数理解析研究所講究録
第 1586 巻 2008 年 74-83

74

Discontinuity of straightening maps
Hiroyuki Inou*

December 10, 2007

1

lntroduction

The straightening theorem for polynomial-hke mappings by Douady and Hubbard [DH]naturally induces a map between a family of renormalizable polynomials and the connectedness locus of polynomials, which we call the straightening map. It is known such a map isalways continuous if it is of degree two and this implies the self-similarity of the boundary of
the Mandelbrot set. However, Douady and Hubbard also showed that straightening maps foranalytic families of polynomial-hke mappings of degree greater than two are not contnuous
in general. In [In3], we gave a much more detailed description how a straightening map be-comes discontinuous and gave an example of a discontinuous$s\alpha aightening$of polynomial-like restrictions of polynomials (of ycater degree).

map for a family

Here we show that it is not continuous or at least not a homeomorphism in general andshow how generally such a discontinuity occurs. To see this, we study parabolic and Misiurcwicz bifurcation to find perturbations which satisfies the assumption of the theorem in[In3] and use Prado-Rzytycki-Urbtski theorem [Pr] to show that if the straightening map iscontinuous (or homeomorphism), then we have local analytic conjugacy between a renornalizable polynomial and the straightening of the renormalization. However, we also show thatsuch alocal analytic conjugacy implies global correspondence [In4], which is impossible forrenormahzable polynomials.

2 Polynomial-like mappings
A polynomial-like mapping is a proper holomorphic map $f$ : $U’arrow U$ such that $U’$ andare topological disks in and $U’GU$ . We denote the filled Julia set by $K(f)=K(- f;U’, U)=$and the Julia set by $J(f)=J(f;U‘, U)=\partial K(f)$ .
For a periodic point $x\in C$ of period for a polynomial-like map (or a polynomial) $f$, let
us denote its multiplier by $mult_{f}(x)$ , i.e.,
$U$

$C$

$\bigcap_{n\geq 0}f^{-n}(U)$$n$$mult_{f}(x)=(fi)’(x)$.

Department of Mathematics, Kyoto University&Laboratoire Emile Picard, Universit6 Paul Sabatier

75

Definition. Let $f$ : $U’arrow U$ and : $V’arrow V$ are polynomial-like mappings and : $Uarrow V$a hybrid conjugacy. We say that preserves multipliers if for any periodic point for $f$ ,we have$g$$be$$\varphi$$x$$\varphi$$|mult_{f}(x)|=|mult_{l}(\varphi(x))|$.

The following theorem is proved by Prado [Pr] for “off-critically hyperbolic“ polynomiallike mappings, and extended to tame rational maps by Rzytycki and Urbanski [PU].

Theorem 2.1 (Prado-Przflych-Urbtsh). Let $f$ : $U’arrow U$ and ; $Varrow V$ are tamepolynomial-like mappings hybrid equivalent. If theoe
ahybrid conjugaq between $f$
and prese ハノ ing mulnpliers, then $tw$ am analytically conjugate on neighborhoods of thefillcd Julia sets.
$g$$e\kappa ists$$g$

We do not give the precise deflnition of tame polynomial-like mappings here, because weneed only the fact that polynomial-like mappings hybrid equivalent to $z+z^{2}$ are tame, whichis guaranteed by the following. (See [Url], [Ur2] and [PU] for details on tame maps.)

Theorem 2.2. Every polynomial-like mapping with no recurrent critical points in its Julia
set (abbr. $NCP$) is tame.
Furthermore, when a given polynomial-like mappings are restrictions of rational maps, we

can say more (see [In4] for more details).

Theorem 23. Let and be two rational maps. Assume
have polynomial-like restrictions :$i=1.2$ analytically conjugate. Then there exist rational maps
andsuch that
andg has a polynomial-like restriction $g:V’arrow V$ analyncallyconjugate to :. In pamcular. and have the same degree.
Funhemore, if and are polynomials, thenand are also polynomials.
Outline ofproof. By shrinking
if necessary, we may assume that there exists an analytic
conjugacy
between and . Let
be the graph of. Define $F:6^{2}arrow 6^{2}$ by $F(z_{1},z_{2})=(f_{1}(z_{1}),f_{2}(z_{2}))$ and let
for $n\geq 1$ . Then it
is easy to see that
. Therefore,
is a connected invariant set under .
Furthermore, we can $*desingul\dot{\bm{t}}ze’\Gamma$ ; namely, there exists a Riemann surface andholomorphic maps : $Xarrow X$ and :such that $\Phi(X)=\Gamma,$ $F\circ\Phi=\Phi\circ g$ , and there
exists an open set such that:is a conformal isomorphism and :isa polynomial-like map analytically conjugate to :, wherebybe the natural projection and.
Since has a chaotic dynamics (e.g.. has a repelling periodic point), cannot be hyperbolic. Therefore is isomorphic to either
or a torus (and we can showthat the case is a torus cannot happen). Since the degree of is flnite, the degree ofis also finite, thus is a rational map. It is not difficult to show thatis also arational map. (It is trivial when
. For other cases, see, e.g., [BE]. Note that can be$\alpha anscendental$
when :
is a polynomial-like mappings of degree one, i.e., andare neighborhoods of a repelling fixed point and is alinearizing coordinate for it.)$f_{1}$$f_{i}$$f_{2}$$t\emptyset$$U_{i}’arrow U_{i},$$g,$$\varphi_{1}$$\varphi_{i}\circ g=f_{i}\circ\varphi_{i}$$\varphi_{2}$$f_{i}$$U_{i}’arrow U_{i}$$f_{1}$$f_{2}$$f_{1}$$f_{2}$$g,$$\varphi_{1}$$\varphi_{2}$$U_{i}$$\phi:U_{1}arrow U_{2}$$f_{1}$$\Gamma_{0}=((z,\emptyset(z));z\in U_{1}I\subset\hat{C}^{2}$$f_{2}$$\phi$$\Gamma_{n}=F^{n}(\Gamma_{0})$$\Gamma_{n}\subset\Gamma_{n+1}$$F$$\Gamma=\bigcup_{n\geq 0}\Gamma_{n}$$X$$Xarrow \mathbb{C}^{2}$$\Phi$$g$$\overline{U}$$\Phi|_{U}$$O-\Gamma_{0}$$g$$f_{j}$$U_{i}’arrow U_{i}$$\tilde{U}’arrow\tilde{U}$$\pi_{i}(z\iota,z_{2})=z_{i}$$\pi_{j}\circ\Phi$$\tilde{U}’=(\pi_{1}\circ\Phi|_{0})^{-1}(U_{1}’)$

$g$

$X$

$g$

$X$

$\hat{C},$$C,$ $C‘=C\backslash \{0\}$

$X$

$F$

$g$

$\phi_{j}=\pi_{j}\circ\Phi$$g$$x\underline{\approx}6$$\phi_{i}$$f_{i}$$U_{i}^{j}arrow U_{i}$$U_{i}’$$U_{i}$

Deflnition. We say two polynomials$\phi_{i}$$f_{1}$

and$f_{2}$$0$

are polynomially semiconjugate up to fnite

76

cover if the conclusion of Theorem 2.3 holds.
Combining the theorems above, we have the following.
Theorem 2.4. Let$f_{i}$:$f_{1}$

and$f_{2}$$n\prime ing$

$U_{i}’arrow U_{i},$Thenand be polynomials. Assume they have polynomial-like restrictions$i=1,2$ which are hybrid conjugate via a conjugacy prese
multipliers.
are polynomially semiconjugate up to finite cover.
$f_{1}$$f_{2}$

3 Analytic families of polynomial-like mappings andstraightening maps
In [In3], we have proved a theorem which relates continuity of straightening maps andmultipliers of periodic points. Before stating the theorem, we introduce some notations anddefinitions. Let $Poly_{d}$ be the set of affine conjugacy classes of polynomials of degree $d(\geq 2)$
and $C_{d}=$ { $f\in Poly_{d};K(f)$ is connected}. By the straightening theorem [DH], for an analytic(in the sense of [DH]), we canfamily of polynomial-like mappings $f=$:
define the straightening map
as follows. Let $C_{f}=$ {is connected}, there exists a uniqueand forhybrid equivalent to . Let
.
$f=$
say
We
is an analytic family ofpolynomial-like mappings witha markzd point if
is an analytic family of polynomial-like mappings, and$x:Aarrow C$ is a holomorphic map such that
. Also for such a fanily of degree $d\geq 2$ ,let
$CK_{i}=$ {is connected and}$(f_{\lambda} :$S_{f}$

U_{\lambda}’arrow U_{\lambda})_{\lambda\epsilon\Lambda})$$C_{i}arrow C_{d}$$\lambda\in Cr$$\lambda\in\Lambda;K(- f_{\lambda})$$g_{\lambda}\in C_{d}$$(f_{\lambda} :$S_{f}(\lambda)=g_{\lambda}$$f_{\lambda}$

U_{\lambda}’arrow U_{\lambda},x_{\lambda})_{\lambda\epsilon\Lambda}$$(f_{\lambda} :U_{\lambda}’arrow U_{\lambda})_{\lambda\epsilon\Lambda}$$f$$x_{\lambda}\in U_{\lambda}$$\lambda\in\Lambda;K\zeta f_{\lambda})$$x_{\lambda}\in K(f_{\lambda})$and we can similarly define the straightenin map$g$$S_{i}$: $CK_{f}arrow M_{d}$as follows. Let $M_{d}=$ {: polynomial of degree $z\in K(f)$ } , where $(f,z)\sim(g,w)$
if there exists an affine map such that $A\circ g=f\circ A$ and $A(w)=z$ For eachis hybrid equivalent to some polynomial of degree with a hybrid conjugacy . Let us$\zeta f,z);f$$d,$$f\sim$$A$$\lambda\in M_{f},$$d$$g_{\lambda}$

defineNow let us state the theorem:$f_{\lambda}$$\psi_{\lambda}$, then it is well-defined.

$S_{f}(\lambda)=(g_{\lambda}, \psi_{\lambda}(x_{\lambda}))\in CK_{d}$Theorem 3.1. Let $f=$
be an analytic family of polynomial-like$d\geq2$
with a marked point. Assume
mappings of degree$(f_{\lambda} :U_{\lambda}’arrow U_{\lambda},x_{\lambda})_{\lambda\epsilon A}$(i) for any $\lambda\in A0$ lies inand it is afixed point for ;(ii)
is a marked repelling periodic point andis a marked critical point for ;(iii) foris a non-degenerate l-parabolicfixed point,lies in the basin of$f_{\lambda}$$U_{\lambda}’$$\alpha_{\lambda}$$f_{\lambda}$$\omega_{\lambda}$$\lambda=\lambda_{0},0$$\omega_{\lambda_{0}}$;and(iv) there exist sequences;$f_{\lambda_{0}}$$x_{\lambda_{0}}=\omega_{\lambda_{0}}$$\lambda_{n}arrow\lambda_{0}$and$\lambda_{n,m}arrow\lambda_{n}$such that$\bullet\lambda_{n},\lambda_{n.m}\in M_{i}$$\bullet\omega_{\lambda_{n}}\neq x_{\lambda_{n}}$$\bullet$

$0$

for

$n\geq 1$;

is a non-degenerate l-parabolicfixed point for$f_{\lambda_{n}}$

;

$0$

for

77

$\bullet$geometrically converges tosuch thatandfixed pointfor ).
as $marrow\infty$ and$f_{\lambda_{nm}}$$g’(\omega_{\lambda_{n}})\neq 0$$g(\omega_{\lambda_{n}})=\alpha_{\lambda_{n}}$as$(f_{\lambda_{n}},g_{n})$$marrow\infty$

for some Lavaurs map(in pamcular,$0$$g_{n}$

is no more a parabolic$f_{\lambda_{\hslash l\prime}}$(v)$S_{i}(\lambda_{n})arrow S_{f}(\lambda_{0})$$S_{i}(\lambda_{n,m})arrow S_{f}(\lambda_{n})$as$narrow\infty$.

Then$|mult_{f\tau_{0}}(\alpha_{\lambda_{0}})|=|mult_{g_{l_{0}}}(\psi_{\lambda_{0}}(\alpha_{4}))|$

where$S(f_{\lambda})=(g_{\lambda},z_{\lambda})$and$\psi_{\lambda}$

is a hybrid conjugacy between,$f_{\lambda}$and$g_{\lambda}$.

4 Rational laminations
We briefly review the notion of rational laminations inffoduced by Thurston [Th] and: $R/Zarrow R/Z$ the d-fold covering
deeply studied by Kiwi [Kil]. Let $d\geq 2$ and denote bymap. A (d-invariant) rational lamination is an equivalence relation on $Q/Z$ suchthat$m_{d}$$tarrow\prime dt$$\lambda$

is closed (in $Q/ZxQ/Z$).
(ii) Each -class is finite.
(i)$\lambda$$\lambda$-classes are pairwise unhnked.
(iv) For a -class $A,$ $m_{d}(A)$ is again a -class.

(iii)$\lambda$$\lambda$$\lambda$.

: $Aarrow m_{d}(A)$ is consecutive-preserving, i.e., if $(s.t)$ is a component of.
then $(ds,dt)$ is a component of.
Here we say two sets $A,B\subset R/Z$ are unlinked if is contained in a component of, let
if the external
For a polynomial
be the rational lamination of i.e.,rays
is the rational lamination
andland at the same point. It is easy to see thatin the above sense. Conversely, Kiwi [Kil] proved that for any rational lamination , there
exists a polynomial
such that. For a rational lamination and a polynomial, we say admits if.
are unlinked,
We say two irrational angles and are -unlinked if any -classA andi.e., is contained in a component ofof unlinked
properties. We state the basic(v)$R/Z\backslash A$$m_{d}|_{A}$$R/Z\backslash m_{d}(A)$$A$$f\in C_{d}$$R_{f}(s)$$f$$\lambda(f)$.

$R/Z\backslash B$$s\sim t$$\lambda(- f)$$R_{f}(t)$$\lambda$$f\in C_{d}$$f\in C_{d}$$f$$\lambda$$\lambda=\lambda(f)$$\lambda$$\lambda\subset\lambda(J)$

$s$

$A$

$\{s.t\}$$\lambda$$\lambda$$t$$R/Z\backslash \{s,t\}$relations here (see [Kil]. See also [Inl] and [In2]).

Lemma 4.1. The -unlinked relation is an equivalence relation. For each -unlinked ckss
$L$ we have the following:$\lambda$$\lambda$$\bullet$$\bullet$

For each -unlinked class $Lm_{d}(L)$ is again a -unlinked class.
A -unlinked class is finite if and only if it is wandering, $i.e.,$$L$$n\neq m$$\bullet$$\lambda$$\lambda$$\lambda$$m_{d}^{n}(L)\neq m_{d}^{m}(L)$.

if

has the $fom$
is infinite, then is a Cantor set and each component of$(s,t)$ with $s,t\in Q/Z$ andis homeomorphic to $R/Z$ and. Furthermore,.
:: $R/Zarrow R/Zfor$ someis conjugate to

If$R/Z\backslash L$$\overline{L}$$\lambda$$\overline{L}/\lambda$$s\sim\lambda t$$m_{d}$$\overline{L}/\lambdaarrow m_{d}(L)/\lambda$

We call a -unlinked clas $sL$ is critical if$\lambda$$\delta=\delta(L)\geq 1$$m_{\delta}$$\delta(L)\succ 1$.

78

We say a polynomial $f$ is subparabolic if each critical point is either eventually periodic or
attracted to anor parabolic periodic point.
$amaC\mathfrak{a}ng$

Lemma 4.2. Let $f$ be a subparabolic polynomial. Then each infinite
corresponds to some bounded Fatou component , i.e.,$\lambda(f)$-unlinked class$L$$\Omega$$\overline{L}=\{\theta\in R/Z:R_{f}(\theta)\in\partial\Omega\}$$Funhemo\prime e,$ $m_{d}(L)$corresponds to$f(\Omega)$and.

is equal to the degree of $f$ :$\delta(L)$$\Omegaarrow f(\Omega)$.

The lemma follows easily from the fact that the Julia set of $f$ is locally connected in thiscase, so it is homeomorphic to $(R/Z)/\lambda(f)$ , whereis the real lamination of $f$ Note thatwhen $J(f)$ is locally connected, thenis simply the landing relation on $R/Z$ (see [Ki2]for general case).

.

$\lambda(f)$$\lambda(f)$Definition. We say a rational lamination is combinatorially renomalizable if there exist
a non-trivial rational sublamination , a critical infinite -unlinked class and $n\geq 1$ suchthat $m_{d}^{n}(L)=L$. We calla combinatorial renomalization of and the smallest such$n\geq 1$
is called the period of the renormalization.
We say a combinatorial renormalization
is of capture type if there exists a cntical-unlmked class such that$\lambda$$\lambda’$$L$$\lambda’$$(\lambda’,L)$$\lambda$$(\lambda’,L)$$L’$$\lambda’$(i) $L’$ is not in the forward orbit of $L$, i.e., $L’\neq m_{d}^{k}(L)$ for every $k\geq 0$.
(ii) $L’$ is captured by $L$ : There exists some $N>0$ such that $m_{d}^{N}(L’)=L$(iii) $\delta(m_{d}^{k}(L’))=1$ for all $0<k<N$.

.

We call such a triple$(\lambda’,L,L^{j})$a capture renonnalization of .
$f$

Let $f\in C_{d}$ and be a rational sublamination of. For an infinite -unlinked class, deflne a compact set $K(f,L)\subset K\wp)$ as follows. By Lemma 4.1, we can write$L=U_{k>0}(s_{k}, t_{k})$ for some $s_{k},t_{k}\in Q/Z$ with
. In particular, the external rays
lands at the same point for each $k>0$ . Let
be the component ofcontaining
for
. Let
$\lambda$$\lambda$$\lambda(\sim f)$$\lambda$$L$$R/Z\backslash$$s_{I}\sim\lambda(f)t_{k}$$R_{f}(s_{k}),R_{f}(t_{k})$$U_{k}$$\overline{(R_{[}(s_{k})UR_{f}(t_{k}))}$$R_{f}(t)$$C\backslash$$t\in(s_{k},t_{k})$$K(J,L)=K(f) \backslash (\bigcup_{k>0}U_{k})$.

Then we have $K(J.m_{d}(L))=f(K,m_{d}(L))$ and $f$ : $K(f,L)arrow K(f,m_{d}(L))$ is a proper map of. See [In2] for more details.
degreeBy Milnor’s thickened puzzle piece argument [Mi, Lemma 1.5], we can relate a combinatorial renormalization ofwith an renormalization of $f$ under some assumption.
Lemma 4.3. Let $f\in C_{d}$ Assume there exists a combinatorial renomalizanon
ofperiod such thatfor each component $(s,t)ofR/Z\backslash L$ the common landing point $xofR_{f}(s)$$andR_{f}(t)$ is not (pre-)parabolic or $lf$ it is (pre-)parabolic, all the basin whose closure contains
is contained in $K_{f}(L)$. $n_{en}$ there exists a polynomial-like restriction: $U’arrow U$ ofsuch that $K\wp;U’,$ ) $=K(f,L)$$\delta(L)$$\lambda\omega$$(\lambda,L)$

$n$

$x$

$U$

.

$f^{n}$$f^{n}$

79

5 Perturbation
5.1

Definition and statements
We want to use Theorem 2.4 and Theorem 3.1 together to show that a straightening map

is discontinuous or not a homeomorphism. To do that, we need to find nice perturbations(satisfying the assumption of Theorem 3.1) for each periodic point of the polynomial-like
resmction.
Deflnition. We say a polynomial $f$ of degree $d\geq 3$ satisfies (C1) if the following hold;(i) is a non-degenerate l-parabolic periodic point of $f$ ;(ii) there exists a quadratic-like restriction: $V’arrow V$ of hybrid equivalent to $z+z^{2}$ ;(iii)
are critical points for $f$ such that $\omega\in V’$ , and $f^{N}(\omega’)=\omega$ for some $N>0$$0$$f^{N}$$f^{N}$$\omega,$.

$\omega’$Furthermore, we say $f$ satisfies (C2) if $f$ satisfies$(i)-(i\ddot{u})$and(iv) every other critical point is eventually periodic and its forward orbit does not intersect$K(f^{N};V’,$(In particular, $f$ is subparabolic.)$\eta$.

Definition. We say $f$ satisfies the condition (C3) if $f$ satisfies the condition (C1) and for any
periodic point of: $V’arrow V$, there exist convergent sequences$\alpha$$f^{N}$$f_{nm}arrow f_{n}marrow\inftyarrow fnarrow\infty$in$C_{d}$

such that the folowing hold; in the following, we denote the continuations of $x=$
for andby and.

$\omega,\omega’,\alpha$(i)$f_{n}$$f_{nm}$$x_{nm}$$x_{n}$

is a periodic point of period $N$ for
periodic point for ;$0$$f_{n}$

and$f_{nm}$. It is a non-degenerate l-parabolic$f_{n}$(ii)(hence we may assume);(iii) the other critical orbit relations of $f$ are preserved for$f_{n}^{N}(\omega_{n}’)\neq\omega_{n}$$f_{n,m}^{N}(\omega_{n,m}’)\neq\omega_{n_{l}n}$$f_{n.m}$(hence also for$f_{n}$).

More precisely, for any critical point $c\in Crit(f)$ , there exists the continuation$c_{n,m}\in Crit(f_{n.m})$ for any $n,m$ (i.e., critical points do not bifUrcate for this perturbations) such that if $c,c’$satisfy $f^{k}(c)=f^{t}(c’)$, then$\in Crit(f)\backslash \{\omega.\omega’\}$$f_{n’ n}^{k}(c_{n,m})=f_{n,m}’(c_{n_{l^{\hslash}}}’)$.

:
are quadratic-like restrictions near ; In;particular,:
are hybrid equivalent to(v)geomebically converges toandassuch that(iv)$f_{n}^{N}$:$V_{n}’arrow V_{n}$$f_{n}^{N}$and$f_{n.m}^{N}$$z+z^{2}$$V_{n}’arrow V_{n}$$(f_{n},g_{n})$.

$0$$V_{n,m}^{j}arrow V_{n;n}$$g_{n}(\omega_{n})=a_{n}$$marrow\infty$$g_{n}’(\omega_{n})\neq$$f_{nm}0$

The main difficulty to find such perturbations
must be in the connectedness
is thatthis,combinatorially andlocus. To do
we start the following polynomial and construct
show the convergence. See \S 5.2.
$f_{n;n}$$f_{nm}$$f_{n,m}$Theorem 5.1. If any polynomial $f$ of degree $d\geq 3$ satisfy (C2) then $f$ satisfies (C3) such that
andadmitsfor any and .
$\lambda(f_{n})=\lambda\omega$$f_{n,m}$$\lambda(f)$

$n$

$m$

80

The proof is given in \S 5.2.
We call a polynomial $f$ Misiurewicz if all critical points are preperiodic.
Theorem 5.2. If a polynomial of degree $d\geq 3$ is Misiurewicz, then there exists a polynomial $f$ arbitrarily close to satisfy ing (C2).
$f_{0}$$f_{0}$

We prove this theorem in \S 5.3.
Therefore, we have the following corollary.

CoroUary 5.3. If a polynomial of degree $d\geq 3$ is Misiurewicz, then there exists a polynomial $f$ arbitrarily close to satisfying (C3).
$f_{0}$$f_{0}$

5.2 Combinatorial construction
Here we give a proof of Theorem 5.1. We first consider perturbations of a quadratic polynomial $Q(z)=z^{2}+1/4$ (which is affinely conjugate to $z+z^{2}$ ), and construct
by “tuning”
the given polynomial $f$ and them.
Take a repelling periodic pointof and let
be the landing angle for. Let
be the landing point of the extemal ray of angle
Then $c_{m}arrow 1/4$ and the critical point$f_{n,m}$$\alpha(Q)$$\theta\neq 0$$Q$$\theta/2^{m}$$c_{m}$$\alpha(Q)$.

is preperiodic forfor sufficiently large . The point $\alpha(Q_{m})=\Psi_{m}(c_{m})=$“ 1 (0), hence it is a repelling periodic point foris a landing point for the external ray
andas. Furthermore,converges to some on the interior
of $K(Q)$ (the parabolic basin) such that $g(O)=\alpha(Q)$ , i.e., $Q_{m}arrow^{gcom}(Q,g)$ and it is easy to seethat $g’(0)\neq 0$ . Let
be the rational lamination for. For later use, we also fix a\d ag ersequenceof periodic angles by the doubling map
such that. Let$J(Q_{m})$be the landing point of the extemal ray. Since is subhyperbolic,is$Q_{m}(z)=z^{2}+c_{m}$

$0$

$m$

$\Psi_{m}$$R_{Q_{l*}}(\theta)$$\alpha(Q_{m})arrow\alpha(Q)$$Q_{m}$$marrow\infty$$\Psi_{m}^{+1}$$g$$Q_{m}$$\lambda_{m}=\lambda_{Q_{n}}$$\theta_{n}\in Q/Z$

$m_{2}$$\alpha_{n}(Q_{m})$$R_{Q_{n}}(\theta_{n})$$t\ovalbox{\tt\small REJECT}\mapsto\rangle$$2t$$\theta_{n}arrow\theta$$Q_{m}$

locally connected and hence $a_{n}(Q_{m})arrow\alpha(Q_{m})$ as.
We construct arational laminationby combinatorial mning’ [In2], which has acertain
desired properties for the rational lamination of. Namely, we construct a laminationsuch that the following hold;$narrow\infty$$\lambda_{n,m}$$f_{n.m}$$\lambda_{nm}\supset\lambda(f)$$\bullet$

let be the -unlinked class such that $K(f,L)$ is the closure of the immediate parabolic
basin of , and let
be a homeomorphism such that.
Then.
let $L’$ be the -unlinked class such that the interior of $K(f,L)$ contains , and let:
be a homeomorphism such that $\phi(m_{d}^{N}(t))=m_{2}(\phi’(t))$ . (Note that$m_{d}^{N}(L’)=L$ and: $L’arrow L$ is 2-to-l.) Then the equivalence relation
is a“lift“ ofbysuch that there exists a unique critical equivalence class (i.e.,
is not one-to-one) with $\# A=2$ and.
$L$$\lambda$$\phi:\overline{L}/\lambdaarrow S^{1}$$0$$\phi(m_{d}^{n}(t))=m_{2}(\phi(t))$$\phi.(\lambda_{n,m}|_{\overline{L}})=\lambda(Q_{m})$$\bullet$$\lambda$$\omega’$$\overline{L’}/\lambdaarrow S^{1}$$\phi’$$m_{d}^{N}$$\lambda(Q_{m})$$\phi’.(\lambda_{nm}|_{\overline{L}},)$$A$$m_{2}$$m_{d}|_{A}$$m_{2}(A)=(\theta_{n})$Then by the theorem of Kiwi [Kil], there exists a postcritically finite polynomial
such that. Then it has a polynomial-like restriction : $V_{nm}’arrow V_{n,m}$ hybrid
equivalent toby Lemma 4.3. Letandbe the critical points which correspondto the critical-class intersecting
respectively. Thenandareperiodic point corresponding toandby a$f_{\hslash ln}\in C_{d}$$\lambda(f_{n})=\lambda_{nm}$$f_{nm}^{n}$$Q_{m}$$\omega_{nm}$$\lambda_{nm}$$\beta_{nm}=f_{nm}^{N+\hslash}(\omega_{nm})$$\omega_{nm}’$$\overline{L}\bm{t}d\overline{L’}$$\alpha(f_{n.m})=f_{nm}^{n}(\omega_{nm})$$repe\mathbb{I}ing$$\alpha(Q_{m})$$\alpha_{n}(Q_{m})$81hybrid conjugacy.
By taking a subsequence, we may assume thatandfor some
and . Let us denote)and soon. Thenwithandare repelling periodic
points withas. This implies that
. Thus, $f$ and are subparabolic maps with the same critical relation and combinatorics.
Furthermore, since all critical points of $f$ other than andare eventually periodic and$f$ is combinatorially rigid, so $f$ must equal to $f$ . This proves the Theorem.
$f_{nm}-marrow\infty f_{n}$$\tilde{f}$$\omega_{n}=\lim_{marrow\infty}\omega_{n,m},\tilde{\omega}=\lim_{marrow\infty}\omega_{n},$$f_{n.m}arrow^{g\infty m}(f_{n},g_{n})$$\alpha\zeta f_{n}$$f_{n}arrow narrow\infty$;$f_{n}$$= \lim_{m\neg\infty}\alpha(\sim f_{nm})$$g_{n}(f_{n}^{N}(\omega_{n}’))=\beta_{n}$$g_{n}(\omega_{n})=\alpha(f_{n})$$marrow\infty$$\beta_{n}-\alpha(f_{n})arrow 0$$0$$narrow\infty$$\beta(\tilde{\omega}’)-\tilde{\omega}=\lim_{narrow\infty}f_{n}^{N}(\omega_{n}’)-\omega_{n}=$$\tilde{f}$$\omega$$\omega’$$f^{N}(\omega’)=\omega,$

5.3 Misiurewicz bifurcations
The proof of Theorem 5.2 is based on a classical normality argument. We first perturb it tomake a cnitical relation between critical points and and make a quadratic-like restriction
by use of the result of McMulen [Mc]. Before the proof, we $in\alpha Muce$ the notion of passive
and active critical points: For a familyof polynomials with marked critical points,
we say is passive at
if {forms a normal family, and is active
otherwise.
In the following, we assume that all of the critical points of are simple just for simplicity.
Choose critical points and of so that no other critical point lies in the backward orbitsof them. Let us con ider a family$\omega$$\omega’$$(f_{\lambda},c_{\lambda})_{\lambda\epsilon A}$$\lambda_{0}\in\Lambda$$c_{\lambda}$$\lambda\ovalbox{\tt\small REJECT}\mapsto f_{\lambda}^{n}(c_{\lambda});n\geq 0)$$c_{\lambda}$$f_{0}$$\omega$$\omega’$$f_{0}$$s$$(f_{\lambda},\omega_{\lambda},\omega_{\lambda}’,c_{1,\lambda}, \ldots,c_{d-3,\lambda})$of polynomials with all critical points marked. We consider a subfamily where all the criticalorbit relations of except on and are preserved. Then since is Misiurewicz, is in
the bifUrcation locus of the family andis passive at . This implies thatis not a normal fammily because the family is two-dimensional. Therefore, there exists some
$N>0$ andarbitranily close to such that$f_{0}$$\omega’$$\omega$$f_{0}$$f_{0}$$0$$c_{j,\lambda}$$\{\lambda\vdash\rangle f_{\lambda}^{n}(\omega_{\lambda}’)\}_{n>0}$$0$$\lambda_{1}$$f_{\lambda_{1}}^{N}(\omega_{\lambda_{1}}’)=\omega_{\lambda_{1}}$.

(1)

Now consider a $fi_{1}rther$ subfamily with the relation (1). Then it is one-parameter family andis not a normal family. Therefore, by [Mc], we can find arbitrarily close{
to such thatlies in the immediate basin of a non-degenerate l-parabolic periodic point.
Therefore, we have proved Theorem 5.2.
$\lambda_{2}$$\lambda\ovalbox{\tt\small REJECT}\mapsto f_{\lambda}^{n}(\omega_{\lambda}))_{n>0}$$\lambda_{1}$$\omega_{\lambda_{2}}$6 Discontinuity
In this section, we apply all the results above to a family of renormalizable polynomials toshow straightening maps are discontinuous, or at least not homeomorphic. Note that all the
results in this section are local properties, so should be understood as a small neighborhoodof a given parameter (which we denote by in the following) in the original parameter space.
$\Lambda$$0$

Theorem 6.1. Let $f=$of degree $d\geq 3$ . Assume$(f_{\lambda} :U_{\lambda}’arrow U_{\lambda})_{\lambda\epsilon\Lambda}$

be an analyticfamily ofpolynomial-like mappings

82

is a restriction of some polynomial of degree $d’\geq d$ ;$\bullet$$f_{0}$$\bullet$$g_{0}=S_{f}(- f_{0})$$\bullet$Thensatisfies (C3);the straightening map:
which is a neighborhood of$S_{f}$$f_{0}$$C_{i}arrow S_{f}(C_{f})\subset C_{d}$$g_{0}$

in$C_{d}$.

is a homeomorphism into its image,and So are afnely conjugate.

It is an easy consequence of Theorem 2.4 and Theorem 3.1.
Furthermore, by Corolary 5.3, we have the foUowing.
Corolary 6.2. Let $d<d’$ and let {$\bullet$$\bullet$for each$f_{0}$$\lambda\in\Lambda,$$f_{\lambda}$

is Misiurewicz.

$f_{\lambda})_{\lambda\epsilon\Lambda}\subset Poly_{d}$be a neighborhood of

has a polynomial-like restriction$f_{\lambda}$:$U_{\lambda}’arrow U_{\lambda}$$f_{0}\in C_{d}$. Assumeof degree ;$d$Then the straightening map:
is not a homeomorphism into its image,or the image does notform a neighborhood of $g_{0}=S(f_{0})$ .
$S_{i}$$C_{i}arrow S_{f}(C_{t})\subset C_{d}$Let us state another version of discontinuity of $s\theta aigtening$ maps:
Theorem 6.3. Let $d<d’$ and let$\bullet$$\bullet$$\{f_{\lambda}\}_{\lambda\epsilon\Lambda}\subset Poly_{d}$be a neighborhood of. Assume

has a polynomial-like restriction :for eachof degree . Let
be the filled Julia set of the polynomial-like restriction;sansff the condition (C2) such that the filled Julia set the quadratic-like restriction$\lambda.\in\Lambda.f_{\lambda}$$f_{\lambda}$$U_{\lambda}’arrow U_{\lambda}$$d$$K_{\lambda}$$f_{0}$

hybrid equivalent to $z+f$ is contained in ; Letbe the analytic continuation ofand in the definition of (C3);:
is an analytic family ofpolynomial-like mappings with amarked critical point.
$K_{0}$$\omega$$\bullet$$f_{0}\in C_{d}$$\omega_{\lambda}’$$\omega’$$f=Cf_{\lambda}$$U_{\lambda}’arrow U_{\lambda}.f_{\lambda}^{N}(\omega_{\lambda}’))$:
is not continuous.
This is also an easy consequence of Theorem 2.4, Thcorem 3.1 and Theorem 5.1.
We can sinilarly prove the following:
Example. Let us consider capture renormalizations of cubic polynomials. The parameter setof renormahizable polynomials for a given combinatorics of capture type can be consideredas the connectedness locus of an analytic family of polynomial-like mappings with marked
point, which is given by the forward orbit of the captured critical point. In [In2], we provedthat for a given combinatorics of a primitive renormalization of capture type for cubic polynomials, the straightening map is bijective. Therefore, the straightening map is not continuous
in any neighborhood of any Misiurewicz map.
T&n the straightening map$S_{f}$$C\iotaarrow S(C_{i})\subset CK_{d}$Note that in the case of disjoint $renor\iota nahzations$ of cubic polynomials, straightening maps
are always continuous because they only consists of straightening maps for quadratic-likefamilies.

83

References
[BE]

[DH]

[Inl]
$m2]$

[In3]

X. Buff, A. L. Epstein. From local to global analytic conjugacies. Ergodic Theory
Dynam. System (2007) 27, $1073-1oe4$ .
A. Douady, J. H. Hubbard, On the $\phi namics$ ofpolynomial-like mappings. Ann. Sci.
\’Ecole $NomL$ Sup. (4) 18 (1985), 287-343.
H. Inou, Limits of renormliwble polynomials. Nonlinearity 19 (2006),.
H. Inou, Combinatorics and topology of straightening maps compacmess andbijectivity. Preprint.
H. Inou, Discontinuity ofthe straightening mapfor afamily of renomalizable polynomials. Complex Dynamics and its Related Topics. RIMS K\^okyfiroku 1537 (2007),$1769-17\mathfrak{B}$$I:$

163-167.

[In4]
[Kil]

[Ki2]
[Mc]

[Mi]

[Pr]

H. Inou, Analytic conjugacy classes ofpolynomial-like mappings. Preprint.
J. Kiwi,laminations of complex polynomials. Laminations and foliations
in dynamics, geometry and topology (Stony Brook, NY, 1998), 111-154. Contemp.
Math., 269, Amer. Math. Soc., Providence, RI, 2001.
J. Kiwi, Real laminations and the topological dynmics of complex polynomials.
Adv. Math., 184 (2004), 207-267.
C. T. McMullen, The Mandelbrot set is universal. The Mandelbrot set, theme andvariations, 1-17, London Math. Soc. Lecture Note Ser., 274, Cambridge Univ. Press,
Cambridge, 2000.
J. Milnor, Local Connectivity of Julia Sets. Expository ltctures. The Mandelbrot
set, theme and variations, 67-116, London Math. Soc. Lecture Note Ser., 274, Cambridge Univ. Press, Cambridge, 2000.
A. Prado, Teichmuller distance for some polynomial-like maps. Stony Brook IMS$Ra\hslash onal$$\cdot$$1996p$

[PU]
$m]$

[Url]
[Ur2]

.

F. Przytycki, M. $Urba\Lambda sb$ , Rigidity of tame rational functions. Bull. Polish Acad.
Sci. Math. 47 (1999), 163-182.
W. Thurston, On the combinatorics of iterated rational maps, manuscript, 1985.
M. Urbatiski, $Ra\hslash onalfimctions$ with no recurrent critical points. Ergodic Theory
Dynam. Systems 14 (1994), 391-414.
M. Urbafiski, Geometry and ergodic theory of confomal non-recurrent dynamics.
Ergodic Theory Dynam. Systems 17 (1997), 1449-1476.