数理解析研究所講究録 1220 巻 2001 年 26-31

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Accumulation of stretching rays for real cubic
polynomials
Shizuo Nakane

Tokyo Institute of Polytechnics(Joint work with Y. Komori)In this note, we investigate the accumulation of stretching rays on the parabolicarc
for the family of real cubic polynomials. We know that stretching rays
with irrational B\"ottcher vectors do not land at any point on $Per_{1}(1)$ . That is, theiraccumulation sets are non-trivial arcs. Here we characterize the accumulation set ofe.arll stretching ray in terms of the Bottcher vector map. As acorollary, it follows
that any point on $Per_{1}(1)$ belongs to the accumulation set of some stretching ray.
$Per_{\mathit{1}}(1)$1Stretching rays :general settings,
be the family of monic centered polynomials of degree $d\geq 2$ . For
be its Bottcher coordinate defined in aneighborhood of . It satisfies
and tangent to identity at . Let
be theGrien function for , which is continued continuously to the whole plal\‘ie by thefunctional equation $h_{P}(P(z))=dh_{P}(z)$ and is harmonic in $\mathrm{C}-K(P)$ , tllc complement of the filled-in Julia set. Put $G(P)= \max$ {is acritical point of }.
Then
can be continued analytically to $U_{P}=\{z; h_{P}(z)>G(P)\}$ . For acomplex$=z|z|^{u-1}$ and we define a
number $u\in H_{+}=\{u=s+it\in \mathrm{C}, s>0\}$ , put$P$by-invariant almost complex structureLet$P\in 7_{d}^{\supset}$$P_{d}$$\infty$$1\mathrm{e}\mathrm{t}.\varphi_{P}$$\varphi_{P}(P(z))=\varphi_{P}(z)^{d}$$h_{P}(z)=\log_{+}|\varphi_{P}(\approx)|$$\infty$$P$$P$$h_{P}(\omega);\omega$$\varphi_{P}$$f_{u}(z)$$\sigma_{u}$$(f_{u}\mathrm{o}\varphi_{P})^{*}\sigma_{0}$$\sigma_{u}=\{$$\sigma_{0}$on $K(P)$ .

on$U_{P}$,is integrated by aqc-map
depends
such that $P_{u}=F_{u}\circ P\circ F_{u}^{-1}\in P_{d}$ . Since the same theorem says
holomorphically on , so does . Thus we define aholomorphic map:
. Thisis equal toby $W_{P}(u)=P_{u}$ . The B\"ottcher coordinateof,
equivalenthybrid
operation is called wringing. Since
istoit holds, the escape locus, we define the, the connectedness locus. Forfor
stretching ray through byThen, by the Measurable Riemann Mapping Theorem,$\sigma_{u}$$F_{u}$$F_{u}$$W_{P}$$P_{u}$$u$$\varphi_{P_{u}}$$P$$P_{u}$$P\in \mathrm{C}_{d}$$H_{+}arrow P_{d}$$f_{u}\circ\varphi_{P}\circ F_{u}^{-1}$$P_{u}$$P_{u}\equiv P$$P\in \mathcal{E}_{d}$$P$$R(P)=W_{P}(\mathrm{R}_{+})=\{P_{s};s\in \mathrm{R}_{+}\}$.

For example, in case $d=2$ , stretching rays coincide with the external rays for theMandelbrot set

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2Stretching rays :special case of real cubic
polynomials

We consider the family of real cubic polynomials in the first quadrant:$P(z)=P_{A,B}(z)=z^{3}-3Az+\sqrt{B};A$ , $B>0$ .

We investigate the accumulation sets of stretching rays above the parabolic arc :$Per_{1}(1)$: $B=4(A+1/3)^{3};0\leq A\leq 1/9$ .

For $Q\in Per_{1}(1)$ ,has aparabolic fixed point $\beta_{Q}=\sqrt{A+1/3}$ with multiplier 1
and both critical points escape toabove $Per_{1}(1)$ .
$Q$$\infty$

We set$\zeta_{P}(z)=\underline{\log\log\varphi_{P}(z)}$$\log 3$and define, for$P\in \mathcal{E}_{3}^{2}$(the real shift locus, i.e. thelocus where both critical points escape), the Bottcher vector $\eta(P)$ by$\eta(P)=.\frac{\log h_{P}(-\sqrt{A})-\log h_{P}(\sqrt{A})}{1\mathrm{o}\mathrm{g}3}.=\zeta_{P}(P(-\sqrt{A}))-\zeta_{P}(P(\sqrt{A}))$

Lemma 2.1 On the stretching ray $R(P)$ through$P\in \mathcal{E}_{3}^{2}$,$\eta(P_{s})$.

is invariant.

Thus each stretching ray in the shift locus
is alevel curve $\eta(P)=\eta$ of the$P\mapsto\eta(P)$mapvector,Bottcher
which we denote by. Thus we have an explicit
description of stretching rays in our family and we can draw their pictures.
$\mathcal{E}_{3}^{2}$$R(\eta)$

3Preliminaries from parabolic implosion
In this section, we introduce some notions from parabolic implosion. For $Q\in$$Per_{1}(1)$ , the immediate basinof the parabolic fixed point
contains bothcritical points
and $J(Q)=\partial B_{Q}$ is aJordan curve. Letandbe theattracting and repelling Fatou coordinates respectively. They satisfy $\phi_{Q,\pm}\circ Q(z)=$$\phi_{Q.\pm}(z)+1$ in the attracting and repelling petals respectively.
Thus, there is anambiguity of additive constant. Appropriately normalized, they are assumed to besy mnetric with respect to the real axis. We define the Fatou vector $\tau(Q)$ of by$B_{Q}$$\beta_{Q}$$\pm\sqrt{A}$$\phi_{Q,-}$$\phi_{Q,+}$$Q$$\tau(Q)=\phi_{Q,-}(-\sqrt{A})-\phi_{Q,-}(\sqrt{A})$.

Note that this definition does not depend on the choice of Fatou coordinates.

Lemma 3.1 The Fatou vector gives a real analytic parametr ization of $Per_{1}(1)$ , $0<$

28Letbe the Fatou coordinates ofabove $Per_{1}(1)$ normalized in thesame way. They are continuous up to $Per_{1}(1)$ . After perturbation, the gate isopen and the incoming Fatou coordinate
can be regarded also as an outgoing
Fatou coordinate and vice versa. Thusanddiffer only by an additive$\tilde{\sigma}(P)=\phi_{P,+}(z)-\phi_{P,-}(z)$constant. We call this difference
the lifted phase andits classinthe phase of . Since all themappings are symmetric with respect to the real axis, the lifted phase is always
real. Roughly speaking, minus the lifted phase is the time needed for the orbits of
to pass through the gate between two fixed points.
$P\in \mathcal{E}_{3}^{R}$$\phi_{P,\pm}$$\phi_{P,-}$$\phi_{P,-}$$\phi_{P,+}$$\sigma(P)=\{\tilde{\sigma}(P)\}\equiv\tilde{\sigma}(P)-[\tilde{\sigma}(P)]$$P$$\mathrm{C}/\mathrm{Z}$$P$$\beta_{P}^{\pm}$Lemma 3.2 The lifted phase
We also define, for$\tilde{\sigma}\in \mathrm{C}$$g_{\overline{\sigma}}=\phi_{Q,+}^{-1}\circ T_{\overline{\sigma}}\circ\phi_{Q,-}$$\tilde{\sigma}(P_{s})$tends $to-\infty$ as, the Lavaurs map ga :, where$T_{\overline{\sigma}}(w)=w+\tilde{\sigma}$

on a stretching ray.

$B_{Q}arrow \mathrm{C}$.

Lemma 3.3 Suppose $P_{n}arrow Q\in Per_{1}(1)$ andsatisfying
lift of . If we takeunifomly on.

of lifted phase$\sigma(P_{n})arrow\sigma\in \mathrm{C}/\mathrm{Z}$, then$N_{n}+\tilde{\sigma}(P_{n})arrow\tilde{\sigma}$$N_{n}arrow \mathrm{o}\mathrm{o}$$\sigma$$sarrow \mathrm{O}$. Let$\tilde{\sigma}$$P_{n}^{N_{n}}arrow g_{\overline{\sigma}}$$\tilde{\sigma}$by

be anylocally$B_{Q}$

Since, in our case, $K(Q)$ is symmetric with respect to the real axis, connected andlocally connected, its image in the repelling Fatou coordinate does not intersect tlle
real axis. Then it follows. Hence we can define the Bottcher
vectorwith phase also for $Q\in Per_{1}(1)$ :$g_{\overline{\sigma}}(\pm\sqrt{A})\in \mathrm{C}-K(Q)$$\eta(Q, \sigma)$$\sigma$.

$\eta(Q, \sigma)=\zeta_{Q}(g_{\overline{\sigma}}(-\sqrt{A}))-\zeta_{Q}(g_{\overline{\sigma}}.(\sqrt{A}))$It depends only on the phase and not on the choice of lifted phase. Note that
this definition depends on the choice of Fatou coordinates. In fact, if we add someconstants to them, this changes the phase. This causes some difficulty in the nextsection. We define the -impression
by the set of points $Q\in Per_{1}(1)$of$P_{n}arrow
Q$ and$P_{n}\inR(\eta)$such that there exists
satisfying. Apparently$I(\eta)=\overline{R(\eta)}-R(\eta)$
the accumulation set,ofis the union of all.
By Lemma 3.3, it easily follows:$R(\eta)$$I_{\eta}(\sigma)$$\sigma$$\sigma(P_{n})arrow\sigma$$R(\eta)$

Lemma 3.4 Under the same assumptions as in Lemma 3.3,Consequently, we have.

$I_{\eta}(\sigma)$$\sigma\in \mathrm{R}/\mathrm{Z}$$\eta(P_{n})arrow\eta(Q, \sigma)$.

$I_{\eta}(\sigma)\subset\{Q\in Per_{1}(1);\mathrm{K}(\mathrm{Q})\sigma)=\eta\}$

We will show that the inclusion above actually is an identity.

4Accumulation sets of stretching rays
In the last conference, we have shown the following.

Theorem 4.1 Suppose is rational. Then the stretching ray
does not land
on $Per_{1}(1)$ . Hence its accumulation set
is a non-trivial arc on $Per_{1}(1)$ .
$\eta$$R(\eta)$$i$$I(\eta)$

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Figure 1: Non-landing stretching rays

In this section, we will characterize the sets
. Note that the map
Bottcher vector map
on the choice of Fatou coordinates.
$\eta(Q, \sigma)$$I_{\eta}(\sigma)$$\eta(Q, \sigma)$andin terms of thedepend
and the set$I(\eta)$$I_{\eta}(\sigma)$

Proposition 4.1 $I_{\eta}(\sigma)=\{Q\in Per_{1}(1);\eta(Q, \sigma)=\eta\}$ .

As acorollary, we have
Corollary 4.1 $R(\eta)$ accumulates at if and only if. Consequently,$Q\in
Per_{1}(1)$, there exist at least one stretching rays accumulating at $Q$ .
for any$Q$$\eta\in\eta(Q, \mathrm{R}/\mathrm{Z})$John Milnor drew pictures of non-landing stretching rays, found that their oscillation is extremely regular and suggested the problems whether the set
consistsof asingle point and whether it depends continuously on and . See Figure 1.
Considering Proposition 4.1, these are true if and only if the map
ismonotone increasing for any .
Tlle proof of Proposition 4.1 relies on the following.
$I_{\eta}(\sigma)$$\eta$$\sigma$$Q\mapsto\eta(Q, \sigma)$$\sigma$Lemma 4.1 Fix aand suppose $Q_{0}\in I$ , a connected component of,. Then there exists a normalization of Fatou coordinates such
satisfiesthat the mapin these new Fatou coordinates is monotone increasing
on I and.
$Per_{1}(1)-\tau^{-1}(\mathrm{Z})$$\eta(Q_{0}, \sigma)=\eta_{0}$$Q\mapsto\tilde{\eta}(Q, \sigma)$$\mathrm{f}\mathrm{j}(\mathrm{Q}0, \sigma)=\eta_{0}$The proof of Lemma 4.1 is done by -deformation of the Bottcher vectors just
as for the proof of Lemma 3.1. But the conclusion is somewhat different: we have tolose the freedom of additive constant for Fatou coordinates, since B\"ottcher vectors
depend on the choice of Fatou coordinates. In Lemma 4.1, the normalization ofFatou coordinates depends on . So we have not solved above problems yet.
proof Suppose $\eta_{0}\in(k, k+1)$ for some. Letand,the annulus $R\leq|z|\leq R^{3}$ in the B\"ottcher coordinate, we take aqc-ma$\mathrm{q}\mathrm{c}$$\sigma$$k\in \mathrm{Z}$$()\mathrm{n}$$R=\varphi_{Q_{0}}(g_{Q_{0},\sigma}(\sqrt{A_{Q_{0}}}))$

30
to mod in the (such that it changes only radial coordinates, maps
coordinate for any $\eta\in(k, k+1)$ and is identity on the boundary. We can extend.
it to the complement of the closed unit disk so that it commutes withis-invariant in the complement of $K(Q_{0})$ . Pulling it backThen, we extend it also inside $K(Q_{0})$ . Actually it extends to the complement ofby
$K(Q_{0}, \sigma)=J(Q_{0}, \sigma)$ , the set of points which do not escape byandandso thatbe the -map integratinghas measure 0. Let.
. Its Bottcher coordinate is expressed by, which is
We normalize the attracting Fatou coordinate bypreserved under -deformation and repelling Fatou coordinates byis preserved under the above -deformation since is. Then,and we have
fixed by . If$\mathrm{Z}$$l=\ell_{\eta}$$\eta$$\eta_{0}$$z\mapsto z^{3}$$Q_{0}$$\sigma_{\eta}=\varphi_{Q_{0}}^{*}\ell^{*}\sigma_{0}$$g_{Q_{0},\sigma}$$Q_{0}$$J(Q_{0}, \sigma)$$\chi=\chi_{\eta}$$\mathrm{q}\mathrm{c}$$G_{Q\mathrm{o},\sigma}$$Q_{\eta}=$$\sigma_{\eta}$$\chi \mathrm{o}Q_{0}\circ\chi^{-1}\in Per_{1}(1)$$\varphi_{Q_{\eta}}=\ell\circ\varphi_{Q_{0}}\circ\chi^{-1}$$\tilde{\phi}_{Q,-}(\sqrt{A_{Q}})=0$$\tilde{\phi}Q,+(\varphi_{Q}^{-1}(R))=$$\mathrm{q}\mathrm{c}$$\tilde{\phi}_{Q,+}^{-1}(\sigma)=\varphi_{Q}^{-1}(R)$$\sigma$$\ell$$\eta=\eta_{0}$$R$$\mathrm{q}\mathrm{c}$$Q_{r\mathfrak{p}}=Q_{0}$.

$\tilde{\phi}_{Q\mathrm{o},+}(\varphi_{Q_{0}}^{-1}(R))=\overline{\phi}_{Q_{0\backslash }+}(g_{Q_{0},\sigma}(\sqrt{A_{Q_{0}}}))=T_{\sigma}(\tilde{\phi}_{Q_{0},-}(\sqrt{A_{Q_{0}}}))=T_{\sigma}(0)=\sigma$

coincides withforHence the original Fatou coordinate
that,
only to show
havewe.
Inordershow
Weof .
is equal to the Lavaurs nldl)-invariant, is holomorphic. Thenis, by definition,
Since
is aholomorphic 11tap ffom attracting petal to the repelling petal. It,
easily follows that commutes with . Then induces acylinderfrom the attracting cylinder onto the repelling one, hence is atranslation. Next we
is preserved by, which completes the proof. The critical pointshow
map and takes 0in the attracting Fatou coordinate. On the other hand, thetomaps,
value, is preserved by . Whenfor any. Hence, this is true also for any and we have$\phi_{Q_{0}.+}$$\tilde{\phi}Q_{0}.+\cdot$$Q_{0}$$g=\chi \mathrm{o}gQ_{()_{\backslash }}\sigma\circ\chi^{-1}$$\tilde{\eta}(Q_{\eta}, \sigma)=\eta$$Q_{\eta}$$gQ_{\eta},\sigma$$T=\tilde{\phi}_{Q_{1},+}.\circ$$g$$g_{Q_{0},\sigma}$$\sigma_{\eta}$$g\circ\tilde{\phi}_{Q_{1},-}^{-1}$

$T$

$T$

$T_{1}$$\mathrm{i}_{\iota}\mathrm{S}0111\mathrm{o}\mathrm{r}\mathrm{p}11\mathrm{i}\mathrm{s}\mathrm{n}1$$\sqrt{A}$$T=T_{\sigma}$$\chi$$\mathrm{q}\mathrm{c}$$\overline{\phi}_{Q,+}^{-1}(\sigma^{)}$$\eta=\eta_{0}$$\chi$$T=T_{\sigma}$$\sigma=\tilde{\phi}_{Q\mathrm{o},+}(\varphi_{Q_{0}}^{-1}(R))$$\eta$$0=\tilde{\phi}_{Q_{0},-}(\sqrt{A_{Q\mathrm{o}}})$$T=T_{\sigma}$$\eta$. Cl

The proof of Proposition 4.1 is now easy. We have only to show { $Q\in Per_{1}(1)$ ;. We apply Lemma 4.1. Then the. Suppose
befor the new Fatou coordinates is monotone increasing. Letmapwith respect to the newand
satisfyingasequence on)lies
is monotone, this. Since
Fatou coordinates. Then
$Q_{1}=Q_{0}$ and we have. Since the repelling Fatou coordinates coincide
inin both normalization, it follows, by the following Lemma 4.2,
at
4.1.
Proposition
the original normalization. This completes the proof offor . They differs only by constants
andTake two Fatou coordinates
depending on . Hence their lifted phases satisfy:$\eta(Q_{0}, \sigma)=\eta_{0}$$\eta(Q, \sigma)=\eta_{0}\}\subset I_{\eta 0}(\sigma)$$P_{n}$$Q\mapsto\tilde{\eta}(Q, \sigma)$$P_{n}arrow Q_{1}$$R(\eta_{0})$$\sigma(P_{n})arrow\sigma$$Q\mapsto\tilde{\eta}(Q, \sigma)$$\tilde{\eta}(Q_{1}, \sigma)=\eta_{0}$$\mathrm{i}\mathrm{m}\mathrm{l}$$\tilde{I}_{\gamma}(h\sigma)=\{Q_{0}\}$$Q\mathrm{o}\in I_{m\mathrm{J}}(\sigma)$$Q_{0}$$\square$$\phi_{P,\pm}^{1}$$\phi_{P,\pm}^{2}(z)-\phi_{P,\pm}^{1}(z)=\delta_{P.\pm}$a2$\phi_{P,\pm}^{2}$

$P$

$P$

$(P)=\phi_{P,+}^{2}(z)-\phi_{P,-}^{2}(z)=\tilde{\sigma}_{1}(P)+\delta_{P,+}-\delta_{P,-}$

And, if $P_{n}arrow Q$ and$\sigma_{1}(P_{n})arrow\sigma_{1}$.

, we have$\sigma_{2}(P_{n})=\sigma_{1}(P_{n})+\delta_{P_{n},+}-\delta_{P_{n},-}arrow\sigma_{2}\equiv\sigma_{1}+\delta_{Q,+}-\delta_{Q,-}$(mod$\mathrm{Z}$).

31
Lemma 4.2 Denote the cor responding impressions by
0, then (. Especially, (, then QE,7.
and only Q6.
and(and QE$I\ovalbox{\tt\small REJECT} 7(\mathrm{a}_{2})$$I_{\ovalbox{\tt\small REJECT}}:(\mathrm{a}.)$$I_{\ovalbox{\tt\small REJECT}}\ovalbox{\tt\small REJECT}((7_{\mathrm{t}}^{\ovalbox{\tt\small REJECT}})$$I\ovalbox{\tt\small REJECT}(\mathrm{a}_{2})$$i\ovalbox{\tt\small REJECT}$$i\ovalbox{\tt\small REJECT}$$5_{Q},.$$\ovalbox{\tt\small REJECT}$$\mathrm{r}_{2}\ovalbox{\tt\small REJECT}$$\mathrm{r}_{1}$If Q$\# 7\ovalbox{\tt\small REJECT}$

(a2)

$E$

$i\ovalbox{\tt\small REJECT}$$I_{\ovalbox{\tt\small REJECT}}\mathrm{S}(\mathrm{c}\mathrm{r}_{1})$References[Br] B. Branner: Turning around the connectedness locus. Topological method inmodern Mathematics, I)p. 391-427.
[M1] J. Milnor: Remarks on iterated cubic maps. Experimental Math. 1(1992), pp.

523.
[W] P. Willumsen: Holomorphic dynamics: On accumulation of stretching rays.
Ph.D. thesis, Tech. Univ. Denmark, 1997