数理解析研究所講究録 1220 巻 2001 年 128-140

128

On perturbations of rational maps andon the Julia sets
construction of semiconjugacies
集合上の半共役の構成について)(有理写像の摂動と Julia

Tomoki KAWAHIRA (川平 友規)
University of Tokyo*

E-mail:kawahira@ms.u-tokyo.ac.jp

Abstract
In this note, we investigate perturbations of parabolic rational mapson the Riemann sphere, and dynamical stability of their Julia sets. A
rational map is called parabolic if every critical point is contained in
the Fatou set. If aperturbation of into another parabolic rational
map is horocyclic, then we can construct asemiconjugacy on their Julia
sets. This means that parabolic rational maps have weak J-stability
$f$

$f$

1J-stability
the space of all
be arational map of degree $d\geq 2$ , andLet :rational maps of degree . The topology of this space is defined by the uniform.
convergence on the sphere measured by the spherical distanceparabolicIn this note, we discuss perturbations of a rational map (especially, and study the dynamical stability of on therational map) within
Julia set $J(f)$ :That is, structural stability of restricted on the Julia set.
Here perturbation of means a family of rational maps
. We represent this family as thesatisfying $f_{0}=f$ and.
form of convergence,
About this, the result below is famous:$f$$\hat{\mathbb{C}}arrow\hat{\mathbb{C}}$$\mathrm{R}\mathrm{a}\mathrm{t}_{d}$$d$$d_{\sigma}(\cdot, \cdot)$

$f$

$f$

$\mathrm{R}\mathrm{a}\mathrm{t}_{\mathrm{d}}$

$f$

$f$

$\{f_{\epsilon}\in \mathrm{R}\mathrm{a}\mathrm{t}_{d} :\epsilon\in[0,1]\}$$d_{\sigma}(f_{\epsilon}, f)arrow \mathrm{O}(\epsilonarrow 0)$$f_{\epsilon}arrow f$-Sad-Sullivan[10])1 If f has a connected neighborTheorem 1.1 (w here the number of attracting cycles is locally constant, thenhood U$\mathrm{M}\mathrm{a}\tilde{\mathrm{n}}\acute{\mathrm{e}}$$\subset \mathrm{R}\mathrm{a}\mathrm{t}_{d}$Research partially supported by JSPS Research Fellowships for Young Scientists$2\mathrm{T}\mathrm{h}\mathrm{e}$original theorem is much better

129
for eachthat is,$f_{\epsilon}\in U$

there exists a quasiconforrmal conjugacyon.

$h_{\epsilon}\circ f_{\epsilon}=f\circ h_{\epsilon}$$h_{\epsilon}$:$J(f_{\epsilon})arrow J(f)_{f}$$J(f_{\epsilon})$This means that the dynamics on the Julia set varies continuously forany perturbation. We say such arational map is -stable. For example,
hyperbolic rational maps are J-stable.
$f$

2

$J$

Parabolic bifurcation

Let be aparabolic periodic point of period 1with multiplier $(f^{l})’(a)=\lambda$
such that. Then we can take alocal coordinate near such that is
mapped to 0and that$a$$\lambda^{q}=1$

$a$

$a$

$f^{lq}(z)=z+z^{p+1}+O(z^{p+2})$

where is amultiple of and is unique for . 2 We call the petal ember ofand is denoted by $p(a)$ . Note that has multiplicity $p+1$ as afixed point of(See the left figure in Figure 2). Then by a perturbation of , aparabolic
cycle may split into $p+1$ cycles (maybe attracting, repelling, indifferent) withmultiplicity, and the dynamics may change not only locally but also globally.
For example, let us consider perturbations of aquadratic polynomial $f(z)=$$z+z^{2}$ , which has a parabolic
fixed point with the petal number 1 at the origin.
$p$

$a$

$q$

$a$

$n$

$p$

$a$

$f^{lq}$$f$(1)$f_{\epsilon}(z)=z+z^{2}-\epsilon(\epsilon[searrow] 0)$(2)$f_{\epsilon}(z)=z+z^{2}+\epsilon(\epsilon[searrow] 0)$

Under the perturbation (1), the parabolic point 0splits into an attracting
fixed point
and arepelling fixed point(In this case, the Julia setsvary continuously). Note that the number of attracting cycles is locally nonconstant.
Under the perturbation (2), the parabolic point 0splits into apair ofrepelling fixed points. (See Figure 1. In this case, the Julia sets vary
discontinuously! [3]Then let us consider:$-\sqrt{\epsilon}$$\sqrt{\epsilon}$$\pm\sqrt{\epsilon}i$$)$

Problem. For a rational map that has parabolic points, find aMSS-liketheorem (or some kind of -stability)by controlling the parabolic bifurcations.
$J$$2\mathrm{S}\mathrm{e}\mathrm{e}$[ , II.5] for basic properties of parabolic points
$1$

130

Figure 1:,$\mathrm{T}1_{1}\mathrm{e}^{s}$

Julia sets of.

$z+z^{\underline{\lambda}}$a11$z+z^{2}+().()\mathrm{t}\mathrm{I}1$.

3Parabolic rational maps and horocyclic pcrturbation
problem, let us introduce theForclass of rational maps
have
. are contained in, if all critical points ofparabolic points. is called$F(f)$ . By Sullivan’s classification of Fatou
ellts and their properties,
aparabolic rational map can havealld parabolic basins, but
no Siegel disks or He rman rings. Especially, hyPe rbolic rational maps
parabolic. Note that ally orbit of $z\in F(f)$ is attracted to an attracting or
parabolic cycle.
Next, to control the parabolic bifurcations, let us introduce some conditions
for perturbations. Tllcll we will be able
control the parabolic bifurcation soparabolic points change gently.
tllat the local dynamics.
Aperturbation is horocyclic if each parabolic point of satisfies following(:$\mathrm{t}1_{1}\mathrm{e}$$\mathrm{t}1_{11}\dot{‘}\mathrm{t}$$\mathrm{s}\mathrm{i}_{111}1^{\mathrm{J}}1\mathrm{c}^{1}\mathrm{s}\mathrm{t}$$p(\iota r.abolic$

$f$

$f$

$(j011\downarrow 1)\mathrm{O}\mathrm{U}^{\cdot}$$(\mathrm{s}\mathrm{u}\mathrm{I})\mathrm{c}^{1}\mathrm{r})\mathrm{a}\mathrm{t}\mathrm{t}\mathrm{r}\mathrm{a}\mathrm{c}^{1}\mathrm{t}\mathrm{i}_{1\mathrm{l}}\mathrm{g}$$\mathrm{a}1^{\cdot}(^{\backslash }$$|_{1}0$$\mathrm{n}\mathrm{t}^{\mathrm{J}},\mathrm{a}\mathrm{I}^{\cdot}$

$a$

$f$

$j011(1\mathrm{i}1.\mathrm{i}\iota)\mathrm{n}.\mathrm{s}^{\backslash }$$(i\iota)$

If$‘\iota$$p\succ \mathrm{t}1_{1}$

(1)

$)$

is period androot of llllity;$\mathrm{T}11\mathrm{t}^{\backslash },1^{\cdot}\mathrm{t}^{1}$$l$.

$11i\iota \mathrm{s}$. are fixed points$‘\iota_{\epsilon}arrow‘\iota\dot{\mathrm{e}}\iota 1\mathrm{l}\mathrm{d}$(c) If we set$\mathrm{c}\mathrm{x}\mathrm{l}$$\lambda_{\epsilon}arrow\lambda$$l^{l}$$1$)($\}\mathrm{t}\dot{‘}\iota \mathrm{l}\mathrm{s}$, its l1lllltil)lit:r$(f^{l})’(‘\iota)=\lambda$$a_{\epsilon}()\mathrm{f}f_{\epsilon}^{l}.\mathrm{w}\mathrm{i}\mathrm{t}]_{1111}\mathrm{m}\mathrm{u}1\mathrm{t}\mathrm{i}\mathrm{p}1\mathrm{i}\mathrm{e}\mathrm{r}\mathrm{s}$is aprimitive$(f_{\epsilon}^{l}.)’(a_{\epsilon})=\lambda_{\epsilon}$satisfying; allCl) $(L_{\epsilon}+i\theta_{\iota}):=\lambda_{\epsilon}/\lambda$, which tends to 1, then$\theta_{\epsilon}^{2}=o(L_{\mathrm{t}})’$.

Horocyclic perturbation is originally defined by C.
as horocyclicof rational lllal) [ , j7-9]. He defined it under more general canditions than the definition above. (For instance, under the original definition,
we need not assume that the multiplier of aparabolic point of with petal is$\mathrm{M}\mathrm{c}\mathrm{j}\mathrm{M}\mathrm{u}11\mathrm{c}_{\mathrm{J}}^{1}11$$\mathrm{J}$$conve7^{\backslash }/‘ er\iota ce$$\mathrm{s}$

$f$

$p$

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aprimitive -th root of unity.) Though we use the stronger conditions for simplicity, we can prove the main theorem in the next section under the original
definition.
Let us consider the effects of horocyclic perturbations on the local dynamics
near , and their representation.
By the condition (b) of horocyclic perturbation, if is sufficiently small,.
of with the same period andis perturbed into aperiodic point
converges to A.
Moreover, the multiplierconverges uniformly to near , for each , we can take alocal
As
coordinate which maps to 0and converges uniformly to that of . Thus we
obtain alocal representation of the convergence;
$p$

$a$

$a$

$\epsilon$$l$$f_{\epsilon}$$a_{\epsilon}arrow a$$a_{\epsilon}$$(f_{\epsilon}^{l})’(a_{\epsilon})=\lambda_{\epsilon}$$f$$f_{\epsilon}$$a$$f_{\epsilon}$$f$$a_{\epsilon}$$f_{\epsilon}^{lq}(z)=\lambda_{\epsilon}z+A_{\epsilon}z^{r}+O(z^{r+1})$$arrow f^{lq}(z)=z+z^{p+1}+O(z^{p+2})$$(\epsilonarrow 0)$.
where $2\leq r\leq p$ and
Now let us consider acoordinate change by$A_{\epsilon}arrow \mathrm{O}$$\zeta=\phi_{\epsilon}(z)=z-B_{\epsilon}z^{r}$,$B_{\epsilon}= \frac{A_{\epsilon}}{\lambda_{\epsilon}(\lambda_{\epsilon}^{r-1}-1)}$.

By the condition (a) of horocyclic convergence (the multiplier Ais aprimitive. Thus we may suppose
for all-th root of unity), we obtain
uniformly near the origin. For each , changing the coordinate
thatby , we obtain.
$\epsilon<<1$$\lambda_{\epsilon}^{r-1}\neq 1$$P$$\phi_{\epsilon}arrow id$$\epsilon$$\phi_{\epsilon}$$\phi_{\epsilon}\mathrm{o}f_{\epsilon}\mathrm{o}\phi_{\epsilon}^{-1}(\zeta)=\lambda_{\epsilon}\zeta+O(\zeta^{r+1})$

So we can continue the discussion replacing with $r+1$ until $r\leq p$ . By composition of the finite number of coordinate changes, we obtain the normalized
form of convergence:$r$$f_{\epsilon}^{lp}(z)=\lambda_{\epsilon}^{p}z+z^{p+1}+O(z^{p+2})arrow f^{lp}(z)=z+z^{p+1}+O(z^{p+2})$

.

(2.1)

This property is important to keep the symmetry of the dynamics for eachpetals.

normalized form as (2.1) even if Ais not a
We can obtainthendoes notprimitive -th root of unity: In fact, if
diverges and thus we can apply this discussion. See [9, \S 7].
Remark

$3.\mathrm{a}$$p$$[perp] \mathrm{u}\mathrm{h}\mathrm{e}$$A_{\epsilon}/(\lambda_{\epsilon}^{p}-1)=O(1)$$B_{\epsilon}$

Next we consider the condition (c) of horocyclic perturbation. Let us set. The geometric meaning of the relationis asfollows: If we fix apair of arbitrary small closed disks which are tangent to the$\lambda_{\epsilon}/\lambda=\exp(L_{\epsilon}+i\theta_{\epsilon})$$\theta_{\epsilon}^{2}=o(L_{\epsilon})$

132
imaginary axis at the origin for the both sides of the axis, then they contain. Equivalently, if
implies. By this relation,
for allthenis aparabolic point of with the same multiplier Aas .
Thus can not have irrationally indifferent periodic points.
near the origin, we obtain following
By solving the equation
three cases:$L_{\epsilon}=0$$\epsilon\ll 1$$L_{\epsilon}+i\theta_{\epsilon}$$|\lambda_{\epsilon}/\lambda|=1$$\theta_{\epsilon}=0$$a$$f_{\epsilon}$$a_{\epsilon}$$f_{\epsilon}$$f_{\epsilon}^{lp}(z)=z$

(1)
(2)

$a_{\epsilon}$is aparabolic point with$p$petals and the multiplieris an attracting point, and there arepoints near ;or$p$$a_{\epsilon}$$\lambda_{\epsilon}=\lambda$;orsymmetrically arrayed repelling$a_{\epsilon}$(3)

is arepelling point, and there arepoints near .
$a_{\epsilon}$$p$

symmetrically arrayed attracting$a_{\epsilon}$with $p=3$
Figure 2: Horocyclic perturbation of aparabolic fixed point of
(1)
right
and theone shows the case (3).
petals. The left figure shows the case$f^{lp}$For (2) and (3), if $p>1$ , these symmetrically arrayed periodic points have
. Moreover, they are contained
the same period and the multipliers. We call them thewith radiusin an open ball centered at,(2)and (3) are equivalent;
itself the planet If $p=1$
satellites of andthat is, splits into apair of attracting and repelling points. Thus we formallydefine the satellite by attracting one and the planet by repelling one. For (1),
we also call the planet, although it has no satellite.
Using these properties, we can obtain akey lemma of horocyclic perturbations. We define the cycle of by the finite orbit of , say$lp$$\approx\lambda_{\epsilon}^{-p^{2}}$$a_{\epsilon}$$a_{\epsilon}$$O((1-\lambda_{\epsilon}^{p})^{1/p})$$a_{\epsilon}$$a$$a_{\epsilon}$

$a$

$a$

$\alpha:=\{a, f(a), \ldots, f^{i-1}(a)\}$.

133
Now we assume that is parabolic and we call a a parabolic cycle.
Let us fix anwhose orbit accumulates to . For an arbitrary small$\delta>0$ ,set. (Hereis the open ball centeredatwith radius measured by the spherical metric.) Then we can take$N_{0}=N_{0}(x, \delta)>>0$ such thatfor any $n\geq N_{0}$ . The key lemma is:$a$$x\in\hat{\mathbb{C}}$$\alpha$$\triangle=\Delta(\delta):=\bigcup_{a\in\alpha}B_{\sigma}(a, \delta)$$B_{\sigma}(a, \delta)$$\delta$$a$$f^{n}(x)\in\triangle$

Lemma 3.1 If
horocyclically and$n\geq
N$ .
that
any
for
$f_{\epsilon}arrow f$$\epsilon\ll 1$, there exists an $N\geq N_{0}$ such$f_{\epsilon}^{n}(x)\in\Delta$

This means that the change of local dynamics by the perturbation is controlled within . This fact is essential for the construction ofmentioned
afterward. The proof is shown in [8].
$\triangle$$\Omega_{\epsilon}$

4

Main results

Our main result is:
Theorem 4.1 (Weak -stability)Let $f$ :be a parabolic rational
map of degree $d\geq 2$ anda horocyclic perturbation.
If sufficiently small, then we can construct a map
with following properties:$J$$\hat{\mathbb{C}}arrow\hat{\mathbb{C}}$$f_{\epsilon}arrow f$$\epsilon\dot{u}$

(i)

$h$

$h_{\epsilon}=h:J(f_{\epsilon})arrow \mathrm{J}(\mathrm{f})$is continuous and surjective.

(ii) For any$x\in J(f_{\epsilon})$,$f\circ h(x)=h\circ \mathrm{f}\mathrm{n}(\mathrm{x})$.

(iii) $If|h^{-1}(y)|\geq 2$ for some $y\in J(f)$ , then the fomard orbit of lands onparabolic periodic point of $f$ , say , and $|h^{-1}(y)|$ corresponds to the petal
number of .
$y$

$a$

$a$

$a$

The properties (i) and (ii) mean that gives asemiconjugacy between
and $J(f)$ . The property (iii) means that the subset ofwhere is not onet0-0ne is either countable or empty. If it is countable non-empty set, $h^{-1}(y)$is consist of the preimages of repelling satellites of an attracting planet whichis generated by the perturbation of . If it is emPty, none of parabolic points
is perturbed into an attracting planet andgives atopological conjugacy
between the Julia sets.
Furthermore, we can conclude the Hausdorff convergence of the Julia sets.
Let us ffix an arbitrary small $r>0$ . If is sufficiently small, we can construct
the semiconjugacy to satisfy $\sup\{d_{\sigma}(h(x), x) : x\in J(f_{\epsilon})\}<r$ . Hence weobtain:$h$$J(f_{\epsilon})$$J(f_{\epsilon})$

$h$

$a$

$h$

$\epsilon$$h$Corollary 4.2 If is parabolic andin the Hausdorff topology$f$$f_{\epsilon}arrow f$horocyclically, then$J(f_{\epsilon})arrow J(f)$134
Remarks.
1. If arational map has no Siegel disks or Herman rings and $f_{n}arrow f$horocyclically, it is known that $J(f_{n})arrow J(f)$ in the Hausdorff topology[7], [9, Thm.9.1]. By this fact, we can obtain Corollary 4.2, because aparabolic rational map has no Siegel disks or Herman rings. However,the proofs are given by different ways.
$f$

2. We call arational map is geometrically finite if every critical point
in $J(f)$ is eventually periodic. Note that parabolic rational maps aregeometrically finite. G. Cui[2] showed that ageometrically finite rati0nal map has aperturbation into acontinuous family of geometricallyfinite rational maps with topological conjugacies on their Julia sets. To
construct this perturbation, he used the technique of pinching deformagave anothertion. For geometrically finite polynomials, P.
approach using -deformation. These results and our Theorem 4.1 partially solve the Goldberg-Milnor conjecture in [5].
$f$$\mathrm{H}\dot{\mathrm{a}}\mathrm{i}\mathrm{s}\mathrm{s}\mathrm{i}\mathrm{n}\mathrm{s}\mathrm{k}\mathrm{y}[6]$$\mathrm{q}\mathrm{c}$5Survey of the proof
In this section, we give asurvey of the proof of Theorem 4.1. See [7] or [8] formore details.
Stepl: Construction of and . Let f be aparabolic rational map and
A the finite set of all parabolic points of f.
$\Omega$$\rho$

Proposition 5.1 There exist a finitely connected compact set
piecewise continuous metriC with following properties:$\Omega\subset\hat{\mathbb{C}}$

and

$a$

$\rho$

1. $\Omega\cap P_{0}(f)=A$ .

2.
S.

$J(f)\subset\Omega$and$f^{-1}(\Omega)\subset \mathrm{I}\mathrm{n}\mathrm{t}(\mathrm{f}\mathrm{i})$$\cup A$.

is defined on Int(fi) and small disk neighborhoods for each parabolic
point of .

$\rho$$f$

4. For every$C^{1}$

curve$\eta\subset f^{-1}(\Omega)$,$1\mathrm{e}\mathrm{n}\mathrm{g}\mathrm{h}_{\rho}(f\circ\eta)>1\mathrm{e}\mathrm{n}\mathrm{g}\mathrm{h}_{\rho}(\eta)$

So

$f$

is expanding for$\rho$.

in the sense of this inequality

135
Proof. To construct , we need to remove the orbits of critical points offrom the sphere. First, remove small disks around the attracting cycles, andsmall attracting flowers around parabolic cycles. Next, remove finite number of
disk-neighborhoods for the critical orbits which have not been removed. Since
we can take such disks and flowers so that the images of them are strictly
contained in themselves, the remained set satisfies the conditions of Q.
One can find the details of construction of the metric in [11, Step 4] (inthe case of geometrically finite rational maps). See also [4, Expose’ No.X] and. Here we only sketch the idea of the construction.
Letbe the Poincar\’e metric of. Since this metric diverges
near , any curve interminating at has infinite length with respect. So we need to modify
toso that such curves have finite lengths.
For sufficiently small $\delta>0$ and each $a\in A$ , we set
and$D$. Note thatis a finite union of narrow cusps near repelling
directions. Thus on each
, we can take asuitable local coordinate
suchthat is expanding from the metricto the metricon.
Furthermore, we take sufficiently large $M>0$ and smaller if necessary, sothat is expanding fromtoon $f^{-1}(\Omega\cap D_{a})-V$ for any $a\in A$ .
Then we can define the metric byon eachand byotherwise.
$\Omega$$f$$\rho$$[1, \mathrm{V}.4.]$$U:=\mathrm{I}\mathrm{n}\mathrm{t}(\Omega)$$\rho u$$A$

$A$$f^{-1}(\Omega)$$\rho_{U}$$\rho u$$D_{a}:=B_{\sigma}(a, \delta)$$\Omega\cap D$$:= \bigcup_{a\in A}D_{a}$$D_{a}$$\zeta_{a}$$f$$|d\zeta_{a}|$$\Omega\cap D$$|d\zeta_{f(a)}|$$\delta$$f$$M|d\zeta_{a}|$$\rho_{U}$$\min\{\rho_{U}, M|d\zeta_{a}|\}$$\rho$$D_{a}$$\rho_{U}$$\blacksquare$: Construction ofand the “0-th” map . Next we constructacompact set
corresponding to , and the correspondence is represented
by the map:.
$\mathrm{S}\mathrm{t}\mathrm{e}\mathrm{p}2$$\Omega_{\epsilon}$$h_{0}$$\Omega$$\Omega_{\epsilon}$$h_{0}$$\Omega_{\epsilon}arrow\Omega$

Proposition 5.2 For, there exist a compact set
uous map:with following properties:$\epsilon\ll 1$$h_{0}$

and a contin-$\Omega_{\epsilon}arrow\Omega$

1.

$\Omega_{\epsilon}\cap P_{0}(f_{\epsilon})$2.

$J(f_{\epsilon})\subset\Omega_{\epsilon}$3.

$h_{0}$:$\Omega_{\epsilon}\subset\hat{\mathbb{C}}$is the set of all parabolic points ofand$\Omega_{\epsilon}arrow\Omega$$f_{\epsilon}^{-1}(\Omega_{\epsilon})\subseteq\Omega_{\epsilon}$$f_{\epsilon}$.

.

is surjective.

4. If there exists

such $that|h_{0}^{-1}(y)|\geq 2$ then is a parabolic point and$|h_{0}^{-1}(y)|=p(y)$ . By the $pe\hslash urbation$,
splits into an attracting planet
and $p(y)$ repelling satellites which coincide with.
$y\in\Omega$

$y$

$y$

$h_{0}^{-1}(y)$Moreover,if we fix an arbitrary small $r>0$ , then we can make.
for all$\sup\{d_{\sigma}(h_{0}(x), x) :x\in\Omega_{\epsilon}\}<r$$h_{0}$

satisfy$\epsilon<<1$

We can constructby modification of near the parabolic cycles. For
this, we need ahelp of the key lemma of horocyclic perturbations. However,the construction is somehow complicated$\Omega_{\epsilon}$$\Omega$

136
For$\Omega$and$\Omega_{\epsilon}$, we set$\Omega_{\epsilon}^{n}:=f_{\epsilon}^{-n}(\Omega_{\epsilon})$;$\Omega^{n}:=f^{-n}(\Omega)$

By the construction of these sets,
andcovering maps. Moreover,$f_{\epsilon}$$\{\Omega_{\epsilon}^{n}\}$:$(n=0,1,2, \ldots)$ .

are
and :
form decreasing sequences as below:$f$$\Omega_{\epsilon}^{n+1}arrow\Omega_{\epsilon}^{n}$$\{\Omega^{n}\}$$\Omega^{n+1}arrow\Omega^{n}$$\Omega_{\epsilon}=\Omega_{\epsilon}^{0}\supset\Omega_{\epsilon}^{1}arrowarrow\cdot\cdotarrow\supset\cdot\supset\Omega_{\epsilon}^{n}arrow\supset\Omega_{\epsilon}^{n+1}\supset\cdot\cdot\supset J(arrow\cdotarrow f_{\epsilon})$$\Omega=\Omega^{0}\supset\Omega^{1}arrowarrow\cdot\cdotarrowarrow\supset\cdot\supset\Omega^{n}\supset\Omega^{n+1}arrow\supset\ldotsarrow\supset J(f)$Step3: Construction of$h_{n}$.

,.

Let us set$A^{-}:=\{y\in\Omega :|h_{0}^{-1}(y)|\geq 2\}$,

which is the finite set of all parabolic points of perturbed into attracting, which is the finite set of allplanets of . Then we set. Note that
repelling satellites generated by the perturbation of
depend on .
and
In addition, we set$f$$\Gamma^{-}:=h_{0}^{-1}(A^{-})$$f_{\epsilon}$$A^{-}\subset A$$\Gamma^{-}$$A^{-}$$\epsilon$$\bullet\Gamma_{n}^{-}:=f_{\epsilon}^{-n}(\Gamma^{-})$, $A_{n}^{-}:=f^{-n}(A^{-})$$\bullet\Gamma_{\infty}^{-}:=\bigcup_{n=0}^{\infty}\Gamma_{n}^{-}$,$A_{\infty}^{-}:= \bigcup_{n=0}^{\infty}A_{n}^{-}$.

have no critical point, and that $f^{n}(y)$ is aparabolic point
andNote that. We thus define the petal number of by $p(y):=p(f^{n}(y))$ .
of foras the first lift of ::First we construct$A_{\infty}^{-}$$\Gamma_{\infty}^{-}$$f$$y\in A_{n}^{-}$$y$$h_{0}$$\Omega_{\epsilon}^{1}arrow\Omega^{1}$$h_{1}$

Proposition 5.3 For the map , there exists a continuous map :,. If
is surjective andsuch thatfor any
is a homeomorphism.
:$h_{1}$$h\circ$$f\circ h_{1}(x)=h_{0}\circ f_{\epsilon}(x)$$h_{1}$$\epsilon<<1$$x\in\Omega_{\epsilon}^{1}$$\Omega_{\epsilon}^{1}arrow\Omega^{1}$$h_{1}$$\Omega_{\epsilon}^{1}-\Gamma_{1}^{-}arrow\Omega^{1}-A_{1}^{-}$

. If
and
, thenProof. Fix an$d_{\sigma}(f(x),f_{\epsilon}(x))<r/2$, thuswe may assume that
. Since is sufficiently far from the;that is,is evenly covered by (If necessary, replace withcritical values,
smaller one and repeat the argument). Thus we can take abranch of, say , such that $g\mathrm{o}f(x)=x$ . This gives the map $h_{1}(x):=$on, which is clearly continuous. The last part of the statement
is not difficult to prove.
$f_{\epsilon}(x)\in\Omega_{\epsilon}$$x\in\Omega_{\epsilon}^{1}$$d_{\sigma}(f_{\epsilon}(x), h_{0}(f_{\epsilon}(x)))\leq r/2$

$d_{\sigma}(f(x), h_{0}(f_{\epsilon}(x)))<$$\epsilon<<1$$r$$h_{0}(f_{\epsilon}(x))\in B_{\sigma}(f(x), r)\cap\Omega$$B_{\sigma}(f(x), r)$$\Omega$$f$$r$$f^{-1}$$B_{\sigma}(f(x), r)$$g$$g\circ h_{0}\circ f_{\epsilon}(x)\in\Omega^{1}$$\blacksquare$inductively as following proposition. The:
Next we define
proof is similar to the case of n $=1$ .
$h_{n}$$\Omega_{\epsilon}^{n}arrow\Omega^{n}$

137
Proposition 5.4 For $n\geq 1$ , suppose that ate have defined the continuous map. Then there:
such that $d_{\rho}(h_{n-1}(x), h_{n}(x))<r_{\epsilon}$ for any:
such that
andexists a continuous map
.
any
that
for
:
is surjective and:Moreover, if
is a homeomorphism for any .
$h_{n}$$x\in\Omega_{\epsilon}^{n}$$\Omega_{\epsilon}^{n}arrow\Omega^{n}$$d_{\rho}(h_{n}(x), h_{n+1}(x))<r_{\epsilon}$$\epsilon\ll 1,$$h_{n}$$h_{n}\mathrm{o}f_{\epsilon}=f\mathrm{o}h_{n+1}$$\Omega_{\epsilon}^{n+1}arrow\Omega^{n+1}$$h_{n+1}$$x\in\Omega_{\epsilon}^{n+1}$$h_{n}$$\Omega_{\epsilon}^{n}arrow\Omega^{n}$$\Omega_{\epsilon}^{n}-\Gamma_{n}^{-}arrow\Omega^{n}-A_{n}^{-}$

$n$

Where

$d_{\rho}(\cdot$,$\cdot$$)$is the distance measured by , and$\rho$$\sup\{d_{\rho}(h_{0}(x), h_{1}(x)) :We can easily check that.
small if$r_{\epsilon}=O(r)$$r_{\epsilon}$x\in\Omega_{\epsilon}^{1}\}$is defined by.

, thus we may suppose that it is sufficiently$\epsilon<<1$

and the proof of $h_{n}arrow h$ . In the proof of the
Step4: The function
convergence of , the expanding property of will play an important role.
For instance, we can easily show the convergence when is hyperbolic:$\tau(s)$$f$$h_{n}$$f$

Proposition 5.5 Suppose that.
formly to the limit on
$h$

$f$

is hyperbolic. For$\epsilon\ll 1$,$h_{n}$

converges uni-

$J(f_{\epsilon})$

Proof. Since has no parabolic point, we may use the Poincare’ metric on
Int (Q) as without modification. Thus there is aconstant such that$>1$ on, there exists aconstant $C>0$ such
. By the definition of
that
$f$

$f^{*}\rho/\rho\geq$$\kappa$$\rho$$\Omega^{1}$$\{h_{n}\}$$\kappa$$d_{\rho}(h_{n}(x), h_{n+1}(x))<C/\kappa^{n+1}$. Thus we can easily follow thatfor any
rapidly to the limit on$x\in J(f_{\epsilon})$$h$$h_{n}$

converges uniformly and$J(f_{\epsilon}).\blacksquare$

However in the case that has parabolic points, is not uniformly expandis very slow. To show the convergence, we willing and the convergence ofuse the idea due to Douady-Hubbard[4, Expose’ No.X] again. See also [11].
(an open ball with
Let $s_{0}>0$ be the supremurn of such that
. We define afunction
for any
respect to ) is evenly covered by
by
:
$f$

$f$

$h_{n}$$B_{\rho}(x, s)$

$s$

$f$

$\rho$$\tau$$x\in\Omega^{1}$$(0, s_{0})arrow \mathbb{R}^{+}$$\tau(s):=\sup_{g}\{d_{\rho}(g(x), g(y)) :

Here
(i)

$g$

$\tau$

ranges over all branches of$f^{-1}$x, y\in\Omega^{1}, d_{\rho}(x, y)\leq s\}$. Then$\tau$

has following properties:

is an increasing and right-continuous function;(ii) $s>\tau(s)$ for any ;and
$s$

.

138
(iii) the function $s\mapsto s-\tau(s)$ is also increasing.
(i) and (ii) are almost clear by the definition, (iii) can be followed by the factthat $\tau(s_{1}+s_{2})\leq\tau(s_{1})+\tau(s_{2})$ .
By using this function, we can prove that:

Proposition 5.6 For
where satisfies$h$$\epsilon\ll 1$,$f\circ h=h\mathrm{o}f_{\epsilon}$

converges uniformly to the limit on. Moreover, for arbitrary small $r>0$ ,$\sup\{d_{\sigma}(h(x), x) :if$\epsilon\ll 1$$h$$J(f_{\epsilon})$$r_{\epsilon}=O(r)$, we may$h_{n}$

x\in J(f_{\epsilon})\}<r$.

Proof. Fix an arbitrary
such thatassume$L$such that $0<L<s\circ\cdot$ Since$\epsilon\ll 1$$d_{\rho}(h_{0}(x), h_{1}(x))<r_{\epsilon}\leq L-\tau(L)$for any $n\geq 1$ .
. We claim that $d_{\rho}(h_{0}(x), h_{n}(x))<L$ on
for any
If $n=1$ , $d_{\rho}(h_{0}(x), h_{1}(x))<L-\tau(L)<L$ . For $n=k$ , let us assume that,$d_{\rho}(h_{0}(x), h_{k}(x))<L$ . Then for any$x\in\Omega_{\epsilon}^{1}$$\Omega_{\epsilon}^{n}$$x\in\Omega_{\epsilon}^{k+1}$$d_{\rho}(h_{0}(x), h_{k+1}(x))\leq d_{\rho}(h_{0}(x), h_{1}(x))+d_{\rho}(h_{1}(x), h_{k+1}(x))$$<d_{\rho}(h_{0}(x), h_{1}(x))+\tau(d_{\rho}(h_{0}(f_{\epsilon}(x)), h_{k}(f_{\epsilon}(x))))$$<L-\tau(L)+\tau(L)=L$ .

We have thus proved the claim by induction on .
. For sufficiently large integer 1 $m$ ,Fix any$n$$x\in J(f_{\epsilon})$$d_{\rho}(h_{l}(x), h_{m+l}(x))<\tau^{l}(d_{\rho}(h_{0}(f_{\epsilon}^{l}(x)), h_{m}(f_{\epsilon}^{l}(x))))$$<\tau^{l}(L)arrow 0$(l$arrow\infty)$.

. By the continuity of
converges uniformly on
Because is arbitrary,defined by is, the limit is also continuous. Since the topology of
convergencealso true with
,
is
this
equivalent to that by the spherical metricrespect to .
The last part of the statement is easily followed by the construction ofand the fact that $d_{\rho}(h_{0}(x), h(x))<L.\blacksquare$$J(f_{\epsilon})$$h_{n}$$x$$\Omega^{n}$$h$$h_{n}$$\rho$$\sigma$$\sigma$$h\circ$

139
Step5: Completing the Proof.
4.1 by the proposition below.
Proposition 5.7 If$\bullet$$\bullet$$\bullet$$h$$\epsilon\ll 1$

Finally we complete the proof of Theorem, $h:J(f_{\epsilon})arrow J(f)$ has following properties.

is surjective.

If $h(x)=h(x’)$ for some different ,$x$$x’\in J(f_{\epsilon})$, then ,$x$$x’\in\Gamma_{\infty}^{-}$

.

For ,
,
as above, there exists an integer $N$ such thatare repelling satellites of an attracting planet which is generated by the
perturbation of a point in
$x$

$x’$

$f_{\epsilon}^{N}(x)$$f_{\epsilon}^{N}(x’)$$a_{\epsilon}$$A^{-}$

Proof. Here we only show the proof of the surjectivity of . Other properties
are shown by using the expanding property of with respect to the Poincare’.
metric ofFix any $y\in J(f)$ . By the surjectivity of , there is asequencesuch that $h_{n}(x_{n})=y$ . For
is compact,
has an accumulate point$x_{n_{k}}arrowx(karrow\infty)$ . Because
and we can take asubsequence
so thatuniformly and is continuous, the inequality
$h$

$f$

$\hat{\mathbb{C}}-P(f_{\epsilon})$$h_{n}$$\Omega_{\epsilon}$$x_{n}\in\Omega_{\epsilon}^{n}$$x\in J(f_{\epsilon})$$x_{n}$$h_{n}arrow h$$x_{n_{k}}$$h$$d_{\rho}(y, h(x))\leq d_{\rho}(h_{n_{k}}(x_{n_{k}}), h(x_{n_{k}}))+d_{\rho}(h(x_{n_{k}}), h(x))$

implies $h(x)=y.@$

By this surjectivity of and an fact thatmaps
to $J(f)-A_{\infty}^{-}$ homeomorphically.
$h$

$h$

$h^{-1}(A_{\infty}^{-})=\Gamma_{\infty}^{-}$, we obtain that

$J(f_{\epsilon})-\Gamma_{\infty}^{-}$

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