数理解析研究所講究録
第 1586 巻 2008 年 26-31

26Some properties of Julia sets of transcendental entire
functions with multiply-connected wandering domains
Masashi KISAKA (木坂 正史)
Department of Mathematical Sciences,

Graduate School of Human and Environmental Studies,
Kyoto University, Yoshida Nihonmatsu-Cho, Kyoto 606-8501, Japan(京都大学大学院人間・環境学研究科数理科学講座)
e-mail: kisaka@math.h.kyoto-u.ac.jp
Abstract
We study Julia components of transcendental entire functions with multiplyconnected wandering domains. Under the assumption that the poet singular
set is contained in the Fatou set, it is shown that every repelling periodic point
satisfies either, where $C(p)$ is the Julia component containing and is an(1)immediate attractive basin.
(2) $C(p)=\{p\}$ and this is a buried singleton component of $J(f)$ .

$p$$C(p)\supset\partial U$

$p$

$U$

\S 1 Introduction
Let be a transcendental entire function, $F(f)$ it8 Fatou set and $J(f)$ its Julia set. The
folowing are some fundamental results on the connectivity of $J(f)$ :$f$Proposition 1 If every Fatou component is bounded and simply connected, then $J(f)\subset$
is connected.

$\mathbb{C}$So it folows that if$J(f)\subset \mathbb{C}$is disconnected, then either

(a)

$f$

has an unbounded Fatou component or(b)$f$has a multiply-connected Fatou component.

For the case (a), the foUowing holds. Note that an unbounded Fatou component
isalways simply connected (see [Bal]) and so we can consider a Riemann map : $Darrow U$of .
$U$$\varphi$$U$

Theorem 2 ( $[K$ , p.192, Main Theorem]) Suppose there exists an unbounded invariant
Fatou component and let us consider the following conditions:$U$27(A)$\infty\in\partial U$

is accessible in$U$(B) There exist a finite point.

with $q\not\in P(f),$ $m_{0}\in N$ and a continuous curve$C(t)\subset U(0\leq t<1)$ with $C(1)=q$ which satisfies $f^{mo}(C)\supset C$ , where$q\in\partial U$$P(f)= \bigcup_{n\simeq 0}^{\infty}f^{n}(sing(f^{-1}))$is the $poet-S\dot{i}$gular set of .
$f$(1) If is either an attractive basin with (A) and (B), or a parabolic basin with (A) and(B), or a Siegel disk with (A), then the set$U$$\Theta_{\infty}$

is dense in$\partial D$$:= \{e^{1\theta}|\varphi(e^{1\theta}):=\lim_{r\nearrow 1}\varphi(re^{j\theta})=\infty\}\subset\partial D$. In particular,$J(f)\subset \mathbb{C}$is disconnected.

(2) If is a Baker domain with (B) and $f|U$ is not univalent, thenis dense in OD
or at least its closure
contains a certain perfect set in. In particular,
is$U$$\Theta_{\infty}$$\overline{\Theta_{\infty}}$$\partial D$$J(f)\subset \mathbb{C}$disconnected.
Next result is a generalization of the above result:

Theorem 3 [BD1, p.439, Theorem 1.1, 1.2, Corollary 1.3]
the assumption (B).
On the other hand,component. For example,$J(f)\subset \mathbb{C}$can be connected nevertheless$Th\infty rem2$$f$holds without

has an unbounded Fatou$f(z)=2-\log 2+2z-e^{z}$

has a Baker domain but $J(f)$ is connected ( $[K$ , p.194, Theorem 4]).

For the case (b), it is known that if has a multiply-connected Fatou component , thenis a wandering $doma\dot{i}$ and bounded (see, [Ba2, Theorem 3.1]) and therefore
is always disconnected. Furthermore $J(f)\cup t\infty$ }is also disconnected imd actually
this is the only case where $J(f)\cup t\infty$ }can be disconnected as follows:
$U$

$f$

$U$

$J(f)\subset \mathbb{C}$$\subset\hat{\mathbb{C}}$$\subset\hat{\mathbb{C}}$

Proposition 4 ( $[K$ , p.191, Theorem 1])
if has a multiply-connected wandering domain.

$J(f)\cup\{\infty\}\subset\hat{\mathbb{C}}$is disconnected if and only$f$

In what follows, we will concentrate on the case (b), that is, the case where has a multiplyconnected wandering domam and investigate some properties of connected componentsof the Julia set, which we call Julia components. We note the following fact (see, [Ba2,p.565, Theorem 3.1]):
$f$

$U$

Proposition 5 If

$U$is a multiply-connected wandering $doma\dot{i}$ , then $f^{n}|Uarrow\infty$ .

28Definition 6(2)$z\in J(f)$(1) We call a connected component of $J(f)$ a Julia component.

is called a buried point if$z$

satisfies$z\not\in\partial U$

for any Fatou component$U$.

(3) We call the set$J_{0}(f):=$the residud Julia set of$f${ $z\in J(f)|z$ is a buried point}.

(4) A Julia component $C$ is called a buried component if $C\subset J_{0}(f)$ .

For rational cases, the folowing are known:, whereExample 7 ([Mc]) Let
is small. Then $J(f)$ is a Cantor set.
of nested quasi-circles. So there are buried components. In particular,$f(z)=z^{2}+ \frac{\lambda}{z^{3}}$$\lambda>0$$J_{0}(f)\neq\emptyset$

Theorem 8 ([Mo, p.208, Theorem 3]) Let
if and only if$f$

be a hyperbolic rational function. Then

$J_{0}(f)\neq\emptyset$(1)$F(f)$ has a completely invariant component,(2)$F(f)$ consists of only two components.

Example 9 ([Mo, p.209])Let$f(z)= \frac{-2z+1}{(z-1)^{2}}$or, then the following hold:(1) The set $\{0,1, \infty\}$ is a super-attracting cycle.
(2)

$f$

is hyperbolic.

(3) Any Fatou component is a preimage of the super-attractive basin above.

(4)$J(f)$ is connected.

So by Theorem 8, we havecomponent.

$J_{0}(f)\neq\emptyset$. But since $J(f)$ is connected, there is no buried

Example 10 ([U]) There exists a rational function whose Julia set is $hom\infty morphic$, but again there is no buried component.
to a Sierpinski gasket. So$f$$J_{0}(f)\neq\emptyset$

Here are some fundamental properties for buried points and residual Julia sets. Note thatneed not be rational and these hold also for transcendental entire functions and even formeromorphic functions.
$f$Proposition 11(1) If $F(f)$ has a completely invariant component, then$J_{0}(f)=\emptyset$(2) If there exists a buried component of $J(f)$ , then $J(f)$ is disconnected.
(3) If$J_{0}(f)\neq\emptyset$, then$J_{0}(f)$is completely invariant, dense in $J(f)$ , and uncountable.

More information on residual Julia sets, see [DF].

.

29\S 2 Results
Main result of this paper is as follows:

Theorem A Let$f$

be a transcendental entire function. Assume that(a)$P(f)= \bigcup_{n=0}^{\infty}f^{n}(sing(f^{-1}))\subset F(f)$

(b)

$f$

,

has a multiply-connected wandering $doma\dot{i}$ .

Then every repeling periodic point$p$satisfies either one of the folowing:, where $C(p)$ is the Julia component $contain\dot{i}gp$ andattractive basin.
(1)$C(p)\supset\partial U$(2)$\{p\}$

is an immediate

is a buried singleton component of $J(f)$ .

Corollary B Let(a), (b) and also
(c)

$U$

$f^{n}(z)arrow\infty$$f$

be a transcendental entire fiiction. Assume the above conditions

for any $z\in F(f)$ .

Then every repelling periodic point
Remark$f$$p$

is a buried singleton component of $J(f)$ .

is called hyperbolic if$dist_{C}(P(f), J(f))>0$ ,whereis the Euclidean distance onweaker than hyperbolicity.
$\bm{i}st_{\mathbb{C}}$$\mathbb{C}$. So the condition (a) in $Th\infty rem$ A is slightly(Outline of the Proof): Let be a repeling periodic point. For simplicity, we assume
that is a fixed point. Suppose that does not satisfy (1). Let $C(p)\subset J(f)$ be the Julia
component containing . Then $f(C(p))=C(p)$ and we can show that $C(p)$ is bounded. If
there exists a Fatou component $U\subset F(f)$ such that, then it $f_{0}nows$ that
Is a wandering domain which satisfies $f^{n}(U)arrow\infty(narrow\infty)$ . Then this contradicts thefact that $C(p)$ is bounded. Hence $C(p)$ is a buried component.
$p$

$p$

$p$

$p$

$C(p)\cap\partial U\neq\emptyset$$U$

Next we can show that the complement of $C(p)$ has no bounded component. Then since$P(f)\subset F(f)$ and $C(p)$ is bounded, we have$dist_{\mathbb{C}}(C(p), P(f))>0$.

Then there exists a simply connected domain $W$ such that $C(p)\subset W$ and there exists
a branchofwhich satisfies $g_{n}(p)=p$ . It is $wen$-known thatis a normalfamily and hence there exists a subsequence : converging to a constant function whichmust be the point . On the other hand, we have $g_{\mathfrak{n}}(C(p))=C(p)$ , so we conclude
that $C(p)=\{p\}$ . This completes the proof of Theorem A. Corollary is an immediate
consequence of $Th\infty rem$ A.
$g_{n}$$f^{-n}$$\{g_{n}\}_{n=1}^{\infty}$$g_{n}$$p$

$B$

$O$

30

\S 3 Examples
There exists an $f(z)$ with the following

Example 12 ([BD2, p.375, Theorem $G]$ )form$f(z)=k \prod_{\mathfrak{n}=1}^{\infty}(1+\frac{z}{r_{n}})$,such that for every repelling periodic point$0<r_{1}<r_{2}<\cdots,$

$p$

$k>0$

is a buried singleton component of $J(f)$ .

Example 13 ([KS])There exists a transcendental entire function with doublyconnected wandering domains, which satisfies the following: Every critical point satisfies $f^{2}(c)=0$ and is a super-attracting fixed point. This implies that this satisfies
the assumptions of $Th\infty rem$ A. Therefore every repelling periodic point satisfies either
for the immediate attractive basin of the super-attractive fixed point or
is a buried singleton component of $J(f)$ .
$f$

$c$

$0$

$f$

$p$

$C(p)\supset\partial U$

$U$

$0$

$\{p\}$

Example 14 ([Be]) By using the similar method as in Example 13, Bergweiler constructed an example of transcendental entire function which has both a simply connected
and a multiply connected wandering domain. Critical points of satisfy the following:
$f$

$f$

(1) Cg $=0<c_{1}<c_{2}<\cdotsarrow\infty$ ,(2) $f(0)=0,$ $f($ 儒 $)=$ 果+1, $i=1,2,$(3)$c_{i}$

is contained in a simply connected wandering domain$f(U_{1}\cdot)=U_{\dot{|}+1}$

So this$f$$\ldots$,$f^{n}|U_{1}\cdotarrow$科科 $(narrow$$U_{i}$

which satisfies

科科).

also satisfies the assumptions (a) and (b) of Theorem A.

Example C We can construct an which satisfies the assumptions (a), (b) and (c) byusing the similar method as in Example 13. Hence every repelhng periodic point is aburied singleton component of $J(f)$ from Corollary B. We omit the details.
$f$

$p$

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