数理解析研究所講究録988 巻 1997 年 57-66

57

Moduli spaces of maps with two critical points
Masayo FUJIMURA
Dept. of Math., College of Sci. and Tech., Nihon Univ.

Abstract

We give directly a defining equation of the symmetry locus, a singularpart of the moduli space of the quadratic rational maps. We show a characterization of this locus. We can expand analogous discussion for the cubic
polynomials and give a “chart” making a comparison between properties ofthese moduli spaces in Appendix A. Moreover, we apply these method to thepolynomials of degree , and give some conjectures.
$n$1

Quadratic rational maps

1.1

Moduli space of quadratic rational mapsLetbe the Riemann sphere andto itself. The groupmaps fromby conjugation,
space$\overline{\mathrm{C}}$the space of all quadratic rationalof M\"obius transformations acts on the$\mathrm{R}\mathrm{a}\mathrm{t}_{2}(\mathrm{C})$$\overline{\mathrm{C}}$$\mathrm{P}\mathrm{S}\mathrm{L}_{2}(\mathrm{C})$$\mathrm{R}\mathrm{a}\mathrm{t}_{2}(\mathrm{C})$for$g\mathrm{o}f\mathrm{o}g^{-1}\in \mathrm{R}\mathrm{a}\mathrm{t}_{2}(\mathrm{C})$$g\in \mathrm{P}\mathrm{S}\mathrm{L}_{2(\mathrm{C})},$$f\in \mathrm{R}\mathrm{a}\mathrm{t}_{2}(\mathrm{C})$.

,are holomorphically conjugate, denoted by
Two maps. The quotient space ofwith
if and only if there exists, and called the moduli space
under this action will be denoted byof quadratic rational maps .
of holomorphic conjugacy classes, let
as follows; for each
Milnor introduced coordinates in$z_{i};\mu_{i}=f’(Z_{i})(1\leq
i\leq
3)$ .
be the fixed points of and the multlpliers ofConsider the elementary symmetric functions of the three multipliers,$f_{1},$$f_{1}\sim f_{2}$$f_{2}\in \mathrm{R}\mathrm{a}\mathrm{t}_{2}(\mathrm{C})$$g\in \mathrm{P}\mathrm{S}\mathrm{L}_{2}(\mathrm{C})$$g\mathrm{o}f_{1^{\mathrm{O}}}g^{-1}=f_{2}$$\mathcal{M}_{2}(\mathrm{C})$$\mathrm{R}\mathrm{a}\mathrm{t}_{2}(\mathrm{C})$$f$$\langle f\rangle$$f\in \mathrm{R}\mathrm{a}\mathrm{t}_{2}(\mathrm{C})$$\mathcal{M}_{2}(\mathrm{C})$$z_{1},$$f$$z_{2,3}z$$\sigma_{1}=\mu_{1}+\mu 2+\mu 3$,$\mu_{i}$$\sigma_{2}=\mu_{1}\mu_{2}+\mu_{2}\mu 3+\mu 3\mu_{1}$These three multipliers determine
only to the restriction that$f$,$\sigma_{3}=\mu 1\mu_{2}\mu_{3}$.

up to holomorphic conjugacy, and are subject$\sigma_{3}=\sigma_{1}-2$.

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is canonically isomorphic toHence the moduli space(Lemma 3.1 in [Mi193]).
andwith coordinates$\mathrm{C}^{2}$$\mathcal{M}_{2}(\mathrm{C})$$\sigma_{2}$$\sigma_{1}$be the set of all conjugacy classes
let
For eachhaving a periodic point of period and multiplier .
andforms a straight lines as follows:
Each of$\mu\in \mathrm{C}$$\mathrm{P}\mathrm{e}\mathrm{r}_{n}(\mu)$$\langle f.\rangle$$n$$\mathrm{P}\mathrm{e}\mathrm{r}_{1}(\mu)$of maps

$f$

which

$\mu$

$\mathrm{P}\mathrm{e}\mathrm{r}_{2}(\mu)$$\mathrm{P}\mathrm{e}\mathrm{r}_{1}(\mu)$$=$$\{\langle f\rangle\in \mathcal{M}_{2}(\mathrm{C});\sigma_{2}=(\mu+\mu^{-}1)\sigma_{1^{-(\mu)\}}}\mu+22-1$$\mathrm{P}\mathrm{e}\mathrm{r}_{2}(\mu)$$=$$\{\langle f\rangle\in \mathcal{M}_{2}(\mathrm{C});\sigma 2=-2\sigma 1+\mu\}$,(Lemmas 3.4 and 3.6 in [Mi193]).

Remarkthat two families coincide.

$\mathrm{P}\mathrm{e}\mathrm{r}_{1}(-1)\subseteq \mathrm{P}\mathrm{e}\mathrm{r}_{2}(1)$by definition. But, in the case of$\mathcal{M}_{2}(\mathrm{C})$, it is clear

Symmetry locus

1.2

By an automorphism of a quadratic rational ap , we will mean
which commutes with . The collectionof all automorphisms offinite group. It is clear thatis isomorphic tofor any
The set$S=$ {
is non-trivial}$f$$\dot{\mathrm{m}}$$f$$g\in \mathrm{P}\mathrm{S}\mathrm{L}_{2}(\mathrm{C})$$\mathrm{A}\mathrm{u}\mathrm{t}(f)$$\mathrm{A}\mathrm{u}\mathrm{t}(\tilde{f})$$\tilde{f}\in\langle f\rangle$$\mathrm{A}\mathrm{u}\mathrm{t}(f)$$\langle f\rangle;\mathrm{A}\mathrm{u}\mathrm{t}(f)$$f$forms a.

$\subset \mathcal{M}_{2}(\mathrm{C})$is called the symmetry locus.
Corollary 1
The symm et locus
ducible alge braic curve as follows;$\mathrm{r}y$$S$of qua$dr\mathrm{a}ii_{C}$ra tion$\mathrm{a}l$maps forms an irre-$S(\sigma_{1}, \sigma_{2})=2\sigma_{1}^{3}+\sigma_{1}^{2}\sigma_{2}-\sigma_{1}^{2}-4\sigma^{2}2-8\sigma 1\sigma_{2}+12\sigma_{1}+12\sigma_{2}-36=0$.

(1)

Proof of Corollary 1.
coincides with the group consisting of all permutations of the fixed points
which preserve the multipliers. In the case of has the three distinct fixed points,
has order 1, 2, or 6 according as three multipliers are distinct, two are equal,or all the three are equal, respectively, while, if has multiple fixed points thenis non-trivial if and only if has a triple fixed point. The multipliers are
the roots of the equation:$\mathrm{A}\mathrm{u}\mathrm{t}(f)$$f$$\mathrm{A}\mathrm{u}\mathrm{t}(f)$$f$$\mathrm{A}\mathrm{u}\mathrm{t}(f)$$f$$\mu^{3}-\sigma_{1\mu^{2}2}+\sigma\mu-\sigma_{1}+2=0$$\mu_{i}$.

(2)

The equation (2) has multiple roots if and only if its discriminant is equal to zero.
Hence we have$(\sigma_{2}-2\sigma_{1}+3)(2\sigma_{1}^{3}+\sigma_{1}^{2}\sigma_{2}-\sigma_{1}^{2}-4\sigma_{2}^{2}-8\sigma_{1}\sigma_{2}+12\sigma_{1}+12\sigma_{2}-36)=0$.

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Figure 1:$\mathcal{M}_{2}(\mathrm{R})$with the real cut of $S$ .

Figure 2: Lines$\{\mathrm{P}\mathrm{e}\mathrm{r}_{1}(\mu)\}$of the moduli spacein the real cut$\mathrm{A}l_{2}(\mathrm{C})$. Considering the line of the first factor
The first factor corresponds with
tangent to the curve of the second factor $(S)$ with tangency of degree
three, the second factor is the required equation.
$\mathrm{P}\mathrm{e}\mathrm{r}_{1}(1)$$(\mathrm{P}\mathrm{e}\mathrm{r}_{1}(1))$

1
The following result is obtained immediately by the definition of the envelope ofthe family of curves.
Corollary 2

The envelope of$\{Pe\mathrm{r}_{1}(\mu)\}_{\mu}$coincides with the symmetry locus.

A quadratic rational map has a non-trivial[Mi193])
Remark (Theorem 5.
automorphism if and only if it is conjugate to a map in the unique normal form$f(z)=k(z+ \frac{1}{z})$ with.
$1.\mathrm{o}\mathrm{f}$$k\in \mathrm{C}\backslash \{0\}$Real moduli space

1.3

be the set of real quadratic rational maps. Then the parameters$\sigma_{i}(1\leq i\leq 3)$ are all real, because the three fixed points and the corresponding multipliers are either all real or one real and a pair of complex conjugate
fornumbers. According to J. Milnor, we define the real moduli space-plane. This notation needs some care when
to be simply the real
the real conjugacy class,, and denote by
used: if we put
is canonically isomorthencorrespondence between, whereas there is a canonicalphic to
and .
:
Let

$\mathrm{R}\mathrm{a}\mathrm{t}_{2}(\mathrm{R})$$\mathcal{M}_{2}(\mathrm{R})$$(\sigma_{1}, \sigma_{2})$$\mathrm{R}\mathrm{a}\mathrm{t}_{2}(\mathrm{R})$$S_{\mathrm{R}}=S\cap \mathcal{M}_{2}(\mathrm{R})$$\langle$$\rangle_{\mathrm{R}}$$( \mathrm{R}\mathrm{a}\mathrm{t}2(\mathrm{R})/\mathrm{P}\mathrm{G}\mathrm{L}2(\mathrm{R}))\backslash \{\langle a(x+\frac{1}{x})\rangle_{\mathrm{R}}, \langle a(x-\frac{1}{x})\rangle \mathrm{R}\}a\in \mathrm{R}^{\cross}$$\mathrm{t}\mathrm{w}\mathrm{o}^{-}\mathrm{t}\mathrm{o}^{-}\mathrm{o}\mathrm{n}\mathrm{e}$$\mathrm{R}^{2}\backslash S_{\mathrm{R}}$$\{\langle a(x\pm\frac{1}{x})\rangle\}_{a\in \mathrm{R}^{\mathrm{x}}}$$S_{\mathrm{R}}$60

2Cubic polynomials

2.1

Moduli space of cubic polynomials
the space of all cubic polynomials fromof affine transformations acts on the space

Let be$\mathrm{P}\mathrm{o}\mathrm{l}\mathrm{y}_{3}(\mathrm{c})$$\mathrm{P}\mathrm{o}\mathrm{l}\mathrm{y}_{3}(\mathrm{c})$$\mathrm{C}$$\mathrm{P}\mathrm{o}\mathrm{l}\mathrm{y}_{3}(\mathrm{c})$$g\mathrm{o}p\mathrm{o}g^{-1}\in \mathrm{P}\mathrm{o}\mathrm{l}\mathrm{y}_{3}(\mathrm{c})$for$g\in \mathrm{P}\mathrm{o}\mathrm{l}\mathrm{y}_{1}(\mathrm{c}),$to itself. The groupby conjugation,$p\in \mathrm{P}\mathrm{o}\mathrm{l}\mathrm{y}_{3}(\mathrm{c})$.

Two maps
are holomorphically conjugate, denoted by,
if and only if there exists
with. The quotient spaceofunder this action will be denoted by, and called the moduli
space of holomorphic conjugacy classesof cubic polynomials .
Doing the same as the case of quadratic rational maps, we introduce coordinates$z_{4}(=\infty)$ be the fixedinas follows; for each
, letpoints of and the multipliers of $z_{i};\mu_{i}=p’(z_{i})(1\leq i\leq 3)$ , and $\mu_{4}=0$ . Consider
the elementary symmetric functions of the four multipliers,$p_{1},p_{2}\in \mathrm{P}\mathrm{o}\mathrm{l}\mathrm{y}_{3}(\mathrm{c})$$p_{1}\sim p_{2}$$g\mathrm{o}p_{1^{\mathrm{O}}}g-1=p_{2}$$g\in \mathrm{P}\mathrm{o}\mathrm{l}\mathrm{y}_{1}(\mathrm{c})$$\mathrm{M}_{3}(\mathrm{C})$$\mathrm{P}\mathrm{o}\mathrm{l}\mathrm{y}_{3}(\mathrm{c})$$\langle p\rangle$$p$$p\in \mathrm{P}\mathrm{o}\mathrm{l}\mathrm{y}_{3}(\mathrm{c})$$\mathrm{M}_{3}(\mathrm{C})$$p$$z_{1},$$z_{2},$$z_{3},$$\mu_{i}$$\sigma_{1}=\mu_{1}+\mu_{2}+\mu_{3}+\mu_{4}=\mu_{1}+\mu 2+\mu 3$$\sigma_{2}=\mu_{1\mu 2}+\mu_{1}\mu_{3}+\mu_{1\mu_{4}+\mu+\mu 2\mu}\mu 234+\mu 3\mu_{4}=\mu_{1\mu_{2}}+\mu_{1\mu\mu_{2}}3+\mu_{3}$$\sigma_{3}=\mu 1\mu 2\mu 3+\mu 1\mu_{2}\mu 4+\mu 1\mu 3\mu 4+\mu 2\mu 3\mu_{4}=\mu_{1}\mu_{2}\mu_{3}$$\sigma_{4}=\mu_{1}\mu 2\mu_{3}\mu_{4}=0$.

These multipliers determine uniquely up to holomorphic conjugacy, and are subject
only to the
that
$p$

$\dot{\mathrm{r}}\mathrm{e}\mathrm{s}\mathrm{t}\mathrm{r}\mathrm{i}\mathrm{C}\mathrm{t}\mathrm{i}_{\mathrm{o}\mathrm{n}}$$3-2\sigma_{1}+\sigma_{2}=0$

Hence the moduli spaceand .

$\mathrm{M}_{3}(\mathrm{C})$.

is canonically isomorphic to$\mathrm{C}^{2}$

with coordinates$\sigma_{3}$Proposition 1

The locus $Per_{1}(\mu)$ forms a$st$raight$li\mathrm{n}$es as follows:$Pe\mathrm{r}_{1}(\mu)=\{\langle f\rangle\in M_{3}(\mathrm{C});\sigma 3=(-\mu^{2}+2\mu)\sigma_{1}+\mu^{3}-3\mu\}$

The locus$\mathrm{p}_{e\mathrm{r}_{2}}(\mu)$$Per_{2}(\mu)$$=$.

forms an algebraic curve of degree three as follows:$\{\langle f\rangle\in \mathcal{M}_{2}(\mathrm{C});\sigma_{3^{+(-}}24\sigma^{2}(1\mu+57)\sigma 1+252)\sigma 3-(4\mu-16)\sigma_{1}^{3}$$+(61\mu-252)\sigma^{2}-1(4\mu^{2}+246\mu-1134)\sigma_{1}-\mu^{3}+51\mu^{2}$$-99\mu-459=0\}$ .

Note that this curve is irreducible if and only if$\mu\neq 1$. In the case of $\mu=1$ ,$Per_{2}(1)=\{Per_{1}(-1)\}\cup\{\langle f\rangle\in M_{2}(\mathrm{C});\sigma_{3}+4\sigma_{1}^{2}-61\sigma_{1}+254=0\}$.

$\sigma_{1}$

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2.2

Symmetry locus

Using conjugation described in above, we can define symmetry locus of this moduli
space as one in, and we obtain next results.
$\mathcal{M}_{2}(\mathrm{C})$

Theorem 1
The symmet
algebraic curve:$\mathrm{r}y$

locus

$S$

of cubic polynomi als forms an irreducible$S(\sigma_{1}, \sigma_{3})=27\sigma 3+(\sigma 1-6)(2\sigma 1-3)^{2}=0$.

(3)

The following result is obtained immediately by the definition of the envelope ofthe family of curves.
Corollary 3

The envelope of

Figure 3:

with the real cut of $S$ .

$\mathrm{M}_{3}(\mathrm{R})$$\{Per_{1}(\mu)\}_{\mu}$coincides with the symmetry locus.

Figure 4: Linesof the moduli space$\{\mathrm{P}\mathrm{e}\mathrm{r}_{1}(\mu)\}$$\mathrm{M}_{3}(\mathrm{C})$in the real cut.

Remark A cubic polynomial has non-trivial automorphism if and only if it isconjugate to a map in the unique normal form $p(z)=z^{3}+az$ .

Real moduli space

2.3
Letcase ofreal

be the set of real cubic polynomials. By the same reason for the, we define the real moduli spaceforto be simply the-plane. This notation needs some care when used: if we put, and denote by
the real conjugacy class, thenis canonically isomorphic to, whereas there
is a canonical-one correspondence between
and .
$\mathrm{P}\mathrm{o}\mathrm{l}\mathrm{y}_{3}(\mathrm{R})$$\mathcal{M}_{2}$$\mathrm{M}_{3}(\mathrm{R})$$\mathrm{P}\mathrm{o}\mathrm{l}\mathrm{y}_{3}(\mathrm{R})$$(\sigma_{1}, \sigma_{3})$$S_{\mathrm{R}}=$$S\cap \mathrm{M}_{3}(\mathrm{R})$$\langle$$\rangle_{\mathrm{R}}$$\{\langle x^{3}+ax\rangle_{\mathrm{R}}, \langle-x^{3}+ax\rangle_{\mathrm{R}}\}_{a\in \mathrm{R}}\mathrm{x}$$\mathrm{t}_{\mathrm{W}\mathrm{O}-}\mathrm{t}_{0}$$(\mathrm{P}\mathrm{o}\mathrm{l}\mathrm{y}_{3}(\mathrm{R})/\mathrm{P}\mathrm{o}\mathrm{l}\mathrm{y}_{1}(\mathrm{R}))\backslash$$\mathrm{R}^{2}\backslash S_{\mathrm{R}}$$\{\langle\pm x^{3}+ax\rangle\}_{a\in \mathrm{n}^{\mathrm{x}}}$$S_{\mathrm{R}}$62

3Polynomials of degree

3.1

Moduli space of polynomials of degree

$n$

$n$

Now we discuss about the moduli space
for the space, Polyn’ of polynomials of degree .
Doing the same as the case of cubic polynomials, we try introducing coordinates$z_{n+1}(=\infty)$ be theinas follows; for each
, let
fixed points of and the multipliers of $z:;\mu_{i}=p’(z_{i})(1\leq i\leq n)$ , and $\mu_{n+1}=0$ .
Consider the elementary symmetric functions of the multipliers,$\mathrm{M}_{n}(\mathrm{C})$$(\mathrm{c}.).$$n$$p(z)\in \mathrm{P}\mathrm{o}\mathrm{l}\mathrm{y}n(\mathrm{c})$$\mathrm{M}_{n}(\mathrm{C})$$p$$z_{1},$$\cdots,$$z_{n},$$\mu_{i}$$n$$\sigma_{n,1}=\mu_{1}+\cdots+\mu_{n}$,$\sigma_{n,2}=\mu 1\mu 2+\cdots+\mu_{n}-1\mu_{n}=\Sigma_{i=}n-1\mu 1i\Sigma_{j}^{n}>i\mu_{j}$$\sigma_{n,n}=\mu_{1}\mu_{2\mu}\ldots n$$\sigma_{n,n+1}=0$

Example 1$\bullet$$\bullet$$\bullet$’

For example, we assume

fixed points:

multiplier:.

$z_{1},$,$p(z)\in \mathrm{P}\mathrm{o}\mathrm{l}\mathrm{y}_{4}(\mathrm{c})$;$z_{2,3}Z,$ $z_{4},$ $\infty$$\mu_{1},$ $\mu_{2},$$\mu_{3},$$\mu_{4},0$elementary symmetric functions:$\sigma_{4,1}=\mu_{1}+\mu_{2}+\mu 3+\mu 4$$\sigma_{4,2}=\mu 1\mu_{2}+\mu 1\mu 3+\mu_{1}\mu_{4}+\mu 2\mu 3+\mu_{2\mu\mu_{3}}4+\mu_{4}$$\{$$\sigma_{4,3}=\mu 1\mu 2\mu 3+\mu 1\mu 2\mu 4+\mu_{1}\mu 3\mu_{4}+\mu 2\mu 3\mu_{4}$$\sigma_{4,4}=\mu_{1}\mu 2\mu 3\mu_{4}$$\sigma_{4,5}=0$Applying Fatou-index theorem to these fixed points;$\frac{1}{1-\mu_{1}}+\frac{1}{1-\mu_{2}}+\frac{1}{1-\mu_{3}}+\frac{1}{1-\mu_{4}}+\frac{1}{1-0}=1$,(4)

where $\mu_{i}\neq 1(1<i<n)$ . Arranging this equation for the form of elementary
symmetric functions;

4–3 $(\mu_{1}+\mu_{2}+\mu 3+\mu 4)+2(\mu_{1\mu_{2}\mu 1}+\mu_{3}+\mu_{1}\mu_{4}+\mu_{2}\mu 3+\mu 2\mu 4+\mu 3\mu_{4})$$-(\mu 1\mu 2\mu 3+\mu 1\mu 2\mu 4+\mu 1\mu 3\mu 4+\mu 2\mu 3\mu 4)=0$.

Hence we have
$4-3\sigma_{4,1}+2\sigma_{4,2}-\sigma 4,3=0$.

For the equation (5), the cases $\mu_{i}=1$ are also allowable.

(5)

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Now we consider a polynomialthat has at least two fixed points. After affine conjugation, we can assume they are
and 1. Then, we will solve the following question: “Do the four multipliers$p(z)=a_{4}z^{4}+a_{3}z^{3}+a_{2}z^{2}+a_{1}z+a_{0}\in \mathrm{P}\mathrm{o}\mathrm{l}\mathrm{y}_{4}(\mathrm{c})$$0$$\mu_{0}=p^{J}(0),$ $\mu_{1}=p’(1),$ $\mu 2=p^{;}(z_{2}),$

where$z_{1},$

of $p(z)?$$z_{2}$

are fixed$po\dot{i}nts$$\mu_{3}=p’(z3)$,of $p(z)$ , determine the five coefficients$a_{4},$$a3,$ $a_{2},$ $a_{1},$ $a_{0}$”In fact, the following equations hold;$a_{0}=0$$f’(\mathrm{O})=\mu_{0}$$a_{1}=\mu_{0}$$a_{2}=a_{4}+3-2\mu_{0}-\mu_{1}$$a_{3}=1-a_{4}-a_{2}-\mu_{0}$and$a_{4}$

because of $f(\mathrm{O})=0$ ,,because of$f’(1)=\mu_{1}$,because of
because of $f(1)=1$ ,is a common root of the following two equations;$A_{1}=(\mu_{2^{-}}^{2}2\mu 3\mu 2+\mu_{3^{-}}\mu^{2}0+22\mu 1\mu_{0}-\mu_{1})24(a_{4}+-4\mu 0+3(4\mu 1+8)\mu 0+2(-4\mu_{1}^{2}-$$8)\mu_{0}+4\mu^{3}1-8\mu^{2}1^{+)((4\mu_{1}+}8\mu 1a_{4}^{3}+-6\mu^{4}0+-28)\mu_{0}+3(4\mu_{1^{+)\mu+}}^{2}4\mu_{1}-4420$$(-4\mu_{1}^{3}+4\mu_{1^{-}}^{2}8\mu_{1}+32)\mu_{0}-6\mu_{1}^{4}+28\mu_{1^{-}}^{3}44\mu^{2}1+32\mu_{1}-16)a_{4}^{2}+(-4\mu_{0}^{5}+$$(-12\mu_{1}+32)\mu_{0}^{4}+(-8\mu_{1}^{2}+64\mu_{1}-96)\mu_{0}^{3}+(8\mu_{1^{-}}^{3}96\mu_{1}+128)\mu^{2}0+(12\mu_{1}^{4}-$$64\mu_{1}^{3}+96\mu_{1^{-}}^{2}64)\mu_{0}+4\mu_{1^{-32}}^{5}\mu_{1}4+96\mu_{1}^{3}-128\mu^{2}1+64\mu_{1})a_{4}-\mu^{6}0+(-6\mu_{1}+$$12)\mu_{0}^{5}+(-15\mu_{1}^{2}+60\mu_{1}-60)\mu_{0}^{4}+(-20\mu_{1}^{3}+120\mu_{1}^{2}-240\mu_{1}+160)\mu_{0}^{3}+$$(-15\mu_{1}^{4}+120\mu_{1}^{3}-360\mu^{2}1+480\mu_{1}-240)\mu 0+2(-6\mu_{1}^{5}+60\mu 1-4240_{\mu_{\iota}^{3}}+480\mu_{1}^{2}-$$480\mu_{1}+192)\mu_{0-}\mu_{1}6+12\mu_{1^{-}}^{5}60\mu 14+160\mu_{1^{-24}}0\mu_{1}32+192\mu 1^{-}64=0$,$A_{2}=(\mu_{2}+\mu_{3}+\mu 0+\mu 1-4)a^{2}4+(2\mu^{2}0-4\mu 0-2\mu_{1}^{2}+4\mu_{1})a_{4}+\mu^{3}0^{+}(3\mu 1-6)\mu^{2}0+$$(3\mu_{1^{-}}^{2}12\mu_{1}+12)\mu_{0}+\mu_{1}^{3}-6\mu_{1}^{2}+12\mu_{1}-8=0$.

satisfy the
Above two equations have common roots if and only if
and they should
are the four multipliers ofequation (5). Since
satisfy the equation (5), the two equations always have common roots. Hence fivecoefficients of $p(z)$ are calculated by its four multipliers, however, this calculation is
not decisive when they have distinct two common roots.
$\mu_{0},$$\mu_{0},$For the case of Polyn
cannot have the term of$\mu_{1},$$(\mathrm{c})$$\mu 2,$$\mu_{1},$$\mu_{2},$$\mu 3$$p(z)\backslash$$\mu_{3}$, it is clear from (4) that the equation corresponds to (5). Hence we can put$\sigma_{n,n}$$c_{0}+c1\sigma n,1+C_{2}\sigma_{n},2+\cdots+c_{n}-1\sigma n,n-1=0$

where $c_{k}(0\leq k\leq n-1)$ are functions of variable.
Paying attention to the form of elementary symmetric functions, we obtain thefollowing equation;$n$$c_{k}=(-1)^{k} \frac{(\begin{array}{ll}n -\mathrm{l} k\end{array})n}{(\begin{array}{l}nk\end{array})}=n-k$

.

64

where

$\sum_{k=0}^{n-1}(-1)k(n-k)\sigma_{n},k=0$

Question Is the moduli spacewith coordinates
isomorphic tofor polynomials of degree, and ?

$\mathrm{M}_{n}(\mathrm{C})$$\mathrm{C}^{n-1}$

3.2$\sigma_{1},$(6).

$\sigma_{2},$$\cdots,$$\sigma_{n-2}$$n$canonically$\sigma_{n}$Symmetry locus

Proposition 2
A polynomial of degree four as a non-trivial automorphism if
and only if it is conjuga to a map in the unique normal form
$h$

$te$

$\{z^{4}+az\}$,$a\in$C.

For a map $p(z)$ in this normal form, $Aut(p)$ is a cyclic group of order three.

Outline of proof.

Let$p(z)\in \mathrm{P}\mathrm{o}\mathrm{l}\mathrm{y}_{4}(\mathrm{c})$.

1. In the case of a map $p(z)$ with multiple fixed points.
(a) The case of $p(z)$ with a fixed point of order four:

is non-trivial.

$\mathrm{A}\mathrm{u}\mathrm{t}(p)$(b) The case of $p(z)$ with a fixed point of order three:$\mathrm{A}\mathrm{u}\mathrm{t}(p)$is trivial.

(c) The case of $p(z)$ with two fixed points of order two: there is not such$p(z)$.

(d) The case of $p(z)$ with a fixed point of order two:$\mathrm{A}\mathrm{u}\mathrm{t}(p)$is trivial.

2. In the case of a map $p(z)$ with four distinct fixed points.
(a) The case of four distinct multipliers:$\mathrm{A}_{\mathrm{u}\mathrm{t}}(p)$is trivial.

(b) The case that only two of multipliers are coincide:$\mathrm{A}_{\mathrm{u}\mathrm{t}}(p)$is trivial.

(c) The case of two pair of same multipliers: there is not such $p(z)$ .

(d) The case of three same multipliers: By an affine conjugation, if three
fixed points (whose multipliers are same) are mapped on the vertices of aregular triangle whose barycenter is the origin and the other fixed pointon the origin, thenis non-trivial. Otherwise
is trivial.
$\mathrm{A}_{\mathrm{u}\mathrm{t}}(p)$$\mathrm{A}_{\mathrm{u}\mathrm{t}}(p)$(e) The case of four same multipliers: there is not such $p(z)$ .

65

Therefore a map $p(z)$ has non-trivial automorphisms if and only if $p(z)$ is in the case. We can check easily that these maps coincide withand the first part ofI
the normal form $\{z^{4}+az\}$ .
$1-(\mathrm{a})$$2-(\mathrm{d})$Conjecture A polynomial of degree has a non-trivial automorphism if and only
if it is conjugate to a map in the unique normal form$n$$\{z^{n}+\sum_{\neq 1),kn-1}A(kk|(n-)z^{k}\}$where $A(k)$ are parameters in C.

References[FN] M. Fujimura and K. Nishizawa. Bifurcations and Algebraic Curves for
Quadratic Rational Functions $\{mf(x)\}_{m}$ . submitted.
[FN96] M. Fujimura and K. Nishizawa. Bifurcations of Nusse-Yorke’s family in thequadratic rational functions. In T. Sugawa, editor, RIMS Kokyuroku 959:
Complex Dynamics and Related Problems, pages 90-102. Kyoto Univ., 1996.
[Mi192] J. Milnor. Remarks on iterated cubic maps. Experimental Mathematics,
1:5-24, 1992.
[Mi193] J. Milnor. Geometry and Dynamics of Quadratic Rational Maps. Experimental Mathematics, ‘2(1):37-83, 1993.

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AComparison between the quadratic rational
maps and cubic polynomials

Masayo FUJIMURA
Dept. of Math., College of Sci. and Tech., Nihon Univ.
1-8, Kanda-Surugadai, Chiyoda-ku, Tokyo, 101 JAPAN-mail: masayo@math.cst.nihon-u.ac.jp$\mathrm{e}$