数理解析研究所講究録
第 1586 巻 2008 年 121-126

121

Blaschke products with a critical point on the unit circle
and rational functions with Siegel disks
Koh Katagata
Interdisciplinary Graduate School of Science and Engineering
Shimane University, Matsue 690-8504, Japankatagata@math.shimane-u.ac.jp
Abstract

We give a brief survey of results on Siegel disks of some rational functions with bounded type rotation number. A Siegel disk of some polynomialwith bounded type rotation number has the quasicircle boundary containing

its critical point. In order to construct such a Siegel disk not of a polynomial
but of a rational $fi_{i}ction$, we consider some Blaschke product and employ thequasiconformal surgery.

1LetResults$P_{\alpha}(z)=z^{2}+e^{2ni\alpha}z$. Then the following theorem holds if is of bounded type.
$\alpha$.

Theorem 1 $(Ghys- Douady- Herman- Shishlkura- Sw1_{1}tek, [6])$ If an irrational
number $\alpha\in[0,1]$ is of bounded type, then the boundary of the Siegel disk of$\Delta$$P_{\alpha}$

centered at the origin is a quasicircle containing its critical $point-P^{ia}/2$ .

Let $Q_{\alpha,m}(z)=e^{2\pi j\alpha}z(1+z/m)^{m}$. Geyer showed the following theorem which is.
extended to some polynomials. Note that is conformally conjugate to$Q_{\alpha,1}$$P_{\alpha}$Theorem 2 (Geyer, [1]). Let $m\geq 1$ be a positive integer. If an irrational number
centered$\alpha\in[0,1]$ isofbounded type, then the boundary ofthe Siegel disk ofat the origin is a quasicircle containing its critical $point-m/(m+1)$ .
$\Delta$

For complex numbers$\lambda$

and with$\mu$$\lambda\mu\neq 1$$Q_{a,m}$and a positive integer , let$F_{\lambda.\mu,m}(z)=z( \frac{z+\lambda}{\mu z+1})^{m}$$m$.

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The origin and the point at infinity are fixed points
respectively. In the case that $\mu=0$ ,$ofF_{\lambda,\mu,m}$$F_{\lambda,0,m}(z)=z(z+\lambda)^{m}$of multiplier$\lambda^{m}$and$\mu^{m}$.

ofdegree $m+1$ is considered as a perturbation
Therefore the rational functionof degree $m+1$ . Note thatis confomally conjugate
ofthe polynomialifto.
$F_{\lambda,\mu,m}$$F_{\lambda,0,m}$$F_{\lambda,0.m}$$Q_{\alpha,m}$$\lambda^{m}=e^{2\pi i\alpha}$

D. If an irrational
Main Theorem. Let $m\geq 1$ be a positive integer and, then there exist suitable pairs
andnumber $\alpha\in[0,1]$ is of boundedwith$\mu\in$$\varphi pe$$e^{2\pi ia}\mu^{m}\neq 1$$\{(\lambda_{j},\mu_{j})\}_{j\cdot 1}^{m}$

(i)

(ii)

$\lambda_{j}^{m}=e^{2nla},$ $\mu_{j}^{m}=\mu^{m}$and$\lambda_{j}\mu_{j}\neq 1$

for$j\in\{1, \ldots,m\}$$\lambda_{j}\neq\lambda_{k}ifj\neq k$such that for each $j\in\{1, \ldots,m\}$ , the boundary of the Siegel disk
centered at the origin is a quasicircle containing its critical point.

$\Delta_{j}$of$F_{\lambda_{J}.\mu_{j}.m}$

Main Theorem contains Theorems 1 and 2. Moreover we obtain the following
corollary.
Corollary. Let $m\geq 1$ be a positive integer, $\alpha\in[0,1]$ be an irrational number ofbe as in Main Theoandwith
bounded $O’pe,$, then the boundaries ofrem. $If\beta\in[0,1]$ is an irrational number of bounded
centered at the origin and the point at infinity
andSiegel disksofnespectively are quasicircles containing one critical point.
$e^{2\pi la}\mu^{m}\neq 1$$\mu^{m}=e^{2\pi\beta}$$\{(\lambda_{j},\mu_{j})\}_{j\cdot 1}^{m}$$\varphi pe$$\Delta_{j}$$\Delta_{j}^{\infty}$$F_{\lambda_{J},\mu_{j},m}$2 Key Theorems
Let $m\geq 1$ be a positive integer. We consider the Blaschke product$B_{\theta,\varphi.m}(z)=F^{m\theta_{Z}}( \frac{z-a}{1-\overline{x}})^{m}(\frac{z-b}{1-bz})^{m}$

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ofdegree $2m+1$ witb

$a\overline{b}\neq 1$

and $0<|a|\leq|b|<\infty$ . Let$x=\{(m+1)^{2}+(m-1)^{2}\mu+2(m^{2}-1)r\cos 2\pi(2\varphi+\theta+\omega)\}^{-1}$$x\{D_{1}\cos 2\pi\varphi+D_{2}\cos 2\pi(\varphi+\theta+\omega)$$+D_{3}$cos$2\pi(3\varphi+\theta+\omega)+D_{4}$

cos$2\pi(3\varphi+2\theta+.2\omega)\}$and$y=\{(m+1)^{2}+(m-1)^{2}r^{2}+2(m^{2}-1)r\cos 2\pi(2\varphi+\theta+\omega)\}^{-1}$$x\{D_{1}\sin 2\pi\varphi-D_{2}\sin 2\pi(\varphi+\theta+\omega)$$+D_{3}\sin 2\pi(3\varphi+\theta+\omega)-D_{4}\sin 2\pi(3\varphi+2\theta+2\omega)\}$

,

where
$D_{1}=(m+1)^{2}(2m+1)-2m(m^{2}-1)r^{2}$ ,$D_{2}=2m(m^{2}-1)r-(m-1)^{2}(2m-1)r^{3}$ ,$D_{3}=-(m+1)^{2}r$,$D_{4}=-(m-1)^{2}’$.

and $b=b(\theta,\varphi)with|a|\leq|b|be$and letTheorem A.
, that is,complex numbers satisffing relations $a+b=x+iy$ andand are the solutions of the equation$Let\mu=rP^{i\omega}\in\overline{D}$$a=a(\theta,\varphi)$$ab=re^{-2ni(\theta*\omega)}$

$a$

$b$

$Z^{2}-(x+iy)Z+re^{-2\pi i(\theta+\omega)}=0$

where andy are as above and$x$(a) In the case$(\theta,\varphi)\in[0,1]^{2}$(\dagger ),. Then the following holds:

that. $r=0$, solutions of the equation (t) are $a=0$ and $b=$.

$(2m+1)e^{2\pi\phi}$(b) In the case that $0<r<1$ , the equation (f) does not have double rvots.

Moreover $0<|a|<1<|b|<\infty$ .
(c) In the case that $r=1$ and $2\varphi+\theta+\omega\equiv 0(mod 1)$ , the equation$(\uparrow)$has

double roots $anda=b=e^{2\pi i\varphi}$.
(d) In the case that $r=1$ and $2\varphi+\theta+\omega\not\in 0(mod 1)$ , the equation

have double roots. Moreover $0<|a|<1<|b|<\infty$ .

$(\uparrow)$does not124(e) In the case (a), (b) or (d),$B_{\theta,\varphi,m}(z)=e^{2nim\theta}z( \frac{z-a}{1-\overline{a}z})^{m}(\frac{z-b}{1-bz})^{m}$is a Blaschke product ofdegree $2m+1$ and the point at $infini\psi$ is afixedpointis a critical point ofwith multiplier . Moreoverofis a homeomorphism, where is the unit circle.
:
and$z=e^{2\pi i\varphi}$$\mu^{m}$$B_{\theta,\varphi.m}$$Tarrow \mathbb{T}$$B_{\theta,\varphi,m}|_{T}$$B_{\theta,\varphi,m}$$\mathbb{T}$be an orientation preserving homeomorphism and denote by
the rotation number of $f$.
Let $f$ :$\mathbb{T}arrow \mathbb{T}$Theorem B. Let $\alpha\in[0,1]$ andin Theorem A Then for the Blaschke product$let\mu=re^{2ni\omega}\in\overline{D},$$a=a(\theta,\varphi)$ $andb=b(\theta,\varphi)$$\rho(f)$be as,$B_{\theta,\varphi,m}(z)= \mathscr{J}^{im\theta_{Z}}(\frac{z-a}{1-\overline{a}z})^{m}(\frac{z-b}{1-bz})^{m}$$B_{\theta,\varphi,m}|_{T}$:$\mathbb{T}arrow \mathbb{T}$is an orientation preserving homeomorphism. Moreover(a) If $O\leq r<1$ , then there exists$(\theta_{0},\varphi_{0})\in[0,1]^{2}$such that(b) $Ifr=1$ and $\alpha+m\omega\not\in O(mod 1)$, then there exists$\rho(B_{h\varphi 0,m}|_{T})=\alpha$and$2\varphi_{0}+\theta_{0}+\omega\not\equiv 0(mod 1)$$\rho(B_{h\cdot\varphi_{0}.m}|_{T})=\alpha$$(\theta_{0},\varphi_{0})\in[0,1]^{2}$.

such that.

3 Proof
Pmof of Main Theorem. By Theorem$B$such that the, there exist. Then there exists a quasisymmetric$(\theta,\varphi)\in[0,1]^{2}$

is $2m+1$ anddegree ofsuch that
homeomorphism :of bounded type. By the theorem of Beurling and Ahlfors,$\rho(B_{\theta,\varphi,m}|_{T})=\alpha$$B_{\theta,\varphi,m}$$h$extension$\mathbb{T}arrow \mathbb{T}$$H:\overline{D}arrow\overline{D}$since ishas a quasiconformal$h\circ B_{\theta,\varphi,m}|_{T}\circ h^{-\downarrow}(z)=R_{\alpha}(z)=e^{2nia}z$$h$with $H(O)=0$ . We define a new map$\mathfrak{B}_{\theta,\varphi,m}=\{\begin{array}{ll}B_{\theta.\varphi,m} on \hat{C}\backslash D,H^{-1_{\circ}}R_{\alpha}oH on D.\end{array}$$\mathfrak{B}_{\theta,\varphi.m}$

as$\alpha$125
The map
is quasiregular on since is an analytic curve. Moreoveris a degree $m+1$ branched covering of . We define a conformal structure$\hat{\mathbb{C}}$$\mathbb{T}$$\mathfrak{B}_{\theta,\varphi,m}$$\mathfrak{B}_{\theta,\varphi,m}$as$\hat{\mathbb{C}}$$\sigma_{\theta,\varphi,m}$$\sigma_{\theta,\varphi,m}=\{\begin{array}{ll}H^{*}\sigma_{0} onD,(\mathfrak{B}_{\theta,\varphi,m}^{n})^{*}\sigma_{0} on \mathfrak{B}_{\theta,\varphi.m}^{-n}(D)\backslash D for all n\in N,\sigma_{0} on \hat{\mathbb{C}}\backslash \bigcup_{n=1}^{\infty}\mathfrak{B}_{\theta,\varphi,m}^{-n}(D),\end{array}$where is the standard confomal structure on . The conformal structure
is invariant underand its maximal dilatation is the dilatation of $H$ since $H$ isis holomorphic. By the measurable Riemam mapping
quasiconformal and:such thattheorem, there exists a quasiconformal homeomorphism
is a rational map of degree $m+1$ . We. Therefore, whereandby $\Psi_{j}(0)=0,$
normalize$\hat{\mathbb{C}}$$\sigma_{\theta,\varphi,m}$$\sigma_{0}$$\mathfrak{B}_{\theta,\varphi.m}$$B_{\theta,\varphi.m}$$\Psi$$\hat{\mathbb{C}}arrow\hat{\mathbb{C}}$$\Psi\circ \mathfrak{B}_{\theta.\varphi,m}\circ\Psi^{-1}$$\Psi\cdot\sigma_{0}=\sigma_{\theta,\varphi,m}$$\Psi_{j}(\infty)=\infty$$\Psi_{j}(b)=-\lambda_{j}$$\Psi=\Psi_{j}$$\lambda_{j}=e^{2nj(a+j)/m}$for $j\in\{1, \ldots,m\}$ .

Lemma. $If\mu\neq 0$, then there exists$\mu_{j}$with$\mu_{j}^{m}=\mu^{m}$such that$F_{\lambda_{j}.\mu_{j}.m}=\Psi_{j}\circ \mathfrak{B}_{\theta.\varphi,m}\circ\Psi_{j}^{-1}$.

issince such. Note thatasunique. Since orders of zeros and poles are invariant under conjugation, we obtain

ProofofLemma. Define$\xi_{j}$$\xi_{j}=-\Psi_{j}(1/a\gamma$$\Psi_{j}$$\lambda_{j}\neq\xi_{j}$that$\Psi_{j^{O}}\mathfrak{B}_{\theta,\varphi.m^{\circ}}\Psi_{j}^{-1}(z)=\eta_{j}z(\frac{z+\lambda_{j}}{z+\xi_{j}})^{m}$.

Since multipliers offixed points are also invariant under conjugation, we obtain that(1)$( \Psi_{j}0\mathfrak{B}_{\theta,\varphi,m}0\Psi_{j}^{-1})’(0)=\frac{\eta_{j}\lambda_{j}^{m}}{\xi_{j}^{m}}=e^{2ni\alpha}$and$\frac{1}{(\Psi_{j}0\mathfrak{B}_{\theta.\varphi,m}0\Psi_{j}^{-1})’(\infty)}=\frac{1}{\eta_{j}}=\mu^{m}$

By the equations (1) and (2), we obtain that
. Therefore
root of unity such that$(\xi_{j}\mu)^{m}=1$$v_{j}$.

(2). Then there exists an m-th$\xi_{j}=v_{j}/\mu$$\Psi_{j^{O}}\mathfrak{B}_{\theta,\varphi,m^{O}}\Psi_{j}^{-1}(z)=\frac{z}{\mu^{m}}(\frac{z+\lambda_{j}}{z+v_{j}/\mu})^{m}=z(\frac{z+\lambda_{j}}{\mu z+v_{j}})^{m}$.

$= \frac{z}{v_{j}^{m}}(\frac{z+\lambda_{j}}{(\mu/v_{j})z+1}]^{m}=z(\frac{z+\lambda_{j}}{\mu_{j}z+1})^{m}=F_{\lambda_{j}.\mu/,m}(z)$

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where

$\mu_{j}=\mu/v_{j}$.

口Let $\mu_{j}=0$ for all $j\in\{1, \ldots,m\}$ if $\mu=0$ . It is easy to check that the pairs
satisfy (i) and (ii). The map
has a Siegel disk $\Delta=\Psi_{j}(D)$ witha critical point. Moreoveris a quasicircle sinceisquasiconformal.
$\{(\lambda_{j},\mu_{j})\}_{j=1}^{m}$$F_{\lambda_{j},\mu_{j}.m}$$\Psi_{j}(e^{2ni\varphi})\in\partial\Delta$$\partial\Delta=\Psi_{j}(T)$$\Psi_{j}$$0$Proofof Corollary. Let $t(z)=1/z$. Then

$F_{\lambda_{J},\mu/’ m}=I\circ F_{\mu_{J},\lambda_{J}.m}\circ I$. Let$\Delta$and$\Delta_{\infty}$

be Siegel disks
centered at the origin and the point at infinity respectively.
By Main Theorem, the boundary of contains a critical point of. On theother hand,is the Siegel disk ofcentered at the origin. By Main
Theorem, the boundary ofcontains a critical point of. Therefore theboundary ofcontains a critical point of.
$ofF_{\lambda_{J},\mu_{J}.m}$$\Delta$$F_{\lambda_{J}.\mu/,m}$$t(\Delta_{\infty})$$F_{\mu_{j}.\lambda_{j},m}$$I(\Delta_{\infty})$$F_{\mu_{j},\lambda_{j},m}$$\Delta_{\infty}$$\square$$F_{\lambda_{J}.\mu_{J},m}$References[1] L. Geyer, Siegel discs, Herman rings and the Amold family, $\pi_{ans}$. Amer.

Math. Soc. 353 (2001), no. 9, 3661-3683.
[2] K. Katagata, Some cubic Blaschke products and quadratic rational functionswith Siegel disks, Int. J Contemp. Math. Sci., to appear.
[3] K. Katagata, Blaschke products and rational functions with Siegel disks, sub-mitted.
[4] J. Milnor, $\emptyset namics$ in One Complex Variable, Vieweg, 2nd edition, 2000.
[5] M. Yampolsky and S. Zakeri, Mating Siegel quadratic polynomials, J. Amer.

Math. Soc. 14 (2001), no. 1, 25-78 (electronic).

.

[6] S. Zakeri, Old and new on quadratic Siegel disks, http: $//www$ math. qc. edu$/\sim zakeri/papers/papers$. html.