\beta)\geq 2d_{\Omega}(v1,v2)$$\ell_{\Omega}(\beta)\geq 4\iota_{\Omega}(v)$$\beta\in \mathcal{T}(v)$$b_{j}(v)\geq 4\iota_{\Omega}(v)$Corollary 2.3. Under the same situation as the Main Theorem, it follows that
$L_{\Omega_{f}} \geq\min\{C_{1}, C_{2}, C_{3}\}(>0)$

,

where
$C_{1}= \min_{\neq v_{1}v_{2}\in f(\mathrm{C}\mathrm{r}\mathrm{i}\mathrm{t}(f))}2d_{\Omega}f(v_{1},v_{2})$$C_{2}=v \in f(\mathrm{C}\mathrm{r}\mathrm{i}\mathrm{t}(\min_{)J)}4\iota\Omega_{t}(v)$$C_{3}= \min_{k=1,\cdots t},L_{A_{k}}$In particular, the Julia set$J_{f}$

,

,

and

.

is uniformly perfect.

Remark 1. As is well-known, any polynomial has no Herman rings. In general, if
there exsits a Herman ring , it is known that the boundary of is contained in theclosure of forward orbits of the critical points of . Therefore, if each critical point of is(pre)periodic or contained in a (super)attracting or parabolic basin, then we can conclude
that 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 memberof the list
.
$A$

$A$

$f$

$f$

$f$

$U_{1},$$\cdots,$$U_{s}$

Remark 2. Letandbe connected components of a cycle of Herman rings.
$l\in$$B_{1}arrow
B_{1}$$B_{2}=f^{l}(B_{1})$Thenfor some
N. Since:
is known to be analyticallyconjugate to an irrational rotation of a round annulus, where $m$ is the period of , wecan see that :is biholomorphic. Hence,is equal to the hyperbolic lengthof the core curve of any component of .
$B_{1}$$B_{2}$$A_{j}$$f^{m}$$A_{j}$$f^{l}$$B_{1}arrow B_{2}$$L_{A_{\mathrm{j}}}$$A_{j}$

Remark 3. A pair of critical values
can accidentally be very close to each other, i.e.,
is very small, while
is not so small. (The phenomenon
can be considered as a limiting case of the above situation.) So, the formulationin the above corollary does not always provide a good estimate for uniform perfectness.
$v_{1},v_{2}$in$\Omega$$\mathcal{T}(v)\neq\emptyset$$d_{\Omega}(v_{1}, v_{2})$$a_{j}(v_{1},v_{2})$43
3. BRANCHED COVERINGS AND UNIFORM PERFECTNESSIn this section, we shall investigate the connection between branched coverings anduniform perfectness. Let : $Uarrow W$ be a holomorphic (possibly branched) covering map
onto another $W$. Precisely speaking,from a (connected) hyperbolic Riemann surface
for each point $w\in W$ there exists an open neighborhood of satisfying the condtion:of $f^{-1}(V)$ there exist a natural number $n\geq 1$ and conformal
For each component
$f$

$U$

$V$

$w$

$\tilde{V}$with $\psi(w)=0$ such thatand :homeomorphisms :$\{|\zeta|<r\}$.
denotes the disk
, whereis freely homotopic to
When a loop is freely homotopic to another in $U,$.
bydefinedbecan. Therefore, the natural homomorphism :First suppose that is unbranched, then by the homotopy lifting property we can see
because is a local
is injective. And moreoverthat the induced map
isometry, therefore we have the next
Proposition 3.1. If : $Uarrow W$ is an unbranched holomorphic covering map, then$\tilde{V}arrow\Delta_{r}$$\psi \mathrm{o}f\mathrm{o}\varphi^{-}(1\zeta)=$$Varrow\Delta_{r^{n}}$$\psi$$\varphi$$(^{n}$$\Delta_{r}$$f*\alpha:=f\mathrm{o}\alpha$$\alpha’$$\alpha$$f_{*}[\alpha]=[f_{*}\alpha]$$C_{U}arrow C_{W}$$f_{*}$$f_{*}\alpha’$

$f$

$f$

$l_{U}[\alpha]=l_{W}[f_{*}\alpha]$$f_{*}$$f$$L_{U}\geq L_{W}$.

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

withbe the set of critical points of and for
Let Crit
section.
previousby the same way as in thedefine the curve families $S(v_{1},v_{2})$ and
And we set, and$a(v_{1},v_{2})=$, $b(v)=$ inf
inf
$f$

$(f)$

$v_{1},$ $v_{2},$$v_{1}\neq v_{2}$$v\in f(\mathrm{c}_{\mathrm{r}}\mathrm{i}\mathrm{t}(f))$$\mathcal{T}(v)$$l_{W}(\beta\rangle$$l_{W}(\beta)$$\beta\in \mathcal{T}(v)$$\beta\in S(v_{1},v_{2})$

inf

$a=$

$a(v_{1},v_{2})$

,

inf

$b=$

$b(v)$.

$v\in f(\mathrm{C}\mathrm{r}\mathrm{i}\mathrm{t}(f))$$v_{1}\neq v_{2}\in f(\mathrm{c}\mathrm{r}\mathrm{i}\mathrm{t}(f))$

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

Lemma 3.2. For a non-trivial loop
follows that$\alpha$in$U$such that$\ell_{U}[\alpha]\geq\min\{a,b\}$$\beta=f_{*}\alpha$is contractible in $W$, it.

then surrounds a puncture. In fact, if
Proof. First we show thatsuch:
injection
holomorphic
of , in other words, there exists a$\epsilon=g(\{|\zeta|=1/2\})$.
for some integer $n\neq 0$ , whereis freely homotopic tothat$W$ and thus
freelyisof$f(g(\Delta*))$puncturea
is a neighborhood of
As is easily seen,of a simple loop around the puncture in $W$. On the othermultiple
non-zerohomotopic tohand, is contractible in $W$, therefore $W$ must be conformally equivalent to the complex. In
plane , but this is impossible because $W$ is hyperbolic. Hence we havetohomotopic
freelyfor the closed geodesic
particulur, we see that$\ell_{U}(\alpha)\leq\min\{a,b\}$case is a smoothinthein . So, for the proof, it suffices to thatcurve. Approximating by another smooth curve if necessary, we may further assume
by thethat does not pass any critical point. Here, we should observe.
Schwarz-Pick lemma:$\ell\sigma[\alpha]>0$$U$$\ell_{U}[\alpha]=0$$\alpha$$g$$\Delta^{*}--\Delta\backslash \{0\}arrow U$$\epsilon^{n}$$\alpha$$\beta$$\beta$$\ell_{U}[\alpha]>0$$\mathbb{C}$$l_{U}[\alpha]=l_{U}(\alpha_{0})$$\alpha$$\alpha_{0}$$\alpha$$U$$\alpha$$\ell_{U}(\alpha)\geq l_{W}(\beta)$$\alpha$$f^{*}\rho_{W}\leq\rho_{U}$44
Let $p:\Deltaarrow W$ be a holomorphic universal covering map of $W$ from the unit disk
and set. Since is contractible, a lift :of via is closed.
$K$Let
be the holomorphically convex hull of
in . In other words,,
whereis the relatively non-compact component ofin .
Now we show that $\#(K\cap C)\geq 2$ . If $K\cap C$ is an empty set, then it is clear that is
homotopic to a point with a homotopy in. Since :is an unbranched covering map, this homotopy can be lifted via to ahomotopy from to a point, but this contradicts the assumption that is non-trivial.
Next, suppose that $K\cap C$ consists of one point . By assumption, we note that
. Then it is not difficult to see that the loop is freely
toin, where is a sufficiently small simple loop around in
and is the winding
number of around . This implies that is freely homotopic to,intherefore is freely homotopic to a loop with
. In particular,
and the length of can be arbitrarily small, therefore, this is not the case. Now
we have proved that $\#(K\cap C)\geq 2$ .
Here we recall that is parametrized by $\mathrm{S}^{1}=\{z\in \mathbb{C} : |z|=1\}$ . For each, wedenote by the hyperbolic segment joining
andin . Now we define positive
numbersbyand$\Delta$$C=p^{-1}(f(\mathrm{C}\mathrm{r}\mathrm{i}\mathrm{t}(f)))$$\tilde{\beta}$$\beta$$\mathrm{S}^{1}arrow\Delta$$\beta$$p$$\Delta$$\overline{\beta}(\mathrm{S}^{1})$$D_{0}$$K=\Delta\backslash D_{0}$$\Delta$$\Delta\backslash \tilde{\beta}(\mathrm{S}^{1})$$\beta$$f$$W\backslash f(\mathrm{C}\mathrm{r}\mathrm{i}\mathrm{t}(f))$$U\backslash f^{-1}(f(\mathrm{c}\mathrm{r}\mathrm{i}\mathrm{t}(f)))arrow$$f$$W\backslash f(\mathrm{c}_{\mathrm{r}}\mathrm{i}\mathrm{t}(f))$$\alpha$$\alpha$$\zeta_{0}\in$$\zeta_{0}$$K\backslash \tilde{\beta}(\mathrm{S}^{1})$$\tilde{\beta}$$\epsilon^{n}$$\mathrm{h}\mathrm{o}\mathrm{m}\mathrm{o}8^{\mathrm{O}}\mathrm{P}^{\mathrm{i}\mathrm{c}}$$\Delta\backslash C$$\zeta_{0}$$\epsilon$$\Delta\backslash C$$n$$\tilde{\beta}$$\zeta_{0}$$p_{*}(\epsilon^{n})$$\beta$$\delta$$\alpha$$W\backslash f(\mathrm{C}\mathrm{r}\mathrm{i}\mathrm{t}(f)\rangle$$f_{*}\delta=p_{*}(\epsilon^{n})$$\ell\sigma[\alpha]\leq P_{U}(\mathit{5})$$\ell_{U}[\alpha]=0$$\delta$$\tilde{\beta}$$\theta\in \mathbb{R}$$\tilde{\beta}(1)$$S_{\theta}$$\Delta$$\tilde{\beta}(e^{i\theta})$$\theta_{-}$$\theta_{+}$$\theta_{\pm}=\max${$\theta\geq 0;S_{\pm u}\cap C=\emptyset$

for all $u\in[0,$$\theta)$}.

Then we see thatand if the equality occurs we have $\#(S_{\theta}+\cap C)\geq 2$ sinceand $\#(K\cap C)\geq 2$ . In any case, there exist distinct two points
andsuch that. We put.
$(n=1,2, \cdots)$ be an increasing sequence of positive numbers which converges
Lettofor each signature, andthe curve obtained fromby replacing its subarcs
by the hyperboic segments, respectively, where
denotes the subintervalof . And set
for each $n=1,2,$. By construction,
is freely homotopic to in, and it holds that. Let
be the lift ofdetermined by $\alpha_{n}(1)=\alpha(1)$ , thenisclosed and homotopic to .
Let
and. Then, we note that $\ell_{W}(\beta’)=\lim\ell_{W}(\beta_{n})\leq\ell_{W}(\beta)$ .
$\beta’\inS(v_{+},v_{-})$
Further, we can see thator
according to thator not.
Therefore, we can compute as follows.
$\theta_{+}+\theta_{-}\leq 2\pi$$K \subset\bigcup_{t\in \mathbb{R}}S_{t}$$\tilde{v}_{-}$$\tilde{v}_{+}$$\tilde{v}_{\pm}\in S_{\pm\theta}\pm\cap C$$v_{\pm}=p(\overline{v}_{\pm})$$u_{n}^{\pm}$$\overline{\beta}_{n}$$\theta_{\pm}$$\overline{\beta}$$\tilde{\beta}|_{I_{n}^{+\tilde{\beta}|}},I_{\overline{n}}$$I_{n}^{\pm}$$s_{u_{n}^{+}’-u_{\overline{n}}}s$$\{e^{i\theta}; \pm\theta\in[0,u^{\pm}]n\}$$\mathrm{S}^{1}$$\beta_{n}$$\beta$$\ell_{\Delta}(\tilde{\beta})=\ell_{W}(\beta)$$\beta_{n}=p_{*}\overline{\beta}_{n}$$l_{W}(\beta_{n})=\ell_{\Delta}(\tilde{\beta}_{n})\leq$$W\backslash f(\mathrm{c}_{\mathrm{r}}\mathrm{i}\mathrm{t}(f))$$\beta_{n}$$\alpha_{n}$$\alpha$$\alpha’=\lim\alpha_{n}$$\cdots$

via.

$f$

$\alpha_{n}$$\beta’=f_{*}\alpha’$$\beta’\in \mathcal{T}(v_{+})$$v_{+}\neq v_{-}$.

$\ell_{U}(\alpha)\geq\ell_{W}(\beta)\geq\ell_{W}(\beta’)\geq\min\{a(v_{+},v-), b(v_{+})\}\geq\min\{a, b\}$If$\beta=f_{*}\alpha$following

is not contractible in $W$, then$\ell_{U}(\alpha)\geq\ell_{W}(\beta)\geq L_{W}$. Whence we have theCorollary 3.3. Let : $Uarrow W$ be a holomorphic branched covering between hyperbolic
Riemann surfaces and W. Then it follows that
$f$

$U$

$L_{U} \geq\min\{L_{W}, a, b\}$where the constants

$a$

and

$b$

are as in the above.

,45
4. PROOF OF THE MAIN THEOREM. In order to prove our main theorem,
curve inLet be a non-trivial$C=\min\{a_{1},\cdots, a_{s}, b_{1}, \cdots, b_{s},L_{A_{1}}, \cdots, L_{A_{t}}\}$ . We,
where
we should show thatof under the n-th iterate of . We note here
the imagedenote byby the Schwarz-Pick lemma. Let be the component
thatof containing . Then, by Sullivan’s No Wandering Domains Theorem, is eventuallyperiodic, i.e., $D=f^{k}(U)$ is a periodic component for some integer . As is well-known, aperiodic component is one of the following:is
1. a (super)attracting immediate basin. In this case, the sequence of curves
is contractible inattracted to a (super)attracting cycle (in ), in particular,.
for sufficiently large
is absorbed by a
2. a parabolic immediate basin. In this case, a subsequence ofin , too, foriscontractible
therefore),(insimply connected attracting petal
sufficiently large .
is of course conis simply connected itself, thus
3. a Siegel disk. In this time,tractible in $D$ .
4. a Herman ring.
is contained in a
is non-trivial for any , thenHence, we can conclude that if
is freely homotopic to afor sufficiently large . In this case,cycle of Herman rings.
non-zero multiple of the core curve of a component of , thus.
In particular, we have$n\geq0$
is trivial
is non-trivial while
such that
Otherwise, there exists an integerbut
non-trivial
words,
is
$n=0$
,
in
other
that
assumemay, wein . Since
is contractible in . If : $Uarrow W:=f(U)$ is unbranched covering, a homotopy
connecting with a constant curve in $W$ can be lifted to a homotopy connecting witha constant curve in via , thus a is contractible in , this is a contradiction. Therefore,, . Now we can apply the: $Uarrow W$ must be branched, i.e., $U=U_{j}$ for some $j=1,$$l_{\Omega}(\alpha)\geq\min\{a_{j’ j}b\}\geq C$,
key lemma in the previous section! By Lemma 3.2, we havethus the proof is now completed.
$\mathrm{c}\mathrm{l}\mathrm{o}\mathrm{S}\mathrm{e}\dot{\mathrm{d}}$$\Omega=\Omega_{f}$$\alpha$$l_{\Omega}(\alpha)\geq C$$f^{n}\mathrm{o}\alpha=(f^{n})_{*}(\alpha)$$\alpha_{n}$$f$$\alpha$$U$$\ell_{\Omega}(\alpha)\geq\ell_{\Omega}(\alpha_{1})\geq\ell_{\Omega}(\alpha_{2})\cdots$$\Omega$$U$$\alpha$$k$$D$$\alpha_{n}$$\Omega$$\Omega$$\alpha_{n}$$n$$\alpha_{n}$$\Omega$$\Omega$$\alpha_{n}$

$n$

$D$

$\alpha_{k}$$n$$\alpha_{n}$$n$$A_{j}$$\alpha_{n}$$\alpha_{n}$$\ell_{\Omega}(\alpha_{n})\geq l_{A_{j}}(\beta)=L_{A_{j}}$$A_{j}$$\beta$$\ell_{\Omega}(\alpha)\geq L_{A_{j}}\geq C$$\alpha_{n+1}$$\alpha_{n}$$\Omega$$\alpha$$l_{\Omega}(\alpha)\geq\ell_{\Omega}(\alpha_{n})$$\Omega$$\beta=f_{*}(\alpha)$$f$$\alpha$$\beta$

$U$

$U$

$f$

$\cdots$$f$$s$

5. APPLICATIONS
We now present some applications of the main result. For a rational map :anddenotes the Fatou set, whereof degree $d\geq 2$ , we have seen that
$C>0$ 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 J\"arvi-Vuorinen [9], while a quantitative version as in the following
can be found in [15].
$f$$\hat{\mathbb{C}}arrow\hat{\mathbb{C}}$$\hat{\mathbb{C}}\backslash J_{f}$$L_{\Omega_{f}}\geq C$$\Omega_{f}$Theorem 5.1. The Hausdorff dimension of the Julia set
estimated as$J_{f}$

of a rational map$\mathrm{H}-\dim(J_{f})\geq\frac{\log 2}{\log(2e^{M_{[mathring]_{f+)}_{\Omega}}}1}\geq\frac{\log 2}{M_{[mathring]_{f}_{\Omega}}+\log 3}\geq\frac{\log 2}{\pi^{2}/L_{\Omega,}+\log 3}$.

$f$

can be46
In particular, any rational map of degree
has always the Julia set of positive Hausdorff dimension. This is a well-known fact and is also shown in [4] by uniform perfectnessof 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.
$\geq 2$$)$

Theorem 5.2 (Pommerenke [12]. See also [15]). Let
be a rational map of degree atleast two. Then, for each point $a\in J_{f}$ and $0<r<$ diam , it holds that Cap$B(a, r))\geq cr$ , where $c\leq 1$ is a constant satisfying, diam stands
for the Euclidean diameter, Cap the logarithmic capacity and $B(a, r)$ is the closed disk
centered at with radius .
$f$$(J_{j}\cap$$J_{f}$$\log 1/c\leq M_{[mathring]_{f}_{\Omega}}+7\log 2$

$a$

$r$

In fact, the above property characterizes uniform perfectness of(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 Poincare’) metric $\rho(z)|dz|=$ofin terms of the distance function$\inf_{a\in j_{f}}|z-a|$ , provided that
. It is always true that $\rho(z)\leq 1/\delta(z)$ . On the other
hand, if is simply connected, it is well-known that $p(z)\geq 1/4\delta(z)$ , while this kind of
inequality need not hold in general, even in the caseis a perfect set. But this is truein our situation, indeed the validity of this inequality characterizes uniform perfectness.
$J_{f}$$\rho_{\Omega}(z)|dz|$$\Omega=\Omega_{f}$$\mathit{5}(z)=\delta_{\Omega}(z)=\mathrm{d}\mathrm{i}\mathrm{s}\mathrm{t}(z, J_{f})=$$\infty\in J_{f}$$\Omega$$\partial\Omega$

Theorem 5.3 (cf. [15]). For a rational map
then we have$f$of degree at least two, we set$L=L_{\Omega_{f}}$. If$\infty\in J_{f}$$\frac{1}{4}\tanh L/2\leq\inf_{;z\in\Omega}\rho\Omega t(Z)\delta\Omega_{f}(Z)\leq\frac{\sqrt{3}L}{\sqrt{\pi^{2}+4L^{2}}}$.

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

6. QUADRATIC POLYNOMIALSIn 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 theuniform perfectness constant(abbreviated by ) of the Jula setof in terms of
the parameter . (For a general rational map , we may estimate
in the similar way
as below, in principle.) For general results of the dynamics of quadratic polynomials, thereader will find a good account in the book [4] by Carleson and Gamelin.
Since
is a polynomial, the point at infinity is a superattracting fixed point of .
And is a unique finite critical point of and is the corresponding critical value. Let
denote the Mandelbrot set {is a bounded sequence}. As is wellknown,
if and only if the Julia set
is connected, in which case $L_{c}=+\infty$ since
is simply connected, thus we have nothing to do. So we assume thatin the sequel. In this case, the Julia set
is a Cantor set, therefore the Fatou
setis connected. In order to estimate
from below, by Corollary 2.3, it is sufficientto estimate
andfrom below. To accomplish it, we may utilize
the monotonicity property of the hyperbolic metrics. If we find a hyperbolic domain$J_{c}$$L_{\mathrm{c}}$$L_{\Omega_{f}}$

$f$

$c$

$f_{c}$$L_{\Omega_{f}}$$f_{c}$$f_{c}$$0$$f_{c}$$\mathcal{M}$$c$$c\in \mathbb{C};(f_{c}^{n}(0))_{n}=1,2,\cdots$$c\in \mathcal{M}$$J_{c}$$\Omega=\Omega_{c}:=\hat{\mathbb{C}}\backslash J_{c}$$c\not\in \mathcal{M}$$J_{c}$$\Omega_{c}$$L_{c}$$d_{\Omega}(c, \infty),$$\iota_{\Omega}(C)$$\iota_{\Omega}(\infty)$$\overline{\Omega}$

47
by thewhich is easier to estimate its hyperbolic metric, thencontaining
for any $a,b\in\Omega$ . On the
Schwarz-Pick lemma. Therefore, it holds that
, but we can avoid this difficulty asother hand, it is not always true that
subdomain of containing
connected
simplyarbitrary
an
be
.
Let
follows. Fix, then we have$\Omega$$\rho_{\Omega}\geq\rho_{\tilde{\Omega}}$$d_{\Omega}(a,b)\geq d_{\tilde{\Omega}}(a, b)$$\iota_{\Omega}(a)\geq\iota_{\tilde{\Omega}}(a)$$D$$a\in\Omega$$\Omega$$a$$\iota_{\Omega}(a)\geq\inf_{w\in\partial D}d_{\Omega}(a, w)\geq\inf_{w\in\partial D}d_{\tilde{\Omega}}(a,w)$.

is thought to be a thrice
The most useful (but not necessarily sufficient) domainpunctured sphere, since it has been studied for a long time and its hyperbolic metric canbe expressed almost explicitly (see, for example, [1] and [3]). Any thrice punctured sphere. The
is conformally (indeed, M\"obius) equivalent to the canonical one:
following is the precise version of Landau’s theorem due to Hempel [6]: The hyperbolic
satisfiesof
metric$D$$D_{0}=\hat{\mathbb{C}}\backslash \{0,1, \infty\}$$\rho_{0}(z)|dz|$$D_{0}$(6.1)$\rho \mathrm{o}(z)\geq\frac{1}{2|z|(|\log|Z||+K)}$,

where $K= \Gamma(\frac{1}{4})^{4}/4\pi^{2}=4.3768796\cdots$ , and the equality occurs if $z=-1$ . Note that this, otherwise we have only to use theestimate is efficient only on the half plane
functional equation $\rho_{0}(1-z)=\rho_{0}(Z)$ .
In order to find out a thrice punctured sphere containing , we have only to specify
in the Julia set, for example, repelling periodic points or their inverse
three points. For example,
images. In our present case, any periodic point is repelling since$(1\pm\sqrt{1-4c})/2$$z^{2}+c=z$
. We
,
thus
equation:
the
of
are
solutions
fixed points ofin the Julia set,
. If we selected the three points
note that if is in , so
$0,1$
, respe,ctively.
and
to
and
M\"obiustransformation mapping
be theletdenotes the hyperbolic, whereThen,
by only theand, however it seems impossible to estimate
distance indata .
For simplicity, we further assume that $c<-2$ for a moment. Set $\alpha=(1+\sqrt{1-4c})/2$
and $\beta=1-\alpha$ , then these are fixed points of $f=f_{c}$ . Then we see that.
, hence
For the later convenience, we set $t=\sqrt{1-4c}-3>0$ . Let. Then $T(\infty)=-(3+t)$ and $T(c)=- \frac{t(3+t)}{4+t}$ . We also note that $T(J_{c})\subset[0, \infty]$ .
Using (6.1), we can calculate as${\rm Re} z \leq\frac{1}{2}$$\Omega$$a_{1},a_{2},a_{3}$$c\not\in \mathcal{M}$$f_{c}$$J_{c}$$\alpha$$\mathrm{i}\mathrm{s}-\alpha$$T$$a_{1},$ $a_{2},$ $a_{3}$$a_{1},$$a_{2}$$\infty$$a_{3}$$d_{0}$$d_{\Omega_{\mathrm{c}}}(c, \infty)\geq d_{\hat{\mathbb{C}}\backslash \{a1,a2,a_{3}\}}(C, \infty)=d_{0}(T(C),T(\infty))$$\iota_{\Omega_{\mathrm{c}}}(\infty)$$\iota_{\Omega_{\mathrm{c}}}(c)$$D_{0}$$a_{j}$$S= \bigcup_{n=1}^{\infty}f^{-n}(\alpha)\subset$$[-\alpha, \alpha]$$J_{\mathrm{c}}\subset[-\alpha, \alpha]\sin \mathbb{C}\mathrm{e}\overline{S}=J_{c}$$T(z)= \frac{(\beta-\alpha)(z+\alpha)}{(\beta+\alpha)(z-\alpha)}=$$(3+t) \frac{\alpha+z}{\alpha-z}$$d_{\Omega_{\mathrm{c}}}(c, \infty)\geq d\mathrm{o}(T(C), \tau(\infty))=\int^{T(}T(\infty c))\rho_{0}(X)dx\geq\frac{1}{2}Q(t)$

,

where
if

$\log\frac{\log(3+t)+K}{K}+\log\frac{\log\frac{4+t}{t(3+t)}+K}{K}$$Q(t)=\{$$\log(3+t)+K$$t\leq t_{0}$

if $t>t_{0}$$\log\log\frac{\overline 4+t}{t(3+t)}+K$

and $t_{0}=0.38297\cdots$ is the positive root of the equation $3+t=(4+t)/t(3+t)$ .
at and . As a preparation, we
Next, we shall estimate the injectivity radii of$h(a)=\iota_{D_{0}}(-a)$
we assume that $0<a\leq 1$ . Let
.
First
$a>0$
for
consider the quantity$\Omega_{c}$$c$$\infty$

48
be the domain defined by $\{\tau\in \mathbb{H};0<{\rm Re}\tau<1, |\tau-\frac{1}{2}|>\frac{1}{2}\}$ , and:the conformal homeomorphism fromonto the upper half plane which maps $0,1,$, respectively. We denote by :to 1,
the inverse map of . Then, as is
well-known, is analytically continued to the universal covering map of
fromby
the reflection principle, in particular,. The map is nothing
but the classical elliptic modular function. For the point $\tau_{0}=(e^{i\theta}+1)/2=g(1+a)\in$$g((1,2])$ $(\pi/2\leq\theta<\pi)$ we can see thatand that
the shortest hyperbolic segment connecting
and $g((\mathrm{O}, 1))=\{ti;t>0\}$ is containedin. Noting that $\lambda(\{\tau\in\Delta;{\rm Re}\tau=\frac{1}{2}\})=\{z\in \mathbb{H};|z-1|=1\}$ , we haveandis contained in.
Denote by the closed curve obtained as the union ofand its complex conjugate,
thenand $2h(a)= \int_{\beta}\rho 0(Z)|dz|$ . Note that $|dz|\geq(|dr|+r|d\theta|)/\sqrt{2}$, where.
Put $a_{0}= \min|\beta|$ , then by (6.1) we have$\Delta$$\Deltaarrow \mathbb{H}$$\lambda$$\Delta$$\infty,$$0$$\mathbb{H}$$\mathbb{H}arrow\Delta$$g$$\infty$$\lambda$$\lambda$$D_{0}$$1/2{\rm Im}\tau=\rho_{0}(\lambda(\mathcal{T}))|\lambda’(\mathcal{T})|$$\mathbb{H}$$\lambda$$d_{\mathbb{H}}(\tau_{0},g((\mathrm{O}, 1)))\leq d_{\mathbb{H}}(\tau_{0},g((-\infty, 0)))$$\gamma$$\tau_{0}$$\{\tau\in\Delta;{\rm Re}\tau\leq\frac{1}{2}\}$$h(a)= \iota D_{0}(-a)=\iota_{D}0(1+a)=\int_{\lambda_{*}\gamma}\rho \mathrm{o}(Z)|dz|$$\{z\in\overline{\mathbb{H}};|z-1|\leq 1\}$$\lambda_{*}\gamma$$\beta$$1-\lambda_{*}\gamma$$|\beta|\leq 1$$z=re^{i\theta}$$2h(a) \geq\int_{\beta}\frac{|dz|}{2|z|(-\log|z|+K)}\geq\int_{\beta}\frac{|dr|+r|d\theta|}{2r\sqrt{2}(-\log r+K)}$$\geq\frac{2}{2\sqrt{2}}\log(\frac{-\log a_{0+K}}{-\log a+K})+\frac{1}{2\sqrt{2}}\frac{2_{T}}{-\log a0+K}$$\geq\frac{\pi/\sqrt{2}}{-\log a+K}$

because $K>\pi$ .
Next, we consider the case $a>1$ . Since the M\"obius transformation $zrightarrow 1/z$ preserves, we have $h(a)=h(1/a)$ , thus $h(a)\geq\pi/2\sqrt{2}(\log a+K)$ . Therefore, for any $a>0$ , wehave
$D_{0}$$\iota_{D_{0}}(-a)\geq\frac{\pi}{2\sqrt{2}(|1o\mathrm{g}a|+K)}$

Letting$D=\mathbb{C}\backslash [0, \infty)$.

, we can estimate the injectivity radius of$\iota_{\Omega_{\mathrm{c}}}(T^{-1}(-a))=\iota\tau(\Omega_{\mathrm{c}})(-a)\geq\inf_{w\in\partial D}$

do$\Omega_{c}$at $T^{-1}(-a)$ as follows:$(-a,w)= \iota_{D_{0}}(-a)\geq\frac{\pi}{2\sqrt{2}(|\log a|+K)}$

Hence,$\min\{2d_{\Omega_{c}}(_{C,\infty}),4\iota\Omega_{\mathrm{c}}(_{C)},4\iota_{\Omega \mathrm{c}}(\infty)\}$(6.2)$\geq R(t):=\min\{Q(t),\frac{\sqrt{2}\pi}{\log(3+t)+K},\frac{\sqrt{2}\pi}{\log\frac{4+}{t(3+t)}+K}‘\}$$\frac{\sqrt{2}\pi}{\log\frac{4+t}{t(3+t)}+K}$$=\{$$\log\frac{\log(3+t)+K}{K}+\log\frac{\log\frac{4+t}{t(3+t)}+K}{K}$$\log\frac{\log(3+t)+K}{\log\frac{4+t}{t(3+t)}+K}$

where $K= \Gamma(\frac{1}{4})^{4}/4\pi^{2}=4.3768796\cdots$ .

if $0<t<0.12626\cdots$ ,if 0.12626 $\cdots<t\leq t_{0}$ ,if $t_{0}=0.38297\cdots<t$ ,.

49from above is rather easy. In the same assumption
In contrast, the estimation ofsince
as the above, we set $\gamma=\sqrt{-c-\alpha}>0$ . Note that$x\in(-\gamma,\gamma)$$f_{c}(x)=x^{2}+c<\gamma^{2}+c=-\alpha$, hence, we see that
. Then, forseparates the. This implies that the annulus
maps onto Teichm\"uller’s extremal
Julia set . The M\"obius transformation, where $r_{1}=(\alpha-\gamma)/2\alpha$ and $r_{2}=2\gamma/(\alpha-\gamma)$ . Thus wedomaindenotes the modulus of Gr\"otzsch’s extremal
have $m(A)=2\mu(\sqrt{r_{1}}/(r_{1}+r_{2}))$ , where
for $0<r<1$ , where denotes the unit disk (see [10]). The behaviour
domainis well understood. Amongst them, it will be useful to record theof the function
following (cf. [10]):.
$L_{c}$$\gamma\in J_{c}\subset[-\alpha, \alpha]$$f_{c}(\gamma)=$$-\alpha\in J_{c}$$A=\hat{\mathbb{C}}\backslash ([-\alpha, -\gamma]\cup[\gamma, \alpha])$$(-\gamma,\gamma)\subset\Omega_{c}$$A$$T(z)= \frac{\gamma+z}{\alpha-z}$$J_{c}$$\hat{\mathbb{C}}\backslash ([-r_{1}, \mathrm{o}]\cup[r_{2}, +\infty])$$\mu(r)$$\mathrm{D}$$\mathrm{D}\backslash [0,r]$$\mu(r)$and$\log\frac{(1+\sqrt{1-r^{2}})^{2}}{r}<\mu(r)<\log\frac{2(1+\sqrt{1-r^{2}})}{r}<\log\frac{4}{r}$$\mu(r)\mu(\frac{1-r}{1+r})=\frac{\pi^{2}}{2}$.

In particular, we can see that$m(A)=2 \mu(\frac{\alpha-\gamma}{\alpha+\gamma})=\frac{\pi^{2}}{\mu(\gamma/\alpha)}$.

Therefore, we have
2.3, we summarize the results obtained above.

$L_{c}\leq\pi^{2}/M_{\Omega_{\mathrm{c}}}\leq\pi^{2}/2\mu((\alpha-\gamma)/(\alpha+\gamma))=\mu(\gamma/\alpha)$

Theorem 6.1. For $c<-2$ , the Fatou set(6.3)$\Omega_{\mathrm{c}}$. Noting Corollaryof $f_{c}(z)=z^{2}+c$ satisfies$R(t) \leq L_{\Omega_{\mathrm{c}}}\leq\frac{\pi^{2}}{2\mu(\frac{\alpha-\gamma}{\alpha+\gamma})}=\mu(\frac{\gamma}{\alpha})$,where $t=\sqrt{1-4c}-3>0,\alpha=(1+\sqrt{1-4c})/2>2,\gamma=\sqrt{-c-\alpha}>0,R(t)$ is theextremal domain
denotes the modulus offunction defined by (6.2) and$Gr\ddot{o}t_{Z}sCh^{f}S$$\mu(r)$$\mathrm{D}\backslash [0,r]$.

Remark. Since $(\alpha-\gamma)/(\alpha+\gamma)=2\alpha/(\alpha+\gamma)^{2}\sim 1/t$ , the upper bound in (6.3) behaves, while $R(t)\sim 4/t\log t$ .
like $\pi^{2}/2\log t$ assince $\gamma/\alpha=\sqrt{t/8}(1+O(t))$ ,When $tarrow+0$ , the upper bound in (6.3)
however $R(t)\sim\sqrt{2}\pi/\log 1/t$ .
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 theJulia set.
$tarrow\infty$$\mathrm{i}_{\mathrm{S}\frac{1}{2}}\log 1/t+O(1)$, we can use the following fundamental
In order to get a result for any
property of the uniform perfectness constants (cf. [15]).
$c\in \mathbb{C}\backslash \mathcal{M}$are quasi-invariant, in other words, if
andProposition 6.2. The constants
is a $K$ -quasiconformal homeomorphism $(K\geq 1)$ then$M_{\Omega}$$L_{\Omega}$$\Omegaarrow\Omega’$$M_{\Omega}/K\leq M_{\Omega’}\leq KM_{\Omega}$

and$L_{\Omega}/K\leq L_{\Omega’}\leq KL_{\Omega}$

.

$f$

:

50
Next we consider a holomorphic motion of the Julia set, which is an important tool
introduced by-Sad-Sullivan. For a subset $E$ of and a pointed hyperbolic Riemannsurface (or, more generally, complex hyperbolic manifold) (X, ) with hyperbolic distance, a map
is called a holomorphic motion of parametrized by (X, )if the following holds:$\hat{\mathbb{C}}$$\mathrm{M}\mathrm{a}\tilde{\mathrm{n}}\acute{\mathrm{e}}$$x_{0}$$F:X\mathrm{x}Earrow\hat{\mathbb{C}}$$d_{X}$$E$$x_{0}$

1.
is holomorphic for each $a\in E$ ,is injective for each $x\in X$ , and
2. $F_{x}:=F(x, \cdot)$ :is the identity map of .
3.
We can state the optimal -lemma proved by Slodkowsky [14] (see also [5]) as in thefollowing form.
$F(\cdot, a):Xarrow\hat{\mathbb{C}}$$Earrow\hat{\mathbb{C}}$$E$$F_{x_{0}}$$\lambda$

Theorem 6.3 (Optimal -lemma). Let $F$ be a holomorphic motion of a subset of the
Riemann sphere parametrized by a simply connected hyperbolic Riemann surface $X$ withbasepoint . $Then_{f}F$ can be extended to a holomorphic motion of the whole sphereparametrized by (X, ) with the following properties.
$E$$\lambda$$\overline{F}$$\hat{\mathbb{C}}$$x_{0}$$x_{0}$

1.
:
is (
$x\in
X$
, the map
2. for eachtrami coefficient
the Teichm\"uller distance$\overline{F}$$X\cross\hat{\mathbb{C}}arrow\hat{\mathbb{C}}$$j_{oin}tl\underline{y)}$continuous,
is a quasiconformal homeomorphism with Bel:denotessatisfying $d_{T}(\mu_{x}, 0)\leq d_{X}(x, x_{0})$ , where$\hat{\mathbb{C}}arrow\hat{\mathbb{C}}$$\underline{F}_{x}$$\mu_{x}=\partial_{\overline{z}}F_{x}/\partial_{z}\overline{F}_{x}$$d_{T}$$d_{T}( \mu, \nu)=\mathrm{e}\mathrm{s}\mathrm{s}Z^{\cdot}\in\sup_{\mathbb{C}}d\mathrm{D}(\mu(Z), \nu(z))=\mathrm{a}\mathrm{r}\mathrm{c}\tanh(||\frac{\mu-\nu}{1-\overline{\nu}\mu}||_{\infty})$.

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 $\Phi_{n}(c):=(f_{c}^{n}(c))^{2^{-n}}$ converges, which is, in turn, a conformal
locally uniformly into a holomorphic function
onto $D:=\{z\in \mathbb{C};|z|>1\}$ , where we take the branch
so as tomapping of$\Phi_{n}(c)=c+O(1)$ as $carrow\infty$ . By the symmetry of the Mandelbrot set A4,
note
that
,
, in particular, $\Phi((-\infty, -2))=(-\infty, -1)$ .
Now we define the function : $H=\{\zeta\in \mathbb{C};{\rm Re}\zeta>0\}arrow D$ by $p(\zeta)=-e^{(}$ , then$q:=\Phi^{-1}\mathrm{o}p$ is a universal covering map of
from the right half plane $H$ . Fix
an arbitrary pointin. Then there exists a in $H$ such that $q(\zeta_{1})=c_{1}$ and$-\pi<{\rm Im}\zeta_{1}\leq\pi$ . Set$c_{0}=q(\zeta_{0})$ and let
be the set of repelling periodic. Then, considering the roots of the equation $f_{c}^{n}(z)=z$ , where $c=q(\zeta)$
points ofand is taken over all positive integers, we can obtain a holomorphic motion of the set
parametrizedby the right half plane $H$ with basepoint because
has no parabolic, thus the roots do not collide. By the optimal -lemma,
periodic points for
we have a holomorphic motion
such thatand thatis theset of repelling periodic points of. (We can take an $F$ compatible with the dynamics,, but we do not this property here.)thus
is a quasiconformal conjugate oftoSince the set of repelling periodic points is dense in the Julia set,holds.
Hence, by Proposition 6.2 and the second property in Theorem 6.3, we have$)$$\Phi(c)$$\mathbb{C}\backslash \mathcal{M}$$\Phi_{n}$$\mathbb{C}\backslash \lambda 4$$\mathrm{w}\mathrm{e}$$\overline{\Phi(\overline{z})}=\Phi(z)$$p$$\mathbb{C}\backslash \mathcal{M}$$c_{1}$$\mathbb{C}\backslash \mathcal{M}$$\zeta_{1}$$\zeta_{0}=|\zeta_{1}|,$$E_{0}$$f_{c_{0}}$$n$$E_{0}$$f_{c}$$\zeta_{0}$$\lambda$$c\in \mathbb{C}\backslash \mathrm{A}1$$F:H\cross\hat{\mathbb{C}}arrow\hat{\mathbb{C}}$$F_{C\mathrm{o}}=\mathrm{i}\mathrm{d}$$F_{\zeta}(E_{0})$$f_{q(\zeta)}$$F_{(}$$f_{c_{0}}$$f_{q(\zeta)}$$F_{\zeta}(J_{c_{0}})=J_{q(\zeta)}$$1/K\leq L_{\Omega_{\mathrm{c}_{1}}}/L_{\Omega_{\mathrm{c}_{0}}},$

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

,$M_{\Omega_{\mathrm{c}_{1}}}/M_{\Omega_{\mathrm{c}_{0}}}\leq K$

51
Now we compute $K$ . We write
calculate as$\zeta_{1}=|\zeta_{1}|e^{i\theta}=z_{0}e^{i\theta}$with $\theta\in(-\pi/2,T/2)$ . Then we can$d_{H}( \zeta_{1}, \zeta_{0})=\mathrm{a}\mathrm{r}\mathrm{c}\tanh(|\frac{\zeta_{1}-\zeta 0}{\zeta_{1}+(_{0}}|)=\mathrm{a}\mathrm{r}\mathrm{c}\tanh(\tan\frac{\theta}{2})=\frac{1}{2}\log(\frac{1+\sin\theta}{\cos\theta})$.

Now we have shown the following.

with $\theta\in(-\pi/2,\pi/2)$, take a point
Theorem 6.4. For an arbitray
such that $-e^{\zeta}=\Phi(c)$ and that $-\pi<{\rm Im}\zeta=r\sin\theta\leq\pi$ . Then, we have the following
estimates:,
andwhere $c_{0}<-2$ is the number determined by $\Phi(c_{0})=-e^{r}$ , and$\zeta=re^{i\theta}$$c\in \mathbb{C}\backslash M$$L_{\Omega_{\mathrm{c}_{0}}}/K\leq L_{\Omega_{\mathrm{c}}}\leq KL_{\Omega_{\mathrm{c}_{0}}}$$M_{\Omega_{\mathrm{c}_{0}}}/K\leq M_{\Omega_{\mathrm{c}}}\leq KM_{\Omega_{\mathrm{c}_{0}}}$$K= \frac{1+\sin\theta}{\cos\theta}$.

We remark that $\log|\Phi(c)|=r\cos\theta$ is Green’s function of the domain
at the infinity.

$\hat{\mathbb{C}}\backslash \mathcal{M}$with pole

REFERENCES[1] AHLFORS, L. V. Conformal Invariants, McGraW Hill, New York (1973).
Springer-Verlag[2] BEARDON, A. F. Iteration of Rational Functions, No. 132 in Grad. Texts Math.,(1991).
London Math.
[3] BEARDON, A. F. and POMMERENKE, CH. The PoinCa.r\’e metric of plane domains, J.
Soc. (2), 18 (1978), 475-483.
[4] CARLESON, L. and GAMELIN, T. W. Complex Dyamics, Springer-Verlag (1993).
autres], Ast\’erisque,[5] DOUADY, A. Prolongement de mouvements holomorphes [d’apr\‘e Siodkowsky et
227 (1995), 7-20.
Landau and[6] HEMPEL, J. A. The Poincar\’e metric on the twice punctured plane and the theorems of
Schottky, J. London Math. Soc. (2), 20 (1979), 435-445.
Cambridge Philos.
[7] HINKKANEN, A. Julia sets of rational functions are uniformly perfect, Math. Proc.
Soc., 113 (1993), 543-559.
(1996), 161[8] HINKKANEN, A. and MARTIN, G. J. Julia sets of rational semigroups, Math. Z., 222

169.

Math.
[9] J\"ARVI, P. and VUORINEN, M. Uniformly perfect sets and quasiregular mappings, J. London
Soc., 54 (1996), 515-529.
, K. I. Quasiconformal Mappings in the Plane, 2nd Ed., Springer-Verlag[10] LEHTO, O. and(1973).
, R. and DA ROCHA, L. F. Julia sets are uniformly perfect, Proc. Amer. Math. Soc., 116[11](1992), 251-257.
. Uniformly perfect sets and the Poincar\’e metric, Ark. Math., 32 (1979), 192-199.
[12] POMMERENKE,
299-321.
[13] POMMERENKE, CH. On uniformly perfect sets and Fuchsian groups, Analysis, 4 (1984),(1991),
111
Soc.,
Math.
Amer.
hulls,
Proc.
polynomial[14] SLODKOWSKY, Z. Holomorphic motions and$\mathrm{v}_{\mathrm{l}\mathrm{R}\mathrm{T}}\mathrm{A}\mathrm{N}\mathrm{E}\mathrm{N}$$\mathrm{M}\mathrm{A}\tilde{\mathrm{N}}\mathrm{E}’$$\circ \mathrm{H}$347-355.

Complex Vari[15] SUGAWA, T. Various domain constants related to uniform perfectness, To appear inables.
[16] SUGAWA, T. Uniform perfectness of the limit sets of Kleinian groups, Preprint (1997).
[17] TSUJI, M. Potential Theory in Modern Function Theory, Maruzen, Tokyo (1959).
DEPARTMENT OF MATHEMATICS, KYOTO UNIVERSITY, 606-8502 KYOTO, JAPAN. jp$E$ -mail address: sugawaQkusm. kyoto-u.
$\mathrm{a}\mathrm{c}$