数理解析研究所講究録920 巻 1995 年 229-235

229

Abstract. We define center curves in the moduli space, consisting of all affine conjugacy classes of

cubic maps and analyse the dynamics of, maps along these curves.

25.1 Moduli space of the complex cubic polynomials.
We consider the family of cubic maps $x\mapsto g(x)=c_{3}x^{s}+c_{2}x^{2}+c_{1}x+c_{\mathrm{O}}$For such a cubic map$g$, we have two normal forms ; $x^{3}-3Ax\pm\sqrt{B},$ $A,$

complex affine conjugacy class of$\mathcal{M}$$g$$(c_{3}\neq 0, c_{\mathfrak{i}}\in \mathrm{C})$$B\in \mathrm{C}$.

. Therefore, thecan be represented by $(A, B)$ . The moduli space, denoted by, consisting of all affine conjugacy classes of cubic maps, can be identified with the coordinate

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space $\mathrm{C}^{2}=\{(A, B)\}([5])$ .

25.1.1 Center in a hyperbolic component.
A complex cubic map , or the corresponding point $(A, B)\in \mathcal{M}$ , belongs to the connectedness$f$locus if the orbits of both critical points$f$$p_{i}$

such that $f’(p.\cdot)=0,$ $i=1,2$ , are bounded. Andis hyperbolic if both of these critical orbits converge towards attracting periodic orbits. The

set of all hyperbolic points in the moduli space

forms an open set.

$\mathcal{M}$

Each connected component of this open set is called a hyperbolic component. By M. Rees $([91)$ ,each hyperbolic component contains a uniquecriticaUy finite complex cubic map. So follow-$\mathrm{p}_{\mathrm{o}\mathrm{s}\mathrm{t}}$ing A. Douady and J. Hubbard (|2]), this map is called a center map orThurston map andthe coordinates $(A, B)$ of$)\mathrm{a}\mathrm{n}\mathrm{d}$$f$will be called a center in the moduli space. Following M.Rees $([9]$J.Milnor $([5])$ , the centers are roughly classified into four diffeent types, as follows. (In thefollowing$t,$$p,$ $q$

denote integers.) A center is of the type

center map coinside and has the period

if$f^{\iota}(p_{1})=p_{2}$$p_{1},$$p_{2}$of the: $f^{p}(p_{1})=p_{1}$ . (In fact, only possible values for$p$

this case are 1, 2.) A center is of the typeif two critical points$A_{p}$$B_{\mathrm{p}+q}$

and $f^{q}(p_{2})=p_{2}$ ; of the type

$D_{p,q}$

if $f^{p}(p_{1})=p_{2}$ and $f^{q}(p_{2})=p_{1}$ ; of the typeif$f^{\mathrm{p}}(p_{1})=p_{1}$

$p$

in

$C_{(t)q}$

and $f^{q}(p_{2})=p_{2}$ .

These exhaust all types of centers. It is clear that there are only a finite number of centers ofa given type.
There exist a unique center of type $A:$ . The corresponding parameter is $(0,0)$ for

Example:$i=1$ and $(0, -1)$ for $i=2$ . There exist three centers of type

$C_{\langle 3\rangle 1}$. The corresponding parametersare$(A, B)=(-.75040, -.18820)$ , $(-.74949, -. 18679)$ , $(-.0924912, -.0614376)$ .

25.2 Center curves in the moduli space.
The center curves$\mathrm{C}\mathrm{D}p,$$\mathrm{B}\mathrm{C}p$, which are algebraic curves, can be defined according to theabove four renormalization-type. We show how the equations of these curves are obtained byinduction on$p$([7] and [8]).

Theorem 2.1 : Deflning equation of a center curve
algebraic curve $CDp$ containing all centers of the typecurve $BCp$ containing all centers of the typethe following curves;$B_{\mathrm{p}+k}$and

For a given , there exist an$p$$C_{(k)p}$$C_{(p)k}$and$D_{k,p}$, and another algebraic. For example, we obtain precisely

231$CDl$:$B=4A(A+ \frac{1}{2})^{2}$,$BC\mathit{1}$:$B=4A(A- \frac{1}{2})^{2}$,$CD\mathit{2}$:$B^{2}-8A^{3}B+4A^{2}B-5AB+2B+16A^{6}-16A^{5}$$-12A^{4}+16A^{3}-4A+1=0$ ,$BC\mathit{2}$:$B^{3}-12A^{3}B^{2}-6AB^{2}+2B^{2}+48A^{6}B+24A^{3}B+21A^{2}B$$-6AB+B-64A^{9}+96A^{7}-20A5-12A^{3}-A=0$ ,Proof:

Let $f(x)=x^{3}-3Ax+\sqrt{B}$, with critical.

$\mathrm{p}_{\mathrm{o}\mathrm{i}\mathrm{n}}\mathrm{t}_{\mathrm{S}}\pm\sqrt{A}$The equation of curve $\mathrm{B}\mathrm{C}1:B=A(2A-1)^{2}$ is obtained by the following equations:$f(\sqrt{A})-(-\sqrt{A})$$=$$(-2A+1)^{\sqrt{A}}+\sqrt{B}=0$$f(-\sqrt{\mathrm{A}})-\sqrt{A}$$=$$(2A-1)\sqrt{A}+\sqrt{B}=0$ .

The equation of curve $\mathrm{C}\mathrm{D}1:B=A(2A+1)^{2}$ is obtained by the following equations:$f(\sqrt{A})-\sqrt{A}$$=$$(-2A-1)^{\sqrt{A}}+\sqrt{B}=0$ ,$f(-\sqrt{A})-(-\sqrt{A})$$=$$(2A+1)\sqrt{A}+\sqrt{B}=0$ .

The equation of curve BC2 is obtained as follows:$f^{2}(\sqrt{A})-(-\sqrt{A})=0$ ,$f^{2}(-\sqrt{A})-\sqrt{A}=0$

Therefore,

BC2 :$B^{3}-12A^{3}B^{2}-6AB^{2}+2B^{2}+48A^{6}B+24A^{3}B+21A^{2}B$$-6AB+B-64A^{9}+96A^{7}-20A5-12A^{3}-A=0$ .
The equation of curve CD2 is obtained by the equations:$f^{2}(^{\sqrt{A}})-\sqrt{A}$,$f^{2}(-\sqrt{A})-(-\sqrt{A})=0$

Thus$B(12A^{3}-3A+1+B)^{2}-A(-8A^{4}+6A^{2}-1-6AB)^{2}..=0$

Fixed points can be also considered as periodic points of period 2. So, this curve contains

Dividing the left-hand side of the last equation by the defining polynomial ofequation of CD2 as follows:

CD2 : $B^{2}-8A^{3}B+4A^{2}B-5AB+2B+16A^{6}-16A^{5}$$-12A^{4}+16A^{3}-4A+1=0$ .

$\mathrm{C}\mathrm{D}1$$\mathrm{C}\mathrm{D}1$.

, we get the232

Suppose now,where

$P_{p},$$Q_{p}$,$f^{\mathrm{p}}(^{\sqrt{A})}$$=$$P_{p^{\sqrt{A}+}}Q_{P}\sqrt{B}$$f^{\mathfrak{n}}(-\sqrt{A})$$=$$-P_{p^{\sqrt{A}\sqrt{B}}}+Q_{p}$are polynomials of $A,$

$B$,

. Then we have,

$P_{\mathrm{p}}$$=$$AP_{p-1}^{3}+3BP_{\mathrm{p}-1}Q_{p-}21-3AP_{p-1}$$Q_{p}$$=$$3AP_{\mathrm{p}-1}^{2}Q_{p-1}+BQ_{\mathrm{p}-1}^{3}-3AQ_{p-1}+1$.

The equation of curve $\mathrm{B}\mathrm{C}p:(P_{p}+1)^{2}A-Q^{2}pB=0$ is obtained as follows:,$f^{p}(\sqrt{A})-(-\sqrt{A})$$=$$(P_{\mathrm{p}}+1)^{\sqrt{A}}+Q_{p^{\sqrt{B}}}=0$$f^{p}(-\sqrt{A})-\sqrt{A}$$=$$(-P_{p}-1)^{\sqrt{A}}+Q_{p}\sqrt{B}=0$

The equation of curve$\mathrm{C}\mathrm{D}p$.

is obtained as follows:,$f^{\mathrm{p}}(\sqrt{A})-\sqrt{A}$$=$$(P_{p}-1)\sqrt{A}+Q_{p}\sqrt{B}=0$$f^{p}(-\sqrt{A})-(-\sqrt{A})$$=$$(-P_{p}+1)^{\sqrt{A}}+Q_{p}\sqrt{B}=0$.

Let$\tilde{\phi}_{p}(A, B)$$:=(P_{p}-1)2A-Q_{p}^{2}B$ .

If $\phi_{q}(A, B)=0$ is the defining equaiton of$\mathrm{C}\mathrm{D}q$, then we have$\tilde{\phi}_{p}(A, B)=\prod_{q1p}\phi_{q}(A, B)$

Therefore if$\{q_{1}, \cdots , q_{n}\}$is the set of all divisors of$\mathrm{C}\mathrm{D}p$Remark:$p$.

except , then$p$: $\phi_{p}(A, B)=\tilde{\phi}_{p}(A, B)/\prod_{=i1}\phi_{q}\hslash.(A, B)=0$ .

The defining equations of Center curves $BCp$ and $CDp,1<p<5$ are obtained by$RISA/ASIR$ (computer algebra system by FUJITSU $CO$ . LTD.)

25.3 Algebraic-geometric properties of center curves
We can embed$\mathrm{C}^{2}$

canonically in$V_{0}=\{(A, B)\in \mathrm{C}^{2} :h(A, B)=0\}$$\mathrm{P}^{2}(\mathrm{C})$: $(A, B)arrow(1 : A : B)$ . Then an affine algebraic curve

uniquely determines a projective algebraic curve $V=\{(C$ :$A$:233$B)\in \mathrm{P}^{2}(\mathrm{C})$: $H(C : A : B)=0\}$ inDeflnition.

For a center curve$V_{0}$$\mathrm{P}^{2}(\mathrm{C})$such that $h(A, B)=H(1 ; A : B)$ and$V\cap \mathrm{C}^{2}=V_{0}$, the corresponding projective algebraic curve $V$ is$\mathrm{c}\mathrm{a}\mathrm{l}\mathrm{l}\mathrm{e}\dot{\mathrm{d}}$projective center curve. We denote by PBCp and PCDp, these curves corresponding toand$\mathrm{C}\mathrm{D}p$.

the$\mathrm{B}\mathrm{C}p$respectively.

We give some algebraic-geometric properties of these curves.

Theorem 3.1 : The interseciton with the line at inflnity ([8]).
curve and the line at infinity,$L_{\infty}$: $C=0$ , intersect at the point$(0$

Each projective center:$0$: 1$)$

only.

This point$(0:0:1)$ is singular and its multiplicity can be calculated explicitly.

Remark.

PCDl and PBCl are both cuspidal cubic. But for $p\geq 2$ , the point $(0:0:1)$ is not a“simple cusp”.

25.3.1 Case $p=1,2$ .
We get the following theorem about the irreducibility of each projective center curve, which isbased on Kaltofen’s algorithms on $RISA/ASIR$ (computer algebra system by FUJITSU$\mathrm{C}\mathrm{O}$. LTD.)([10], [6]).

Theorem 3.2 : Irreducibility and.
.

$\dot{\mathrm{S}}$ingurarity ([8]).

For projective center curves PCDiand PBCi $(i=1,2)$ ,and PBCl.

$PCD1$

These two are rrreducible curves of degree 3. Hence, no other singular poznts exist.

PCD2.

It is an irreducible curve of degree 6. It has one 4-fold point $(0:0:1)$ and one ordinary double.

point (0.25, -0.4375).

PBC2.

It is anxrreducible curve of degree 9. It has$one\mathit{6}^{-}-\vec{f}\mathrm{r}o^{\vee}ld$

point$(0:\mathrm{o}:1)$

and four ordinary double$point_{S}.\cdot$$(-0.1341351918179714, - 1.37344484910264)$ ,$(-0.5531033117555605, - 0.6288238268413773)$ ,$(0.3041906503790061*i+0.3436192517867655$ ,$0.6886343379400248-\mathrm{o}.04267412324347224*i)$ ,$(0.3436192517867655-0.3041906503790061*i$ ,$0.04267412329900053*i+0.6886343379735695)$ ,234

25.3.2 Genus of center curves.
Deflnitio$\mathrm{n}$([3]).

To calculate genus$g$of each projective center curve$\Gamma$, we determine the principal part at$(0:0:1)$ of the curves by using Newton Polygons and apply the Pl\"ucker’s formula. I am gratefulto Y. Komori $([4])$ for helpful suggestions on the genus.

Lemma 3.3 : Principal part of the center curves.

The prrncipal part at$(0$:$0$: 1 of$)$

PCDI and of PBCl is $(C^{2}-4A^{3})^{1}$ , of PCD2 is $(C^{2}-4A^{3})^{2}$ , and of PBC2 is $(C^{2}-4A^{3})^{3}$ .
Pl\"ucker’s formulaLetthe set of singular points$P_{i}$Let , be the multiplicity of$r$$\Gamma$of$P_{i}$be an irreducible curve of degree$\Gamma$$n$

and of its strict transform obtained by blowing up several times.

. Then,$g= \frac{(n-1)(n-2)}{2}-\sum_{1i=}^{k}\frac{r_{i}(r.-1)}{2}.$Theorem 3.4 : Genus.

. Let $Sing\Gamma=\{P_{1}, \cdots , P_{k}\}$ be.

The curves PCDl and PBCl are rational. Hence the genus is$\mathit{0}$.

The genus of PCD2 is 1. The genus of PBC2 is 3.
We would like to state the following conjectures for the projective center curves:..
.

Conjectures.

All projective center curves are irreducible.

All singular points except $(0:0:1)$ are ordinary double points.

Especially, for real graph of center curves, the sigular point exists only in$\mathcal{R}_{1}$.

The pricipale part at $(0 : 0:1)$ of every projective center curve has a form $(C^{2}-4A^{3})^{k}$ .

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