数理解析研究所講究録 1269 巻 2002 年 110-123

110

APPLICATIONS OF MANE’S THEOREMYESHUN SUN AND YONGCHENG YIN
ABSTRACT. In this paper, we show that the Julia set of acritically
non-recurrent rational map on Riemann sphere is either shallow

with respect to spherical metric or itself the whole sphere. Rigidity
and ergodicity results are also obtained for such maps.
Key words and phrases. Jula set, critically non-recurrent rational

map, distortion, shallow, rigidity, ergodicity.

1. INTRODUCTION
Let

$f$

:

$\overline{\mathbb{C}}arrow\overline{\mathbb{C}}$be arational map with degree $\deg(f)\geq 2$ . Apointis said to be aperiodic point if $f^{k}(z)=z$ for some $k\geq 1$ . The
minimal of such is called the period of . For aperiodic point ,denote the multiplier of by A $=(f^{k})’(z_{0})$ . The periodic point
is
$z$

$k$

$z$

$z_{0}$$z_{0}$$z_{0}$

either attracting, indifferent or repelling according to $|\lambda|<1$ , $|\lambda|=1$

or$|\lambda|>1$. In the indifferent case, we say$z_{0}$

is parabolic if Ais the rootof unity.

The Julia set, denoted by $J(f)$ , is the closure of set of repelling
periodic points. Its complement is called Fatou set, denoted by $F(f)$ . A
connected component of $F(f)$ is called aFatou component D. Sullivan

proved that each Fatou component $U$ is preperiodic, i.e., there exist

integers $m\geq 1$ , $n\geq 0$ so that $f^{m+n}(U)=f^{n}(U)$ and every periodic

Fatou component is either an attracting basin, aparabolic basin, aSiegel disk or aHerman ring. For the classical results of complex
dynamics, see [B], [Mi] and [CG].
This work is supported by National Natural Science Foundation of China, ProjectNo. 10171090

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YESHUN SUN AND YONGCHENG YIN

The post-critical set $P(f)$ of , i.e., the closure of the forward orbits$f$of critical points, will play the crucial rule in the study of complex dythere exists
of is {
namics. Recall that the -limit set$\omega(c)$$\omega$such that$\infty$$z= \lim_{karrow\infty}f^{n_{k}}(c)\}$$z\in\overline{\mathbb{C}}|$$c$$n_{k}arrow$.

Definition 1.1. Arational map$f$:$\overline{\mathbb{C}}arrow\overline{\mathbb{C}}$is called critically non-recurrent if each critical point $c\in J(f)$ is non-recurrent, i.e.,
We equip Riemann sphere$\frac{2}{1+|z|^{2}}|dz|$$\overline{\mathbb{C}}$$c\not\in\omega(c)$.

with the standard spherical metric $ds=$. If we identify the unit sphere$S^{2}$

in$\mathbb{R}^{3}$

with the Riemannsphere, then this metric coincides with the spherical distance and isinvariant under 50(3) as well as $1/z$ . Restricted to any proper compact

subset of Riemann sphere, this metric is equivalent to the Euclidean

metric. Denote$B(x, \epsilon)$

spherical metric on$\epsilon$

the ball centered atandin the Euclidean metric on C.

$D(x, \epsilon)$$\overline{\mathbb{C}}$$x$and of radius

the disk centered at$x$$\epsilon$

in theand of radius

The following definition is given by C. McMullen[Mc].

is shallow if there exists
Definition 1.2. Acompact subset$0<k<1$ such that any ball $B(z, r)$ , where $z\in X$ and $0<r<1$ ,$X\subset\overline{\mathbb{C}}$contains asmaller ball$B(y, kr)\subset\overline{\mathbb{C}}\backslash X$

It’s easy to prove that$2$$\mathrm{i}\mathrm{f}X$.

is shallow, then its box dimension

and its Hausdorff dimension$\mathrm{H}\mathrm{D}(X)\leq \mathrm{B}\mathrm{D}(X)<2$$\mathrm{B}\mathrm{D}(X)<$.

Our first result on critically non-recurrent rational maps is the following.

Theorem 1.1. For a critically non-recur rent rational map f :,$\overline{\mathbb{C}}arrow\overline{\mathbb{C}}$the Julia set $J(f)$ is either shallow or $J(f)=\mathbb{C}$ .

An immediate consequence of Theorem 1.1 is:Corollary 1.1 (Cf. [U]). For a critically non-recurrent rational map$f$:, $HD(J(f))<2$ or$\overline{\mathbb{C}}arrow\overline{\mathbb{C}}$$J(f)=\overline{\mathbb{C}}$.

112
APPLICATIONS OF$\mathrm{M}\mathrm{A}\overline{\mathrm{N}}\acute{\mathrm{E}}’\mathrm{S}$THEOREM

Remark 1.1. C. McMullen [Mc] and F. Przytycki and S. Rohde [PR]proved the same theorems for quadratic polynomials with Siegel diskof bounded type rotation numbers and CoUet-Eckmann rational maps

without parabolic points.

Qusiconformality is arelaxation of conformality in complex analysis.
It turns out to be very useful in the study of complex dynamics andcan be defined in more general setting.
Definition 1.3. Ahomeomorphism $\phi:Xarrow X$ of metric space $(X, d)$is called quasiconformal if there exists aconstant $K$ such that$\lim_{rarrow}\sup_{0}\frac{\max_{d(y\rho)=r}d(\phi(y),\phi(x))}{\min_{d(y,x)=r}d(\phi(y),\phi(x))}\leq K$

for all $x\in X$ .

Aconjugacy quasiconformal off $J(f)$ being actually globally quasiconformal reflects the rigidity of the systems. The next result is about
the rigidity of critically non-recurrent rational maps.

Theorem 1.2. Let be a critically non-recurrent rational map. If theorientation preserving homeomorphism :conjugates to$f$$\phi$

rational map , then$g$$\phi$$\overline{\mathbb{C}}arrow\overline{\mathbb{C}}$

$f$

$a$

is isotopic relative to $J(f)$ to a quasiconformal

conjugacy.

Remark 1.2. P. Hai’ssinsky [H] obtained the same result for uniformly

weakly hyperbolic rational maps defined therein in different way. F.
Przytycki and S. Rohde [PR1] proved it for topological CoUet-Eckmann
rational maps.
Recall that ahomeomorphism $\phi:Xarrow X$ is isotopic to$\psi$: $Xarrow X$

relative to a subset $A\subset X$ if there exists acontinuous map $H$ :$X\cross$$[0,1]arrow X$ such that $H(x, 0)=\phi$ and $H(x, 1)=\psi$ for all $x\in X$

and$H(x,t)=x$ for all $x\in A$ and $0\leq t\leq 1$ .

The Lebesgue measure on Riemann sphere induced by the sphericalmetric is denoted by mes(X) for measurable subset
the full measure of the Riemann sphere is$4\pi$.

. Therefore$X\subset\overline{\mathbb{C}}$113
YESHUN SUN AND YONGCHENG YIN

Definition 1.4. Arational map
set

$A$

$f$

:

$\overline{\mathbb{C}}arrow\overline{\mathbb{C}}$is ergodic if any measurable

satisfying $f^{-1}(A)=A$ has zero or full measure in the sphere.

Now we come to our ergodicity result on critically non-recurrent

rational maps.
,
Theorem 1.3. For a critically non-recurrent rational rnap :either the Julia set $J(f)$ is the whole sphere and is ergodic or the$f$$\overline{\mathbb{C}}arrow\overline{\mathbb{C}}$$f$

Hausforff dimension of the Julia set $J(f)$ is strictly less than 2.
The following notations will be used in the subsequent sections.
(1) Reserve ($d$$d_{E}$

distance on, resp.) to denote the spherical(Euclidean, resp.)$\overline{\mathbb{C}}$( , resp.) and diam $X$ (responding diameter of the subset(2) The distortion Dist(X, )
$z$

set

$X\subset\overline{\mathbb{C}}$($\mathrm{d}\mathrm{i}\mathrm{a}\mathrm{m}_{E}X$$\mathbb{C}$$X\subset \mathbb{C}$$X\subset\overline{\mathbb{C}}$$(\mathrm{D}\mathrm{i}\mathrm{s}\mathrm{t}_{E}(X, z)$(, resp.) the cor-

$X\subset \mathbb{C}$, resp.).

, resp.) of acompact sub-, resp.) about $z\not\in X$ is$\frac{\max_{y\in X}d(y,z)}{\min_{y\in X}d(y,z)}$($\frac{\max_{y\in X}|y-z|)}{\min_{y\in X}|y-z|}$, resp.).

and $B$ are $K$-commensurable or simply
commensurable if $K^{-1}\leq A/B\leq K$ for some $K\geq 1$ indepen-(3) Two positive numbers

dent of$A$

and B.

$A$$A_{\wedge}\vee B$

means

$A$

and

$B$

are commensurable.

2. BACKGROUND MATERIALS

2.1. Distortion lemmas. The distortion theorem of univalent maps isapowerful tool in the study of complex dynamics. The following distortion theorem of the version for -valent maps is well-known(See [CJY]or Appendix$\mathrm{C}$in [ST] .
$)$Lemma 2.1. Let$g:U_{1}arrow U_{2}$$U_{1}$

and$U_{2}$

be simply connected domains in$\overline{\mathbb{C}}$andbe a proper holomorphic map of $degg\leq p$ . Then$\{w|\rho\sigma_{2}(w, g(z_{0}))\leq C^{-1}\}\subset g(\{z|\rho_{U_{1}}(z, z_{0})\leq r\})\subset\{w|\rho_{U_{2}}(w,g(z_{0}))\leq \mathrm{l}$for any$U_{2}$, andare hyperbolic metrics ofis a constant depending only on and .

$z_{0}\in U_{1}$$C$, where$\rho_{U_{1}}$and$\rho_{U_{2}}$

$p$

$r$

$U_{1}$and114
APPLICATIONS OF$\mathrm{M}\mathrm{A}\overline{\mathrm{N}}$\’E’S THEOREM

Denote the hyperbolic metric in $D(0,$ 1) and$\rho$$B_{\rho}(0,$r) the hyperbolicball centered at 0of radius r.

Lemma 2.2. Let $h:D(0,1)arrow D(0,1)$ be a proper holomorphic mapof degree at mostdepending only on$p$

and $h(0)=0$ . Then there exists a constant

$p$

and

$R$

$R^{*}$

such that $D(0, R^{*})\subset h(B_{\rho}(0, R))$ and thecomponent of $h^{-1}(D(0, R^{*}))$ containing 0is a subset of $B_{\rho}(0, R)$ .

Proof Assume $\deg h=ff$$\leq p$

. In the case

$ff$

$=1$

, the conclusion fol-lows immediately from Koebe one-quarter theorem of univalent maps.
Now we suppose $p’\geq 2$ . Let $0=\infty$ , ,, $\#-1$ be the preimages of$z_{1}$$h(z_{0})=0$ and$h_{\mathrm{j}}$: $D(0,1)arrow D(0,1)$ be the univalent map so that$h_{j}(zj)=0,0\leq j\leq ff$ $-1$

. Then$h(z)=e^{\theta}.

for some$\theta\in \mathrm{R}$$\cdots$\prod_{=\mathrm{j}0}^{d-1}h_{j}(z)$.

There are at most $d$ points of $\{z_{j}|0\leq j\leq d -1\}$ in the closed
hyperbolic ball0in

$B_{\rho}(0, R)$$\overline{B_{\rho}(0,R)}$. We can find ahyperbolic circle 7centered atsuch that$\rho(z, z_{j})\geq\frac{R}{2(ff-1)}$

for any$z\in\gamma$

and $0\leq j\leq d$ $-1$ . Then for all$z\in\gamma$,.

$\rho(h_{j}(z), 0)=\rho(h_{j}(z),h_{j}(z_{j}))=\rho(z, z_{j})\geq\frac{R}{2(ff-1)}$

Going back to the Euclidean metric, we have$d_{E}(h_{\mathrm{j}}(z),0) \geq\frac{e^{\frac{R}{2(/-1)}}-1}{e^{\frac{R}{2y-1)}}+1}=C_{1}$.

Then$d_{E}(h(z), 0)= \prod_{j\ovalbox{\tt\small REJECT}}^{p’-1}d_{E}(h_{j}(z),0)\geq C_{1}^{t}=R^{*}(p’, R)$.

and $R^{*}= \min\{R^{*}(ff, R) : 1\leq d \leq p\}$ is the constant needed.

From Lemma 2.2, we have auseful corollary in spherical metric$\square$115
YESHUN SUN AND YONGCHENG YIN

Corollary 2.1. Fix the positive integer

connected subset of$B(w_{0},2\delta)$$\delta<\underline{\pi}-A^{\mathit{6}}2^{\cdot}$$\overline{\mathbb{C}}$

$p$

and

$\epsilon_{0}>0$. Let

missing a spherical disk of radius$\epsilon\circ$$U$

be a simply

and $g:Uarrow$be a proper holomorphic map of $degg\leq p,w_{0}=g(z_{0})$ , and, then there exist
If maps $B(z_{0}, r)\subset U$ into$\mathrm{D}(\mathrm{z}\mathrm{o}, \delta)$$g$constants $K$ depending only on

$p$

and

$\epsilon\circ$, and $r^{*}\geq diamg(B(z_{0}, r))/K$

so that $B(w_{0}, r^{*})\subset g(B(z_{0},r))$ and the component of $g^{-1}(B(w_{0}, r^{*}))$
containing
is a subset of $B(z_{0}, r)$ .
$z_{0}$

Proof. By rotating

$U$

and

$B(w_{0},2\delta)$

disk at infinity and of radius

metric first.
Assumeoint of$\epsilon_{0}$respectively to avoid the spherical, we prove the statement in Euclidean$(\partial g(D(z_{0}, r)))\cap(\partial D(w_{0}, \delta))\neq\emptyset$

containing$g^{-1}(D(w_{0}, \delta))$$\mathrm{p}$Therefore,$\mathrm{d}\mathrm{i}\mathrm{a}\mathrm{m}_{\rho U}D(z_{0}, r)\vee\wedge 1$$z_{0}$

at first. Let $U’$ be the com-

. Then mod$(U\backslash U’)\geq\underline{1}02\pi^{\frac{2}{p}}\mathrm{B}$.

and, from distortion lemmas for uni-

for any point $x\in\partial D(z_{0}, r)$ , for aconstant $C(p)$ depending only on . Take $R= \min_{x\in\partial D(z_{0},r)\rho u(x,z\circ)}$ . By

valent maps,$\rho_{U}(x, z_{0})\wedge\vee 1$$p$such that $D(w_{0}, r^{*})\subset g(D(z_{0}, r))$and the component of $g^{-1}(D(w_{0}, r^{*}))$ containing is asubset of $D(z_{0}, r)$ .

Lemma 2.2, there exists$r^{*}=2\delta R^{*}$$z_{0}$

Denote $K=1/R^{*}$ . ThenIn the case$\leq\delta$$r^{*}\geq \mathrm{d}\mathrm{i}\mathrm{a}\mathrm{m}g(D(z_{0}, r))/K$$\partial g(D(z_{0}, r))\cap\partial D(w_{0}, \delta)=\emptyset$. We replace$U$.

, take $\delta’=d(w_{0}, \partial g(D(z_{0}, r)))$by the component of $g^{-1}(D(w_{0},2\delta’))$ containing$z_{0}$.

Using the same method as above, we get the constant $K$ .
Coming back to the spherical metric, we know the constant $K$ depends also on$\epsilon_{0}$.

$\square$proved a2.2. Mane’s theorem. In [Ma](see also [ST]), R.
beautiful theorem. It is the key lemma in the proof of our main results.
$\mathrm{M}\mathrm{a}\tilde{\mathrm{n}}\acute{\mathrm{e}}$Lemma 2.3 ($\mathrm{M}\mathrm{a}\tilde{\mathrm{n}}$\’e’s theorem). Let $f$ :$\overline{\mathbb{C}}arrow\overline{\mathbb{C}}$be a rational map of

an integer depending only on . If a point$x\in J(f)$ is not a parabolic periodic point and is not contained in the
degree at least two and$\omega$$p$-limit set of a recur rent critical point, then for all$f$$\epsilon>0$

there exists

116
APPLICATIONS OF$\delta>0$$\mathrm{M}\mathrm{A}\overline{\mathrm{N}}$\’E’S THEOREM, such that for each $n\geq 0$ and each connected component $U$ of$f^{-n}(B(x, \delta))$,(1) the spherical diameter of$p$$U$ $is<\epsilon$

and $deg(f^{n} : Uarrow B(x,\delta))\leq$, and(2) for all$\epsilon_{1}>0$

there exists $n_{0}>0$ such that if $n\geq n_{0}$ , thespherical diameter of component

$U$

of

$f^{-n}(B(x,\delta))$$is\leq\epsilon_{1}$.

Note that for acritically non-recurrent rational map, acommon
can be taken in above lemma for all points in acompact subset of Julia$\delta$

set containing no parabolic points.

2.3. Dynamics near parabolic points. Let$f_{0}$:$\overline{\mathbb{C}}arrow\overline{\mathbb{C}}$be rational

map with anon-degenerate parabolic fixed point 0, i.e., $f_{0}’(0)=1$

and $f_{0}’(0)\neq 0$ . Choose a neighborhoodof 0so that
homeomorphically onto aneighborhood .
$U_{0}$$f_{0}$

maps$U_{0}$$U_{0}’$

Lemma 2.4 (Fatou coordinates). Take ,andas above. Then
there exist simply connected domains
compactly contained in, whose union forms a punctured neighborhood of 0, satisfying$f_{0}$$U_{0}$$U_{0}’$$D_{\pm}$$U_{0}\cap$$U_{0}’$$f_{0}^{\pm}(\overline{D}_{\pm})\subset D_{\pm}\cup\{0\}$andMoreover, there exist univalent maps$\bigcap_{n\geq 0}f_{0}^{\pm n}(\overline{D}_{\pm})=\{0\}$$\Phi_{\pm}:$$D_{\pm}arrow \mathbb{C}$.

such that(1) $\Phi_{\pm}(f_{0}(z))=\Phi_{\pm}(z)+1$ ,(2) $Range( \Phi_{+})\supset\{\zeta|-\frac{3}{4}\pi<\arg(\zeta-R) <\frac{3}{4}\pi\}$ and Range(\Phi -)\supset$\{\zeta|\frac{1}{4}\pi<\arg(\zeta+R_{0})<\frac{7}{4}\pi\}$

The domains$D_{+}$

and$D_{-}$

for some $R_{0}>0$ .

in Lemma 2.4 are called attracting petal

and repelling petal respectively. We will make use of this coordinates
to deal with the parabolic points in critically non-recurrent rational

maps.
Set

$\tilde{V}_{0}^{*}=\{\zeta|{\rm Re}\zeta<-R\},\tilde{V}_{0}=\{\zeta|{\rm Re}\zeta<-\mathrm{R}\mathrm{o}, |{\rm Im}\zeta|>R\},\tilde{\Omega}_{0}=$$\{\zeta|-R_{0}-1<{\rm Re}\zeta<-\mathrm{R}\mathrm{o}, |{\rm Im}\zeta|<R_{0}\},\tilde{\Omega}_{j}=\tilde{\Omega}_{0}-j,j\geq 1$$V_{0}^{*}=\Phi_{-}^{-1}(\tilde{V}_{0}^{*})$,$V_{0}=\Phi_{-}^{-1}(\tilde{V}_{0}),\Omega_{j}=\Phi_{-}^{-1}(\tilde{\Omega}_{j}),j\geq 0$

and. From Lemma 2.4

117
YESHUN SUN AND YONGCHENG YIN$V_{0}$

is contained in the Fatou set

and$F(f_{0})=\overline{\mathbb{C}}\backslash J(f_{0})$$f_{0}$:$\Omega_{j+1}arrow\Omega_{j}$is ahomeomorphism for $j\geq 0$ .

2.4. Definition of quasiconformality. In [KK], S. Kallunki and P.
Koskela improved the remarkable result about the definition of quasiconformal map in Euclidean space in [HK]. We state it in Riemman
sphere case with the original inequality replaced by an equivalent condition. This is because that the original inequality appeared in [KK]

holds when$rarrow \mathrm{O}$. Then we may use this inequality in spherical metric.

Moreover, we can replace the basis of disks and its image by that of

uniformly bounded distortion.
By definition, asubset$E\subset\overline{\mathbb{C}}$of$\sigma$-finite length is the countable

union of subsets of finite one dimensional Hausdorff measure in spherical metric(Cf. [G]).
Lemma 2.5. Let:$\phi$$\overline{\mathbb{C}}arrow\overline{\mathbb{C}}$If there areis a homeomorphism.

-finite length and a constant such that there exists
a basis of neighborhoods $\{U(x)\}$ of with Dist(U(x), ) $<K$ and, then is quasiconformal) $<K$ for eachDist((/)(U(x)),
a set $E$ of$K$$\sigma$$x$$x$$\phi(x)$$x\in\overline{\mathbb{C}}\backslash E$$\phi$

3. Proofs OF MA1N RESULTS

3.1. Proof of Theorem 1.1. Suppose thataspherical disk in$\overline{\mathbb{C}}$of radius$\epsilon_{0}$$J(f)\neq \mathbb{C}$. One can finddisjoint from $J(f)$ .

For $x\in J(f)$ , $r>0$ , define$h(x, r)= \sup$Then{ there is $B(y,$ $\xi)\subset F(f)\cap B(x$ , }.

$h:J(f)\cross \mathbb{R}_{+}arrow \mathrm{R}_{+}$

Since int(J(/))$=\emptyset$$r)$$\xi|$

is continuous.

,$h(r)= \inf\{h(x, r)|x\in J(f)\}>0$

for any $r>0$ .

is acritically non-recurrent rational map
having no parabolic periodic points. From Lemma 2.3, there are $0<$

First of all, we suppose

$f$

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APPLICATIONS OF

THEOREM$\mathrm{M}\mathrm{A}\tilde{\mathrm{N}}\acute{\mathrm{E}}’\mathrm{S}$and $p<\infty$ so that for any $x\in J(f)$ , $n>0$ and any$\delta_{0}<\frac{\pi-}{2}\epsilon\Delta$ofaspherical disk of radius ofcomponent$V$$f^{-n}(B(x, 2\delta_{0}))$$\epsilon_{0}$$\deg(f^{n} :,$V$

is simply connected domain missing, andVarrow B(x, 2\delta_{0}))\leq p$.

For any $z_{0}\in J(f)$ and $0<r\leq\delta_{0}$ , look at the forward images$B_{m}=f^{m}(B(z_{0}, r))$Let$n_{0}$, $z_{m}=f^{m}(h)$ , $m\geq 0$ .

be the smallest integer so that$l_{0}\delta_{0}\leq \mathrm{d}\mathrm{i}\mathrm{a}\mathrm{m}B_{n_{0}}\leq\delta_{0}$$\mathrm{d}\mathrm{i}\mathrm{a}\mathrm{m}B_{n\mathrm{o}+1}>\delta_{0}$. Thenfor some $0<l_{0}<1$ . By Corollary 2.1, there are aconstant $K$ depending only on and , and disk $B(R, r_{0})\subset B_{n_{0}}$ withsuch that the component of $f^{-n0}(B(\varphi,r_{0}))$containing
is asubset of $B(h, r)$ . There exists$p$$\epsilon$$r_{0}\geq \mathrm{d}\mathrm{i}\mathrm{a}\mathrm{m}B_{n0}/K\geq l_{0}\delta_{0}$$B(y_{0}, \frac{1}{2}h(r_{0}))\subset$$z_{0}$$B(z_{n_{0}}, r_{0})\cap F(f)$. Lettained in $B(z_{0}, r)$ ,$D_{0}$$y\in D_{0}\cap f^{-n_{0}}(y_{0})$depending only on$M$

be acomponent of$p$and diam. Then$0<k’<1$ which does not depend on$D_{0}\subset B(z_{0}, r)\cap F(f)$$\mathrm{D}\mathrm{i}\mathrm{s}\mathrm{t}(\partial D_{0}, y)\leq M$$D_{0}\vee\wedge r$$z_{0}$

confor some$f^{-n_{0}}(B(y_{0}, \frac{1}{2}h(r_{0})))$. Therefore, there exists

and

$r$

so that $B(y, k’r)\subset$.

Now let be anon-recurrent rational map having parabolic periodic points. For simplicity, we suppose has only one non-degenerate
$f$

$f$

parabolic fixed point 0.

Take$\Omega_{j}(j\geq 0)$

as in Section 2.3. The set$X_{0}=J(f) \backslash (\bigcup_{j\geq 1}\Omega_{j}\cup\{0\})$is acompact subset of $J(f)$ . By Lemma 2.3, there exist$p_{1}<\infty$$V$of $f^{-n}(B(x, 2\delta))$ , $V$ is simply connected andLet$K_{1}$

Varrow B(x, 2\delta))\leq p_{1}$be the constant depending only on$p_{1}$.

and$\epsilon_{0}$Suppose both diam $D$ and diam(f(D)) are less thandisk

$B(z, \epsilon)$with$\epsilon\geq \mathrm{d}\mathrm{i}\mathrm{a}\mathrm{m}$$D/K_{1}$$\delta$in Corollary 2.1.

. If

$B(z, \epsilon)$$K_{1}$, where c’and$\epsilon_{0}$.

$\geq \mathrm{d}\mathrm{i}\mathrm{a}\mathrm{m}$$D$, then $f(D)$ contains adisk

such that the component of $f^{-1}(B(f(z), \epsilon^{*}))$ containing

on

and

so that for all $x\in X_{0}$ , $n>0,0<\delta\leq\delta_{1}$ and any component$\deg(f^{n} :of$\delta_{1}>0$$f(D)/K_{0}$for aconstant$K_{0}$$z$contains a$B(f(z), \epsilon^{*})$

is asubset

depending only

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YESHUN SUN AND YONGCHENG YIN

Choose $N$ large enough so that $N\geq 128K_{0}R\circ$ andfor asuitable
for $n\geq N$ . Take$d(0, \partial\Omega_{n})>$$\delta_{2}=\mathrm{r}\mathrm{j}\mathrm{d}\mathrm{i}\mathrm{a}\mathrm{m}\Omega_{N}$$4K_{0}\mathrm{d}\mathrm{i}\mathrm{a}\mathrm{m}\Omega_{n}$$\eta$

so that$B(x, 2 \delta_{2})\subset V_{0}^{*}=V_{0}\cup(\bigcup_{j\geq 0}\Omega_{j})$for any$x \in\bigcup_{j=1j}^{N}\Omega$.

Now we prove that the Julia set $J(f)$ is shallow.
For any $z_{0}\in J(f)$ and $0<r \leq\delta_{0}=\min(\delta_{1}, \delta_{2})$ , denote $B_{m}=$be the smallest integer$f^{m}(B(z_{0}, r))$ , $z_{m}=f^{m}(z_{0})$ , $m\geq 0$ .
Let$n_{0}$

such that$\mathrm{d}\mathrm{i}\mathrm{a}\mathrm{m}B_{n\mathrm{o}+1}>\delta_{0}$$\mathrm{d}\mathrm{i}\mathrm{a}\mathrm{m}B_{n0}\wedge\vee\delta_{0}$. Then$\mathrm{d}\mathrm{i}\mathrm{a}\mathrm{m}B_{m}\leq\delta_{0}$for $m\leq n_{0}$ and.

If $z_{n0}\in X_{0}$ , then the same argument as in no parabolic periodic
points case shows there exists $B(y, k_{1}r)\subset B(z_{0}, r)\cap F(f)$ for some$0<k_{1}<1$ .

Supposeand $m_{0}\geq 0$ is the smallest integer suchcontains a
for all $m_{0}\leq j\leq n_{0}$ . Thensuch that the component
with$z_{n_{0}} \in(\bigcup_{j\geq 1}\Omega_{j})\cup\{0\}$that$z_{j} \in(\bigcup_{j\geq 1}\Omega_{j})\cup\{0\}$disk$W=B(z_{m_{0}}, \epsilon^{*})$$B_{m_{0}}$$\epsilon^{*}\geq \mathrm{d}\mathrm{i}\mathrm{a}\mathrm{m}B_{m_{0}}/K_{0}$of $f^{-m0}(W)$ containing$z_{0}$

is asubset of $B(z_{0}, r)$ .

We claim that there exists adisk $B(y_{0}, r_{0})\subset B_{m0}\cap F(f)$ such that$f^{-m0}(B(y_{0}, r_{0}))$ has acomponent contained in $B(z_{0}, r)$ for some$r_{0}\wedge\vee$diam$B_{m_{0}}$.

Case 1. If $z_{m0}=0$ , takeforCase 2. If$r_{0}= \frac{1}{2}\epsilon^{*}$$z_{n0}\in\Omega_{j\mathrm{o}}$$B(z_{n_{0}}, \mathrm{z}\mathrm{n}\mathrm{o})\subset V_{0}^{*}$

containing$B_{m0}$.

$1\leq j\mathrm{o}\leq N$, then. Let $B’$ be acomponent of. Then $f^{n0-m0}$ :$B_{n0}\subset B(z_{n_{0}}, \delta_{0})$and$f^{-(n_{0}-m_{0})}(B(z_{n_{0}},2\delta_{0}))$$B’arrow B(z_{n0},2\delta_{0})$is univalent and

l0g2. By distortion theorem of univalent map,,
for some
and there is$\mathrm{m}\mathrm{o}\mathrm{d} (B’\backslash B_{m0})\geq\frac{1}{2\pi}$$\mathrm{D}\mathrm{i}\mathrm{s}\mathrm{t}(\partial W_{1}, z_{n0})\wedge\vee 1$$B(y_{0}^{*}, r_{0}^{*})\subset W_{1}\cap F(f)$$r_{0}^{*}\vee\wedge\delta_{0}$where $W_{1}=f^{r\iota 0-m0}(W)$ . Hence there is $B(y_{0}, r_{0})\subset W\cap F(f)$ for some.
$r_{0\wedge}\vee \mathrm{d}\mathrm{i}\mathrm{a}\mathrm{m}B_{m0}$for $j_{0}>N$,
Case 3. If
When diam$z_{n0}\in\Omega_{j\mathrm{o}}$$\mathrm{r}\mathrm{g})$take$z_{m0}\in\Omega j\mathrm{o}+(n_{0}-m_{0})=\Omega:_{0}$$B_{m_{0}}>2K_{0}\mathrm{d}\mathrm{i}\mathrm{a}\mathrm{m}\Omega:_{0}$$r_{0}= \frac{1}{4}d(z_{m_{0}}, \partial W)$

.

,

.

$d(z_{m_{0}}, \partial W)>2\mathrm{d}\mathrm{i}\mathrm{a}\mathrm{m}\Omega:_{0}$,120
APPLICATIONS OF(3b). When diam$V^{*}$

. Let

Then

$B_{m0}\leq 2K_{0}\mathrm{d}\mathrm{i}\mathrm{a}\mathrm{m}\Omega_{\dot{\infty}}$$\mathrm{D}\mathrm{i}\mathrm{s}\mathrm{t}_{E}(\partial\tilde{W},\tilde{z}_{m_{0}})\leq 16$

such that$W\cap F(f)$If,$D=\mathrm{B}(\mathrm{z}\mathrm{O}, 2\mathrm{d}\mathrm{i}\mathrm{a}\mathrm{m}B_{m_{0}})\subset$$\tilde{D}=\Phi_{-}(D),\tilde{W}=\Phi_{-}(W),\tilde{W}_{1}=\Phi_{-}(W_{1})$$\mathrm{I}\mathrm{f}d_{E}(\tilde{z}_{m_{0}}, \partial\tilde{W})>4R_{0}$$\subset\tilde{V}_{0}$\’E’S THEOREM$\mathrm{M}\mathrm{A}\tilde{\mathrm{N}}$for

Dist(C?lY

diam$d_{E}(\tilde{z}_{m_{0}},\partial\tilde{W})\leq 4R_{0}$$B_{m_{\mathrm{O}}}$$\tilde{B}=D(\tilde{y}_{0}, \frac{1}{2}d_{E}(\tilde{z}_{m_{0}},\partial\tilde{W}))$

contains adisk $B(y_{0},r_{0})\subset W\cap V_{0}\subset$.

, then$\tilde{W}\subset D(\tilde{z}_{m_{0}},64K_{0}R)$and$\tilde{W}_{1}=\tilde{W}+$. Since$(n_{0}-m_{0})\subset D(\tilde{z}_{n_{0}},64K_{0}R\Uparrow),\tilde{z}_{n0}=\tilde{z}_{m0}+(n_{0}-m_{0})=\Phi_{-}(z_{n_{0}})$$\tilde{z}_{n_{0}}\in\tilde{\Omega}_{j_{0}}$,$j_{0}>N\geq 128\mathrm{K}\mathrm{o}\mathrm{R}\mathrm{o}$$D’=\Phi_{-}^{-1}(\Pi)\sim\subset V^{*}$

Hence Dist
$16^{2}K_{0}$,.

$\tilde{z}_{m_{0}}=\Phi_{-}(z_{m0})$) $\leq 16K_{0}$ .

, then there is adisk$\Phi_{-}^{-1}(\tilde{B})=B$$r_{0\wedge}\vee$$z_{m_{0}}$and, we have$B_{n_{0}}\subset D’$$\subset\tilde{V}_{0}^{*}$,$\mathrm{m}\mathrm{o}\mathrm{d} (D \backslash B_{n_{0}})\geq\frac{1}{2\pi}\log 2$.

$\tilde{D}’=D(\tilde{z}_{n_{0}},128\mathrm{K}\mathrm{o}\mathrm{R}\mathrm{o},$and$(\partial W_{1}, z_{n_{0}})\leq 16\mathrm{D}\mathrm{i}\mathrm{s}\mathrm{t}_{E}(\partial\tilde{W}_{1},\tilde{z}_{n_{0}})=16\mathrm{D}\mathrm{i}\mathrm{s}\mathrm{t}_{E}(\partial\tilde{W},\tilde{z}_{m_{0}})\leq$.

Let $D’$ be the component of$f^{n\mathrm{o}-m_{\mathrm{O}}}$

containing$f^{-(n_{\mathrm{O}}-m_{0})}(D’)$$B_{m_{0}}$. Then: $D’arrow D’$ is univalent. By distortion theorem, there is adisk$B(y_{0},r_{0})\subset W\cap F(f)$for some$r_{0\wedge}\vee$diam$B_{m_{0}}$.

The proof of our claim is completed.
Now we come back to the proof of the main theorem.
If $m_{0}=0$ , we are done. If $m_{0}\geq 1$ , thencontains adisk with radius$r_{0}’\wedge\vee \mathrm{d}\mathrm{i}\mathrm{a}\mathrm{m}B_{m\mathrm{o}-1}$$B_{m\mathrm{o}^{-1}}\cap f^{-1}(B(y_{0}, \mathrm{r}\mathrm{O})$. The same argument as inno parabolic points case shows that there is $B(y, k_{2}r)\subset B(z_{0},r)\cap F(f)$for some $0<k_{2}<1$ . Take $k’= \min(k_{1}, k_{2})$ . Then for any$z_{0}\in$$J(f)$ ,$r>0$ , $B(z_{0},r)$ contains adisk $B(y, k’r)\subset F(f)$ and hence $J(f)$is shallow.

3.2. Proof of Theorem 1. 2. Suppose that and are two critically
non-recurrent rational maps which are conjugated by ahomeomor$f$

phism$\psi$$J(f)$ to. By aroutine argument on Fatou set,anew conjugacy

We shall prove thatLet$\phi$$\phi$$g$$\psi$

is isotopic relative towhich is quasiconformal on Fatou set $F(f)$ .

is actually aglobally quasiconformal map.

$X_{0}(f)\subset J(f)(\mathrm{r}\mathrm{e}\mathrm{s}\mathrm{p}.

X_{0}(g)\subset J(g))$

fined in the proof of Theorem 1.1. Let$\delta_{f}$be the compact sets de( , resp.)$\delta_{g}$$< \frac{\pi}{2}$(We will appl

121
YESHUN SUN AND YONGCHENG YIN

Corollary 2.1 for$\epsilon_{0}=\frac{\pi}{2}$) and integers $p(f)$ ($p(g)$ , resp.) be the num-bers guaranteed by Mane’s theorem for $X_{0}(f)$ ( $X_{0}(g)$ , resp.) and for. In cases$X_{0}(f)=J(f)$ .
$\epsilon=\frac{\pi}{2}$Take asmall$B( \phi(x), \frac{1}{2}\delta_{g})$$J(f)=\overline{\mathbb{C}}$and $J(f)$ containing no parabolic points, let$0<r_{0}< \frac{1}{2}\delta_{f}$such that for all $x\in J(f)$ ,$\phi(B(x, r_{0}))\subset$.

be acomponent of $f^{-n}(B(f^{n}(x),r_{0}))$ condoes not depend on such thattaining . There is aconstant. If is not eventually mapped to the)Dist(\phi (B(y, )),parabolic periodic points, it visits $X_{0}(f)$ infinitely many times. It fol-

Let $x\in J(f)$ and$U_{n}(x)$$K_{1}$$x$$r_{0}$$\leq K_{1}$$\phi(y)$

$y$

$x$

lows from the distortion theorem for -valent version that there are
and aconstant $K$ independent of such that
subsequence
$p$

$x$

$\{n_{k}\}$$\mathrm{D}\mathrm{i}\mathrm{s}\mathrm{t}(U_{n_{k}}(x),x)\leq K$

and$\mathrm{D}i\mathrm{s}\mathrm{t}(\phi(U_{n_{k}}(x)), \phi(x))\leq K$

It then follows from Lemma 2.5 that$\phi$:.

$\overline{\mathbb{C}}arrow\overline{\mathbb{C}}$is quasiconformal.

The proof is complete.

3.3. Proof of Theorem 1.3. By Corollary 1.1, we assume the Julia
set of is the whole sphere.
be completely invariant subset. Suppose that mes(X)
Let
$f$

$>$

$X\subset\overline{\mathbb{C}}$$0$. We shall prove that$X$

has the full measure.

Let $x_{0}\in X$ be adensity point of $X$ . Let $x_{n}=f^{n}(x)$ , $n>0$ . We
converging to . Let $A=\{x_{n_{k}}\}\cup$
can choose asubsequence. Then is aproper compact subset of $J(f)$ . Let and be$x^{*}$$\{x_{n_{k}}\}$$\{x^{*}\}$$A$$\delta$in Mane’s theorem with -neighborhood ofbe the component ofandmissing aspherical disk of radius

the numbers given for
$A$

$p$

$f^{-n_{k}}(B(x_{n_{k}}, \frac{1}{2}\delta))$$\mathrm{D}\mathrm{i}\mathrm{s}\mathrm{t}(U_{n_{\mathrm{k}}}, x_{0})\leq K$$\epsilon$$\mathrm{e}$$U_{n_{\mathrm{k}}}$$\epsilon_{0}$containing$x_{0}$. Then$U_{n_{k}}$, $K$ depending only on$p$shrinks toand$\lim_{karrow\infty}\frac{\mathrm{m}\mathrm{e}\mathrm{s}(U_{n_{k}}\cap X)}{\mathrm{m}\mathrm{e}\mathrm{s}(U_{n_{k}})}=1$,$\epsilon_{0}$$x_{0}$

nicely since. Therefore,

122
APPLICATIONS OF$\mathrm{M}\mathrm{A}\tilde{\mathrm{N}}\acute{\mathrm{E}}’\mathrm{S}$THEOREM

which is equivalent to$\lim_{karrow\infty}\frac{\mathrm{m}\mathrm{e}\mathrm{s}(U_{n_{\mathrm{k}}}\cap(\overline{\mathbb{C}}\backslash X))}{\mathrm{m}\mathrm{e}\mathrm{s}(U_{n_{k}})}=0$.

It folows from distortion theorem that,,$\lim_{karrow\infty}\frac{\mathrm{m}\mathrm{e}\mathrm{s}(B(x_{n\iota},\frac{1}{2}\delta)\cap(\overline{\mathbb{C}}\backslash X))}{\mathrm{m}\mathrm{e}\mathrm{s}(B(x_{n_{\mathrm{k}}},\frac{1}{2}\delta))}=0$

which is equivalent to,$\lim_{karrow\infty}\frac{\mathrm{m}\mathrm{e}\mathrm{s}(B(x_{n_{k}},\frac{1}{2}\delta)\cap X)}{\mathrm{m}\mathrm{e}\mathrm{s}(B(x_{n_{k}},\frac{1}{2}\delta))}=1$

and hence$\frac{\mathrm{m}\mathrm{e}\mathrm{s}(B(x^{*},\frac{1}{2}\delta)\cap X)}{\mathrm{m}\mathrm{e}\mathrm{s}(B(x^{*},\frac{1}{2}\delta))}=1$

This implies that

Since

$B(x^{*}, \frac{1}{2}\delta)\subset X\mathrm{a}.\mathrm{e}$$B(x^{*}, \frac{1}{2}\delta)\subset\overline{\mathbb{C}}=J(f)$. Hence$f^{k}(D(x^{*}, \frac{1}{2}\delta))=\overline{\mathbb{C}}$.

.

, there exists an integer$k$. This means that$\overline{\mathbb{C}}\subset X\mathrm{a}.\mathrm{e}$such that$X$

has fallmeasure and completes the proof.
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MATHEMATICS, ZHEJIANG UNIVERSITY, HANGZHOU, ZHE-JIANG 310027, CHINA-mail address: sunQmath. zju. edu.cn$E$

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