数理解析研究所講究録 1269 巻 2002 年 143-166

143

Porosity of Julia sets of semi-hyperbolic fiberedrational maps and rational semigroups
Hiroki Sumi
Department of Mathematics,
Tokyo Institute of Technology,
2-12-1, Oh-0kayama, MegurO-ku, Tokyo, 152-8551, Japane-mail ; sumi@math.titech.ac.jp
Abstract
We consider fiber-preserving complex dynamics on fiber bundles
whose fibers are the Riemann spheres and whose base spaces are compact metric spaces. We define the semi-hyperbolicity of dynamics onfiber bundles. We will show that if adynamics on fiber bundle is semihyperbolic, then we have that the fiberwise Julia sets are k-porous andthat the dynamics has akind of weak rigidity. We also show that theJulia set of rational semigroup(semigroup generated by rational mapson ) which is semi-hyperbolic except at most finitely many points in
the Julia set and satisfies the open set condition is porous or is equal tothe closure of the open set. Note that if aset in is porous thenthe upper Box dimension of the set is less than $2-c(k)$ where $c(k)$is apositive constant depending only on . Further we get an upper
estimate of the Hausdorff dimension of the Julia set.
$\overline{\mathbb{C}}$$J$$\overline{\mathbb{C}}$

$k$

$J$

$k$

1Introduction
To investigate random one-dimensional complex dynamics, dynamics of semigroups generated by rational maps on the Riemann sphere and fiberpreserving holomorphic dynamics on fiber bundles which appear in complex
dynamics in in several dimensions, we consider the dynamics of fibered rar
tional maps, that is, fiber-preserving complex dynamical systems on fiber
bundles whose bers are the Riemann spheres and whose base spaces are
general compact metric spaces. The notion of dynamics of fibered rationalmaps, which was ageneralized notion of ‘dynamics of fibered polynomial
maps’ by O.Sester([Sel], [Se2], [Se3]), was introduced by M.Jonsson in [J2].
The research on dynamics of semigroups generated by rational maps on
the Riemann sphere ([HM1], [HM2], [HM3] [GR], [Bo], [Stl], [St2], [St3],[SI], [S2], [S3], [S4], [S5] , the research of random iterations of rational
functions([FS], [BBR]) and the research on polynomial skew products on$\overline{\mathbb{C}}$$\dot{\mathrm{f}}\mathrm{i}$

$)$

144
are directly related to this subject. For the research$\mathbb{C}^{2}([\mathrm{H}1], [\mathrm{H}2], [\mathrm{J}1])$of polynomial skew products (dynamics of fibered polynomials) whose basespaces are general compact metric spaces, see O.Sester’s works [Sel], [Se2]and [Se3]. In [Se3] he investigated the quadratic case in detail. In particular,
he developed acombinatorial theory for quadratic fibered polynomials andconstructed an abstract space of combinatorics. Moreover he showed somereadability and rigidity for an abstract combinatorics.

1.1

Notations and definitions

Definition 1.1. ([J2]) Atriplet
1.

$\mathrm{Y}$2.

$\pi:\mathrm{Y}arrow X$$(\pi,\mathrm{Y},X)$

is called a-bundle’if$‘\overline{\mathbb{C}}$and $X$ are compact metric spaces,

is acontinuous and surjective map,

3. There exists an open covering $\{U\}$ of $X$ such that for each there exists a::satisfying that $i
is a Mobius map for::each, under the identification.
$i$$\mathrm{h}\mathrm{o}\mathrm{m}\infty \mathrm{m}\mathrm{o}\mathrm{r}\mathrm{p}\mathrm{h}\mathrm{i}\mathrm{s}\mathrm{m}\Phi$$\overline{\mathbb{C}})=\pi^{-1}(x)\mathrm{m}\mathrm{d}$$x\in U_{\dot{1}}$$\Phi_{j}^{-1}0\Phi$$U_{\dot{1}}$$\cross\overline{\mathbb{C}}arrow\pi^{-1}(U_{\dot{1}})$$\{x\}$$\cross\overline{\mathbb{C}}arrow\{x\}$$\cap Uj$$(\{\mathrm{x}\}\cross$$\cross\overline{\mathbb{C}}$$\{x\}$$\cross\overline{\mathbb{C}}\cong\overline{\mathbb{C}}$

Remark: By the condition 3, each fiber
has acomplex structure. We also have that given $x_{0}\in X$ we may find acontinuous family:of homeomorphisms for close to . Such a family
will becase‘local parameterization’. Since $X$ is compact, we may assume that
there exists acompact subsetof the set of Mobius transformations ofsuch thatfor any two local parametrizatios
and. In
this paper we always assume that.
Moreover in this paper we always assume the folowing condition:
there exists asmooth $(1, 1)$ form
inducing ametric on
andis continuous. That is, if
is alocal parametrization,then the pull backis apositive smooth form on
dependingcontinuously on .
$\mathrm{Y}_{x}:=\pi^{-1}(x)$$i_{x}$$\overline{\mathbb{C}}arrow \mathrm{Y}_{x}$$x$$\{i_{x}\}$$\mathrm{x}\mathrm{o}$$\mathrm{a}$$M_{0}$$\overline{\mathbb{C}}$$i_{x}\circ j_{x}^{-1}\in M_{0}$$\{i_{x}\}$$\{j_{x}\}$$\omega_{x}>0$$\bullet$$x$$\vdasharrow\omega_{x}$$\mathrm{Y}_{x}$$\{i_{x}\}$$i_{x}^{*}\omega_{x}$$\overline{\mathbb{C}}$$x$

Definition 1.2. Let
be a -bundle. Let
and $g:Xarrow$$X$ becontinuous maps. We say that is afibered rational map over (orarational map fibered over ) if$(\pi, \mathrm{Y}, X)$$\overline{\mathbb{C}}$$f:\mathrm{Y}arrow \mathrm{Y}$

$f$

$g$

$g$

1.

2.

$\pi \mathrm{o}f=g\mathrm{o}\pi$:

is rational map for any $\in X$ .That is,
is arational map ffom to itself for any local parametrization$\in X$ andat $g(x)$ .
$f|_{\mathrm{Y}_{\mathrm{f}}}$$\mathrm{Y}_{x}arrow \mathrm{Y}_{g(x)}$$x$$(i_{g}.)^{-1}\mathrm{o}f\mathrm{o}i_{x}$$\overline{\mathbb{C}}$

$x$

at

$i_{g(x)}$Notation:then we put

If:

is afibered rational map over $g:Xarrow X$,
for any $\in X$ and. Furthermore we put$=\deg(f_{x}^{n})$ and $d(x)=d_{1}(x)$$\inX$
for any
and.
$f$$\mathrm{Y}arrow \mathrm{Y}$$f_{x}^{n}=f^{n}|\mathrm{Y}_{l}$$d_{n}(x)$$i_{x}$

$x$

$n$

$x$

$\in \mathrm{N}$$n$$\in \mathrm{N}$145
is afibered ra-bundle. Let :beDefinition 1.3. Let
tional map over : $Xarrow X$ . Then for any $x\in X$ we denote bysatiswhich has aneighborhood of in
the set of points
is anormal family in , that is, $y\in F_{x}$ if and only iffying that) isof rational maps on ( denotes
the family:note that by remark in the definition of -bundle, thisnormal neardoes not depend on the choices of local parametrizations at and . Stillof mappings
is the open subset of where the familyequivalently,.
into is local equicontinuous. We put
fromFurthermore, we put$\mathrm{Y}arrow \mathrm{Y}$$f$$\mathrm{a}\overline{\mathbb{C}}$$(\pi, \mathrm{Y}, X)$$F_{x}(f)\langle \mathrm{s}\mathrm{i}\mathrm{m}\mathrm{p}\mathrm{l}\mathrm{y}$

$g$

$U$

$y\in \mathrm{Y}_{x}$$F_{x})$$y$$\mathrm{Y}_{x}$$U$$\{f_{x}^{n}\}_{n\in \mathrm{N}}$$g^{n}(x)$$\overline{\mathbb{C}}$$Q_{x}^{n}=i_{x_{n}}^{-1}\circ f_{x}^{n}\circ i_{x}$$x_{n}$$\overline{\mathbb{C}}$$i_{x}^{-1}(y)$$x$$F_{x}$$\mathrm{Y}_{x}$$\{f_{x}^{n}\}$$\mathrm{Y}_{x}$$J_{x}(f)(\mathrm{s}\mathrm{i}\mathrm{m}\mathrm{p}1\mathrm{y}J_{x})=\mathrm{Y}_{x}\backslash F_{x}$$\mathrm{Y}$$\tilde{J}(f)=\overline{\cup J_{x}x\in X}’\tilde{F}(f)=\mathrm{Y}\backslash \tilde{J}(f)$and$x_{n}$$\hat{J}_{x}(f)(\mathrm{s}\mathrm{i}\mathrm{m}\mathrm{p}1\mathrm{y}\hat{J}_{x})=\tilde{J}(f)\cap \mathrm{Y}_{x}$,for each $x\in X$ .

satysfying that
Remark 1. There exists afibered rational map :is NOT compact.
We give some notations and definitions on dynamics of rational semi$f$$\mathrm{Y}arrow \mathrm{Y}$$\bigcup_{x\in X}J_{x}$groups,For aRiemann surface , let End(S) denote the set of all holomorphic endomorphisms of . It is asemigroup with the semigroup oper\^ation being composition of maps. Arational semigroup is asubsemigroup of
without any constant elements. We say that arational semigroupis apolynomial semigroup if each element of is apolynomial. The researches on dynamics of rational semigroups were started by A.Hinkkanen
and GJ.Martin ([HM1]), who were interested in the role of dynamics ofpolynomial semigroups in the research of various one-complex-dimensional
moduli spaces for discrete groups, and F.Ren’s group([GR]).
$S$

$S$

$\mathrm{E}\mathrm{n}\mathrm{d}(\overline{\mathbb{C}})$

$G$

$G$

Definition 1.4. Let$F(G)=$ {$z\in\overline{\mathbb{C}}|G$$G$

be arational semigroup. We set

is normal in aneighborhood of },$z$$J(G)=\overline{\mathbb{C}}\backslash F(G)$.

is called the Fatou set for and $J(G)$ is called the Julia set for .
The backward orbit $G^{-1}(z)$ of and the set of exceptional points $E(G)$ aredefined by: $G^{-1}(z)= \bigcup_{g\in G}g^{-1}(z)$ and $E(G)=\{z\in\overline{\mathbb{C}}|\# G^{-1}(z)\leq 2\}$ . For, we set $G^{-1}(A)= \bigcup_{g\in}cg^{-1}(A)$ . We denote by
any subset A.
the rational semigroup generated by the family
is genbe a rational semigroup and assume
Lemma 1.5 ([S4]). Let
erated by a precompact subset Aof End(C). Then
$F(G)$

$G$

$G$

$z$

$\langle h_{1}, h_{2}, \ldots\rangle$$\mathrm{o}\mathrm{f}\overline{\mathbb{C}}$$\{h:\}$$G$$G$$J(G)=\cup f^{-1}(J(G))=\cup h^{-1}(J(G))f\in\Lambda h\in\overline{\Lambda}$.

In particular if Ais compact then we have $J(G)= \bigcup_{f\in\Lambda}f^{-1}(J(G))$ .
We call this property the backward self-similarity of the Julia set

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Remark 2. By the backward self-similarity, the research on the Julia setsof rational semigroups may be considered as akind of generalization of theresearch on self-similar sets constructed by some similitudes ffom to itself,
which can be regarded as the Julia sets of some rational semigroups. It iseasily seen that the Sierpinski gasket is the Julia set of arational semigroupwhere $h_{i}(z)=2(z-p:)+p_{\dot{1}},i=1,2,3$ withbeing aregular triangle.
$\mathbb{C}$$G=\langle h_{1},h_{2}, h_{3}\rangle$$p_{1}p_{2}p_{3}$Example 1.6.
Let

1. ([S4].) Let ,,be non-constant rational maps.
be the space of one-sided infinite sequences of$m$ symbols and:be the shift map: that is, is definedby $g((w_{1},w_{2}, \ldots))=(w2,w_{3}, \ldots)$ . Let $X$ be acompact subset ofsuch that $g(X)\subset X$. Letand :be the naturalprojection. Then
is a -bundle. Let :be amap
defined by: $f((w, y))=(g(w), h_{w_{1}}(y))$ . Then :is afiberedrational map over $g:Xarrow X$.
In the above if $X=\Sigma_{m}$ then we say that :is the fiberedrational map associated with the generator systemof the rational semigroup,, ) . Then by Proposition 3.2
in [S5](See also \S 8:N0te in [S7]) we have$h_{1}$$\ldots$$h_{m}$$\Sigma_{m}=\{1, \ldots,m\}^{\mathrm{N}}$$\Sigma_{m}arrow\Sigma_{m}$

$g$

$g$

$\Sigma_{m}$$\mathrm{Y}=X\cross\overline{\mathbb{C}}$$(\pi, \mathrm{Y}, X)$$\mathrm{Y}arrow X$$\pi$$\overline{\mathbb{C}}$

$f$

$f$

$f$

$\mathrm{Y}arrow \mathrm{Y}$$\mathrm{Y}arrow \mathrm{Y}$$\mathrm{Y}arrow \mathrm{Y}$$\{h_{1}, \ldots h_{m}\}$$G=\langle h_{1}$$\ldots$$h_{m}$$\pi_{\overline{\mathbb{C}}}(\tilde{J}(f))=J(G)$where$\pi_{\overline{\mathbb{C}}}$:$\mathrm{Y}arrow\overline{\mathbb{C}}$,is the projection. See [S4] for more details.

2. Letbe a ruled surffice over a Riemann sur face $X$ :that is,isasmooth projective variety of complex dimension 2which is also aholomorphic-bundle over $X$ . Every
has aunique conformal
structure and apositive form
, where is the K\"ahler formon Y. Letbe the projection. Then
is a.
Dabija [D] showed that (almost) every holomorphic selfmap of isafibered rational map over aholomorphic map $g:Xarrow X$ .
$\mathrm{Y}$$\mathrm{Y}$$P^{1}(\mathbb{C})$$\mathrm{Y}_{x}$$\omega_{x}=\omega|\gamma_{ae}$$\omega$$\pi:\mathrm{Y}arrow X$$(\pi, \mathrm{Y}, X)$$\overline{\mathbb{C}}\mathrm{b}\mathrm{u}\mathrm{n}\mathrm{d}\mathrm{l}\mathrm{e}$$f$3. Let$f$:

be a polynomial with degree at least two andapolynomial of the form: $q(x,y)=y^{n}+a_{1}(x)y^{n-1}+\cdots$ . Let
be amap defined by$p(x)\in \mathbb{C}[x]$$\mathbb{C}[x,y]$$\mathrm{Y}$$\mathrm{q}(\mathrm{x},$$\in$$\mathbb{C}^{2}arrow \mathbb{C}^{2}$$f((x,y))=(p(x), q(x,y))$ .

This is called apolynomial skew product in . Dynamics of maps ofthis form were investigated by S.-M.Heinemann in [H1] and [H2] andby M.Jonsson in [J1].
Let $X$ be acompact subset of such that $p(X)\subset X$. (e.g. the Julia
set of ) Let
be a trivial -bundle. Then the map$\tilde{f}((x,y))=(p(x),q(x,y))$defined by
is afibered rational
map over $p:Xarrow X$ .
$\mathbb{C}^{2}$$\overline{\mathbb{C}}$$p.$$\tilde{f}:\mathrm{Y}arrow \mathrm{Y}$$(\pi, \mathrm{Y}=X\cross\overline{\mathbb{C}}, X)$$\overline{\mathbb{C}}$147
Notation :$\bullet$Let
andbe two topological spaces and :For any subset of Z2, we denote by $c(g, A)$ the set ofcomponents of $g^{-1}(A)$ .
$Z_{2}$$Z_{1}$$g$$Z_{1}arrow Z_{2}$$A$$\bullet$$\mathrm{a}\mathbb{I}$be amap.
connected

for any
and $>0$ , we put $B(y, \delta)=\{y’\in\overline{\mathbb{C}}|d(y,y’)<\delta\}$ ,weandwhere is the spherical metric. Similarly, for any$y\in\overline{\mathbb{C}}$$\delta$$d$$\delta>0$$y\in \mathbb{C}$

put $D(y, \delta)=\{y’\in \mathbb{C}||y-y’|<\delta\}$ .
$\bullet$Let$(\pi, \mathrm{Y},X)$

be a -bundle. For any$\overline{\mathbb{C}}$$y\in \mathrm{Y}$and $r>0$ we set$\tilde{B}(y,r)=\{y’\in \mathrm{Y}_{\pi(y)}|d_{\pi(y)}(y’,y)<r\}$where for each $x\in X$ we denote by
the form .

$d_{x}$,

the metric on$\mathrm{Y}_{x}$induced by$\omega_{x}$Now we define the semi-hyperbolicity of fibered rational maps.

Definition 1.7. (semi-hyperbolicity on fibered rational maps) Let:
be afibered rational map overbe a -bundle. Let: $Xarrow X$ . Let. We denote by $SH_{N}(f)$ the set of points
satisfying that there exists apositive number , aneighborhood of,
in such that for any $x\in U$, any
and alocal parametrization, we have
any $x_{n}\in g^{-1}(x)$ and any $V\in c(i_{x}(B(i_{\pi(z)}^{-1}(z), \delta)),$$(\pi, \mathrm{Y}, X)$$f$$\overline{\mathbb{C}}$$\mathrm{Y}arrow \mathrm{Y}$$z\in \mathrm{Y}$$N\in \mathrm{N}$

$g$

$U$

$\delta$$\{i_{x}\}$$U$$\pi(z)$$n\in \mathrm{N}$$f_{x}^{n})$$\deg(f_{x}^{n} :Varrow i_{x}(B(i_{\pi(z)}^{-1}(z), \delta)))\leq N$.

We set$UH(f)=\mathrm{Y}\backslash \cup SH_{N}(f)N\in \mathrm{N}^{\cdot}$

Apoint $z\in SH_{N}(f)$ is called asemi-hyperbolic point of degree $N$. We say. This isthat is semi-hyperbolic (along fibers) if.
for some
equivalent to$\tilde{J}(f)\subset\bigcup_{N\in \mathrm{N}}SHN(f)$$f$$\tilde{J}(f)\subset SH_{N}(f)$$N\in \mathrm{N}$

Similarly we define the semi-hyperbolicity on rational semigroups.
Definition 1.8. (semi-hyperbolicity on rational semigroups) Let
be arational semigroup and $N$ apositive integer. We denote by $SH_{N}(G)$the set of points
satisfying that there exists apositive number suchthat for any $g\in G$ and any $V\in c(B(z, \delta)$ , $g)$ , we have$G$$z\in\overline{\mathbb{C}}$$\delta$$\deg(g :Varrow B(z, \delta))\leq N$.

. Apoint 26 $SH_{N}(G)$ is
Further we set
called asemi-hyperbolic point of degree $N$. We say that is semi-hyperbolic. This is equivalent to $J(G)\subset SH_{N}(G)$ for some
if
$UH(G)= \overline{\mathbb{C}}\backslash (\bigcup_{N\in \mathrm{N}}SH_{N}(G))$$G$$—J(G) \subset\bigcup_{N\in \mathrm{N}}SH_{N}(G)$148
xample 1.9.
$X$ . We set

1. Let$f$:$\mathrm{Y}arrow \mathrm{Y}$be arational map fibered over$P(f)=\cup\cup f_{x}^{n}$( critical points of$f_{x}$$g$: $Xarrow$).

$n\in \mathrm{N}x\in X$

This is called the fiber post critical set of fibered rational map . If:is hyperbolic along fiberes: that is,, thenis semi-hyperbolic along fiberes with the constant $N=1$ .
$f$

$f$

$\mathrm{Y}arrow \mathrm{Y}$$P(f)\subset \mathrm{p}(\mathrm{f})$$f$2. In Corollary 6.7 of [Se3] O.Sester showed that any ‘non-reccurent
quadratic fibered polynomials’ with connected fiberwise filled-in Julia sets are semi-hyperbolic.
be non-constant rational functions on . Let $f$ :be the fibered rational map in Example 1.6.1. By easy arguments
we can show that :is semi-hyperbolic along fiberes if andonly if is semi-hyperbolic.
In [S4], if is afinitely generated rational semigroup, then asufficientcondition to be semi-hyperbolic for apoint $z\in J(G)$ was given, which
gives ageneralization of’s work([Ma]). Further in [S4], thefollowing statement was shown: Assume that there exists an elementof with the degree at least two, that each element of Aut
this is not empty) is loxodromic and that
. Then is semihyperbolic if and only if all of the following conditions are satisfied.

3. Let$\{h_{1}, \ldots, h_{m}\}$$\overline{\mathbb{C}}$$\mathrm{Y}arrow$$\mathrm{Y}$$f$$\mathrm{Y}arrow \mathrm{Y}$

$G$

$G$

$\mathrm{R}.\mathrm{M}\mathrm{a}\tilde{\mathrm{n}}\acute{\mathrm{e}}$$G$$\overline{\mathbb{C}}\cap \mathrm{G}(\mathrm{i}\mathrm{f}$$G$$J(G)\neq\overline{\mathbb{C}}$(a) for each $z\in J(G)$ there exists aneighborhood $U$ ofin suchthat for any sequence $(g_{n})\subset G$ , any domain
in and any
point ( $\in U$, we have that the sequence
does NOT converge
to locally uniformly on(b) for each $j=1$ ,, $m$ each $c\in C(f_{j})\cap J(G)$ satisfies
$V$

$z$

$\overline{\mathbb{C}}$$\overline{\mathbb{C}}$$(g_{n})$$V$$\zeta$$\ldots$$d(c, (G\cup\{id\})(f_{j}(c))).>0$

From this fact it was shown in [S4] that if we assume that there exists
an element of with the degree at least two, that each element ofAut(if this is not empty) is loxodromic, that there is no super
attracting fixed point of any element of in $J(G)$ and $F(G)\neq\emptyset$ ,then is semi-hyperbolic.
By this theorem we know that $G=(z^{2}+2, z^{2}-2)$ is semi-hyperbolic.
This is NOT hyperbolic. See [S4].
$G$$\overline{\mathbb{C}}\cap G$

$G$

$G$

We need some technical conditions.

finition 1.10 (Condition(Cl)). Let
be a. Let :
be arational fibered over : $Xarrow X$. We say that satisfies theidition (C1) if there exists afamilyof topological disks with
, $x\in X$ such that the following conditions are satisfied$(\pi, \mathrm{Y},X)$$arrow \mathrm{Y}$

$f$

$g$

$\{D_{x}\}_{x\in X}$$\subset$$\mathrm{Y}_{x}$$\overline{\mathbb{C}}\mathrm{b}\mathrm{u}\mathrm{n}\mathrm{d}\mathrm{l}\mathrm{e}$

$f$

149
1. for each x $\in X$ there exists apoint,such that$z_{x}\in \mathrm{Y}_{x}$and apositive number$r_{x}$$D_{x}=\tilde{B}(z_{x}, r_{x})$2.

$\overline{\bigcup_{x\in X}\bigcup_{n\geq 0}f_{x}^{n}(D_{x})}\subset\tilde{F}(f)$,3. for any $x\in X$ , we have that$\mathrm{d}\mathrm{i}\mathrm{a}\mathrm{m}(f_{x}^{(n)}(D_{x}))arrow 0$, as $narrow\infty$ , and

4. $\inf_{x\in X}r_{x}>0$ .
be a -bundle. Let :Definition 1.11 (Condition(C2)). Let
be afibered rational map over $g:Xarrow X$ . We say that satisfies
the condition (C2) if for each $x_{0}\in X$ there exists an open neighborhood, $x\in O$ suchof topological disks withand afamilyofthat the folowing conditions are satisfied:$f$$\overline{\mathbb{C}}$$(\pi, \mathrm{Y}, X)$$f$$\mathrm{Y}arrow \mathrm{Y}$$O$$D_{x}\subset \mathrm{Y}_{x}$$\{D_{x}\}_{x\in O}$$x_{0}$

1. for each $x\in O$ there exists apoint,such that$z_{x}\in \mathrm{Y}_{x}$and apositive number$r_{x}$$D_{x}=\tilde{B}(z_{x}, r_{x})$2.

$\overline{\bigcup_{x\in O}\bigcup_{n\geq 0}f_{x}^{n}(D_{x})}\subset\tilde{F}(f)$,3. for any $x\in O$ , we have that4.

$x\mapsto\succ D_{x}$$\mathrm{d}\mathrm{i}\mathrm{a}\mathrm{m}(f_{x}^{(n)}(D_{x}))arrow 0$, as$narrow\infty$, andis continuous in $O$ .

be non-constant rational functions
1. LetExample 1.12.
$\deg(h_{1})\geq
2$
be the fibered rational map as. Let :on withsociated with the generator system $\{h_{1}, \ldots, h_{m}\}$ of rational semigroup, which is described in Example 1.6.1. Suppose thatis notis semi-hyperbolic along fibers and that
equal to the Riemann sphere. Then we have that satisfies the condition (C2). Actually, there exists an attracting fixed point of someelement of $G$ in $F(G)$ . Since $G$ is semi-hyperbolic, we have that setting$D_{x}=D(a, \epsilon)$ for eachwhere is apositive number, satis. For more
fies the condition (C2) with the family of disks
details, see Theorem 1.35 and Remark 5in [S4].
$\{h_{1}, \ldots h_{m}\}$$f$$\overline{\mathbb{C}}$$\mathrm{Y}arrow \mathrm{Y}$$G=\langle h_{1}, \ldots, h_{m}\rangle$$\pi_{\overline{\mathbb{C}}}(\tilde{J}(f))=J(G)$

$f$

$f$

$a$

$x\in\Sigma_{m}$$f$$\epsilon$$(D_{x})_{x\in\Sigma_{m}}$be abe atrivial -bundle. Let :2. Letfibered rational map over : $Xarrow X$ satisfying that is apolynomial
mapping of degree at least two for each $x\in X$ . Then setting $D_{x}=D$
where $D$ is asmal neighborhood of infinity for each $x\in X$ , thefibered rational map satisfies the condition (C2) with the family of.
disks$\overline{\mathbb{C}}$$(\pi, \mathrm{Y}=X\cross\overline{\mathbb{C}}, X)$

$g$

$f$

$\mathrm{Y}arrow \mathrm{Y}$$f_{x}$$f$$(D_{x})_{x\in X}$

We give the definition of ‘conical’ set in the Julia set.
Definition 1.13. (conical set for fibered rational maps) Let
be afibered rational map over $g:Xarrow X$ .
be a -bundle. Let :$(\pi, \mathrm{Y}, X)$$\overline{\mathbb{C}}$$f$$\mathrm{Y}arrow \mathrm{Y}$

150
Let N
and r $>0$ . We denote byN,r) the set of points z
satisfying that for any, there exists apositive integer n such thatthe element U
r),containing z satisfies the following
conditions:$\tilde{J}_{cm}(f,$$\in \mathrm{N}$$\in\tilde{J}(f)$$\epsilon>0$$\in c(\tilde{B}(f^{n}(z),$1. diam2.

$U<\epsilon$$f^{n}|\mathrm{Y}_{\pi(z)})$,is simply connected, and$U$

3. $\deg(f^{n} : Uarrow\tilde{B}(f^{n}(z),r))\leq N$.
We set$\tilde{J}_{cm}(f,N)=\bigcup_{t>0}\tilde{J}_{\mathrm{c}m}(f,N,r)$and$\tilde{J}_{\mathrm{c}m}(f)=\bigcup_{N\in \mathrm{N}}\tilde{J}_{\mathrm{c}m}(f, N)$.

Definition 1.14. (conical set for rational semigroups) Let be arational semigroup. Let
and $>0$ . We denote by $J_{em}(G,N,r)$ theset of points $z\in J(G)$ satisfying that for any, there exists an ele
$g\in
G$
ment
such that $g(z)\in J(G)$ and the element $U\in c(B(g(z),’), g)$containing satisfies the following conditions:$G$$N\in \mathrm{N}$$r$$\epsilon>0$$z$

1. diam2.

$U$$U<\epsilon$,

is simply connected, and

3. $\deg(g:Uarrow B(g(z), ’))\leq N$.
We set $J_{cm}(G, N)= \bigcup_{r>0}J_{eon}(G, N,r)$ and$J_{\omega n}(G)= \bigcup_{N\in \mathrm{N}}J_{eon}(G,N)$.

Definition 1.15. (good points for fibered rational maps) Let
be a -bundle. Let :be afibered rational map over : $Xarrow X$ .
We set$(\pi, \mathrm{Y},X)$$\overline{\mathbb{C}}$$f$$\mathrm{Y}arrow \mathrm{Y}$$g$$\tilde{J}_{good}(f)=\{z\in\tilde{J}(f)|\lim_{narrow}\sup_{\infty}d(f^{n}(z), UH(f))>0\}$.

Definition 1.16. (good points for finitely generated rational semigroups) Let ( ,,be arational semigroup. Let:
be the fibered rational map associated with the generator system$h_{1}$$\ldots$$h_{m}\rangle$$f$$\Sigma_{m}\cross\overline{\mathbb{C}}arrow$$\Sigma_{m}\cross\overline{\mathbb{C}}$. Then we set. Note that this definition does not depend on the choice of any generator system of which
consists of finitely many elements.
$\{h_{1}, \ldots, h_{m}\}$$J_{gM}(G)=\pi_{\overline{\mathbb{C}}}(\tilde{J}_{good}(f))$$G$

2Results on Fibered Rational Maps
In this section we state some results on dynamics of fibered rational maps
which are deduced by semi-hyperbolicity, except Theorem 2.6. The proofs
are given in \S 4.

Theorem 2.1. (measure zero) Letbe-bundle. Let :be a fibered rational map over : $Xarrow X$ . Suppose all of the follow $ing$
conditions:$(\pi, \mathrm{Y}, X)$$\mathrm{Y}$$g$$a\overline{\mathbb{C}}$$f$$\mathrm{Y}arrow$

151
1. f satisfies the condition (Cl),

2. for each $x\in X$ , the boudary of $\hat{J}_{x}(f)\cap UH(f)$ inpoints in ,$\mathrm{Y}_{x}$does not separate$\mathrm{Y}_{x}$3.

$\tilde{J}(f)\backslash \bigcup_{n\in N}f^{-n}(UH(f))\subset\tilde{J}_{good}(f)$andand each open neighborhood of indoes not tend to zero as
we have that the diameter of

4. for each$V$$z\in\tilde{J}(f)\cap UH(f)$$z$$narrow\infty$$f_{\pi(z)}^{n}(V)$

Then

.

and for each $x\in X$ , the 2-dimensional Lebesgue
is equal to zero.

$\tilde{J}(f)=\bigcup_{x\in X}J_{x}$measure of$\mathrm{Y}_{\pi(z)}$$J_{x} \backslash \bigcup_{n\in N}f^{-n}(UH(f))$

be ametric space. Let be aconstant with
Definition 2.2. Let$0<k<1$ . Let be asubset of Y. We say that is -porous if for any $x\in J$
and any positive number there exist aball in $\{y\in \mathrm{Y}|d(y, x)<r\}\backslash J$$k$$(\mathrm{Y}, d)$

$J$

$J$

$k$

$r$

with the radius at least $kr$ .

and is the Euclidean metric,
Remark 3. If is the Euclidean spaceis less than $n-c(k, n)$ ,the Box dimension of any -porous bouded set in.
where $c(k, n)$ is apositive constant which depends only on andbebundle Let :be
Theorem 2.3. (porosity) Let
fibered rational map over $g:Xarrow X$ . Suppose that satisfies the conditionis semi-hyperbolic. Then there exists a constant with(Cl) and that$0<k<1$ such that is -porous in for each $x\in X$ . In particular, there
exists a constant $0\leq c<2$ such that for each $x\in X$ ,$d$$\mathbb{R}^{n}$$\mathrm{Y}$

$J$

$k$

$\mathbb{R}^{n}$$k$$(\pi, \mathrm{Y}, X)$$f$$a\overline{\mathbb{C}}$$n([\mathrm{P}\mathrm{R}])$$\mathrm{Y}arrow \mathrm{Y}$

$a$

$f$

$k$

$f$

$k$

$J_{x}$$\mathrm{Y}_{x}$$\dim_{H}(J_{x})\leq\dim_{B}(J_{x})\leq c$,

where $\dim_{H}$ denotes the Hausdorff dimension and $\dim_{B}$ denotes the Box
is the form ininduced bydimension with respect to the metric on
the remark in Definition 1.1).
$\omega_{x}(\omega_{x}$$\mathrm{Y}_{x}$be two -bundles.
andTheorem 2.4. (a rigidity) Let
be a fibered rational map over $g:Xarrow X$ andLet :be a homeomorphism. Let
fibered rational map over :. satisfies thatwhich is a bundle conjugacy be tween and$\tilde{f}\circ u=u\circ f$ . Suppose that
$XarrowX$andfor some homeomorphism :is semi-hyperbolic along fiberes and satisfies the condition (C1). For each$w\in X$ , let
be the restriction of . Let $x\in X$ be a point.
:
we have, for each
is $K$ -quasiconformal on
Then if
is $K$ -quasiconformal on the whole .
:that$(\tilde{\pi},\tilde{\mathrm{Y}},\tilde{X})$$(\pi,\mathrm{Y}, X)$$\tilde{f}:\tilde{\mathrm{Y}}:arrow\tilde{\mathrm{Y}}a$$\mathrm{Y}arrow \mathrm{Y}$$f$$\overline{\mathbb{C}}$$u:\mathrm{Y}arrow\tilde{\mathrm{Y}}$$\tilde{X}arrow\tilde{X}$$\tilde{g}$$\tilde{f}:i.e$$f$$\tilde{\pi}ou=v\circ\pi$

$u$

$f$

$v$

$u_{w}$$a\in\overline{\bigcup_{n\in}\mathrm{z}\{g^{n}(x)\}}$$F_{x}$$u_{x}$$u_{a}$$u$$\mathrm{Y}_{w}arrow\tilde{\mathrm{Y}}_{v(w)}$$\mathrm{Y}_{a}$$\mathrm{Y}_{a}arrow\tilde{\mathrm{Y}}_{v(a)}$Definition 2.5. Let be apositive number. Let $K$ be aclosed subset of. We say that $K$ is -uniformly perfect if for any doubly connected domain
in satisfying that separates $K$ i.e. both two connected components(the modulus of .
have non-empty intersections with $K$,ofless than .
For the definition, see
$C$

$C$

$\overline{\mathbb{C}}$$A$$\overline{\mathbb{C}}$$A$$\mathrm{m}\mathrm{o}\mathrm{d} A$$\overline{\mathbb{C}}\backslash A$$[\mathrm{L}\mathrm{V}])\mathrm{i}\mathrm{s}$

$C$

$A$

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Remark 4. Uniform perfectness implies many good properties
This term was introduced in [Po]. In [Su], there is asurvey on uniform perfectness.

$([\mathrm{B}\mathrm{P}],[\mathrm{P}\mathrm{o}],[\mathrm{S}_{1}$Theorem 2.6. (uniform perfectness Let
be-buundle. Let
be a fibered rational map over $g:Xarrow X$ satisfying that $d(x)\geq 2$
for any $x\in X$ . Then we have the following.
$(\pi,\mathrm{Y}, X)$$a\overline{\mathbb{C}}$$f:\mathrm{Y}arrow \mathrm{Y}$1. There exists a positive constant $C$ such that for any $x\in X$ ,that
andare $C$ -unifomly perfect$J_{x}$

2. Suppose further
is an open map).

$f(\tilde{F}(f))\subset\tilde{F}(f)$$(f_{\pi(z)}^{n})’(z)=0$

interior oftoe have$\hat{J}_{x}$If a point

for some$\hat{J}_{\pi(z)}$(for example, assume that$z\in \mathrm{Y}$: $Xarrow X$
satisfies that $f_{\pi(z)}^{n}(z)=z$ andand, thenwith respect to the topology of.
$n\in \mathrm{N}$$z\in\hat{J}_{\pi(z)}$

$z$

$g$

belongs to the$\mathrm{Y}_{\pi(z)}$3Results on Rational Semigroups
In this section we state some results on dynamics of semigroups generated
by rational functions on the Riemann sphere. The proofs are given in \S 4.

Definition 3.1. Let
be afinitely generated rationalsemigroup. Let
be an open set in C. We say thatsatisfies the open
set condition with with respect to the generator systems
if for each $j=1$ ,, $m$ ,
and $\{h_{j}^{-1}(U)\}j=1,\ldots,m$ are mutually
disjoint.
$G=\langle h_{1}, h_{2}, \ldots, h_{m}\rangle$

$U$

$G$

$U$

$\ldots$$\{h_{1},h_{2}, \ldots, h_{m}\}$$h_{j}^{-1}(U)\subset U$

Theorem 3.2. (porosity) Let
be a rational semigroupwith an element of degree at least two. Suppose all of the following conditions;$G=\langle h_{1}, \ldots, h_{m}\rangle$

1.

$G$

satisfies the open set condition with an open set

the generator system$\{f_{1}, \ldots, f_{m}\}$,$U$

with respect toZ. $\#(UH(G)\cap J(G))<\infty$ and

3. $UH(G)\cap J(G)\subset U$.
Then we have that $J(G)=\overline{U}$ or that $J(G)$ is porous
so the Box
dimension of $J(G)$ is strictly less than 2). Moreover, the fibered rational map:
associated with the generator systemthatsatisfies$f$$\Sigma_{m}\cross\overline{\mathbb{C}}arrow\Sigma_{m}\cross\overline{\mathbb{C}}$$\{h_{1}, \ldots, h_{m}\}$$\tilde{J}(f)=\bigcup_{x\in\Sigma_{m}}J_{x}$.

153
Definition 3.3. Let G be arational semigroup andber. We say that aBorel probability measure on
for each g $\in G$ and for each Borel measurable set$\delta$$\overline{\mathbb{C}}$$\mu$anon-negative numis -subconformal if$\delta$$A$$\mu(g(A))\leq\int_{A}||g’(z)||^{\delta}d\mu$,

the norm of the derivative with respect to thewhere we denote by
and each real number we set
spherical metric. For each$||\cdot$$||$$x\in\overline{\mathbb{C}}$$s$$S(s, x)= \sum_{g\in G}\sum_{g(y)=x}||g’(y)||^{-s}$counting multiplicities and$S(x)= \inf\{s|S(s, x)<\infty\}$

If there is not$s$.

such that $S(s, x)<\infty$ , then we set

so (G) $= \inf\{S(x)\}$ ,$\mathrm{s}(\mathrm{G})=\inf${$G$

we set: -subconformal measure}$\delta$$\delta|\exists\mu$

We have an estimate on so(G) when$S(x)=\mathrm{o}\mathrm{o}.\mathrm{A}1\mathrm{s}\mathrm{o}$satisfies the open set condition.

be a rational semigroup. When
Proposition 3.4. Let$m=1$ , toe assume thatis neither identity nor an elliptic M\"obius transformation. Suppose satisfies the open set condition with an open set with.
. Suppose also that
respect to the genercstor system
included in $U\cap F(G)$ such that for almost
Then there exists an open set$x\in V$ with respect to the 2-dimensional Lebesgue measure, we have
$G=\langle h_{1}, \ldots h_{m}\rangle$$h_{1}$$U$$G$$J(G)\neq\overline{U}$$\{h_{1}, \ldots, h_{m}\}$$V$$S(2, x)<\infty$.

In particular, so $(G)\leq 2$ .
be a rational
Theorem 3.5. (Hausdorff dimension) Let
semigroup. Under the same assumption as that of Theorem 3.2, we have that
$G=\langle h_{1}, \ldots h_{m}\rangle$$\dim_{H}(J(G))\leq s(G)\leq \mathrm{s}\mathrm{o}(\mathrm{G})$where $\dim_{H}$ denotes the Hausdorff dimension with respect to the sphericalmetric in C.

Example 3.6. Let $h_{1}(z)=z^{2}+2$ , $h_{2}(z)=z^{2}-2$ and $U=\{|z|<2.\}$ .
. Let be aThen we have $h_{1}^{-1}(U)\cup h_{2}^{-1}(U)\subset U$ andpolynomial which is conjugate to by an affine map , whereis anumber large enough. Taking appropriately, we haveand
. Taking large enough, we havesatisfies the conditions in the. Then$UH(G)\cap
J(G)$ is the parabolic fixedassumption of Theorem 3.2. In this casepoint of h$. By Theorem 3.2, we get that $J(G)$ is porous and in particular,
the Box dimension is strictly less than 2.
$h_{1}^{-1}(U)\cap h_{2}^{-1}(U)=\emptyset$$\alpha$$h_{4}^{n}$$n\in \mathrm{N}$$h_{4}(z)=z^{2}+ \frac{1}{4}$$\alpha$$J(h_{3})\subset U\backslash (h_{1}^{-1}(\overline{U})\cup h_{2}^{-1}(\overline{U}))$$U\backslash (h_{1}^{-1}(\overline{U})\cup h_{2}^{-1}(\overline{U}))$$h_{3}$$n$$G=\langle h_{1}, h_{2}, h_{3}\rangle$$h_{3}^{-1}(U)\subset$

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4

Tools and Proofs

4.1

Tools

To show theorems in \S 2 and \S 3, we need the followings. For the researchon semi-hyperbolicity of usual dynamics of rational functions, see [CJY] and[Ma].
Notations.

1. Let $X$ be acompact set inDist$\overline{\mathbb{C}}$

and

$z$

be apoint in$\overline{\mathbb{C}}\backslash X$. Then we set$(X, z)= \max d(y, z)/\mathrm{m}\dot{\mathrm{m}}d(y, z)y\in Xy\in X^{\cdot}$

2. For two positive numbers and ,for some constant $K$ independent of
$A$

$B$

means $K^{-1}\leq A/B\leq K$$A_{\wedge}\vee B$

$A$

and

$B$

.

Lemma 4.1 ([CJY]). (distortion lemma for proper maps) For any
positive integer $N$ and real number with $0<r$ $<1$ , there eists a constant$C=C(N, r)$ such that if : $D(0,1)arrow D(0,1)$ is a proper holomorphic map
with $\deg(f)=N$ $and/(0)=0$ , then
$r$

$f$

$D(f(z_{0}), C)\subset f(D(z_{0},r))\subset D(f(z_{0}),r)$

for any $z0\in D(0,1)$ . Here we can take $C=C(N,r)$ independent of .
$f$

The folowing is ageneralized distortion lemma for proper maps.

.

Lemma 4.2Let
be a domain in , $K$ a continuum inwith diamsK $=a$ . Assume. Let : $Varrow D(0,1)$ be a proper
holomorphic map of degree N. Then there exists a constant $r(N,a)$ depending
only on $N$ and such that for each with $0<r$ $\leq r(N,a)$ , there exists
constant $C=C(N, r)$ depending only on $N$ and satisfying that for eachconnected component of $f^{-1}(D(0,r))$ ,$([\mathrm{S}4],[\mathrm{S}6])$$V$$\overline{\mathbb{C}}$$\overline{\mathbb{C}}$$V\subset\overline{\mathbb{C}}\backslash K$

$a$

$f$

$r$

$a$

$r$

$U$

$d:am_{S}U\leq C$,

where we denote by diams the spherical diameter. Also we have $C(N, r)$as$r$ $arrow 0$.

$arrow 0$

The folowing lemma is aslightly modified version of Lemma 2.15 in [S4].

Lemma 4.3 ([S4]). Letbe-bundle. Let $f$ :be$XarrowX$rationalmap
over :. Assume satisfies the condition
fibered(Cl). Assume $z_{0}\in SH_{N}(f)$ for some
. Then there uish a positive
number such that for each 6withthere
a neighborhood$X$ satisfying
inthat for each, each $x\in U$ and eachof
$x_{n}\in p^{-n}(x)$ ,we have that each element ofis simply
connected$(\pi, \mathrm{Y},X)$$a\overline{\mathbb{C}}$$\mathrm{Y}arrow \mathrm{Y}$

$f$

$g$

$N\in \mathrm{N}$$\delta_{0}$$U$$x0:=\pi(z\mathrm{o})$$b$$0<\delta<\delta \mathrm{p}$$n\cdot s\hslash$$n\in \mathrm{N}$$c(i_{x}i_{x_{0}}^{-1}\tilde{B}(z_{0}, \delta),$$f_{x_{n}}^{n})$

$a$

155
The following theorem says about what happens if there exists anonconstant limit function on acomponent of afiber-Fatou set. This is the key
to state other results.

-bundle. Let
beTheorem 4.4 ([S4]). (Key theorem I) Let$Xarrow
X$. Assume satisfies:
be a fibered rational map over :be a local
be a point with $z\in F_{\pi(z)}$ . Let
the condition (Cl). Let
in C.
parametrization. Let be a connected open neighborhood of
Suppose that there exists a sequence (nj) of such thatconverges to a non-constant map uniformly on as $jarrow\infty$ .
$a\overline{\mathbb{C}}$$(\pi, \mathrm{Y}, X)$$\mathrm{Y}arrow \mathrm{Y}$

$f$

$f$

$g$

$(i_{x})$$z\in \mathrm{Y}$$U$$i_{\pi(z)}^{-1}(z)$$Rj:=i_{\pi f^{n_{j}}(z)}^{-1}\circ$$\mathrm{N}$$U$$\phi$$f_{\pi(z)}^{n_{j}}\circ i_{\pi(z)}$converges to a point

fibrther suppose$1\leq i\leq j$ . We set$f_{\pi(z)}^{n_{\mathrm{j}}}(z)$. Let$z0\in \mathrm{Y}$.for$S_{\dot{l},j}=f_{g^{n}\cdot\pi(z)}^{n_{\mathrm{j}}-n}.\cdot$$V= \{a\in \mathrm{Y}_{\pi(z_{0})}|\exists\epsilon>0, \lim\sup \sup d(S_{\dot{\iota},j}\circ\varphi(\xi), \xi)=0\}$,$:arrow\infty j>i\xi\in\tilde{B}(a,\epsilon)$

where is a map from. Thention aroundwe have that

onto
defined by the local parametrizaand
is a non-empty open proper subset of$\mathrm{Y}_{g^{n}:\pi(z)}$$\mathrm{Y}_{\pi(z_{0})}$$\varphi$$\pi(z_{0})$$V$$\mathrm{Y}_{\pi(z\mathrm{o})}$$\partial V\subset\hat{J}_{\pi(z\mathrm{o})}(f)\cap UH(f)$.

be a fibered-bundle. Let :Corollary 4.5. Letberational map over : $Xarrow X$ . Assume satisfies the condition (Cl). Asdoes
sume also that for each $x\in X$ , the boundary of $\hat{J}_{x}(f)\cap UH(f)$ inwith $z\in F_{\pi(z)}$ , we have thatnot separate points in Yx. Then for each$W$$diamf_{\pi(z)}^{n}(W)arrow 0$ as $narrow\infty$ for each open connected neighborhoodofand that $d(f_{\pi(z)}^{n}(z), UH(f))arrow 0$ as $narrow\infty$ .
in$(\pi, \mathrm{Y}, X)$$\mathrm{Y}arrow \mathrm{Y}$$f$$a\overline{\mathbb{C}}$

$f$

$g$

$\mathrm{Y}_{x}$$z\in \mathrm{Y}$$z$$\mathrm{Y}_{\pi(z)}$4.2

Proofs of results on fibered rational maps

We start with the following.
be a fibered-bundle. Let :be
Proposition 4.6. Let
has no interior points forrational map over : $Xarrow X$ . Suppose thatiseach $x\in X$ . Then the two dimensional Lebesgue measure ofequal to zero for each $x\in X$ .
$(\pi, \mathrm{Y}, X)$$f$$a\overline{\mathbb{C}}$$\mathrm{Y}arrow \mathrm{Y}$$\hat{J}_{x}$$g$$\tilde{J}_{\omega n}(f)\cap J_{x}$Proof. Fix$N\in \mathrm{N}$$\tilde{J}_{cm}(f, N)\cap J_{x}$. Suppose that there exists apoint $x\in X$ such that

(this is an open set in$J_{x}$) has positive measure. Then there ex-. Let $y_{m}=f_{x}^{m}(y)$ and $x_{m}=$ists aLebesgue density point$g^{m}(x)$ for any
. Let
. Let
be anumber such that
containing,,,
be the elements of, there exists asubsequence (n) in
respectively. Since
is simply connected,
such thatwith$y\in\tilde{J}_{\omega n}(f, N)\cap J_{x}$$U_{m}$$y\in\tilde{J}_{cm}(f, N, \delta)$$\delta>0$$m\in \mathrm{N}$$c(\tilde{B}(y_{m}, \delta/2)$$U_{m}’$$f_{x}^{m})$$c(\tilde{B}(y_{m}, \delta),$$y\in\tilde{J}_{cm}(f, N, \delta)$$y$$narrow\infty$$U_{n}’$$f_{x}^{m})$$\mathrm{N}$$\deg(f_{x}^{n} :U_{n}’arrow\tilde{B}(y_{n}, \delta))\leq N$

156
for each$n$and$\mathrm{d}\mathrm{i}\mathrm{a}\mathrm{m}U_{n}’arrow 0$

parametrization$i_{x}$

as$narrow\infty$,. By Corollary 1.9 in [S4] for any local$\lim_{narrow\infty}\frac{m(i_{\overline{x}}^{1}(U_{n}\cap J_{x}))}{m(i_{\overline{x}}^{1}(U_{n}))}=1$,(1)where $m$ denotes the spherical measure of C. Using an argument in the proofof Theorem 4.4 in [S4], from (1) we can show that$\lim_{narrow\infty}\frac{m(1_{x_{n}}^{-1}(\tilde{B}(y_{n},\delta/2)\cap F_{x_{n}}))}{m(i_{\overline{x}_{n}}^{1}(\tilde{B}(y_{n},\delta/2)))}=0$,(2)

where $iXn$ denotes alocal parametrization. There exists asubsequence (nj)of (n), apoint
and apoint $x_{\infty}\in X$ such thatand$jarrow\infty$as. By (2) we have that
. On the other
hand, by the assumption we have that for any $a\in X$ ,has no interior
point. This is acontradiction.
$y_{\infty}\in \mathrm{Y}$$y_{n_{j}}arrow y_{\infty}$$\tilde{B}(y_{\infty}, \delta/2)\subset\hat{J}_{x_{\infty}}$$x_{n_{\mathrm{j}}}arrow x_{\infty}$$\hat{J}_{a}$$\square$Proposition 4.7. Letbe-bundle. Let :be a fiberedrational map over $g:Xarrow X$ . Suppose satisfies the condition (Cl). Then
we have the following.
$(\pi, \mathrm{Y},X)$$a\overline{\mathbb{C}}$$f$$\mathrm{Y}arrow \mathrm{Y}$$f$1.
$J_{g\infty d}(f)\cap\cup J_{x}\subset\tilde{J}_{cm}(f)x\in X^{\cdot}$

2. If we assume further that for each $x\in X$, the boundary of$UH(f)$ in
does not separate points in , then$\mathrm{Y}_{x}$$\hat{J}_{x}(f)\cap$$\mathrm{Y}_{x}$$J_{good}(f)\subset\tilde{J}_{em}(f)\cap\cup J_{x}x\in X^{\cdot}$

Proof First we$\mathrm{w}\mathrm{i}\mathrm{U}$show the &st statement.

Let$z \in\bigcup_{x\in X}J_{x}$be apoint satisfying that
. For each
let $z_{m}=f^{m}(z)$ and $x_{m}=\pi f^{m}(z)$ . For eachand each $r>0$ let$U_{m}(r)$ , $U_{m}’(r)$ be the elements of,contaking respectively. There exists apositive number , positive integer$N$ and asequence (n)in such that.
By Lemma 4.3, taking 6smal enough we can assume thatis simply$\lim\sup_{narrow\infty}d(f^{n}(z), \mathrm{U}\mathrm{H}(\mathrm{f})>0$$m\in \mathrm{N}$$m\in \mathrm{N}$$c(\tilde{B}(z_{m}, \mathrm{J}/2)$$f_{\pi(z)}^{m})$$z$$c(\tilde{B}(z_{m},r),$$f_{\pi(z)}^{m})$$\delta$$\mathrm{N}$$\deg(f_{\pi(z)}^{n} :U_{n}’(\delta)arrow\tilde{B}(z_{n}, \delta))\leq N$$U_{n}’(\delta)$

connected.
Suppose that diam
does not tend to zero as $narrow\infty$ in (n). Thenby distortion lemma for proper maps there exists asubsequenceof (n)
withand apositive number such thatfor each. Hence$(U_{n}(\delta))$$(n_{j})$$n_{j}arrow\infty$$r$$U_{n_{j}}(\delta)\supset\tilde{B}(z, r)$$j$$f^{n_{\mathrm{j}}}(\tilde{B}(z,r))\subset\tilde{B}(f_{n_{\mathrm{j}}}(z), \delta)$

(3)

157
for each . By condition (C1), if we take small enough (3) contradicts toas $narrow\infty$ in (n).
. Hence we get that diamthat.
Hence we get that
The second statement follows from Corollary 4.5 and the first statement.
$\delta$$j$$U_{n}(\delta)arrow 0$$z \in\bigcup_{x\in X}J_{x}$$z\in\tilde{J}_{\omega n}(f)$$\square$be a fibered-bundle. Let :and thatrational rnap over : $Xarrow X$ . Suppose thatsatisfies the condition (Cl). Then for each $x\in X$ we have that

Corollary 4.8. Let$(\pi, \mathrm{Y}, X)$

be$\mathrm{Y}arrow \mathrm{Y}$$f$$a\overline{\mathbb{C}}$$f$$\tilde{J}(f)=\bigcup_{x\in X}J_{x}$$g$$d(f_{x}^{n}(y), UH(f))arrow 0$, as$narrow\infty$,with respect to the Lebesgue measure in .
for almost every
has no interior points for each $x\in X$ .
Proof. By condition (C1)$\mathrm{Y}_{x}$$y\in J_{x}$$\hat{J}_{x}=J_{x}$$\mathrm{B}\mathrm{y}\square$

Proposition 4.6 and Proposition 4.7, we get the statement.

satisfyProof. of Theorem 2.1. Suppose that there exists apoint. By Corollary 4.5, For each open connected neighborhood
ing that$z\in\tilde{J}(f)$$z\in F_{\pi(z)}$we have diam $f^{n}(W)arrow 0$ and $d(f^{n}(z), UH(f))arrow 0$ as $narrow$of in. But by condition 3and 4in the assumption of our theorem, it causes a. By Corollary 4.8
contradiction. Hence we have shown that
we get that the 2-dimensional Lebesgue measure ofis equal to zero.
$W$

$z$

$F_{\pi(z)}$$\infty$$\tilde{J}(f)=\bigcup_{x\in X}J_{x}$$J_{x} \backslash \bigcup_{n\in \mathrm{N}}f^{-n}(UH(f))$$\square$

Proof. of Theorem 2.3. For any$y’ \in\bigcup_{x\in X}J_{x}$and $r>0$ , we set$h(y’, r)= \sup\{s|\exists y’’\in J_{\pi(y’)},\tilde{B}(y’’, s)\subset F_{\pi(y’)}\}$

and $h(r)= \inf\{h(y’, r)|y’\in\bigcup_{x\in X}J_{x}\}$ . By Theorem 2.1, we havefor any $x\in X$ . Hence we. By the condition (C1) we have$\tilde{J}(f)=$$\mathrm{i}\mathrm{n}\mathrm{t}J_{x}=\emptyset$$\bigcup_{x\in X}J_{x}$get that $h(r)>0$ for any $r>0$ .
Since is semi-hyperbolic and satisfies the condition (C1), by Lemma 4.3
such
we have that there exists apositive number and anumberofand any component
that for any $y’\in\tilde{J}(f)$ , $0<\delta\leq\delta_{1}$ ,, is simply connected and $\deg(f^{n} : Varrow\tilde{B}(y’, 2\delta))\leq N$ .
and $r>0$ .We set $B_{n}=f^{n}(\tilde{B}(y,r))$ and $y_{n}=f^{n}(y)$ for eachLet, we have that there exists the smallest positive integer. Since. Then there exists aconstant such thatsuch that diam
. By Corollary 2.3 in [Y], there exists aconstant $K$ depending
,
such
with
only on $N$ and aball.
containing is asubset ofthat the component of.
included in
There exists aball$f$$N\in \mathrm{N}$$\delta_{1}$$V$$n\in \mathrm{N}$$(f^{n})^{-1}(\tilde{B}(y’, 2\delta))$$V$$y\in\tilde{J}(f)$$n\in \mathrm{N}$$y\in J_{\pi(y)}$$n_{0}$$l_{0}$$B_{n\mathrm{o}+1}>\delta_{1}$$l_{0}\delta_{2}<\mathrm{d}\mathrm{i}\mathrm{a}\mathrm{m}B_{n0}$$\tilde{B}(y_{n0}, r_{0})\subset B_{n0}$$(f^{n0})^{-1}(\tilde{B}(y_{n_{0}},r\mathrm{o}))$$\tilde{B}(y_{l}’, \frac{2}{3}h(r_{0}))$$r_{\mathrm{Q}}\geq \mathrm{d}\mathrm{i}\mathrm{a}\mathrm{m}B_{n0}/K\geq\underline{l}_{0}K\mathrm{p}\delta$$y$$\tilde{B}(y_{n0},r_{0})\cap F_{\pi(y_{n_{0}})}$$\tilde{B}(y,r)$

158
Let
be acomponent ofcontained in.
We have that
. Let $y’\in D_{0}\cap(f^{n_{0}})^{-1}(y’)$ be apoint. Then by
Corollary 1.8 and 1.9 in [S4], Dist $(\partial D_{0},y’’)\leq M$ for some $M$ depending
only on $N$ and diam. Hence there exists aconstant $0<k<1$ which
does not depend on and such that.
$(f^{n0})^{-1}( \tilde{B}(y’, \frac{1}{2}h(r_{0})))$$D_{0}$$\tilde{B}(y,r)$$D_{0}\subset F_{\pi(y)}$$D_{0\wedge}\vee r$

$r$

$y$

$\mathrm{B}(\mathrm{y}", kr)$$\subset D_{0}\subset F_{\pi(y)}$$\square$

4.3

Proofs of results on rational semigroups

Notation Throughout this subsection, for agenerator systemlet :be the fibered rational map over the shift map:, where, associated with the generator
system. We set
for any.

$\{h_{1}, \ldots h_{m}\}$$f$$\sigma$$\Sigma_{m}\cross\overline{\mathbb{C}}arrow\Sigma_{m}\cross\overline{\mathbb{C}}$$\Sigma_{m}arrow\Sigma_{m}$$\Sigma_{m}=\{1, \ldots, m\}^{\mathrm{N}}$$\{h_{1}, \ldots, h_{m}\}$$q_{x}^{(n)}(y)=\pi_{\overline{\mathbb{C}}}(f_{x}^{n}(y))$$(x,y)\in\Sigma_{m}\cross\overline{\mathbb{C}}$

Lemma 4.9. Let $E$ be a finite subset of . Let ( ,,be a rationalsemigroup. Then for any number $M>0$ there exists a positive integer
such that for any
withwhich satisfies all of
the following conditions:$\overline{\mathbb{C}}$$h_{1}$$\ldots$$h_{m}\rangle$$n_{0}$$(n,x, y)\in \mathrm{N}\cross\Sigma_{m}\cross E$

1. $q_{x}^{(j)}(y)\in E$ for $j=0$ ,2.

$(q_{x}^{(n)})’(y)\neq 0$

3. for any
then$i\in \mathrm{r}\mathrm{n}$$\ldots$,$n\geq n_{0}$

$n$

and

and

with $i+j\leq n$ ,$j\in \mathrm{N}$$|(q_{\sigma(x)}^{(j)}\dot{.})’(q_{x}^{(\dot{1})}(y))|>1$

if$q_{\sigma(x)}^{(j)}.\cdot$$(qi()(y))=q_{x}^{(\dot{1})}(y)$,we have that $|(q_{x}^{(n)})’(y)|>M$.

Pmof. This lemma can be shown by induction on

as that in Lemma 1.32 in [S4].

$\# E$

using the same method$\square$Lemma 4.10. Let
be a finetely generated rational semigroup.
Suppose $\#(UH(G)\cap J(G))<\infty$ and $UH(G)\cap J(G)\neq\emptyset$ . Then for each$z\in UH(G)\cap J(G)$ there exists an element $g\in G$ , an element $h\in G$ and$p_{\mathit{0}\dot{l}}ntw\in UH(G)\cap J(G)$ such that $h(w)=z$ ,$g(w)=w$ and $|g’(w)|\leq 1$ .
$(h_{1}, \ldots, h_{m})$$a$Proof. Suppose that there exists apoint $z\in UH(G)\cap J(G)$ for which there

exists no $(g, h, w)$ in the conclusion of our lemma. Then by Lemma 4.9 andthe Koebe distortion theorem, we can easily see that for arbitrarily small
there exists apositive number 6and apositive constant $N$ such that
if apoint $w_{0}\in UH(G)\cap J(G)$ and an element $g0\in G$ satisfy
then the diameter of the component ofcontaining
is less
than and $\deg(g_{0} : Varrow B(z, \delta))\leq N$ . Then taking small enough, since
is finitely generated and $\#(UH(G)\cap J(G))<\infty$ we can easily deduce that
there exists apositive constant $N’$ such that for any element $g_{1}\in G$ andany component $W$ of, we have that $\deg(g_{1} : Warrow V)\leq N’$ . Thisimplies that $z\in SH_{N+N’}(G)$ and this contradicts to that $z\in UH(G)$ . Cl$\epsilon>0$$g\mathrm{o}(w_{0})=z$$V$$g_{0}^{-1}(B(z, \delta))$$\epsilon$$\epsilon$$g_{1}^{-1}(V)$$w_{0}$$G$

159
Lemma 4.11. Under the assumpstion of Theorem 3.2, there exists a disk
D in $F(G)$ such that1.

$\overline{\bigcup_{g\in G}g(D)}\subset F(G)$2. diam$q_{x}^{(n)}(D)arrow 0$

and

as

unifomly on$narrow\infty$

In particular, the fibered rational map$f$$x\in\Sigma_{m}$satisfies the condition (C2).

Proof. Let $h\in Ci$ be an element of degree at least two. Since$\emptyset\neq\overline{\mathbb{C}}\backslash \overline{U}\subset$

and $UH(G)\cap J(G)\subset U$, we have that there exists an attractingU}$
it follows. Since $UH(G)\cap J(\underline{G)\subset again,
inperiodic point$D$ around$\bigcup_{g\inG}g(D)\subset
F(G)$ . Bysuch thatthat there exists adisk
Lemma 1.30 in [S4], the statement of our lemma follows.
$F(G)$$z_{0}$$F(G)\backslash \mathrm{U}$$z\circ$$\square$Lemma 4.12. Under the assumption of Theorem 3.2, if $UH(G)\cap J(G)\neq\emptyset$then for each point $z\in UH(G)\cap J(G)$ there exists the unique element $h\in G$. Further we have that is a parabolic
satisfying that $h^{n}(z)=z$ for each
fixed point of .
$z$$n\in \mathrm{N}$$h$

Proof. By Lemma 4.10 and the open set condition, there exists the unique. Further we must haveelement $h\in G$ with $h^{n}(z)=z$ for each$|h’(z)|\leq 1$ .
$n\in \mathrm{N}$If $\deg(h)=1$ , then by Lemma 4.11 it follows that is arepelling fixed
point of . This is acontradiction. If $\deg(h)\geq 2$ , then since we are assumingthat $\#(UH(G)\cap J(G))<\infty$ we have that is an attracting or parabolic
fixed point of . Suppose is an attracting fixed point of . Then there
exists an open neighborhood $V$ of in $U$ such that $h(V)\subset V$. Let
for each , where $x=(x_{1},x_{2}, \ldots)$ .
be the point such thatweand any
Then by the open set condition for any$G$. Hence we have that is normal in $V$
have thatand this is acontradiction.
$z$

$h$

$z$

$h$

$h$

$z$

$x\in\Sigma_{m}$

$z$

$n$

$h_{x_{n}}\circ\cdots\circ h_{x_{1}}=h$$n\in \mathrm{N}$$x’\in\Sigma_{m}\backslash \{x\}$$h_{x_{\acute{n}}}\circ\cdots h_{x_{\acute{1}}}(V)\subset\overline{\mathbb{C}}\backslash U$$\square$Lemma 4.13. Under the assumption of Theorem 3.2, we have that for each$(x, y)\in\pi_{\overline{\mathbb{C}}}^{-1}(G^{-1}(J(G)\backslash UH(G)))$

Proof. Let $(x,y)$ be apoint in$J(G)\backslash UH(G)$ for each.

,$\lim\sup_{narrow\infty}d(q_{x}^{(n)}(y), UH(G))>0$$\pi_{\overline{\mathbb{C}}}^{-1}G^{-1}(J(G)\backslash UH(G))$. Then.

$q_{x}^{(n)}(y)\in$$n\in \mathrm{N}$Assume that $\lim_{narrow\infty}d(q_{x}^{(n)}(y), UH(G))=0$ . We will deduce acontradiction. For each $z\in UH(G)\cap J(G)$ , let be the element of in the statementof Lemma 4.12 Let $H=\{g_{z}|z\in UH(G)\cap J(G)\}$ . Then we have $\#(H)<\infty$ .
be asmall number such that if apoint $z\in UH(G)\cap J(G)$ and anLetelement $h\in H$ satisfy $h(z)=z$ , then$G$$g_{z}$$\epsilon>0$.

$h(B(z, \epsilon))\subset U$

(4)

160
Let

be the -neighborhood of $UH(G)\cap J(G)$ in C. Then there exists a
numbersuch thatfor each.
For each $n\geq n0$ , let $z_{n}\in UH(G)\cap J(G)$ be the unique point suchthat
. Since $g(UH(G))\subset UH(G)$ for each $g\in G$ we may
assume that
$A_{\epsilon}$$\epsilon$$q_{x}^{(n)}(y)\in A_{\epsilon}$$n_{0}\in \mathrm{N}$$n\geq n_{0}$$d(z_{n},q_{x}^{(n)}(y))<\epsilon$$q_{\sigma^{n}(x)}^{(1)}(z_{n})=z_{n+1}$

for each.
Since $\#(UH(G)\cap J(G))<\infty$ , there exists apositive integer
andsuch that. Let $g_{1}\in Ci$ be the unique element such that
. Let $w\in\{1, \ldots,m\}^{l}$ be the word such that.
Then by (4) and the open set condition we have that
. Since$d(q_{x}^{(n)}(y),UH(G))arrow0$
we are assuming, byas
we getthatas. Hence by Lemma 4.12 we must have thatis aparabolic fixed point of and, where $W$
belongs to$P$
is asmal neighborhood ofin ,is the union of attracting petals ofatandis alarge positive number with. Then there exists
an open neighborhood of such that $q_{x}^{(n_{2})}(V)\subset W\cap P$ . Taking $W$ sosmall andso large we may assume thatfor any: N. Since
for each $j=1$ ,, $m$ , we getforeach. By the open set condition, for any
we have thatfor each. Hence we get that is normal in andthis contradicts to that.
Cl$n\geq n_{0}$$n_{1}\geq n_{0}$$l\in \mathrm{N}$$z_{n_{1}+l}=z_{n_{1}}$$g_{1}(z_{n_{1}})=z_{n_{1}}$$h_{w_{l}}\circ\cdots\circ h_{w_{1}}=g_{1}$$\sigma^{n_{1}}(x)=w^{\infty}$$narrow\infty$$g_{1}^{k}(q_{x}^{(n_{1})}(y))arrow z_{n_{1}}$$karrow\infty$$z_{n_{1}}$$q_{x}^{(n_{2})}(y)$$g_{1}$$W\cap \mathcal{P}$$U$$z_{n_{1}}$$g_{1}$$n_{2}\geq n_{1}$$n_{2}$$z_{n_{1}}$

$V$

$y$

$g_{1}^{s}(q_{x}^{(n_{2})}(V))\subset W\cap P$$n_{2}$$h_{j}^{-1}(U)\subset U$$s$$z_{n_{1}+l}=z_{n_{1}}$$q_{x}^{(n)}(V)\subset U$$\ldots$$n\in \mathrm{N}$$x’\in\Sigma_{m}\backslash \{x\}$$q_{x}^{(n)},(V)\subset\overline{\mathbb{C}}\backslash U$$G$$n\in \mathrm{N}$$V$$y\in \mathrm{J}\{\mathrm{G})$Now we will give aproof of Theorem 3.2.

Pmof. of Theorem 3.2. Suppose$J(G)\neq\overline{U}$. Then by Proposition 4.3 in[S4], we have intJ(G)=\emptyset . For any $d$ $\in J(G)$ and$r$$>0$ ,

we set$h(y’,r)= \sup\{s|\exists y’\in\overline{\mathbb{C}}, B(y’, s)\subset F(G)\cap B(y’,r)\cap U\}$and $h(r)= \inf\{h(\oint,r)|\nu \in J(G)\}$ . Then since intJ(G)=\emptyset , we have$h(r)>0$ for any $r>0$ .
Let
be small number. Let $B$ be the -neighborhood of $UH(G)\cap$$J(G)$ in C. By Lemma 4.3 and Lemma 4.11, we have that there exists a
positive number and anumber
such that for any $y’\in J(G)\backslash B$ , $0<$$V$$g^{-1}(B(y’,2\delta))$ ,$V$
and any component ofis simply connected and$\deg(g:Varrow B(y’,2\delta))\leq N$ . By Lemma 4.13 and Theorem 2.1, we have$\delta_{0}>0$$\delta_{0}$$\delta_{1}$$N\in \mathrm{N}$$\delta\leq\delta_{1}$$\tilde{J}(f)=\bigcup_{x\in\Sigma_{m}}J_{x}$Let $y\in J(G)$ be apoint. Since(5) we have that there exists apoint.

(Proposition 3.2 in [S5]), bysuch that $y\in J_{x}$ .

$\pi_{\overline{\mathbb{C}}}\tilde{J}(f)=J(G)$$x\in\Sigma_{m}$

(5)

161
. Let be apositive number. We set $B_{n}=q_{x}^{(n)}(B(y, r)$Let. Since $y\in J_{x}$ , we have that there exists theand $y_{n}=q_{x}^{(n)}(y)$ for each
. Then there exists asuch that diamsmallest positive integer.
constant such that.
Case 1.
By Corollary 2.3 in [Y], there exists aconstant $K$ depending only on$N$ and aball $B(y_{n0}, r_{0})\subset B_{n0}$ withsuch that thediam
containing is asubset of $B(y, r)$ . There
component of
included in $B(y_{n_{0}},r\mathrm{o})\cap F(G)\cap U$.
exists aballcontained in $B(y, r)$ .
be acomponent ofLet
By the open set condition, we have $g^{-1}(U\cap F(G))\subset U\cap F(G)$ for each $g\in G$ .
be apoint. Then
Hence we have $D_{0}\subset F(G)\cap U$. Let
by Corollary 1.8 and 1.9 in [S4], Dist $(\partial D_{0}, y’)\leq M$ for some $M$ depending. Hence there exists aconstant $0<k<1$ which
only on $N$ and diam
does not depend on and such that $B(y’, kr)\subset D_{0}\subset F(G)\cap B(y, r)$ .
.
Case 2.
small enough
By Lemma 4.12 and that $UH(G)\cap J(G)\subset U$, taking
and using the method in pp286-287 in [Y] we can show that there exists a
is aconstant with $0<k’<1$ball $B(y’, k’r)$ in $B(y, r)\cap F(G)$ where
which does not depend on and .
$\delta_{2}=\min\{\delta_{0}, \delta_{1}\}$$r$$n\in \mathrm{N}$$B_{n_{0}+1}>\delta_{2}$$n_{0}$$l_{0}\delta_{2}<\mathrm{d}\mathrm{i}\mathrm{a}\mathrm{m}B_{n0}$$l\circ$$y_{n_{0}}\in J(G)\backslash B$$B_{n_{0}}/K\geq \mathrm{m}l\delta K$$r_{0}$$(q_{x}^{(n\mathrm{o})})^{-1}(B(y_{n0},r_{0}))$$y$$B(y’, \frac{2}{3}h(r_{0}))$$(q_{x}^{(n_{0})})^{-1}(B(y’, \frac{1}{2}h(r\mathrm{o})))$$D_{0}$$y’\in D_{0}\cap(q_{x}^{(n\mathrm{o})})^{-1}(y’)$$D_{0}\vee\wedge r$

$y$

$r$

$y_{n_{0}}\in B$$\delta_{0}$

$k’$

$y$

$r$

$\square$Now we$\mathrm{w}\mathrm{i}\mathrm{l}$show Proposition 3.4.

.
Proof, of Proposition 3.4. By the open set condition, we have
We will show the folowing.
Claim 1: There exists an open set $V’$ included in $U\cap F(G)$ such that$h^{-1}(V’)\cap V’=\emptyset$ for each $h\in G$ .
Before showing this claim, we remark that we can easily show the following claim.
,
Claim 2: If there exists apoint $z\in U\cap F(G)$ such that
$V’$ of.
then the claim 1holds with an small open neighborhood
To show the claim 1, by the open set condition we have
$J(G)\subset\overline{U}$$z\in\overline{\mathbb{C}}\backslash \overline{G(z)}$$z$$j=1\cup h_{j}^{-1}(U\cap F(G))m\subset U\cap F(G)$.

(6)

Suppose the equality does not hold in (6). Then there exists apoint $z\in$, $m$ . Hence by the open
$U\cap F(G)$ such thatfor each $j=1$ ,. By the claim 2, the claim 1holds
set condition, we get that$h_{j}(z)\in\overline{\mathbb{C}}\backslash U$$z\in\overline{\mathbb{C}}\backslash \overline{G(z)}$$\ldots$162
Hence we may assume that
$j=1\cup h_{j}^{-1}(Um\cap F(G))=U\cap \mathrm{F}(\mathrm{G})$.

(7): $U\cap F(G)arrow U\cap F(G)$ be the map defined as: $\alpha(z)=h_{j}(z)$ if$z\in h_{j}^{-1}(U\cap F(G))$ . This is well defined by (7) and the open set
condition.
Let $z\in U\cap F(G)$ be apoint. If, then by the claim 2wehave the claim 1. Hence we may assume
i.e.
Let

$\alpha$$z\in\overline{\mathbb{C}}\backslash \overline{G(z)}$$z\in\overline{G(z)}$$z\in\cup\{\alpha^{n}(z)\}n=0\infty$.

(8)

Let $W$ be the connected component of $U\cap F(G)$ containing . By (8) there
exists the smalest positive integer with
. By (8) and the open
set condition, we have one of the folowing cases 1and 2.
Case 1: $W$ is included in an attracting basin of an element $g\in G$ , isthe attracting fixed point in the basin and $g|W=\alpha^{n}|_{W}$ .
Case 2: $W$ is included in aSiegel disk or aHerman ring of an element$g\in G$ of degree at least 2and.
If we have the case 1, then there exists an open set $V’$ included in $W$
withfor each
i.e. $h^{-1}(V’)\cap V’=\emptyset$ for each $h\in G$ .
If we have the case 2, then taking $V’$ in aconnected componentofwith $A\cap W=\emptyset$ , we havefor eachi.e.
$h^{-1}(V’)\cap V’=\emptyset$
for each $h\in G$ .
Hence we have shown the claim 1. Let $V’$ be an open set included in$U\cap F(G)$ such that $h^{-1}(V’)\cap V’=\emptyset$ for each $h\in G$ . Then by
the open set
condition we have $g^{-1}(V’)\cap h^{-1}(V’)=\emptyset$, if , $h\in G$ and $g\neq h$ . Further
the post critical set of
$z$

$n$

$\alpha^{n}(W)\subset W$$z$$g|_{W}=\alpha^{n}|_{W}$$\alpha^{-l}(V’)\cap V’=\emptyset$$l\in \mathrm{N}$$A$$\alpha^{-n}(W)$$\alpha^{-l}(V’)\cap V’=\emptyset$$l\in \mathrm{N}$

$g$

$G$

$P(G):=\cup$ { critical values of $g$ }$g\in G$

does not accumulate in $V’$ . Let
Then we have that$V$

be an open disk included in$\int_{V}\sum_{h\in G}\sum_{\alpha}||\alpha’(z)||^{2}dm(z)<\infty$$V’\backslash \mathrm{F}(\mathrm{G})$.

,where runs over aU well-defined inverse branches of on . Hence foralmost every $x\in V$ with respect to the Lebesgue measure, we have $5(2,x)$$\alpha$$\infty$.

Now we will show Theorem 3.5. we need some lemmas

$h$

$V$

$<$

$\square$163
Lemma 4.14. Let be a rational semigroup. Assume that $\infty\in F(G)$ andfor each $x\in E(G)$ there exists an element $g\in G$ such that $g(x)=x$ andbe a subset of $J(G)$ . Suppose that there exist positive$|g’(x)|<1$ . Let, and with $0<c<1$ such that for each $x\in A$ , there exist
constantsofandtwo sequencesof positive real numbers and a sequence
elements of satisfying all of the following conditions:
$G$

$A$

$c$

$a2$

$a_{1}$$(r_{n})$$(g_{n})$$(R_{n})$$G$1.

$r_{n}arrow 0$

and for each n,$0< \frac{f}{R}\mathrm{n}-<\mathrm{C}n$and $g_{n}(x)\in J(G)$ .

2. for each n, $g_{n}(D(x, R_{n}))\subset D(g_{n}(x), a_{1})$ .
3. for each $ng_{n}(D(x,r_{n}))$ :) $D(g_{n}(x), a_{2})$ .
Then$dim_{H}(A)\leq s(G)$.

Proof. We may assume that $\#(J(G))\geq 3$ . Let

be anumber anda -subconformal measure. By the method in the proof of Lemma 5.5
in [S4], we can show that there exists aconstant $d>0$ not depending on
and $x\in A$ such that$\delta\geq s(G)$$\delta$$\mu$$n\in \mathrm{N}$$\frac{\mu(D(x,r_{n}))}{r_{n}^{\delta}}\geq c’$.

Prom this and Theorem 7.2 in [Pe], we get $\dim_{H}A\leq\delta$.

$\square$Proposition 4.15. Let be a rational semigroup. Assume thatand that for each $x\in E(G)$ , there exists an element $g\in G$ such that$g(x)=x$ and $|g’(x)|<1$ . Then we have
$F(G)\neq\emptyset$$G$$dim_{H}(J_{con}(G))\leq s(G)$.

Proof. We have only to show the following:

and $r>0$ , $\dim_{H}(J_{con}(G, N,r))\leq s(G)$ .
Claim: For fixed
We will show this. We can assume $\infty\in F(G)$ . Let $x\in J_{con}(G, N,r)$ be
wein such that for eachapoint. Then there exists asequence
have $g_{n}\in J(G)$ ,$\deg(g$ : $V_{n}(r)arrow D(g_{n}(x),r)\leq N$
is simply connected and diam $V_{n}(r)arrow 0$ as $narrow\infty$ , whereand: $D(0,1)arrow V_{n}(r)$containing . Let
is tne element of $c(D(g_{n}(x), r)$ ,be the Riemann map such that $\varphi_{n}(0)=x$ . By the Koebe distortion theorem
we have for each ,$N\in \mathrm{N}$$(g_{n})$$n\in \mathrm{N}$$G$$V_{n}(r)$$V_{n}(r)$$g_{n})$$x$$\varphi_{n}$$n$$V_{n}(r) \supset D(x, \frac{1}{4}|\varphi_{n}’(0)|)$.

By Lemma 4.1 and the Koebe distortion theorem, there exists an,that for each$n\in \mathrm{N}$$V_{n}( \epsilon r)\subset D(x, \frac{1}{8}|\varphi_{n}’(0)|)$.

$\epsilon>0$

such

164
Since diam
, we haveas nLemma 4.14, we obtain the claim.
$V_{n}(r)$ $arrow 0$$arrow\infty$$|\varphi_{n}’(0)|arrow 0$

as n$arrow\infty$. ApplyingCl

Now we will show the following theorem.

Theorem 4.16. Let
be a finitely generated rational semigrvup with
. Let :
be the fibered rational map associated with the generator system, where. Supposethat satisfies the condition (Cl) and that for each, the bound$ary$ of $\hat{J}_{x}(f)\cap UH(f)$ in
does not separate points in . Then we have
$J_{good}(G)\subset J_{cm}(G)$ and$G=\langle h_{1}, \ldots, h_{m}\rangle$$F(G)\neq\emptyset$$f$$\mathrm{Y}arrow \mathrm{Y}$$\{h_{1}, \ldots, h_{m}\}$$\mathrm{Y}=\Sigma_{m}\cross\overline{\mathbb{C}}$$f$$x\in\Sigma_{m}$$\mathrm{Y}_{x}$$\mathrm{Y}_{x}$$\dim_{H}(J_{good}(G))\leq s(G)\leq s_{0}(G)$.

Proof. We may assume $\#(J(G))\geq 3$ . First we will show the folowing:

Claim: If, then for each $x\in E(G)$ there exists an element$g\in G$ such that $g(x)=x$ and $|g’(x)|<1$ .
If there exists an element $h\in G$ with $\deg(h)\geq 2$ , then this claim istrivial. Suppose that each element of
is of degree 1. By Lemma2.3 in[S5], we have $\#(E(G))\leq 2$ . Since satisfies the condition (C1) for each,
is loxodromic. Since $h_{:}(E(G))=E(G)$ for each , we must have thateach $x\in E(G)$ is fixed byfor each . Let $x\in E(G)$ be apoint. Suppose
for each . Then we get $J(G)=\{x\}$ and this is acontradiction
since we are assuming that $\#(J(G))\geq 3$ . Hencefor some . Hencethe claim holds.
The statement of our theorem follows ffom the claim, the second statement in Proposition 4.7, Proposition 4.15 and Theorem 4.2 in [S2].
0$E(G)\neq\emptyset$

$G$

$f$

$i$

$h_{:}$$i$$h_{:}$$|h_{\dot{1}}’(x)|>1$

$i$

$i$

$|h_{\dot{1}}’(x)|<1$

Now we$\mathrm{w}\mathrm{i}\mathrm{l}$$i$show Theorem 3.5.

Proof. of Theorem 3.5. This follows from Lemma 4.11, Lemma 4.13 andTheorem 4.16.

$\square$

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