数理解析研究所講究録
1042 巻 1998 年 68-77

68

Dynamics of Sub-hyperbolic and Semi-hyperbolic
Rational Semigroups and Conformal Measures ofRational Semigroups
Hiroki Sumi
Graduate School of Human and Environmental Studies,
Kyoto University, Kyoto 6068315, Japan-mail; sumi@math.h.kyoto-u.ac.jp*$\mathrm{e}$October 1997
Abstract
We consider dynamics of semigroups of rational functions on Riemannsphere. First, we will define hyperbolic rational semigroups and show the
metrical property. We will also define sub-hyperbolic and semi-hyperbolic
rational semigroups and show no wandering domain theorems. By using
these theorems, we can show the continuity of the Julia set with respect
to the perturbation of the generators. By using a method similar tothat in [Y], we can show that if a finitely generated rational semigroup issemi-hyperbolic and satisfies the open set condition with the open set
satisfying $\#(\partial O\cap J(G))<\infty$ , then 2-dimensional Lebesgue measure of
the Julia set is equal to .
Next, we will consider constructing -subconformal measures. If arational semigroup has at most countably many elements, then we canconstruct -subconformal measures. We will see that if a finitely generated
rational semigroup is semi-hyperbolic, then the Hausdorff dimension of theJulia set is less than the exponent .
Considering conformal measures in a skew product, with a method of
the thermodynamical formalism, we can get another upper estimate of
the Hausdorff dimension of the Julia sets of finitely generated expanding
$O$

$0$

$\delta$$\delta$$\delta$semigroups.

In more general cases than the cases in which semigroups are hyperbolic or satisfy the strong open set condition, we can construct generalized
Brolin-Lyubich’s invariant measures or self-similar measures in the Julia
sets and can show the uniqueness. We will get a lower estimate of themetric entropy of the invariant measures. With these facts and a generalization of
result, we get a lower estimate of the Hausdorff dimension of any finitely generated rational semigroups such that the backward
images of the Julia sets by the generators are mutually disjoint.
$\mathrm{M}\mathrm{a}\tilde{\mathrm{n}}\acute{\mathrm{e}}’ \mathrm{s}$For a Riemann surface , let End $(S)$ denote the set of all holomorphic endomorphisms of . It is a semigroup with the semigroup operation being compositionof functions. A rational semigroup is a subsemigroup of End
without any
constant elements.
$S$

$S$

$(\overline{\mathbb{C}})$s Partially

supported by JSPS Research Fellowships for Young Scientists.

69
Definition 0.1. Let$F(G)=${$G$be a rational semigroup. We set

is normal in a neighborhood of },$z\in\overline{\mathbb{C}}|G$$z$$J(G)=\overline{\mathbb{C}}\backslash F(c)$$F(G)$ is called the Fatou set for $G$ and $J(G)$ is called the Julia set for $G$$J(G)$ is backward.

.

invariant under but not forward invariant in general. If$G=\{f_{1},$is a finitely generated rational semigroup, then $J(G)$ has the
backward self-similarity. That is, we have $J(G)= \bigcup_{i=1}^{n}fi(-1J(G))$ . The Julia
set of any rational semigroup is a perfect set, backward orbit of any point of
the Julia set is dense in the Julia set and the set of repelling fixed points of thesemigroup is dense in the Julia set. For more detail about these properties, see[ZR], [GR], [HM1], [HM2], [S1] and [S2].
$G$$f_{2},$ $\ldots f_{n}\rangle$

1

Sub-hyperbolic and Semi-hyperbolic Rational
Semigroups

Definition 1.1. Let$G$

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

We call $P(G)$ the post critical set of . We say that is hyperbolic if $P(G)\subset$$F(G)$ . Also we say thatis sub-hyperbolic if $\#\{P(G)\cap J(G)\}<\infty$ and $P(G)\cap$$F(G)$ is a compact set.
$G$

$G$

$G$

.

Theorem 1.2
Let
be a finitely generated hyperbolic
rational semigroup. Assume that contains an element with the degree at least
two and each M\"obius transformation in is neither the identity nor an elliptic
element. Let $K$ be a compact subset. Then there are a positive$\lambda>1$,
number a number
and a conformal metric on an open subsetofwhich contains $K\cup J(G)$ and is backward invariant under such thatfor each$([\mathrm{S}4])$$G=\langle f_{1}, f_{2}, \ldots f_{n}\rangle$

$G$

$G$

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

$c$

$V$

$\rho$$G$$\overline{\mathbb{C}}\backslash P(G)$$k$$\inf\{||(fi_{k^{\mathrm{O}\cdots 0}}f_{i_{1}})^{l}(z)||\rho|z\in(f_{i}k\mathrm{O}\cdots \mathrm{O}fi_{1})^{-1}(K), (i_{k}, \ldots :$\geq c\lambda^{k}$, here we denote by

metric$\rho$$||\cdot||_{\rho}$

to it.

i_{1})\in\{1, \ldots, n\}^{k}\}$the norm of the derivative measured from the

Now we will show the converse of Theorem 1.2.

.

Theorem 1.3
Let
be a finitely generated rationalsemigroup. If there are a positi. $venumberc\backslash$ ’ a number
and a conformal
metric on an open subset containing $J(G)$ such that for each$([\mathrm{S}4])$$G=\langle f_{1}, f_{2}, \ldots f_{n}\rangle$$\lambda>1$$U$$\rho$$k$$\inf\{||(f_{i}k^{\mathrm{O}}\ldots \mathrm{o}fi1)’(Z)||\rho.|z\in(f_{i}k^{\mathrm{O}\cdots 0}f_{i_{1}})^{-1}(J(c)), (ik, \ldots,\dot{i}1)\in\{1, \ldots,n\}k\}$, where we denote by
the norm of the derivative measured from theto it, then is hyperbolic and for each $h\in G$ such that $\deg(h)$metric on
is one the map is not elliptic.
$\geq c\lambda^{k}$$||\cdot||_{\rho}$$\rho$

$V$

$G$

$h$

70
Remark 1. Because of the compactness of $J(G)$ , we can show, with anargument, which is familiar to us in the iteration theory of rational functions,that even if we exchange the metric to another conformal metric , theenequality of the assumption holds with the same number
and a different$\mathrm{e}\mathrm{a}s\mathrm{y}$$\rho_{1}$$\rho$$\lambda$.

$\mathrm{c}\mathrm{o}\mathrm{n}\dot{\mathrm{s}}\mathrm{t}\mathrm{a}\mathrm{n}\mathrm{t}_{C_{1}}$Definition 1.4. Let
group. We say that is expanding if$G=\langle f_{1}, f_{2}, \ldots f_{n}\rangle$$G$

be a finitely generated rational semiassumption in Theorem 1.3 holds.

$\mathrm{t}\mathrm{h}\mathrm{e}\sim$a ball of center and radius in the spherical metric.
We denote by
Also for any rational map , we denote bya connected component of$B(x, \epsilon)$$x$$B_{g}(X,\epsilon)$$g$$g^{-1}(B(x, \epsilon))$$\epsilon$.

Definition 1.5. Let be a rational semigroup. We say that is semi-hyperbolic(resp. weakly semi-hyperbolic) if there are positive number and positive integer $N$ such that for any $x\in J(G)$ (resp.
) , any element $g\in G$ and any$g^{-1}(B(x,\delta))$,connected component
of
$G$

$G$

$\delta$$\partial J(G)$$B_{g}(x,\delta)$$\deg(g:B_{g}(X, \delta)arrow B(x,\delta))\leq N$

Remark 2.

1. If

.

is semi-hyperbolic and $N=1$ , then

$G$

$G$

is hyperbolic.

2. If is sub-hyperbolic and for each $g\in G$ , there is no super attracting
fixed point of in $J(G)$ , then is semi-hyperbolic.
$G$

$G$

$g$

3. For a rational map

with the degree at least two,is semi-hyperbolic
if and only if has no parabolic orbits and each critical point in the Julia
set is non-recurrent. If is semi-hyperbolic, then there are
no indifferent cycles and Hermann rings.
$f$$\langle f\rangle$$f$$([\mathrm{C}\mathrm{J}\mathrm{Y}], [\mathrm{Y}])$$\langle f\rangle$

Definition 1.6. Let be a rational semigroup and $U$ be a component of $F(G)$ .
For every element of , we denote by
the connected component of $F(G)$$U$
containing $g(U)$ . We say that is a wandering domain if
is infinite.
$G$

$G$

$g$

$U_{g}$$\{U_{g}\}$

Theorem 1.7. Let be a rational semigroup. Assume that is weakly semihyperbolic and there is a point $z\in F(G)$ such that the closure of the G-orbit
is included in $F(G)$ . Then for each $x\in F(G),$and there is
no wandering domain.
$G$

$G$

$\overline{G(x)}\subset F(G)$$\overline{G(z)}$With this result, we get
Theorem 1.8. Let$\bullet$$\bullet$$G$

be a rational semigroup. Assume that$G$

is weakly semi-hyperbolic and for each $g\in G,$ $\deg(g)\geq 2$ , or$G$

is semi-hyperbolic and there is an element $h\in G$ such that $\deg(h)\geq 2$ .

Then for each $x\in F(G),$$\overline{G(x)}\subset F(G)$and there is no wandering domain.

By Theorem 1.8, we can show the following result.

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Theorem 1.9. Let
be a finitely generated rational semigroup which is subhyperbolic or semi-hyperbolic. Assume that, there is an element $g\in G$
such that $\deg(g)\geq 2$ and for each M\"obius transformation in
is loxodromicorrbolic. Then there is a non-empty compact subset $K$ of $P(G)\cap F(G)$such that $K$ is an attractor $i.e$ . for any open neighborhoodof $K$ and each$z\in F(G),$ $g(z)\in U$ for all but finitely many $g\in G$ .
$G$

$F(G)\neq\emptyset$$G$$hype\backslash$$U$

Theorem 1.10. Let be a finitely generated rational semigroup which contains
an element with the degree at least two. Assume that $\# P(G)<\infty$ and $P(G)\subset$$J(G)$ . Then.
$G$

$J(G)=\overline{\mathbb{C}}$By Theorem 1.9 and Theorem 2.3.4 in [S3], we get the following result.
Theorem 1.11. Let $M$ be a complex manifold. Let
be a holomorphic family of rational semigroups (See the definition in) where $G_{a}=$. We assume that for a point $b\in M$,
is sub-hyperbolic orsemi-hyperbolic, contains an element of the degree at least two and each M\"obiustransformation in is hyperbolic or loxodromic. Then the map$\{G_{a}\}_{a\in M}$$l^{s\mathit{3}}J$$\langle f_{1,a}, \cdots , f_{n,a}\rangle$$G_{b}$$G_{b}$$aarrow*J(c_{a})$is continuous at the point $a=b$ with respect to the Hausdorff metric.

Definition 1.12. Let
be a finitely generated rational semigroup. We say that satisfies the open set condition with respect to the generators
if there is an open set such that for each $j=1,$ $\ldots,m,$
andare mutually disjoint.
$G=\langle f_{1}, f_{2}, \ldots f_{m}\rangle$

$G$

$O$

$f_{1},$ $f_{2},$ $\ldots f_{m}$$O$$f_{j}^{-1}(O)\subset$$\{f_{j}^{-1}(O)\}_{j=}1,\ldots,m$

Proposition 1.13. Let $G=\{f_{1},$be a finitely generated rational semigroup. Assume that satisfies the open set condition with respect to the generators
and
where
is the open set in the definition$J(G)$
has empty interior points.
of open set condition. Then$f_{2},$ $\ldots f_{m}\rangle$$G$$f_{1},$$f_{2},$$O$$O\backslash J(G)\neq\emptyset$$\ldots f_{m}$We get the next lemma by a modification of the arguments in [Y] or [CJY].
Lemma 1.14. Let
be a finitely generated rational semigroup which is semi-hyperbolic and satisfies the open set condition with respect
to the generators. Let be the open set in Definition 1.12 andbe a number in the definition of semi-hyperbolicity. Then for any there is apositive integer
such that for each $g\in G$ with the word length greater than
$G=\langle f_{1}, f2, \ldots fm\rangle$$f_{1},$ $f_{2},$ $\ldots f_{m}$$O$$\delta$$\epsilon$$n_{0}$$\frac{1}{2})ieshan\epsilon.\delta)$$g^{-1}’(B(n0eaChpointy \in y,\frac{1}{2}\delta)),theJ(Gdiameter)\backslash B(\partial o_{B_{g}}of’(y,\frac{1}{2}\delta\delta)andeac_{Slst}hConnected_{C}omponentB_{g}(y,$ofBy Proposition 1.13, Lemma 1.14 and a modification of the arguments in[Y], we get the next result.
Theorem 1.15. Let
be a finitely generated rational semigroup which is semi-hyperbolic and satisfies the open set condition with respect
to the generators. Let be the open set in Definition 1.12. Assume
that $\#(\partial O\cap J(G))<\infty$ . Then the 2-dimensional Lebesgue measure of $J(G)$ isequal to .
$G=\langle f_{1}, f_{2}, \ldots f_{m}\rangle$$f_{1},$ $f_{2},$ $\ldots f_{m}$

$0$

$O$

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2

$\delta$-subconformal measure

Definition 2.1. Let be a rational semigroup and be a non-negative number.
We sa.y that a probability measure on is -subconformal if for each $g\in G$
and for each measurable set$G$$\delta$$\overline{\mathbb{C}}$$\delta$

$\mu$

$A$

$\mu(g(A))\leq\int_{A}||g’(z)||\delta d\mu$

For each$x\in\overline{\mathbb{C}}$and each real number$s$.

we set$S(s, x)= \sum_{cg\in g(y}\sum||g’(y)||)=x-s$

counting multiplicities and$S(x)= \inf\{s|S(s, x)<\infty\}$

If there is not

such that $S(s, x)<\infty$ , then we set$s$$s_{0}(G)= \inf\{S(x)\},$ $s(G)= \inf${$\delta|\exists\mu$.

$S(x)=\infty.\mathrm{A}\mathrm{l}\mathrm{s}\mathrm{o}$we set: -subconformal measure}$\delta$. Let be a rational semigroup which has at most countTheorem 2.2
ably many elements. If there exists a pointsuch that $S(x)<\infty$ then$S(x)$
there is a
-subconformal measure.
$G$$([\mathrm{S}4])$$x\in\overline{\mathbb{C}}$

Proposition 2.3
measure for where. Let

be a rational semigroup and a -subconformal
is a real number. Assume that $\# J(G)\geq 3$ and for each$x\in E(G)$ there exists an element $g\in G$ such that $g(x)=x$ and $|g’(x)|<1$ .
Then the support of contains $J(G)$ .
$([\mathrm{S}4])$

$G$

$G$

$\delta$$\tau$$\delta$$\tau$

By Theorem 1.9 and Proposition 2.3, we can show the next result.

Theorem 2.4. Let
be a finitely generated rational semigroup. Assume thatis sub-hyperbolic, for each $g\in G$ there is no super
attracting fixed point of in $J(G)$ , there is an element of with the degree
at least two and each M\"obius transformation inis hyperbolic or loxodromic.
Then$G=\langle f_{1}, f_{2}, \ldots f_{n}\rangle$

$G$

$G$

$g$

$G$

$\dim_{H}(J(G))\leq S(G)\leq S0(G)$3.

Conformal Measures in a Skew Product

Let $m$ be a positive integer. We denote byis$\Sigma_{m}$the one-sided word space, that$\Sigma_{m}=\{1, \ldots, m\}^{\mathrm{N}}$and denote by$\sigma$:$\Sigma_{m}arrow\Sigma_{m}$the shift map, that is$(w_{1}, \ldots)rightarrow(w_{2}, \ldots)$

.

73
Let

$G=\langle f_{1},$ $f2,$ $\ldots fm$

map) be a finitely generated rational .semigroup. We define aby$\tilde{f}:\Sigma_{m}\cross\overline{\mathbb{C}}arrow\Sigma_{m}\cross\overline{\mathbb{C}.}$$\tilde{f}((w,x))=(\sigma w, f_{w}1x)$.

is finite-to-one and open map. We have that a point
satisfies
if and only if is a homeomorphism in a small neighborhoodof $(w,x)$ . Hence the map has infinitely many critical points. We set. Then by definition,). Also from the backward
self-similarity of $J(G)$ , we can show that $\pi(J)=J(G)$ where :is the second projection.
, $m$ , let be a H\"older continuous function on $f_{j}^{-1}(J(G))$ .
For each $j=1,$
We set for each $(w,x)\in\tilde{J},$ $\varphi((w,x))=\varphi_{w_{1}}(x)$ . Then is a H\"older continuousfunction. We define an operater on{ :continuous}$\dot{\mathrm{a}}$$(w, x)\in\Sigma_{m}\cross\overline{\mathbb{C}}$$\tilde{f}$$f_{w_{1}}’(x)\neq 0$$\tilde{f}$$\tilde{J}=$$\tilde{f}$$\bigcap_{n=0}^{\infty}(\Sigma_{m}\cross J(G))$$=\tilde{J}$$\tilde{f}^{-1}\mathrm{t}^{\tilde{J}}$$\Sigma_{m}\cross\overline{\mathbb{C}}arrow\overline{\mathbb{C}}$$\pi$$\ldots$$\varphi_{j}$$\varphi$$C(\tilde{J})=$$L$$\mathrm{o}\mathrm{n}.\tilde{J}$$\psi$$\tilde{J}arrow \mathbb{C}|$by$L\psi((w, x))=$$\sum$$\frac{\exp(\varphi((w,yJ)))}{\exp(P)}\psi((w^{l},y))$,$\tilde{f}((w’,y))=(w,x)$

counting multiplicities, where we denote by$(\tilde{f}|_{\tilde{J}}, \varphi)$$P=P(\tilde{f}|_{\tilde{J}}, \varphi)$.

the pressure ofbeLemma 3.1. With the same notations as the above, leta finitely generated expanding rational semigroup. Then for each set of H\"older
continuous functions’ there exists a unique probability measureon such that
$G=\langle f_{1}, f_{2}, \ldots f_{m}\rangle$$\{\varphi_{j}\}j=1,\ldots,m$$\tau$$\tilde{J}$$\bullet L^{*}\tau=\tau$$\bullet$,for each$\psi\in C(\tilde{J}),$$\lim\iotaarrow\infty^{L^{l}}(1)\in C(\tilde{J})$$\bullet$$\alpha\tau$$||L^{n}\psi-\mathcal{T}(\psi)\alpha||_{\tilde{J}}arrow 0,$

and we denote byis an equilibrium state for$(\tilde{f}|_{\tilde{J}}, \varphi)$, where we set
the supremum norm on ,$narrow\infty$$\alpha=$$\tilde{J}$$||\cdot||_{\overline{J}}$.

be a finitely generated expanding rational
Lemma 3.2. Let
semigroup. Then there exists a unique number
such that if we set
$m$
$P=0$
, , then$-\delta\log(||fj’(x)||),j=1,$.
$G=\langle f_{1}, f2, \ldots fm\rangle$$\delta>0$$\varphi_{j}(x)=$$\ldots$

From Lemma 3.1, for this there exists a unique probability measuresuch thatwhere
is an operator on
defined by$\delta$$\tilde{J}$$L_{\delta}^{*}\tau=\tau$$L_{\delta}\psi((w,X))=$

Also$\delta$$\sum$$\tilde{f}((w’,y))=(w,x)\frac{\psi((w’,y))}{||(f_{w_{1}},)(y)||\delta},$on.

satisfies that$\delta=\frac{h_{\alpha\tau}(\tilde{f})}{\int_{\tilde{J}}\tilde{\varphi}\alpha d\tau}\leq\frac{\log(\sum_{j=1}^{m}\deg(fj))}{\int_{\tilde{J}}\tilde{\varphi}\alpha d\tau}$,, we denote bywhere
the metric entropy ofand is a function on defined by $\tilde{\varphi}((w,x))=\log(||f’w1(x)||)$ .
By these argument, we get the following result.
$\alpha=\lim_{larrow\infty}L_{\delta}\iota(1)$$\tilde{J}$$\tilde{\varphi}$$\tau$$C(\tilde{J})$$L_{\delta}$$h_{\alpha\tau}(\tilde{f})$$(\tilde{f}, \alpha\tau)$

74
be a finitely generated expanding ratioTheorem 3.3. Let
nal semigroup and the number in the above argument. Then
$G=\langle f_{1}, f2, \ldots fm\rangle$$\delta$.

$\dim_{H}(J(c))\leq s(c)\leq\delta$

Moreover, if the sets $\{f_{j}^{-1}(J(c))\}$ are mutually disjoint, then $\dim_{H}(J(G))=$$\delta<2$ and $0<H_{\delta}(J(G))<\infty$ , where we denote by
the -Hausdorff measure.
$\delta$$H_{\delta}$

Corollary 3.4. Let
nal semigroup. Then

$G=\langle f_{1}, f_{2}, \ldots f_{m}\rangle$

be a finitely generated expanding ratio-$\dim_{H}(J(G))\leq\frac{\log(\sum_{j}^{m}=1\deg(f_{j}))}{\log\lambda}$

where$\lambda$,

denotes the number in Definition 1.4.

Generalized Brolin-Lyubich’s Invariant Measure, Self-Similar Measure

4

With the same notation as the previous section, we define an operator
by$\tilde{A}$on$C(\tilde{J})$$\sum$$\tilde{\psi}((wy’,))$$\tilde{A}\tilde{\psi}((w, x))=\frac{1}{\sum_{j=1}^{m}\deg(f_{j})}\tilde{[}((w’,y))=(w,x)$

and an operator$A$

on $C(J(G))=$ { :$\psi$$\tilde{\psi}\in C(\tilde{J})$,continuous} by$J(G)arrow \mathbb{C}|$$A \psi(x)=\frac{1}{\sum_{j=1}^{m}\deg(f_{j})}\sum_{j=1}^{m}\sum_{f\mathrm{j}(y)=x}\psi(y)$, for each

, for each $\psi\in C(J(G))$ .

, wheredefinedThen
is the map from $C(J(G))$ to
by $(\pi^{*}\psi)((w, X))=\psi(x)$ . Note that since $\pi(\tilde{J})=J(G)$ , we have that for each$\tilde{A}0\pi^{*}=\pi^{*}\mathrm{o}A$$\psi\in c(J(G))$$C(\tilde{J})$$\pi^{*}$,$||\pi^{*}\psi||_{\tilde{J}}=||\psi||_{J(G})$.

(1)

Now we consider a condition such that the invariant measures are unique.

Definition 4.1. Let $G=(f_{1},$be a finitely generated rational semi.
section, we say that satisfies
thesamenotationasthepreviousWith
gro.u, for any
, there exists a positive integer
for any
such that$f_{2},$$\ldots f_{m}\rangle$$G$$\mathrm{p}$$\mathrm{c}\mathrm{o}\mathrm{n}\mathrm{d}\mathrm{i}\mathrm{t}\mathrm{i}_{0}\mathrm{n}*\mathrm{i}\mathrm{f}$$z\in\tilde{J}\backslash _{\mathrm{P}^{\mathrm{e}\mathrm{r}}}(\tilde{f})$$\epsilon>0$$n_{0}=n_{0}(z, \epsilon)$$\#\{\tilde{f}^{-n_{0}}(z)\cap Z_{\infty}\}$$\overline{(\sum_{j=1}^{m}\deg(f_{j}))^{n0}}<\epsilon$,(2)

counting multiplicities, where we set$Z_{\infty}= \bigcup_{n=1}^{\infty}\tilde{f}^{n}$({ critical points of$\tilde{f}\}\cap\tilde{J}$

).

(3)

75
$f2,$
) be a finitely generated rational semigroup.
Remark 3. Let.
In each case of the following, the$\ldots fm$$G=\langle f_{1},$$\mathrm{c}\mathrm{o}\mathrm{n}\mathrm{d}\mathrm{i}\mathrm{t}\mathrm{i}_{0}\mathrm{n}*\mathrm{h}\mathrm{o}\mathrm{l}\mathrm{d}\mathrm{s}$$\bullet$$\bullet$$\bullet$.There exists an element
The sets$f$such that for each $j=1,$$\{f_{i}^{-1}(J(G))\}_{i}=1,\ldots,m\mathrm{a}r\mathrm{e}$$J(G) \backslash \bigcup_{g\in G}${

values of$\mathrm{c}\mathrm{r}\mathrm{i}\mathrm{t}\mathrm{i}\mathrm{c}\mathrm{a}1$$g$}$\ldots,$ $m,$$f_{j}=f$.

mutually disjoint.
$\cap J(G)\neq\emptyset$.

Therefore we have many finitely generated rational semigroups satisfying
if a finitely generated. It seems to be true that the$J(G)\cap
E(G)=\emptyset$$E(G)$ denotes the
,
where
that
rational semigroup satisfies.
exceptional set of , that is$\mathrm{c}\mathrm{o}\mathrm{n}\mathrm{d}\mathrm{i}\mathrm{t}\mathrm{i}0\mathrm{n}*\mathrm{h}\mathrm{o}\mathrm{l}\mathrm{d}\mathrm{s}$$\mathrm{c}\mathrm{o}\mathrm{n}\mathrm{d}\mathrm{i}\mathrm{t}\mathrm{i}_{0}\mathrm{n}*$$G$$E(G)= \{z\in\overline{\mathbb{C}}|\#\{\bigcup_{g}\in Gg^{-1}(z)\}<\infty\}$$G$

be a finitely generated rational semiTheorem 4.2. Let$F(H)\supset
J(G)$, where we set $H=\{g^{-1}\in Aut(\overline{\mathbb{C}})|g\in$group. Assume that, and $conditi_{\mathit{0}}n*holds$ . Then we have the following:$G=\langle f_{1}, f_{2}, \ldots f_{m}\rangle$$Aut(\overline{\mathbb{C}})\cap G\}$

1. There exists a unique probability measure$||\tilde{A}^{n}\tilde{\varphi}-\tilde{\mu}(\tilde{\varphi})1_{\tilde{j}}||_{\tilde{J}}arrow 0,$$\tilde{\mu}$$narrow\infty$on$\tilde{J}$such that, for any$\tilde{\varphi}\in C(\tilde{J})$,the constant function on taking its value 1, andwhere we denote byexists a unique probability measure on $J(G)$ such that$\tilde{J}$$1_{\tilde{J}}$$\mu$$||A^{n}\varphi-\mu(\varphi)1J(c)||_{J(G)}arrow 0,$

where we denote by1.

2.

$\pi_{*}\tilde{\mu}=\mu$

3.

$(\tilde{f},\tilde{\mu})$

4.
5.

$\mu$

and

$1_{J(G)}$$narrow\infty$, for any $\varphi\in C(J(G))$ ,the constant function on $J(G)$ taking its value

is -invariant.
$\tilde{f}$$\tilde{\mu}$is exact. In particular,$\tilde{\mu}$is ergodic.

is non-atomic. supp (p) is equal to $J(G)$ .

$h( \tilde{f}|_{\tilde{J}})\geq h_{\tilde{\mu}}(\tilde{f})\geq\log(\sum_{j}^{m}=1\deg(f_{j}))$entropy of$\tilde{f}$on$\tilde{J}$.

, where$h(\tilde{f}|_{\tilde{J}})$

denotes the topological.

Proof. We will show the statement in the similar way to [L]. By [HM3], thefamily of all holomorphic inverse branches of any elements of in any open set
which has non-empty intersection with $J(G)$ is normal in . With this fact,,
is almost periodic, i.e. for each
we can show that the operator
is the direct sum of. Hence, by [L],is relative compact in
the attractive basin of for and the closure of the space generated by unit
eigenvectors. It is easy to see that 1 is the unique eigenvalue and the eigenvectors
are constant. Therefore 1. holds.
fact, wein $F(G)$ . With
$G$

$U$

$U$

$\tilde{\psi}\in C(\tilde{J})$$\tilde{A}$$\{\tilde{A}^{n}\tilde{\psi}\}_{n}$$0$$\tilde{A}$$\mathrm{S}\mathrm{h}\mathrm{o}\mathrm{W}\mathrm{B}\mathrm{e}\mathrm{c}\mathrm{a}\mathrm{u}\mathrm{s}\mathrm{e}\mathrm{o}_{\mathrm{i}- \mathrm{a}\mathrm{t}}\mathrm{f}\mathrm{t}\mathrm{t}\mathrm{h}\mathrm{a}\mathrm{t}\mu \mathrm{S}\mathrm{n}\mathrm{o}\mathrm{n}\mathrm{o}\mathrm{m}\mathrm{i}_{\mathrm{C},\mathrm{W}\mathrm{h}\mathrm{i}_{\mathrm{C}}}\mathrm{h}\mathrm{i}\mathrm{m}\mathrm{p}\mathrm{h}\mathrm{e}\mathrm{C}\mathrm{o}\mathrm{n}\mathrm{d}\mathrm{i}\mathrm{t}\mathrm{i}\mathrm{o}\mathrm{n}*,E(G)\mathrm{i}_{\mathrm{S}}\mathrm{i}\mathrm{n}\mathrm{C}1\mathrm{i}\mathrm{e}\mathrm{S}51\mathrm{u}$Remark 4. If$C(\tilde{J})$$C(\tilde{J})$$\tilde{f}|_{\tilde{J}}$.dedis expansive,(in particular, if$\backslash \mathrm{t}\mathrm{h}\mathrm{e}$$\mathrm{c}\mathrm{a}\mathrm{n}$$G$

is expanding, ) then$h( \tilde{f}!\tilde{J})=h(\tilde{\mu}\tilde{f})=\log(\sum_{j}m=1\deg(f_{j}))$.

76
Remark 5. We can also construct self-similar measures on $J(G)$ and show theuniqueness under a similar assumption to. For example, in each caseof the Remark after Definition 4.1, we can show that.
$\mathrm{c}\mathrm{o}\mathrm{n}\mathrm{d}\mathrm{i}\mathrm{t}\mathrm{i}_{0}\mathrm{n}*$Now we consider a generalization of.

$\mathrm{M}\mathrm{a}\tilde{\mathrm{n}}\acute{\mathrm{e}}’ \mathrm{s}\mathrm{r}\mathrm{e}\mathrm{s}\mathrm{u}\mathrm{l}\mathrm{t}([\mathrm{M}\mathrm{a}])$Theorem 4.3. Let $G=(f_{1},$be a finitely generated rational semigroup. Assume that the sets $\{f_{i}^{-1}(J(G))\}_{j}=1,\ldots,m$ are mutually disjoint. $We$
define a map : $J(G)arrow J(G)$ by $f(x)=f_{i}(x)$ if $x\in f_{i}^{-1}(J(G))$ . If is anergodic invariant probability measure for : $J(G)arrow J(G)$ with $h_{\mu}(f)>0$ , then$f_{2},$$\ldots f_{m}\rangle$

$f$

$\mu$

$f$

$\int_{J(G)}\log(||f’||)d\mu>0$and,$HD( \mu)=\frac{h_{\mu}(f)}{\int_{J(G})\log(||f||)d\mu}$where we set$HD( \mu)=\inf\{\dim_{H}(\mathrm{Y})|\mathrm{Y}\subset J(G), \mu(\mathrm{Y})=1\}$.

Proof. We can show the statement in the same way as [Ma]. Note that the
Ruelle’s inequality
also holds for the map $f:J(G)arrow J(G)$ .
$\square$$([\mathrm{R}\mathrm{u}])$From the remark after Definition 4.1, Theorem 4.2 and Theorem 4.3, we get
the following result. This solves the Problem 12 in [Re] of F.Ren’s.
Theorem 4.4. Let
be a finitely generated rational semigroup. Assume that the sets $\{f_{i}^{-1}(J(c))\}_{j}=1,\ldots,m$ are mutually disjoint. Then
$G=\langle f_{1}, f_{2}, \ldots f_{m}\rangle$,$\dim_{H}(J(c))\geq\frac{\log(\sum^{m}j=1\deg(f_{j}))}{\int_{J(G)}\log(||f||)d\mu’}$

where

$\mu$

denotes the probability measure in Theorem 4.2 and $f(x)=f_{i}(x)$ if$X\in f_{i}^{-}1(J(G))$.

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