数理解析研究所講究録988 巻 1997 年 98-112

98

On Dynamics of Hyperbolic Rational Semigroupsand Hausdorff Dimension of Julia sets
Hiroki Sumi
Graduate School of Human and Environmental Studies,
Kyoto U.n iversity.
-mail: sumi@math.h.kyoto-u.ac.jp$\mathrm{e}$December 12, 1996
Abstract
We consider about subsemigroups of. We can define Julia sets,
Fatou sets, etc. We show the backward sel&imilarity of Julia sets of finitely
generated rational semigroups. If the post critical set of the semigroupis contained in a domain of Fatou set, the Julia set is a self similar set.
Next, hyperbolic rational
have no wandering domains undera general assumption. If the hyperbolic rational semigroup is finitely generated and satisfies some conditions, the limit functions of the semigroupon the Fatou set are only constant functions that take their values on
post critical set. When the generators of a finitely generated hyperbolic
rational
are perturbed, the hyperbolicity is kept and the Jilia
sets depend cotiniously on the generators. Further more, if the finitely
semigroup is hyperbolic and if the inverse images bygenerated
the generators of the Julia set are mutually disjoint, then the Julia set
moves by holomorphic motion.
Next we consider about finitely generated rational semigroups satisa strong open set condition. We show that if a semigroup satisfies
the strong open set condition, the Julia set has no interior points, andfurthur more, if the semigroup is hyperbolic, the Hausdorff dimension ofJulia set is strictly lower than 2. The value of the dimension coincides
with the unique value that allows us to construct -conformal measure.
The -Hausdorff measure of the Julia set is a finite value strictly bigger
than zero.
Next we coustruct the generalized Lyubich measure on Julia set when
the semigroups are hyperbolic or satisfy strong open set condition. The
measure also can be considered as the generalized BernouUi measure on
self similar set. And using that measure we try to estimate the Hausdorff$\mathrm{E}\mathrm{n}\mathrm{d}\overline{\mathbb{C}}$$\mathrm{s}\mathrm{e}\mathrm{m}\mathrm{i}\mathrm{g}_{\mathrm{l}\mathrm{o}\mathrm{u}\mathrm{p}\mathrm{s}}$$\mathrm{s}\mathrm{e}\mathrm{m}\mathrm{i}\mathrm{g}_{1}\cdot \mathrm{o}\mathrm{u}\mathrm{p}$$1\mathrm{a}\mathrm{t}\mathrm{i}_{\mathrm{o}\mathrm{n}\mathrm{a}}1$$\mathrm{f}\mathrm{y}\mathrm{i}_{1\mathrm{l}}\mathrm{g}$$\delta$$\delta$$\delta$dimension of Julia set from below.

1

Introduction

For a Riemann surface , let Elld $(S)$ denote the set of all holomorphic endomorphisms of . It is a semigroup with the semigroup operation being functional
composition. A rational semigroup is a subsemigroup of End
without any. Similarly, an entire semigroup is a subsemigroup of End(C)constant. A rational semigroup is called a polynomial
without any constantsemigroup if each $g\in G$ is a polynomial. When a ratio nal or elltire semigroupis generated by $\{f_{1}, f2, \ldots fn’\ldots\}$ , we denote this situation by
$S$

$S$

$(\overline{\mathbb{C}})$$\mathrm{e}\mathrm{l}\mathrm{e}\mathrm{n}\mathrm{l}\mathrm{e}\mathrm{n}\mathrm{t}\mathrm{s}$$G$$\mathrm{e}\mathrm{l}\mathrm{e}\mathrm{n}\mathrm{l}\mathrm{e}\mathrm{n}\mathrm{t}_{\mathrm{S}}$$G$$G=\langle f_{1}, f2, \ldots f_{1\iota}, \ldots\rangle$.

99
The rational or entire semigroup gellerated by a single function, th iterate of$(g)$ . We denote the
by .
$f$

$f|$

Definition 1.1 Let$G${$F(G)^{\mathrm{d}\mathrm{f}}=^{\mathrm{e}}$$g$is denoted by$f^{n}$

be a rational semigroup.
$z\in\overline{\mathbb{C}}|G$

is normal in a neighborhood of$z$}$J(G)^{\mathrm{d}\mathrm{e}}=^{\mathrm{f}}\overline{\mathbb{C}}\backslash F(G)$is called Fatou set for
and $J(G)$ is called Julia set for G. Similarly,
Fatou set and Julia set for entire semigroup are defined.

$F(G)$$G$$\mathrm{D}\mathrm{e}\mathrm{f}\mathrm{i}\mathrm{n}\mathrm{i}\mathrm{t}\mathrm{i}\mathrm{o}\mathrm{n}backwardofb1.2Leito-(zt)of_{Z}anhesetoGbear_{dt}ational_{S}emigroupaCfexepti_{\mathit{0}}nalp_{oi}nendzb.apEtS(c)areointof\overline{\mathrm{c}d}.efine\tau hed$$by$:$O^{-}(z)^{\mathrm{d}\mathrm{f}}=^{\mathrm{e}}$

{.

$w$

there is some $g\in G$ such that $g(w)=z$ },$\in\overline{\mathbb{C}}|$.

$E(G)^{\mathrm{d}}=^{\mathrm{f}}\mathrm{e}\{z\in\overline{\mathbb{C}}|\#^{o}-(Z)\leq 2\}$

Definition 1.3 A subsemigroup $H$ of a semigroupis said to be of finite
index if there is a finite collection of elementsof such that
$G= \bigcup_{i=1}^{n}g:H$ .
Similarly we say that a subsemigroup $H$ of
has cofinite index
if there is a finite collection of elements $\{g1, g2, \ldots, gn\}$ of such that for every$g\in G$ there is $j\in\{1,2, \ldots, n\}$ such that $g_{j}g\in H$ .
$G$$\{g_{1},g_{2}, \ldots , g_{n}\}$

$G$

$G$

$G$

Lemma 1.1 Let$G$

be a rational semigroup.

1. For any $f\in G$ ,$f(F(G))\subset F(G),$ $f^{-1}(J(G))\subset J(G)$$F(G)\subset F((f\rangle), J(\langle f))\subset J(G)$2. If$G=\langle f_{1}, \ldots , f_{n}\rangle$, then$F(G)= \bigcap_{i=1}^{n}fi^{-}1(F(G)),$$J(G)= \bigcup_{i=1}^{\hslash}f_{i}-1(J(G))$, we say tllat $K$ has backward self
If a set $K$ satisfies thatsimilarity. Next lemma was shown in [HM1].
$K= \bigcup_{i=1}^{\hslash}f^{-}i(1Iir)$

Lemma 1.2 Let$G$

be a rational semigroup.

1. If a subsemigroup $H$ of$G$

is of finite or cofinite index, then
$J(H)=J(c)$ .

In particular, $u’ henG$ is a rational semigroup generated by finite elements\dag er $f2,$} and $m$ is an integer, if we set$\ldots fn$$f_{1},$$H_{m}=$$I_{m}=$

then{$g=f_{j_{1}}\cdots f_{j_{k}}\in G|m$devides },$k${ $g\in G|g$ is a product of some dements of word length }$m$$J(G)=J(H_{m})=J(I_{m})$ .

Here we say an $e\dot{l}ementf\in G$ is word length $m$ if $m$ is the minimum
integer such that
$f=f_{j_{1}}\cdots f_{jn}$.

100
2. If $J(G)$ contains at least three points, then $J(G)$ is a perfect set.

3. If there$\dot{u}$some $g\in G$ such that $\deg(g)\geq 2$ or there is some $g\in G$ suchand order of is infinite, thenthat $\deg(g)=1$$g$$E(G)=\{z\in\overline{\mathbb{C}}|\# O^{-}(Z)<\infty\},$$\# E(G)\leq 2$.

is not in $E(G)$ , then for every $x\in J(G),$ belongs to
In particular if a point belongs to $J(G)\backslash E(G)$ , then

4. If a point’$\overline{O^{-}(z)}$$x$$\sim$.

$z$.

$\overline{o-(\approx)}=J(G)$5. If there is some $g\in G$ such that $\deg(g)\geq 2$ or there is some $g\in G$ suchthat $\deg(g)=1$ and order of is infinite and $J(G)$ contains at least threepoints, then $J(G)$ is the smallest closed backward invariant set containing
at least three points. Here we say that a set is backward invariant under
$g$

$A$

$G$

if for each$g\in G,$ $g^{-1}(A)\subset A$.

6. If $J(G)$ contains at least three points, then
$J(G)=\overline{\{z\in\overline{\mathbb{C}}|z}$isarepelling fixed point of some $g\in G$ }Proof [HM1].
Remark A similar result of 6. for entire semigroup can also be stated.

$\square$, we set $g_{1}(z)=2(z-p_{1})+p_{i},$ $i=$
Example 1.1 For a regular triangle$1,2,3$ . And let, not as a group.
be a rational semigroup generated by$J(G)$
Then
is the Sierpi\’{n}ski Gasket.
$p_{1}p_{2}p_{3}$$G$$\{g_{1}\}$

2

Dynamics of Hyperbolic Rational Semigroups

2.1

Limit Functions

Definition 2.1 Let$G$

be a rational semigroup. We set$P(G)= \bigcup_{g\in G}$

and we say that$G$

{ critical values of }$g$is hyperbolic if and only if $P(G)\subset F(G)$ .

Definition 2.2 Let be a rational semigroup and be a component of $F(G)$ .
For every element of , we denote by
the connected component of $F(G)$containing $g(U)$ . We say thatisis a wandering domain if and only if
$G$

$g$

$U$

$G$

$U_{g}$$U$$\{U_{g}\}$infinite.

Theorem 2.1 Let be a rational semigroup andbe a wandering domain.
Then there is a constant limit function of on taking its value in $J(G)$ .
$G$

$U$

$G$

$\varphi$$U$$\zeta$Now we show a sufficient condition such that there is no wandeling domain.
be a rational semigroup andbe a wandering domain.
Theorem 2.2 Let
Also let be a constant limit function of on taking its value in $J(G)$ . If
there is an element of such that the degree is at least two, then the value isin $P(G)$ .
$G$

$U$

$G$

$\varphi$

$G$

$U$

$\zeta$$\zeta$101
Corollary 2.1 If is a hyperbolic rational semigroup containing an elementof degree at least two, then there is no wandering domain of $F(G)$ .
$G$

Similarly we can show$\mathrm{t}1_{1}\mathrm{e}$following result.

Theorem 2.3 In the same situation as Theorem 2.1, assume that every ele, we denote the closure ofment of is of degree one. For every point
orbit of by $A(x)$ . Then for all
but at most two points of -fixed points,
belongs to $A(x)$ .
$G$

$G$

$x\in\overline{\mathbb{C}}$$G$$x\in\overline{\mathbb{C}}$$x$$\zeta$Corollary 2.2 If every element of is degree one and there is a pointsuch that $A(x)$ contains at least two points and is included in $F(G)$ , then there
is no wandering domain of $F(G)$ .
$G$$x\in\overline{\mathbb{C}}$Next we consider about limit functions of a hyperbolic rational semigroup on
the Fatou set. By Theorem 2.2, we get one of main results.
Theorem 2.4 Let be a finitely generated hyperbolic rational semigroup whichcontains an element of degree at least two and assume that every element ofdegree one is not elliptic. Then for every compact subset $K$ of $F(G)$ , theorbit of can accumulate only to $P(G)$ and every limit function of on $F(G)$is a constant function that takes its value in $P(G)$ .
$G$

$G$

$G$

$I\iota’$

2.2

Continuity of Julia sets

Definition 2.3 et $E$ be a metric space. We denote by $C_{omp^{*}}(E)$ the set of
non-empty compact subsets of E. For every $A,$
we set$\dot{L}$$B\in \mathrm{C}\mathrm{o}\mathrm{m}\mathrm{p}^{*}(E)$$\partial(A, B)=\sup\{d(x, B)|x\in A\}$and$d_{H}(A, B)= \max\{\partial(A, B), \partial(B, A)\}$

It is well knoum thatmetric.

$d_{H}$.

is a metric on $c_{omp^{*}}(E)$ . We call it the Hausdorff

Next we consider if a Julia set depends continuously on the generators.

Definition 2.4 Let $M$ be a complex manifold. Suppose the map$(z, a.)\in\overline{\mathbb{C}}\mathrm{X}M\mapsto f_{j,a}(z)\in\overline{\mathrm{c}}$.

,is holomorphic for each $j=1,$
We set $G_{a}=(f_{1,a},$say thatis a holomorphic family of rational semigroups.
$\ldots$$n$$\cdots,$$f_{n,a}\rangle$. Then we$\{G_{a}\}_{a\in M}$

Definition 2.5 Let be a rational semigroup. We say that a compact subset
of $F(G)$ is a confinement set of if and only if for every $z\in F(G)$ , for all
$G$

$G$

$I\mathrm{i}’$but finite elements

Theorem 2.5 Let

$g$

of

$G$

the point $g(z)$ is included in $IC$.

$\{G_{a}\}_{a\in M}$$u’ hereG_{a}=\langle f_{1,a}, \cdots, f_{n,a}\rangle$$con,finement$ set $Ii$of

$G_{b}$

be a holomorphic family of rational semigroups. We assume that for a point $b\in M$ there is a

. Then the map.

$arightarrow J(G_{a})\in \mathrm{C}\circ \mathrm{m}\mathrm{p}^{*}(\overline{\mathbb{C}})$is continuous at the point $a=b$ with respect to Hausdorff metric.

102
By Theorems 2.4, 2.5, we get the following result.

be a holomorphic family of rational semigroups
Theorem 2.6 Let$G_{a}=(f_{1,a},$we assume that for every $d_{j}=\deg(f_{j},a)$ is
.
And
where
constant independent of a. Then$\{G_{a}\}_{a\epsilon M}$$j,$$f_{n},a\rangle$$\cdots,$

is hyperbolic. And also assume
1. Let be a point of M. Assume tha.t
is at least two and for every $g\in G_{b}$ such that $\deg(g)$ is equal tothat
one is not elliptic. Then there is an open neighborhood $W$ of suchis hyperbolic and the mapthat for every $a\in W$ the rational semigroup$arightarrow J(G_{a})$ is continuous with respect to the
Hausdorff metric.
$G_{b}$$b$$d_{1}$$b$$g$$G_{a}$

tuallyare
2. Under the same assumption as 1, if the sets
disjoint, then there is an open neighborhood of and a continuous map$(f_{j,b}^{-1}(j(c)))_{i}$$V$$m\mathrm{u}$$b$the map $arightarrow i(z, a)$ is holosuch that for every:$i(z, a)$ is a quasi conformal
morphic, and for every $a\in V$ the map
and maps $J(G_{b})$ onto $J(G_{a})$ .
homeomorphism$z\in\overline{\mathbb{C}}$$\overline{\mathbb{C}}\mathrm{x}Varrow\overline{\mathbb{C}}$$i$$zarrow\rangle$$of\overline{\mathbb{C}}$

Self Similarity of Julia Sets

2.3

When is generated by a rational function , we know that if all the criticalpoints are in the immediate attractive basin of a fixed point, then the Julia set
is a Cantor set. Now we consider about the folloing situation similar to that.
$G$

$f$

be a finitely generated rational semigroup.
Theorem 2.7 Let $G=(f_{1},$$\deg(f_{1})$
at
two
and
is least
Assume thatfor each such that $\deg(f_{j})$ is one, the$P(G)$
is included in a connected component of $F(G)$ ,map is not elliptic. If,. and mappings
then there are simply connected domains$W$ such that$j,$metrichyperbolic
map
the
a
each
is
to
for
from
contraction from to some , with the contraction rate bounded by a constantandstrictly less than one throughout$f_{n}\rangle$$\ldots,$

$j$

$U$

$f_{j}$$V_{1},$$\ldots,$$i$$W= \bigcup_{j}V_{j}$$h_{1},$$V_{k}$$h_{\epsilon}$$\ldots$$h_{j}$$V_{:}$$V_{*}$$V_{i}$$J(G)\subset W,$$\bigcup_{j}h_{j(j(G))=J()}G$

Example 2.1 Let $G_{c}=(z^{2}+c,$
sufficiently large positive number .

$z^{2}+ci\rangle$. Then.

$J(G_{\mathrm{c}})$is a Cantor set for$c$3Hyperbolicity and Strong Open Set Condition

be a finitely generated rational semiDefinition 3.1 Let
group. We say that satisfies strong open set condition if and only if there isis included in andan open neighborhood of $J(G)$ such that each set$G=\langle f_{1}, f2, \ldots f_{1}1\rangle$

$G$

$O$

is

$mutually\backslash$$f_{j}^{-1}(O)$$O$

disjoint.

The Julia set of rational semigloup may have interior points in general. For, itexample, the Julia set of $(z^{2},2z)$ is the closure of the unit disc.
super
was shown that if is a finitely generated rational semigroup, then each
attracting point of ally element of $g\in G$ does not belong to the boundary of
the Julia set. So we can construct many examples such that the Julia set has
interior points. Here we show a sufficient condition such that the Julia set has$\ln[\mathrm{H}\mathrm{M}2]$$G$

no interior points.

103
be a finitely generated rational semiTheorem 3.1 Let $G=(f_{1},$group. We assume that for each $(i,j)$ such that $i\neq j$ the set $f_{1}^{-1}.(J(G)\cap$$f_{j}^{-1}(J(G))$ has at most countable points. Then the Julia set $J(G)$ has no interi or points.
$f_{2},$$\ldots f_{\hslash}\rangle$

be a component ofAssume $J(G)$ has its interior points and let
Proof
is dense in . So if
int $(J(G))$ . From Lemma 1.1.2,(int $J(G)$ ) is dense in andis not empty, thenof int $(J(G))$ . In this way, we can take a sequenceis included in a component. such that for each the number is in $(1, \ldots, n)$ and
$U$

$U$

$U \mathrm{n}\bigcup_{j=1}\hslash f_{j}^{-1}(\mathrm{i}\mathrm{n}\mathrm{t}J(c))$$U$$U\cap f_{\dot{2}1}^{-}1$$U\cap f_{i_{1}}^{-1}(\mathrm{i}\mathrm{n}\mathrm{t}J(c))$$f_{i_{1}}(U)$$U_{1}$$k$$(i_{k}.)_{k}$$i_{k}$$f_{i_{k^{\circ\cdots\circ}}}f_{1}.1(U)\subset Uk$,

of int $(J(G))$ . Now let
be a sequence of elements
where . is a
, thenof . If the sequence contains infinite elements of
is anormal family on . If
contains a subsequencesuch that for each the, then for each the set
map does not belong to the sequence {is a normal family on . So is included
is included in $F(G)$ and so
in $F(G)$ and this is a contradiction.
$U_{k}$$(g_{j})$$\mathrm{C}\mathrm{O}\ln_{\mathrm{P}^{\mathrm{o}\mathrm{n}\mathrm{e}\mathrm{n}\mathrm{t}}}$$G$$(g_{j})$$(f_{i_{k}}\mathrm{o}\cdots \mathrm{o}fi1)$$U$$(h_{l})$$(g_{j})$

$l$

$l$

$f_{1}k\mathrm{o}\cdots \mathrm{o}fi1)$$h_{l}$$h_{l}.(U)$$U$$(g_{j})$$U$$\mathrm{O}$be a polynomial semigroup. We denote by $K(G)$ theDefinition 3.2 Let
such that for each $z\in Ii_{1}’(c)$ there is a sequenceclusure of a set
consisting of mutually distinct elements ofand the sequence $(g_{m}(z))_{m}$ isbounded. $K(G)$ is called the filled-in Julia set of .
$G$$(g_{m})_{m}$$I\mathrm{i}_{1}’(c)$

$G$

$G$

the inverse image $g^{-1}(K(G))$ is included in $K(G)$Remark For$J(G)\subset
K(G)$$G=\{f_{1},$. Ifand) is a finitely generated polynomial
semigroup, then$\mathrm{e}\mathrm{a}\mathrm{c}1_{1}g\in G$$f_{2},$ $\ldots f_{n}$$K(G)=\cup^{n}f^{-}j(K(G))j=11$.

) be a finitely generated polynomial semiTheorem 3.2 Letgroup. Assume that for each $(i,j)$ such that $i\neq j$ the set $f_{i}^{-1}(J(c)\cap f_{j}^{-1}(J(G))$
has at most countable points. Then
$G=\langle f_{1},$$f_{2},$ $\ldots f_{n}$$\partial(K(c))=J(G)$

Proof

.

By remark and a similar argument in the proof of Theorem 3.1.

$\square$Now we consider about the expanding property of hyperbolic rational semi-groups.
be a finitely generated hyperbolic rational
Theorem 3.3 Let $G=(f_{1},$semigroup. Assume that $\deg(f_{1})$ is at least two and for each $g\in G$ such that$\deg(g)$ is one the map.
is not elliptic. Let $K$ be a compact subset
such that for each
Then there is a positive number and is a number$f_{2},$$\ldots f_{\hslash}\rangle$$of\overline{\mathbb{C}}\backslash P(c)$$g$$\lambda>1$

$c$

$k$

$\inf\{||(f_{i_{k^{\mathrm{O}}}}\cdots \mathrm{O}f:1)’(z)|||z\epsilon(f|.k^{\circ}\ldots \mathrm{o}fi_{1})^{-1}(I\iota’), (ik, \ldots, i1)\in(1, \ldots, n)^{k}\}$.
, here we denote $by||||$ the hyperbolic metric on an open subset$W$
is a simply connected domain which is a relative compact
In particular, if
and $A$ is a family of maps on $W$ such that each element
subdomain
where is an element of $G$ , then eachof $A$ is a well defined branch oflimit function of $A$ on $W$ is a constant function such that the constant value isin $J(G)$ .
$\geq c\lambda^{k}$$of\overline{\mathbb{C}}\backslash P(c)$$of\overline{\mathbb{C}}\backslash P(c)$$h$$g^{-1}$

$g$

104
Proof
We take a relatively compact open subset ofsuch that
contains $K\cup J(G)$ alld for each element of the inverse image $g^{-1}(V)$ isincluded in . We take the hyperbolic metric in each component of . From [S3],on $F(G)$ is a constant function such that the constantevely lilnit function ofvalue is in $P(G)$ . So for each large number the closure ofis included
in whereis any element of in the form
. So the inclusion
map fromtosatisfies that $||i’(\approx)||<1$ for each $z\in g_{k}(-1V)$ fromthe Schwartz lemma, where we denote by
the hyperbolic metric on $g_{k}(-1V)$ .
The nlap
is a covering map fromto and is an isometry between thehyperbolic metric onand . So $||g_{k}’(Z)||>1$ for each $z\in g_{k}^{-1}(V),$
we denote bythemetric on .
$V$

$V$

$\overline{\mathbb{C}}\backslash P(G)$

$G$

$g$

$V$

$V$

$G$

$g_{k}^{-1}(V)$

$k$

$V$

$G$

$g_{k}$$V$$g_{k}^{-1}(V)$$i$$f_{1}k\mathrm{o}\cdots \mathrm{o}f_{i_{1}}$$||||$$V$$g_{k}^{-1}V$$g_{k}$$V$$g_{k}^{-1}V$$\mathrm{w}\mathrm{h}\mathrm{e}\mathrm{r}\coprod \mathrm{e}$$||||$$V$$1_{1}\mathrm{y}\mathrm{p}\mathrm{e}\mathrm{r}\mathrm{b}_{0}1\mathrm{i}\mathrm{C}$By Theorems 3.1, 3.3, we get the following result.

Theorem 3.4 Let $G=(f_{1},$be a finitely generated hyperbolic rationalsemigroup satisfying strong open set condition. Then $m(J(G))=0$ , where wedenote by $m$ the Lebesgue measure on .
$f_{2},$$\ldots f,.\rangle$$\overline{\mathbb{C}}$4$\delta$-Conformal Measure

We construct -conformal mbasure on Julia sets of rational semigroups.
measure on Julia sets of rational function is considered in [Sul].
$\delta$$\delta$con-$\mathrm{f}_{\mathrm{o}\mathrm{r}\mathrm{l}\mathrm{n}\mathrm{a}}1$$f2,$
Theorem 4.1 Let) be a finitely generated hyperbolic rationalsemigroup satisfying strong open set condition. We assume that when is equal
to one the degree ofis at least two. And assume that $\infty\in F(G)$ . Letbethe open set in Definition 3.1. Then there is a number $0<\delta\leq 2$ and there is a
probability measure such that the support of the measure is included in $J(G)$and if is a measurable set included insuch thatis injective on ,$G=\langle f_{1},$$\ldots fn$

$n$

$O$

$f_{1}$

$\mu$

$A$

$f_{j}^{-1}(O)$$\mu(f_{j(A))=}\int_{A}|f_{j}’(z)|^{\delta}d\mu$say that a probability measure issatisfies the above. And we set
Also$u\prime e$$\delta(G)=\inf${$\delta\geq 0|$

there is a$\delta$$\delta$$A$$f_{j}$.

-confomal if and only if the measure-confomal measure on $J(G)$ }.

Then $\delta(G)>0$ .
Theorem 4.2 Let) be a finitely generated hyperbolic rationalsemigroup satisfying strong open set condition. We assume that when is equal
to one the degree ofis at least two. And assume that $\infty\in F(G)$ . Let bea number satisfying that $0<\delta\leq 2$ and assume that there is a -conformal
measure on $J(G)$ . Thenand
$G=\langle f_{1},$$f_{2},$ $\ldots f_{n}$$n$$\delta$$f_{1}$$\delta$$\delta=\delta(G)$$\mu$$\mathrm{d}\mathrm{i}_{\ln_{H}}(j(G))=\delta(G),$where$\mathrm{d}\mathrm{i}_{\mathrm{l}}\mathrm{n}_{H}$$0<H_{\delta(G)(}J(G))<\infty$

is the Hausdorff dimension and$H_{\alpha}$

is the$\alpha$,-Hausdorff measure.

By Theorem 3.4, Theorem 4.1 and Theorem 4.2, we get one of main results.
Corollary 4.1 Let
be a finitely generated hyperbolic rationalsemigroup satisfying strong open set condition. We assume that when is equal
to one the degree of is at least two. Then
$G=\langle f_{1}, f_{2}, \ldots f_{\hslash}\rangle$$n$$f_{1}$$0<\dim_{H}(j(G))<2$.

105
And if we set $\alpha=\dim(J(G))$ , then$0<H_{\alpha}(J(c))<\infty$.

be a finitely generated hyperbolic rational
Theorem 4.3 Let
semigroup satisfying strong open set condition. We assume that when is equal
to one the degree of is at least two. Let be the number in Theorem 3.3 when
$K=J(G)$ . Then
$G=\langle f_{1},$ $f_{2,\ldots f_{n}\rangle}$$n$$\lambda$$f_{1}$$\dim_{H}(J(c))\leq\frac{\log(\sum_{j}\deg(f_{j}))}{\log\lambda}$.

$n(z-$
Example 4.1 Let be a positive integer such that $n\geq 4$ . We set$4)+4\rangle$ . Then
is a finitely generated hyperbolic rational semigroup satisfying
strong open set condition and$G=\langle z^{n},$

$n$

$G$

$1 \leq\dim_{H}J(c)\leq\frac{\log(n+1)}{\log(n)}$5.

Invariant Measure

We introduce some notations and results from [L]. Let be a bounded operator
in the complex Banach space . The operator is called almost periodic if theconditionally compact. The
isof any vector
orbit
eigenvalue and related eigenvector are called unitary if $|\lambda|=1$ . The set of. We denoteunitary eigenvectors of the operator will be denoted by
the closure of the linear span of the unitary eigenvectors of the operatorby. And we set$B0=\{\varphi|A^{m}\varphiarrow 0(marrow\infty)\}$ ,
$A$

$A$

$B$

$\varphi\in B$$\{A^{m}\varphi\}_{m=1}\infty$$\mathrm{s}\dot{\mathrm{t}}\mathrm{r}\mathrm{o}\mathrm{n}\mathrm{g}\mathrm{l}\mathrm{y}$$\lambda$$A$$\mathrm{s}_{\mathrm{P}}\mathrm{e}\mathrm{c}_{u}A$$B_{u}$$A$

here the convergence is assumed to be strong.

: $Barrow B$ is an almost periodic operator in the complex
Theorem 5.1 If
Banach space , then
$A$

$B$

$\mathcal{B}=B_{u}\oplus \mathcal{B}0$.

Corollary 5.1 Let $A:Barrow B$ be an almost periodic operator in the complex
Banach space B. Assume that $spec_{u}A=\{1\}$ and the point $\lambda=1$ is a simple
eigenvalue. Let $h\neq 0$ be an invariant vector of the operator A. Then there exists$\mu(h)=1$ , such that
invariant functionalan$\mu\in B^{*},$$A^{*}$$A^{m}\varphiarrow\mu(\varphi)h$$marrow\infty$.

Proof [L].

$\square$

We now construct invariant llleasures on Julia sets of hyperbolic rational
be a finitely generated rational semigroup.
semigroups. Let
by $C(K)$ all continuous complex valued
denotewecompactset
For eachisfunctions on . It is a Banach space with suprelnum norm. Assume thatset
backward invariant under . For each and for each element we$G=\langle f_{1},$ $f_{2,\ldots f_{n}\rangle}$$K\mathrm{o}\mathrm{f}\overline{\mathbb{C}}$$IC$$I\iota’$

$G$

$j$

$\varphi$$(A_{j\varphi})(Z)= \frac{1}{\deg(f_{j})}\zeta\in f\sum_{j}\varphi(\zeta)-1(z)$ ’

any point of . Then
operator on $C(K)$ . We set$\mathrm{w}\mathrm{h}\mathrm{e}\mathrm{r}\mathrm{e}\approx \mathrm{i}\mathrm{s}$$IC$$A_{j\varphi}$

is an element of$C(I\zeta)$

and$\mathcal{W}=\{(a_{1}, \ldots, a_{\mathrm{r}}.,)\in \mathrm{R}^{n}|\sum_{j}a_{j}=1, a_{j}\geq 0\}$.

$A_{j}$

is a bounded.

$\cdot$106
And for eacll$a\in \mathcal{W}$

we set.

$(B_{a} \varphi)(\approx)=j1\sum_{=}^{n}a_{j(A_{i^{\varphi}}})(_{Z})$Then$B_{a}$

is a bounded operator on $C(K)$ .

be a finitely generated hyperbolic rational
Theorem 5.2 Let
be aanelementhas
semigroup. Assumeof degree at least two. Let$a:\neq
0$
elliptic
that
is
not
an
and
point satisfying that there is a number such
. Then there is a probability measure on such that forelement ofeach compact set $Ii’$ included in$G=\langle f_{1}, f_{2}, \ldots f_{n}\rangle$$G$$a\in \mathcal{W}$$i$$f_{i}$$\overline{\mathbb{C}}$$Aut(\overline{\mathbb{C}})$$\mu_{a}$$\overline{\mathbb{C}}\backslash P(G)$$marrow\infty$$||B_{a}^{m}\varphi-\mu a(\varphi)1||_{K}arrow 0$,(1)where we denote by 1 the constant function taking its value 1 and $||||K$ issupremum norm on $Ii’$. Also

supp$(\mu_{a})=J(\langle f_{i}1’\ldots, f_{i}k))$,where $\{i_{1}, \ldots, i_{k}\}=\{j|a_{j}\neq 0\}$.

To prove Theorem 5.2, we need the following two lemmas.
Lemma 5.1 If is backward invariant compact subset
is an almost periodic operator on $C(K)$ .
$I\mathrm{i}’$Lemma 5.2 Let$B_{a}\varphi=\lambda\varphi,$$a$be backward invariant compact subset of, thenand is constant. That is,$I\mathrm{i}’$$|\lambda|=1$$of\overline{\mathbb{C}}\backslash P(G)$$a$$\lambda=1$, then$\overline{\mathbb{C}}\backslash P(G)$$B_{a}$. If$(C(K))_{u}=\overline{\mathbb{C}}\cdot 1$$\varphi$.

Proof of Lemma 5.1 By Ascoli Arzela theorem it is sufficient to show that foris equicontinuous on $K$ because
each element $\varphi\in C(K)$ the family
is uniformly bounded.
for each and so the family
Let be a point of $K$ and let be a simply connected open neighborhood of. Then for each $g\in G$ we can take well defined branches ofin$g\in G$ } is normal ononon . The family { $S|$ a branch ofis.
equicontinuous on . So
Proof of Lelllma 5.2 Let be a point of such that$\{B_{a}^{m}\varphi\}_{m}$$m$$||B_{a}^{m}\varphi||_{K}\leq||\varphi||_{K}$$\{B_{a}^{m}\varphi\}_{m}$

$U$

$z$

$\mathrm{i}\mathrm{n}\mathrm{c}\mathrm{l}\mathrm{u}\mathrm{d}\mathrm{e}\mathrm{d}$$-\mathrm{l}$$z$$\overline{\mathrm{c}}\backslash P(c)$

$U$

$g$

$g^{-1}$$U,$$U\mathrm{a}\mathrm{n}\mathrm{d}$$U$$\{B_{a}^{m}\varphi\}_{m}$$\mathrm{e}\mathrm{q}\mathrm{u}\mathrm{i}_{\mathrm{C}\mathrm{O}}11\mathrm{t}\mathrm{i}\mathrm{n}\mathrm{U}\mathrm{o}\mathrm{u}\mathrm{S}$$I\mathrm{i}^{r}$$z$$|\varphi(Z)|=$$\sup_{\prime,w\epsilon I\mathrm{i}}|\varphi(w)|$

.

Then
$=$

$|\varphi(Z)|$$=$$|(B_{a}\varphi)(Z)|$$| \sum_{j}a_{j}(Aj\varphi)(z)|$$\leq$$\sum_{j}a_{j^{\frac{1}{\deg(f_{j})}}}\zeta\in f_{j}\sum_{z()}|\varphi(\zeta)-1|$$\leq$$\sum_{j}a_{j}|\varphi(z)|=\varphi(Z)$tllen
So if is a point of, thenis a point ofsuch that foris a sequence$f_{i}^{-1}(Z),$$\zeta$$f_{i}^{-r}(Z)$$(\zeta_{m})_{m}$.

. Since if
and so. Now for each point (of $J((f_{i}))$ there
andbelongs to
the point$|\varphi(\zeta)|=|\varphi(z)|$$\varphi(\zeta)=\lambda^{r}\varphi(z)$$\mathrm{e}\mathrm{a}\mathrm{c}1_{1},?1$$\zeta_{m}arrow\zeta$$\varphi(\zeta)=\lambda\varphi(z)$$\zeta_{m}$.

$f_{i}^{-m}(Z)$$\zeta$107
Then$marrow\infty$$\lambda^{m}\varphi(\approx)=\varphi(\zeta_{m})arrow\varphi(\zeta)$So$\lambda=1$. Now we$\mathrm{w}\mathrm{i}\mathrm{U}$show that$\varphi$

is constant. We put$B_{a}(\Re_{\varphi)\Re}=\varphi,$

Let

$z$

be a point of$I\iota’$$B_{a}(\Im_{\varphi)\Im}=\varphi$.
$\varphi=\Re\varphi+i\Im\varphi$. Then.

such tllat$| \Re\varphi(z)|=\sup_{w\in I\mathrm{f}}|\Re\varphi(w)|$.

By a similar argument we can show thatfor each. Let
point ofbe. Let
be a sequence such that for each $m$ thepointbelongs toand. Thenso.
In the same way we can show that if is the minimum point of the function, then, where is any point of. So is constantby the salne argument
is also constant. Whence is constant.
$\varphi(\zeta)=\varphi(z)$$J((f\dot{|}\rangle)$$\zeta$$\mathrm{a}\mathrm{l}\tau \mathrm{y}$$f_{:}^{-m}(\approx)$$\zeta_{m}$$\zeta\in f_{\dot{8}}^{-1}(Z)$$(\zeta_{m})_{m}$$\varphi(z)=\varphi(\zeta)$$\varphi(\zeta_{m})arrow\varphi(\zeta)$$\zeta_{m}arrow\zeta$$x$$\varphi(x)=\varphi(\zeta)$$\Re\varphi$$J((f:\rangle)$$\zeta$$\Re\varphi$$\mathrm{a}\mathrm{n}\mathrm{d}\square$$\Im\varphi$$\varphi$

Proof of Theorem 5.2 we can assume that for each $j,$
. By Lemma 5.1,
Lemma 5.2, and Corollary 5.1, if is a backward invariant compact subset oftllen there is a probability measure
on $K$ sucll that for each$a_{j}\neq 0$$I\iota’$$\overline{\mathbb{C}}\backslash P(G),$$\mu_{a,K}$$\varphi\in C(K)$$||B_{a}^{m}\varphi-\mu a,K(\varphi)1||_{K}arrow\infty$$marrow\infty$.

(2)

We consider ,as a probability measure on . Thenis independent of
which is backward invariant under
and is included inbecause$J(G)$ is included in $K$ by Lemlna 1.2.5 and (2). For eachwe
. is a probability measureput.
. Now let be any
compact subset of. There is a compact subset $K$ of
which
contains and is backward invariant under . Then by (2),$\overline{\mathbb{C}}$$\mu_{a,K}$$\mu_{a,K}$$G$$I\iota^{-}$$\overline{\mathbb{C}}\backslash P(G)$$\varphi\in C(\overline{\mathbb{C}})$$1_{1}\mathrm{o}\mathrm{l}\mathrm{d}\mathrm{s}$$\mu_{a}(\varphi)=\mu_{a,K}(\varphi)$$L$$\mathrm{o}\mathrm{n}\overline{\mathbb{C}}$$\mathrm{T}1_{1\mathrm{e}}\mathrm{n}\mu_{a}$$\overline{\mathbb{C}}\backslash P(G)$$\overline{\mathbb{C}}\backslash P(G)$

$L$

$G$

$marrow\infty$$||B_{a\varphi-}^{m}\mu_{a}(\varphi)1||_{L}arrow\infty$

Now we will show that. To prove
So suppeach
for each$\mathrm{s}\mathrm{u}\mathrm{p}\mathrm{p}\mu_{a}=J(G)$$\mu_{a}.\subset J(G)$$\approx\in J(G)\mathrm{a}\mathrm{l}\mathrm{l}\mathrm{d}$. If we set $K=J(G)$ , then$\mathrm{s}\mathrm{u}\mathrm{p}\mathrm{p}\mu_{a}\supset J(G)$$\varphi\in C(\overline{\mathbb{C}})$.

such that$\int_{\overline{\Phi}}\varphi d\mu_{a}>0$$\mu_{a}=\mu_{a,K}$.

it is sufficient to show that for$\varphi\geq 0,$$\varphi(z)>0$.

We set$U=\{\zeta\in j(c)|\varphi(\zeta)>0\}$.

By Lellmla 1.2.6, there is a
is a repelling fixed pointsuch thatof included inof an element $g\in G$ . Then there is an open neighborhood
is included. When $\deg(g)$ is at least two, thensuch that$P(G)\subset
p(G)$
in
and
$\mathrm{p}\mathrm{o}\mathrm{i}\mathrm{n}\mathrm{t}\approx 0\in U$$z_{0}$$U_{0}$$U$$g(U_{0})\supset U_{0}$$z_{0}$$E(\langle g\rangle)$$J(G) \subset\bigcup_{m=1}g^{m}(U0)\infty$.

Whence there is a positive integer $N$ such that for each $\zeta\in J(G)$$g^{-N}(\zeta)\cap U\neq\emptyset$.

(3)

When $\deg(g)$ is equal to one if we change to some another element of , wethat (3) holds for an element $g\in G$ and for an integer $N$. Now for
call
$G$

$g$

$\mathrm{s}\mathrm{l}_{\mathrm{l}\mathrm{O}\mathrm{W}}$$\mathrm{e}\mathrm{a}\mathrm{c}11\approx\in j(G)$$B_{a}^{N}\varphi(\approx)>0$,

108
so there is a positive llunll)er such that for each $z\in J(G)$$c$$B_{a}^{N}\varphi(z)\geq C$$\mathrm{W}\mathrm{l})\mathrm{e}\mathrm{n}\mathrm{C}\mathrm{e}$.

for each integer $m\geq N$ and for each $z\in J(G)$$B_{a}^{m}\varphi(z)\geq C$,

and so$\int\varphi d_{l^{\iota}a}\geq c>0$.
$0$Theorem 5.3 Let
Then

1.

$/t_{a}.= \sum a_{j}A_{j}^{*}\mu a$$\overline{\mathbb{C}}\backslash P(G)$2. Letandbe the probability measure constructed in Theorem 5.2.

$\mu_{a}$. And if is a probability measure$on\overline{\mathbb{C}}$$\mu$$\mu=\sum a_{j}A_{j}^{*}\mu$, Then $\mu=\mu_{a}$ .

such that supp$\mu\subset$

be a point ofand assume that there is an integer such thatand is not an elliptic element of. Then the map
is continuous at urith respect to the weak topology.
$b$$a:\neq 0$$i$$\mathcal{W}$$Aut\overline{\mathbb{C}}$$f_{i}$$arightarrow\mu_{a}$$b$3. Letbe a point. If there is an integerj such thatis at least two, thenis non atomic.
$a\in \mathcal{W}$$a_{j}\neq 0$

and $\deg(f_{j})$$\mu_{a}$Theorem 5.4 Let be a point of . Assume that there is a number
thatandis not elliptic element of. Then$\mathcal{W}$$a$$a_{j_{0}}\neq 0$1.

$j_{0}$

such$Au\varpi$$f_{j_{0}}$$/\iota_{a}(f_{j}^{-}1(J(G)))\geq a_{j}$, for each number .
$j$

2. Assume that for each $(i,j)$ such that $i\neq j,$ $f_{i}^{-1}(j(G))\cap f_{j}^{-1}(J(G))=\emptyset$ .
Then $\mu_{a}(f_{j}^{-1}(j(G)))=a_{j}$ , for each number .
$j$S. Assume that there is a number such that .
and $\deg(f_{k})\geq 2$ . Also
assume that for each $(i,j)$ such that $i\neq j$ , the set $f_{i}^{-1}(J(G))\cap f_{j}^{-1}(J(G))$
has at most countable points. Then $\mu_{a}(f_{j}^{-1}(J(G)))=a_{j}$ , for each number
$k$,

$j$

$a_{k}$$\neq 0$.

Theorem 5.5 In the same assumption as Theorem 5.2,

1. if for each number the map
is not elliptic element ofand foreach $(i,j)$ such that $i\neq j$ the set $f_{i}^{-1}(J(G))\cap f_{j}^{-1}(J(G))$ is empty, thenthe map$i$$Aut\overline{\mathbb{C}}$$f_{i}$$arightarrow\mu_{a}$is topological embedding from
into the space of all probability measures
on with $fes_{l)ect}$ to the weak topology.
$\mathcal{W}$$\overline{\mathbb{C}}$

2. if

and for each $(i,j)$ such that $i\neq j$ the set $f^{-1}.\cdot(J(G))\cap$has at most countable points, then the map$G\cap Aut\overline{\mathbb{C}}=\emptyset$$f_{j}^{-1}(J(G))$$arightarrow\mu_{a}$is topological embedding from
into the space of all probability measures
on with respect to the weak topology.
$\mathcal{W}$$\overline{\mathbb{C}}$Proof

By 2., 3. of Theorem 5.4.

$\square$109
Theorem 5.6 Let $M$ be a complex manifold. Suppose for each $j=1,$$\ldots$

,

$n$

the

map

$(z, u)\in\overline{\mathbb{C}}\cross Mrightarrow f_{j,u}(z)\in\overline{\mathrm{c}}$. And we assume that for every
is holomorphic. We set $G_{u}=(f_{1,u},$$d_{j}.=\deg(f_{j},u)$ is constant independentof . Let be a point of M. Assume
is a hyperbolic rational semigroup not included inthatand each $g\in$
is a hyperbolic element. Let be a point of . Then there is an
open neighborhood of in $M$ such that for each $u\in V$ we can construct thein Theorem 5.2 with respect to the hyperbolic rational
probabilty measure
and the map
semigroup$\cdots,$$f_{\hslash,u}\rangle$

$j,$

$u$

$v$

$Aut\overline{\mathbb{C}}$$G_{v}$$G_{v}\cap Aut\overline{\mathbb{C}}$$\mathcal{W}$

$a$

$V$

$v$

$\mu_{a,u}$$G_{u}$.

$urightarrow l^{\iota_{a,u}}$is continuous from to the space of all probability measures onto the weak topology.
$V$$\overline{\mathbb{C}}$with respect

Now we construct invaliant measures on Julia sets of rational semigroups
satisfying strong open set condition.
Theorem 5.7 Let $G=(f_{1}, f_{2}, \ldots f_{n})$ be a finitely generated rational semigroupsatisfying strong open set condition. When $n=1$ , we assume thatis not. Let be an open set in Definition 3.1. Then for each
elliptic element ofon such that for each compact subset
there is a probability measure
which is backu’ard invariant under. of$f_{1}$$O$$Aut\overline{\mathbb{C}}$$a\in \mathcal{W}$$I\iota’$$\overline{\mathbb{C}}$$\mu_{a},$

$G$

$O$

$||B_{a}^{m}\varphi-l‘\cdot a(\varphi)1||_{K}arrow 0,$

u’here$\varphi\dot{r}s$

any element of$C(\overline{\mathbb{C}})$$marrow\infty$,. Alsosupp $(\mu_{a})=J(\langle fi1’\ldots, fi_{k}))$ ,u’here $\{i_{1}, \ldots, i_{k}\}=\{j|a_{j}\neq 0\}$. Also$\mu_{a}=\sum_{=j1}^{n}ajA^{*}\mu j.a$and the map$a\mapsto\mu_{a}$into the space of all probability measures on
is a topologacal embedding from$ect$
topology.
to
the
weak
with resp.
$\mathcal{W}$$\overline{\mathbb{C}}$The proof is similar to that of Theorem 5.2 so we only have toProofis equicontinuous on $Ii’$ for each $\varphi\in C(K)$ .
that the family$\mathrm{s}\mathrm{h}\mathrm{o}\mathrm{w}\square$$\{B_{a}^{m}\varphi\}_{m}$be a finitely generated rational semigroup
Theorem 5.8 Let $G=(f_{1},$is notsatisfying strong open set condition. When $n=1$ , we assume that
. Let be a point of W. Assume that there is a number
elliptic element of
is non atomic.
and $\deg(f_{j})\geq 2$ . Thensuch that$f_{2,\ldots f_{n}\rangle}$$f_{1}$$Aut\overline{\mathrm{C}}$$j$6$a_{j}\neq 0$$a$$\mu_{a}$Estimate of Hausdorff Dimension of Julia sets

Using invariant measures in Theorem 5.7, we get one of main results which givesa lower estimate of Hausdorff dimension of the Julia set of a rational semigroupwhich satisfies strong open set condition.

110
Theorem 6.1 Let $G=(f_{1}, f_{2}, \ldots f_{n})$ be a finitely generated rational semigroupis notsatisfying strong open set condition. When $n=1$ , we assume that$\infty\in
F(G)$
we
set
and
that
. Assume
elliptic element of$f_{1}$$Aut\overline{\mathbb{C}}$$M=$$\max$$\max$$|f_{j}’(z)|$.

$j=1,\ldots n1z\in f_{J}-1(j(G))$

Then$\dim_{H}J(c)\geq\frac{\log(\sum_{j_{-}1}^{n}-\deg(f_{j}))}{\log M}$

where

,

denote by $\dim_{H}$ the Hausdorff dimension.

$u\prime e$

Let $\mu=\mu_{a}$ be the probability measure constructed in Theorem 5.7

Proof
where

$a_{j}= \frac{\deg(f_{j})}{\sum_{i=1}^{n}\deg(f_{i})}$.

We fix a number satisfying$t$$0<t< \frac{\log(\sum_{j_{-}1}^{n}-\deg(f_{j}))}{\log M}$

and we take a$\mathrm{n}\mathrm{u}\mathrm{m}\mathrm{l}$)$\mathrm{e}\mathrm{r}$$a$such that$t<a< \frac{\log(\sum_{j_{-}1}^{n}-\deg(f_{j}))}{\log M}$.

the neighborhood
be a slnall number and for each we denote byLet$z\in f_{j}^{-1}(J(G))$
each
for
and
of $f_{j}^{-1}(J(G))$ . We take small such that for eacll$j$$\epsilon>0$$J_{j,\epsilon}$$\epsilon$$j$$\epsilon$$|f_{j}’(Z)|<( \sum_{j=1}^{n}\deg(f_{j}))^{\frac{1}{l}}$.

We set$C= \bigcup_{j=1}^{n}$and{$\prime \mathrm{c}\mathrm{r}\mathrm{i}\mathrm{t}\mathrm{i}\mathrm{C}\mathrm{a}1$points of$f_{j}$}$C’=c\cup J(G)$ .

we can as sulne that.

$C’=\cup jn=1J_{j},\epsilon\cap C$

We fix any positive integer . Because G. satisfies strong open set condition,
there is no super attracting fixed point of any element of in $J(G)$ . So there
alud if $w\in C’$ is a critical point of all
is a positive number such thatelement $g\in G$ of word length $l<p$ , then$\mathrm{p}$$G$$\eta<\epsilon$$\eta$$|w-g(w)|>\eta$.

Also there is a positive nunuber such that $\rho<q2$ and for eachpoint $z\in D(w, \rho)$
point of , then for$\rho$

$j$

if

$w$

is a critical$\mathrm{e}\mathrm{a}\mathrm{c}1_{1}$$f_{j}$$|f_{j}’(z)|<1$

the$\delta<e2$ and for eachLet)

$C_{\rho}’1$$\mathrm{e}$.

of $C’.$ Tllel.e is a positive number such thatis injective on
and for each the map$\delta$$\rho \mathrm{n}\mathrm{e}\mathrm{i}\mathrm{g}\mathrm{h}\mathrm{b}_{0}1^{\cdot}\mathrm{h}\mathrm{o}\mathrm{o}\mathrm{d}$$z\in J(G)\backslash C_{\rho}’$$j$$f_{j}$

111. Now letbe any point of $J(G)$ alldtake a positive integer such that

$D(z, \delta)$$\zeta$$r$

be. a small positive nu.mber. We$s$.

$\delta((\sum_{j=1}\deg(fj))*n)^{-\epsilon-}11<?\cdot<\delta((\sum^{n}j=1\deg(f_{j}))*)1-\epsilon$

There is the unique element $g\in G$ of word lengtll such that $g(\zeta)\in J(G)$ . We
call as sulne that $\deg(f_{1})=\max_{j}\deg(f_{j})$ . Then the equation $g(z)=g(\zeta)$ has at$s$$\mathrm{n}\mathrm{l}\mathrm{o}s\mathrm{t}$( $\sum_{\mathrm{j}}\hslash-22$
degt ) $),1$$(\deg(f1))=1$

roots in$D(\zeta, f\cdot)$$f\mathrm{j}$counting multiplicities. Then for each$\mu^{g(\zeta)}\theta+m(D(\zeta, ’\cdot))$$\leq$

Let$\uparrow?l\cdotarrow\infty$$\frac{(\deg(f1))^{(}\sum^{n}\mathrm{j}_{--\mathrm{l}}\mathrm{e}2\mathrm{d}\mathrm{g}(f\mathrm{j})-2)\frac{}{p}}{(\sum_{j=1}^{n}\deg(fj))\epsilon}$.
$\frac{(\deg(f1))^{(}\sum_{\mathrm{j}\mathrm{l}}*--2\deg(f_{\mathrm{j}})-2)\overline{\mathrm{p}}}{(\sum_{j=1}^{n}\deg(fj))\epsilon}$$\leq$$( \sum_{1j=}^{n}\deg(f_{j}))^{-}\epsilon(1-\cdot\frac{\dot{\sum}^{n}--1\mathrm{z}\mathrm{d}_{l}(\text{ノ}\mathrm{j}^{)-}2}{p}.)$$\leq$$p$.

and we get$\mu(D(\zeta, ’\cdot))$

If we take.

$\frac{(\sum_{j=1}^{n}\deg(f_{j}))^{m}\cdot(\deg(f1))^{(}\sum^{n}j--1-22\deg(f.\mathrm{j}))\frac{}{p}}{(\sum_{j=1}^{n}\deg(f_{j}))^{\epsilon+m}}$.
$=$$m\in \mathbb{N}$.

such that$a(1- \frac{\sum_{j=1}^{n}2\deg(f_{j})-2}{l}))>t$,

then we get$l^{\iota(D(\zeta},$$r \cdot))\leq(\sum_{=j1}\deg n(fj))^{-\frac{t}{a}}.$.

HellCe(4)$\mu(D(\zeta, ’\cdot))\leq(\frac{r}{\delta})^{t}(j=\sum_{1}^{n}\deg(f_{j}))^{\frac{t}{l}}$and the statement of our theorem follows.

$\square$Next we consider the case a hyperbolic rational semigroup does not satisfy
strong open set condition but atisq open set condition and $J(G)$ is finitely
raniified. Ill this case we also consider the lower es timate of Hausdorff dimensionof the Julia set. Here we get the next interesting example in which the Julia set
Sierpitski Gasket.
to
is
$G$

$s$

$\mathrm{s}\mathrm{i}_{1}\mathrm{n}\mathrm{i}\mathrm{l}\mathrm{a}\mathrm{r}$$\mathrm{t}1_{1}\mathrm{e}$Example 6.1 Let $G=(f_{1},$$f_{1}(z)=z^{2},$$G’$$f_{2},$$f_{3}\rangle$

where$f_{2}(z)=3(Z-4)+4,$$f\mathrm{a}(z)=3(z-4i)+4i$

is hyperbolic. For small positive number we set$c$$f_{1,\mathrm{c}}(\approx)=z^{2},$$f_{2,\mathrm{c}}(z)=(3+C)(\approx-4)+4,$$G_{\mathrm{c}}=(f_{1,\mathrm{c}},$$f_{2,\mathrm{c}},$.
$f_{3,c}\rangle$

where$f_{3,\mathrm{c}}(z)=(3+C)(_{Z}-4i)+4i$

.

112
Then

$G_{c}$

satisfies.

is also hyperbolic.

$G$does not satisfy strong open set condition but
and by Theorem 6.1,

$G_{\mathrm{c}}$$J(G_{\mathrm{c}})=J((\{f_{i},c^{\mathrm{O}}fj,c\}_{i,j}\rangle)$$\dim_{H}J(G_{c})$$\geq$$\frac{\log 16}{\log\max_{i,j\in(f}\max_{z\mathrm{c}^{\circ f).(J(c))}}-1|\mathfrak{i},\mathrm{j}.\mathrm{c}(f_{1c}\mathrm{o}f_{j,c})’(z)|}.$,$\underline{\log 16}carrow 0$$arrow$$\log 12$

By

’

.

, the map $crightarrow J(Gc)$ is continuous with respect to the Haus-$Theorem\mathit{2}.\mathit{6}.\mathit{1}$dorff metric. And by Theorem 5.6 and (4) in Theorem 6.1, we get$\mathrm{d}\mathrm{i}\mathrm{n}1_{H}J(G)\geq\frac{\log 16}{\log 12}$.

References[Bel] A.F.Beardon, Symmetries of Julia sets, Bull.London.Math.Soc.22(1990), 576-582.
[Be2] A.F.Bealdoll, Iteration of Rational Functions, Springer-Verlag GTM132.

[D] A. Douady, Does a Julia set depend continuously on the $Polynomial^{q}$, Complex Dynan)ic al System, The Mathematics behind the Malldelbrot andJulia Sets, (editor R. Devaney), p91-p138, Proceedings of Symposia
in Applied Mathematics, vol.49, AMS.

[HM1] A.Hinkkallen, G.J.Martin, The dynamics of semigroups of rational functions I, Proc.London Math.Soc. (3) $73(1996)$[HM2] A.Hinkkanen,
appear$\mathrm{G}.\mathrm{J}.\mathrm{M}\mathrm{a}\mathrm{r}\mathrm{t}\mathrm{i}_{1}1$, Jilia sets of rational semigroups , Math.Z. to[L] M.J.Lyubich, Entropy Properties of Rational Endomorphisms of the Riemann Sphere, Ergod.th.&Dynam.Sys. (1983), 3, 351-385[MTU] S.Morosawa, M. Tanigudli and T. Ueda A Primer on Complex Dynamics, (Japanese velsiou; Baihuukan, 1995) English version, in preparation.
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241-273.

[Sul] D.Sullivan, Conformal Dynamical System, in Geometric Dynamics,
Springer Lecture Notes 1007(1983), 725-752.
[S1] H.Sumi,

On Limit Functions and Continuity of Julia Sets of Rational
Semigroups, In T. Sugawa, editor, RIMS Kokuroku 959: Complex Dynamics and Related Problems, page s 59-72. Kyoto Univ., 1996.

[S3] H.Sumi, On Dynamics of Hyperbolic Rational Semigroups, preprint.
[S4] H.Sumi, On Hausdorff Dimension of Julia Sets of Hyperbolic Rationalsemigroups, preprint.

[S5] H.Sumi, Invariant measures on Julia Sets of Rational Semigroups, preprint.