000This work was supported by the Research Institute for Mathematical
Sciences, a Joint Usage/Research Center located in Kyoto University.

# Dynamics of Chebyshev endomorphisms on some affine algebraic varieties

Keisuke Uchimura  Department of Mathematics, Tokai University, Hiratsuka,
259-1292, Japan<EMAIL_ADDRESS>
###### Abstract.

The Chebyshev polynomials $T_{d}$ in one variable are typical chaotic maps on
${\mathbb{C}}$. Chebyshev endomorphisms
$P_{A_{n}}^{d}:{\mathbb{C}}^{n}\to{\mathbb{C}}^{n}$ are also chaotic. We
consider the action of the dihedral group $D_{n+1}$ on ${\mathbb{C}}^{n}$. The
endomorphism $P_{A_{n}}^{d}$ maps any $D_{n+1}-$orbit of ${\bf
z}\in{\mathbb{C}}^{n}$ to a $D_{n+1}$-orbit of $P_{A_{n}}^{d}({\bf z})$. The
endomorphism $P_{A_{n}}^{d}$ induces a mapping on ${\mathbb{C}}^{n}/D_{n+1}$.

Using invariant theory we embed ${\mathbb{C}}^{n}/D_{n+1}$ as an affine
subvariety $X$ in ${\mathbb{C}}^{m}$. Then we have morphisms $g_{d}$ on $X$.
We study the cases $n=2$ and $3$. In these cases the morphisms $g_{d}$ are
defined over ${\mathbb{Z}}$. We find a class of affine subvarieties $V$ of $X$
which are invariant under $g_{d}$. These varieties are concerned with branch
loci or critical loci.The class contains ${\mathbb{C}}^{2}$, a cuspidal cubic,
a parabola, a quadric hypersurface in ${\mathbb{C}}^{4}$, an affine algebraic
surface in ${\mathbb{C}}^{4}$ which is birationally equivalent to an affine
quadric cone in ${\mathbb{C}}^{3}$, and others. For each affine variety $V$ in
the class, there exists a polynomial parametrization $P_{V}$ satisfying
$g_{d}\mid_{V}(P_{V}(y_{1},\dots,y_{k}))=P_{V}(T_{d}(y_{1}),\dots,T_{d}(y_{k}))$,
where $T_{d}(z)$ is a Chebyshev polynomial in one variable. Then we determine
the set of bounded orbits of $g_{d}\mid_{V}$ in each invariant set $V$ and
give relations between them.

###### Key words and phrases:

Dynamical system, Chebyshev map, affine algebraic variety, orbit variety, Lie
algebra.

###### 2010 Mathematics Subject Classification:

Primary 14A05, 37F45; Secondary 37F10.

##  1\. Introduction

Chebyshev endomorphisms $P_{A_{n}}^{d}$ of degree $d$ on ${\mathbb{C}}^{n}$
are originally defined by Veselov [26] and Hoffman and Withers [7]. The
endomorphism $P_{A_{n}}^{d}$ is associated to the Lie algebra of type $A_{n}$
. We studied dynamics of $P_{A_{2}}^{d}$ on ${\mathbb{C}}^{2}$ and
$P_{A_{3}}^{d}$ on ${\mathbb{C}}^{3}$ in [22] and [23], respectively.

In this paper, we study the action of the dihedral group $D_{n+1}$ of order
$2(n+1)$ on ${\mathbb{C}}^{n}$.

Petersen and Sinclair [14] studied conjugate reciprocal polynomials $f$:

$f(x)=x^{N}+\sum_{i=1}^{N}c_{i}x^{N-i}\quad\mbox{where}\quad c_{N}=1,\quad
c_{N-i}=\bar{c}_{i}\quad\mbox{for}\quad 1\leq i\leq N-1.$

They defined the set $W_{N}\subset{\mathbb{R}}^{N-1}$ such that $W_{N}$ is one
to one correspondence with the set of conjugate reciprocal polynomials of
degree $N$ with all roots on the unit circle. Theorem 1.3 in [14] states that
the group of isometries of $W_{N}$ is isomorphic to the dihedral group $D_{N}$
of order $2N$.

We find that the set $W_{n+1}$ coincides with the set of bounded orbits of the
Chebyshev endomorphism $P_{A_{n}}^{d}$. The set $K(f)$ of bounded orbits of a
regular polynomial endomorphism $f$ on ${\mathbb{C}}^{n}$ is defined by

$K(f):=\\{z\in{\mathbb{C}}^{n}:\quad\\{f^{k}(z):k=1,2,\dots\ \\}\quad\mbox{is
bounded}\\},$

where $f^{k}$ denotes the $k$-th iterate of $f$. See [21], [22] and [23]. We
call $K(f)$ the $K$ set of $f$.

In this paper we extend this group action. We consider the action of $D_{n+1}$
on ${\mathbb{C}}^{n}$. We find that $P_{A_{n}}^{d}$ maps any $D_{n+1}-$orbit
of ${\bf z}\in{\mathbb{C}}^{n}$ to a $D_{n+1}$-orbit of $P_{A_{n}}^{d}({\bf
z})$. Then we study maps on the orbit space ${\mathbb{C}}^{n}/D_{n+1}$ .
Indeed. The dihedral group $D_{n+1}$ is generated by two elements $R$ and $C$.
That is, $D_{n+1}=<R,C>$ . We will show that

(1.1) $R^{d}\circ P_{A_{n}}^{d}=P_{A_{n}}^{d}\circ R\quad\mbox{and}\quad
C\circ P_{A_{n}}^{d}=P_{A_{n}}^{d}\circ C.$

Hence we can define the induced map of $P_{A_{n}}^{d}$ on
${\mathbb{C}}^{n}/D_{n+1}$ .

We define a subspace $R_{n}$ of ${\mathbb{C}}^{n}$ by

$R_{n}:=\\{(z_{1},z_{2},\dots,z_{n})\mid z_{j}\in{\mathbb{C}},\quad
z_{n+1-j}=\bar{z}_{j},\quad\mbox{for any}\quad j=1,\dots,n\\}.$

Note that $R_{n}\simeq{\mathbb{R}}^{n}.$ We will show that $P_{A_{n}}^{d}$,
$R$ and $C$ admit an invariant subspace $R_{n}$. These maps restricted to
$R_{n}$ satisfy the equality (1.1). The generators $R$ and $C$ restricted to
$R_{n}$ can be represented by matrices in $GL(n,{\mathbb{R}})$. Hence we
regard $D_{n+1}$ as a finite matrix group in $GL(n,{\mathbb{R}})$. We denote
the map $P_{A_{n}}^{d}$ restricted to $R_{n}$ by $f_{d}(x_{1},\dots,x_{n})$.
The map $f_{d}(x_{1},\dots,x_{n})$ is a polynomial map over ${\mathbb{Z}}$.

Next, we consider complexification. We regard $x_{1},\dots,x_{n}$ as complex
variables and study the orbit space ${\mathbb{C}}^{n}/D_{n+1}$ where $D_{n+1}$
is the finite matrix group in $GL(n,{\mathbb{R}})$. We use invariant theory.
See [3], [5] and [20]. We consider the set of all polynomials which are
invariant under the action of $D_{n+1}$ :

${\mathbb{C}}[x_{1},\dots,x_{n}]^{D_{n+1}}:=\\{f\in{\mathbb{C}}[x_{1},\dots,x_{n}]:\forall\pi\in
D_{n+1}(f=f\circ\pi)\\}.$

Then the orbit space ${\mathbb{C}}^{n}/D_{n+1}$ can be regarded as an affine
algebraic variety whose coordinate ring is
${\mathbb{C}}[x_{1},\dots,x_{n}]^{D_{n+1}}$. The affine variety
${\mathbb{C}}^{n}/D_{n+1}$ is called an orbit variety and it is embedded as an
affine subvariety into ${\mathbb{C}}^{m}$, where $m$ is the number of
generators of ${\mathbb{C}}[x_{1},\dots,x_{n}]^{D_{n+1}}$. By selecting a
basis $\\{p_{1},\dots,p_{m}\\}$ of
${\mathbb{C}}[x_{1},\dots,x_{n}]^{D_{n+1}},$ we can construct a map
$g_{d}(p_{1},\dots,p_{m})$ on ${\mathbb{C}}^{n}/D_{n+1}$ from the map $f_{d}$.
The map $g_{d}$ is really a morphism defined over ${\mathbb{R}}$. In Sections
3 and 4, we will study morphisms $g_{d}$ on ${\mathbb{C}}^{2}/D_{3}$ and
${\mathbb{C}}^{3}/D_{4}$. We will show that these are morphisms defined over
${\mathbb{Z}}$.

In Section 3, we consider the dynamics of $g_{d}$ on
${\mathbb{C}}^{2}/D_{3}\simeq{\mathbb{C}}^{2}.$ We will give a polynomial
parametrization $P_{{\mathbb{C}}^{2}}$ of ${\mathbb{C}}^{2}$ satisfying

(1.2)
$g_{d}(P_{{\mathbb{C}}^{2}}(p,q))=P_{{\mathbb{C}}^{2}}(T_{d}(p),T_{d}(q)),$

where $T_{d}(z)$ is the $d^{th}$ Chebyshev polynomial in one variable.

In the theory of complex dynamics, it is important to analyze the orbit of the
set of critical points of a map. In our case, Chebyshev endomorphism
$P_{A_{n}}^{d}$ is postcritically finite. More precisely, the set of critical
values of $P_{A_{n}}^{d}$ is invariant under $P_{A_{n}}^{d}$. Inspired by this
fact, we consider the branch locus of a morphism $g_{d}$ for $d=2,3$. Then we
find that these two branch loci coincide and the branch locus consists of two
affine curves. Each of these affine curves is invariant under the morphisms
$g_{d}$ for any $d\in{\mathbb{N}}$.

We see that $g_{2}$ is a finite morphism and its branch locus consists of two
curves $C_{C}$ and $C_{D}$. (The same holds for $g_{3}$ .) The curve $C_{C}$
is the cuspidal cubic $V(p_{1}^{3}-p_{2}^{2})$ and $C_{D}$ is a parabola
$V(p_{1}^{2}+18p_{1}-8p_{2}-27)$ which corresponds to the relative orbit
variety (deltoid)$/D_{3}$. The deltoid is the boundary of $K(P_{A_{2}}^{d})$.
And it is equal to the set of critical values of $P_{A_{2}}^{d}$ in ${R}_{2}$.
See [22, p.998].

We will show that

$g_{d}(C_{C})=C_{C},\enskip\mbox{and}\quad g_{d}(C_{D})=C_{D}\quad\mbox{for
any}\quad g_{d}\enskip(d\in{\mathbb{N}}).$

We will show the dynamics of $g_{d}$ on such affine algebraic curves $V$ where
$V=C_{C}$ or $C_{D}$. For each affine curve $V$, there exists a polynomial
parametrization $P_{V}$ satisfying

(1.3) $g_{d}(P_{V}(z))=P_{V}(T_{d}(z)).$

From (1.2) and (1.3), we can determine the sets $K(g_{d}\mid_{V})$, for
$V={\mathbb{C}}^{2},\ C_{C},$ and $C_{D}$. The set
$K(g_{d}\mid_{{\mathbb{C}}^{2}})$ is a closed region on the real slice
${\mathbb{R}}^{2}$ of ${\mathbb{C}}^{2}$. It is invariant under $g_{d}$. The
$K$ sets $K(g_{d}\mid_{C_{C}})$ and $K(g_{d}\mid_{C_{D}})$ are two arcs with
same end points. Then we connect these arcs and get a Jordan curve $\gamma$ on
which $g_{d}$ is invariant and chaotic. The Jordan curve $\gamma$ is the
boundary of the closed region $K(g_{d}\mid_{{\mathbb{C}}^{2}})$ in
${\mathbb{R}}^{2}$.

Let $\bar{C_{C}}$ be the projective closure of $C_{C}$. The morphism $g_{d}$
restricted to $C_{C}$ can be extended to a morphism $\bar{g_{d}}$ from
$\bar{C_{C}}$ to $\bar{C_{C}}$. We can extend this morphism to a morphism from
${\mathbb{P}}^{2}({\mathbb{C}})$ to ${\mathbb{P}}^{2}({\mathbb{C}})$. The same
holds for $\bar{C_{D}}$.

The curves $C_{C}$ and $C_{D}$ are invariant under
$g_{d}\enskip(d\in{\mathbb{N}})$. We will also show that there is an infinite
number of invariant affine algebraic curves under
$g_{d}\enskip(d\in{\mathbb{N}})$.

In Section 4, we study the orbit variety ${\mathbb{C}}^{3}/D_{4}$ and consider
the dynamics of morphisms $g_{d}$ on ${\mathbb{C}}^{3}/D_{4}$. We can choose a
fundamental system of invariants of ${\mathbb{C}}[x_{1},x_{2},x_{n}]^{D_{4}}$
such that

${\mathbb{C}}[x_{1},x_{2},x_{3}]^{D_{4}}={\mathbb{C}}[p_{1},p_{2},p_{3},p_{4}],$

with $p_{j}\in{\mathbb{Z}}[x_{1},x_{2},x_{3}]$ and that the syzygy ideal
$I_{F}$ for $F=(p_{1},p_{2},p_{3},p_{4})$ is written as
$I_{F}=(p_{2}p_{4}-p_{3}^{2})$. Then ${\mathbb{C}}^{3}/D_{4}\simeq V(I_{F})$.
That is , ${\mathbb{C}}^{3}/D_{4}$ has the structure of the affine quadric
hypersurface $V(I_{F})$ in ${\mathbb{C}}^{4}$. We denote $V(I_{F})$ by
$X_{Q}$. We will show that $g_{d}:X_{Q}\to X_{Q}$ is a morphism defined over
${\mathbb{Z}}$.

We may choose any other fundamental system of invariants of
${\mathbb{C}}[x,y,z]^{D_{4}}$. Let $F^{\prime}$ be the ideal generated by that
fundamental system of invariants. Then we have another syzygy ideal
$I_{F^{\prime}}$, an affine algebraic set $V(I_{F^{\prime}})$ and morphisms
$g^{\prime}_{d}$ on $V(I_{F^{\prime}})$. We will show that there is an
isomorphism $\varphi:V(I_{F})\to V(I_{F^{\prime}})$ such that $\varphi\circ
g_{d}=g^{\prime}_{d}\circ\varphi$.

As in (1.2), we will give a polynomial parametrization $P_{X_{Q}}$ of $X_{Q}$
satisfying

(1.4) $g_{d}(P_{X_{Q}}(u,v,w))=P_{X_{Q}}(T_{d}(u),T_{d}(v),T_{d}(w)).$

The morphism $g_{2}$ is finite and deg $g_{2}=8$ . The hypersurface $X_{Q}$ is
normal. As in the case ${\mathbb{C}}^{2}/D_{3}$, we consider the branch locus
of $g_{2}$. The branch locus of $g_{2}$ consists of three subvarieties;
(1) a line $L_{1}$, (2) an affine algebraic surface $S_{P}$, (3) an affine
algebraic surface $S_{A}$.
The subvarieties are written in the form

$L_{1}=V(p_{2},p_{3},p_{4}),S_{P}=V(p_{1}^{2}-p_{4},p_{2}p_{4}-p_{3}^{2}),S_{A}=V(A_{h},p_{2}p_{4}-p_{3}^{2}),$

where $A_{h}$ is a polynomial in ${\mathbb{Z}}[p_{1},p_{2},p_{3},p_{4}]$.
Branch divisors $S_{P}$ and $S_{A}$ have the similar properties as those in
the former case. The line $L_{1}$ is preperiodic for any $g_{d}$
$(d\in{\mathbb{N}})$. Set $C_{1}=g_{2}(L_{1})$. Then the affine curve $C_{1}$
is invariant under $g_{d}$ for any $d\in{\mathbb{N}}$.

We will show the dynamics of $g_{d}$ on the affine varieties $V$, where
$V=S_{A}$ or $S_{P}$. For each affine variety $V$, there exists a polynomial
parametrization $P_{V}$ satisfying

(1.5) $g_{d}(P_{V}(w,t))=P_{V}(T_{d}(w),T_{d}(t)).$

Then, we will show that $g_{d}(S_{P})=S_{P}$ and $g_{d}(S_{A})=S_{A}$ for any
morphism $g_{d}$. The singular locus of $S_{P}$ is the line
$L_{2}=V(p_{1},p_{3},p_{4})$. The line $L_{2}$ is also preperiodic for any
$g_{d}$ $(d\in{\mathbb{N}})$. Set $C_{2}=g_{2}(L_{2})$. Then the affine curve
$C_{2}$ is invariant under $g_{d}$ for any $d\in{\mathbb{N}}$.

From (1.4) and (1.5), we can determine the set $K(g_{d}\mid_{V})$ with $d\geq
2$, for $V=X_{Q},\ S_{A}$ and $S_{P}$. The union $K(g_{d}\mid_{S_{A}})\cup
K(g_{d}\mid_{S_{P}})$ is a real 2- dimensional surface and is invariant under
$g_{d}$. We have $S_{A}\cap S_{P}=C_{1}\cup C_{2}$ and $\
K(g_{d}\mid_{S_{A}})\cap K(g_{d}\mid_{S_{P}})=K(g_{d}\mid_{C_{1}})\cup
K(g_{d}\mid_{C_{2}}).$

The surface $S_{A}$ has relation to the relative orbit variety
$\mathcal{A}/D_{4}$, where $\mathcal{A}$ is an astroidalhedron defined in [23,
p.205]. It is shown in [23] that $\mathcal{A}$ is a ruled surface in the sense
of differential geometry. The projective closure of $S_{A}$ is a birationally
ruled surface. Indeed. The affine algebraic surface $S_{A}$ is birationally
equivalent to an affine quadric cone $S_{C}$. The projective closure of
$S_{C}$ is a birationally ruled surface. Let $\psi:S_{C}-\to S_{A}$ and
$\phi:S_{A}-\to S_{C}$ be rational maps such that $\phi\circ\psi=id_{S_{C}}$
and $\psi\circ\phi=id_{S_{A}}$. We will show that the map $\phi\circ
g_{d}\circ\psi$ on $S_{C}$ is a morphism defined over ${\mathbb{Z}}$ and is
described by Chebyshev polynomials in one variable. It can be extended to a
morphism on ${\mathbb{P}}^{3}({\mathbb{C}})$.

The singular locus of $S_{A}$ is an affine algebraic curve $C_{A}$. The curve
$C_{A}$ corresponds to the relative orbit variety $\alpha_{3}/D_{4}$, where
$\alpha_{3}$ is the astroid curve in ${\mathbb{R}}^{3}$. See [23, p.206]. We
will show that $g_{d}(C_{A})=C_{A}$, for any $d\in{\mathbb{N}}$.

We consider the dynamics of on the affine curves $V$, where $V=C_{1},C_{2}$ or
$C_{A}$. For each affine curve $V$, there exists a polynomial parametrization
$P_{V}$ satisfying

(1.6) $g_{d}(P_{V}(z))=P_{V}(T_{d}(z)).$

We explain our main ideas of the proofs. We studied complex dynamical system
of $P_{A_{n}}^{d}$ on ${\mathbb{C}}^{n}$ in [22, 23]. In this paper, we make
use of these results for the research of morphisms on affine algebraic
varieties. In particular we construct parametric representation $P_{V}$ of
affine algebraic varieties $V$ by using parametrizations of several sets
appeared in complex dynamical system of $P_{A_{n}}^{d}$ on ${\mathbb{C}}^{n}$.
From (1.2), (1.3), (1.4), (1.5) and (1.6), we see that the morphisms $g_{d}$
on the affine algebraic varieties behave well in accordance with these
parametric representations.

We also use the theory of Gröbner bases of ideals. See [3] .

Küçüksakalli [8, 9] studied Chebyshev maps $P_{A_{2}}^{d}$ over finite field,
based on [22]. Our results will have some relations to arithmetic dynamics.

##  2\. From holomorphic maps to morphisms on varieties

###  2.1. Chebyshev endomorphisms and dihedral groups

Chebyshev endomorphism $P_{A_{n}}^{d}$ of degree $d$ on ${\mathbb{C}}^{n}$ is
associate to the Lie algebra of type ${A_{n}}$. For short, we denote
$P_{A_{n}}^{d}$ by $T_{d}$.

We begin with constructing Chebyshev endomorphisms $T_{d}$ on
${\mathbb{C}}^{n}$.
Let $t_{j},\ 1\leq j\leq n+1,$ be variables satisfying
$t_{1}t_{2}\cdot\cdot\cdot t_{n+1}=1$.
Let $z_{j},\ 1\leq j\leq n,$ be the $j$ -th elementary symmetric polynomial in
$t_{1},t_{2},\dots,t_{n+1}$.

We define a map $\Phi$ from $({\mathbb{C}}^{*})^{n+1}$ to ${\mathbb{C}}^{n}$
by

(2.1) $\Phi(t_{1},\dots,t_{n+1})=(z_{1},\dots,z_{n}).$

Let

$(z_{1}^{(d)},\dots,z_{n}^{(d)}):=\Phi(t_{1}^{d},\dots,t_{n+1}^{d}),\ d\in\
{\mathbb{N}}.$

Since $z_{j}^{(d)}$ is a symmetric polynomial in $t_{1},t_{2},\dots,t_{n+1}$
with coefficients in ${\mathbb{Z}}$, each $z_{j}^{(d)}$ can be expressed as a
polynomial in $z_{1},\dots,z_{n}$ with coefficients in ${\mathbb{Z}}$. We
define Chebyshev morphism $T_{d}$ on ${\mathbb{C}}^{n}$ of degree $d\geq 1$
over ${\mathbb{Z}}$ by

(2.2) $T_{d}(z_{1},\dots,z_{n})=(z_{1}^{(d)},\dots,z_{n}^{(d)}).$

See [22, 23] for the case that $n=2$ or $3$.

Note that

$z_{n+1-k}=\sum t_{i_{1}}\cdot\cdot\cdot t_{i_{n+1-k}}=\sum
1/(t_{j_{1}}\cdot\cdot\cdot t_{j_{k}}),$

where $\\{i_{1},\dots,i_{n+1-k},j_{1},\dots,j_{k}\\}=\\{1,2,\dots,n+1\\}.$
Considering the correspondence of $t_{i}$ and $1/t_{i},i=1,\dots,n+1$, we have

$z_{n+1-j}^{(d)}(z_{1},\dots,z_{n})=z_{j}^{(d)}(z_{n},\dots,z_{1}).$

If $z_{j}=\bar{z}_{n+1-j}$, for any $j=1,\dots,n$, then

$\overline{z_{j}^{(d)}(z_{1},\dots,z_{n})}=z_{j}^{(d)}(z_{n},\dots,z_{1}).$

Hence if $z_{j}=\bar{z}_{n+1-j}$ for any $j=1\dots,n$, then
${z_{n+1-j}^{(d)}}=\overline{z_{j}^{(d)}},$ for any $j=1\dots,n.$
Then we consider a subspace $R_{n}$ of ${\mathbb{C}}^{n}$ given by

$R_{n}:=\\{(z_{1},\dots,z_{n})\in{\mathbb{C}}^{n}:z_{j}=\bar{z}_{n+1-j},\mbox{for
any}\ j=1,\dots,n\\}.$

The maps $T_{d}$ admit the invariant subspace $R_{n}$. The subspace $R_{n}$
appears also in [2, 21].

The set of bounded orbits of $T_{d}$, denoted by $K(T_{d})$, is defined by

$K(T_{d}):=\\{z\in{\mathbb{C}}^{n}:\\{T_{d}^{m}(z):m=1,2,\dots\\}\ \mbox{is
bounded}\\},$

where $T_{d}^{m}$ denotes the $m$-th iterate of $T_{d}$. It is known that
$K(T_{d})\subset R_{n}$ in [23]. Since

$\Phi(t_{1}^{-1},\dots,t_{n+1}^{-1})=(z_{n},z_{n-1}\dots,z_{1}),$

we can define $T_{-d}(z_{1},\dots,z_{n})$ and have that

$T_{-d}(z_{1},\dots,z_{n})=T_{d}(z_{n},z_{n-1}\dots,z_{1}).$

The endomorphisms $T_{-d}$ admit also the subspace $R_{n}$. Note that

(2.3) $T_{j}\circ T_{k}=T_{jk}\quad\mbox{for any }\quad
j,k\in{\mathbb{Z}}\setminus\\{0\\}.$

Next we define actions $R$ and $C$ on ${\mathbb{C}}^{n}$. Set $\zeta=e^{2\pi
i/(n+1)}$. We define

$R(t_{j})=\zeta t_{j},\enskip t_{j}\in{\mathbb{C}}^{*},\enskip
j=1,\enskip\dots,n+1.$

Note that $R(t_{1})R(t_{1})\cdots R(t_{n+1})=1$. Since

$z_{k}=\sum t_{j_{1}}\cdots t_{j_{k}}\ \mbox{where}\
\\{j_{1},\dots,j_{k}\\}\subset\\{1,2,\dots,n+1\\},$

we have

$R(z_{k})=\zeta^{k}z_{k},\quad k=1,\dots,n.$

We set, for $(z_{1},\dots,z_{n})\in{\mathbb{C}}^{n},$

$R(z_{1},\dots,z_{n}):=(R(z_{1}),\dots,R(z_{n}))=(\zeta z_{1},\
\zeta^{2}z_{2},\ \dots,\ \zeta^{n}z_{n}).$

######  Proposition 2.1.

For any $d\in\mathbb{Z}\setminus\\{0\\},$ we have

(2.4) $R^{d}\circ T_{d}=T_{d}\circ R.$

###### Proof.

We have

$R^{d}\circ T_{d}(z_{1},\dots,z_{n})=(\zeta^{d}z_{1}^{(d)},\
\zeta^{2d}z_{2}^{(d)},\ \dots,\ \zeta^{nd}z_{n}^{(d)}),\
(z_{1},\dots,z_{n})\in{\mathbb{C}}^{n}.$

Any element $(z_{1},\dots,z_{n})$ determines a set $\\{t_{1},\dots,t_{n+1}\\}$
satisfying (2.1) as roots of the following equation ;

$t^{n+1}+(-1)z_{1}t^{n}+\ ...\ +(-1)^{n}z_{n}t+(-1)^{n+1}=0.$

Since

$z_{k}^{(d)}=\sum t_{j_{1}}^{d}\cdots t_{j_{k}}^{d}\ \mbox{where}\
\\{j_{1},\dots,j_{k}\\}\subset\\{1,2,\dots,n+1\\},$

we have

$\zeta^{kd}z_{k}^{(d)}=\sum(\zeta t_{j_{1}})^{d}\cdots(\zeta t_{j_{k}})^{d}.$

On the other hand, the element $R(z_{1},\dots,z_{n})$ determines a set
$\\{\zeta t_{1},\dots,\zeta t_{n+1}\\}$ satisfying (2.1) . Then the $k$-th
element of $T_{d}\circ R(z_{1},\dots,z_{n})$ is equal to

$\sum(\zeta t_{j_{1}})^{d}\cdots(\zeta t_{j_{k}})^{d}.$

∎

Next we define an action $C$ on ${\mathbb{C}}^{n}$. We define

$C(t_{j})=\bar{t}_{j},\ t_{j}\in{\mathbb{C}}^{*},\ j=1,\dots,n+1.$

Note that

$C(t_{1})\ \cdots\ C(t_{n+1})=1.$

Clearly $C(z_{k})=\bar{z}_{k}$. Since $z_{k}^{(d)}$ is a polynomial in
$z_{1},\ \dots\ ,z_{n}$ over ${\mathbb{Z}}$, it follows that

$\overline{T_{d}(z_{1},\dots z_{n})}=T_{d}(\bar{z}_{1},\dots,\bar{z}_{n}).$

Set

$C(z_{1},\dots,z_{n}):=(C(z_{1}),\dots,C(z_{n}))=(\bar{z}_{1},\ \dots,\
\bar{z}_{n}).$

Then we have the following.

######  Proposition 2.2.

For any $d\in\mathbb{Z}\setminus\\{0\\},$ we have

(2.5) $C\circ T_{d}=T_{d}\circ C.$

Clearly

$CRC(z_{1},\dots,z_{n})=(\bar{\zeta}z_{1},\bar{\zeta}^{2}z_{2},\dots,\bar{\zeta}^{n}z_{n})=({\zeta}^{-1}z_{1},{\zeta}^{-2}z_{2},\dots,{\zeta}^{-n}z_{n})$
$=({\zeta}^{n}z_{1},{\zeta}^{2n}z_{2},\dots,{\zeta}^{n^{2}}z_{n})=R^{n}(z_{1},\dots\
,z_{n})=R^{-1}(z_{1},\ \dots\ ,z_{n}).$

Hence $CRC=R^{-1}$. Then the group generated by the two elements $R$ and $C$
is the dihedral group $D_{n+1}=<R,C>$ of order $2n+2$.

We consider the $D_{n+1}$-orbit of ${\bf z}\in{\mathbb{C}}^{n}$ defined by

$D_{n+1}\cdot{\bf z}:=\\{\pi{\bf z}:\pi\in D_{n+1},\ {\bf
z}\in{\mathbb{C}}^{n}\\}.$

Then by Propositions 2.1 and 2.2, we see that $T_{d}$ maps an orbit
$D_{n+1}\cdot{\bf z}$ to another orbit $D_{n+1}\cdot{T_{d}}({\bf z})$. Hence
we can define an induced map of $T_{d}$ on the orbit space
${\mathbb{C}}^{n}/D_{n+1}$.

Recall that $R_{n}$ is the subspace of ${\mathbb{C}}^{n}$. Clearly,

$R(R_{n})\subset R_{n}\ \mbox{and}\ C(R_{n})\subset R_{n}.$

Hence the maps $T_{d},\ R$ and $C$ admit the subspace $R_{n}$ . Then from
(2.4) and (2.5) we have

(2.6)
$(R\mid_{R_{n}})^{d}\circ(T_{d}\mid_{R_{n}})=(T_{d}\mid_{R_{n}})\circ(R\mid_{R_{n}}),$
(2.7)
$(C\mid_{R_{n}})\circ(T_{d}\mid_{R_{n}})=(T_{d}\mid_{R_{n}})\circ(C\mid_{R_{n}}).$

Note that $R_{n}\simeq{\mathbb{R}}^{n}$. Sometimes we regard $R_{n}$ as
${\mathbb{R}}^{n}$. We set

$z_{1}=x_{1}+ix_{2},\ z_{2}=x_{3}+ix_{4},\dots,\ \ z_{n/2}=x_{n-1}+ix_{n}\
\mbox{ if}\ n\ \mbox{ is\enskip even},$

and

$z_{1}=x_{1}+ix_{2},\dots,\ z_{(n-1)/2}=x_{n-2}+ix_{n-1},\ z_{(n+1)/2}=x_{n}\
\mbox{ if}\ n\ \mbox{is\enskip odd}.$

Then we have

(2.8) ${}^{t}(z_{1},\dots,z_{n})=M_{n}\ ^{t}(x_{1},\dots,x_{n}),$

where $M_{n}$ is a non-singular matrix.

We denote the map $T_{d}$ restricted to $R_{n}$ by $f_{d}(x_{1},\dots,x_{n})$.
Note that $f_{d}(x_{1},\dots,x_{n})$ is a polynomial map defined over
${\mathbb{Z}}$. Properties of $f_{d}(x_{1},x_{2})$ are studied in [21] and
[22]. The generators $R$ and $C$ restricted to $R_{n}$ can be represented by
matrices in $GL(n,{\mathbb{R}})$. Those are written in the following block-
diagonal matrices. The generator $R\mid_{R_{n}}$ is represented in the form:

$R\mid_{R_{n}}=\left(\begin{array}[]{cccccc}R_{11}&&&&&\\\
&R_{22}&&&$\mbox{\Huge 0}$&\\\ &&&&&\\\ &&\ddots&&\\\ &&&&&\\\ &$\mbox{\Huge
0}$&&&&\\\ &&&&&R_{mm}\end{array}\right).$

If $n$ is even, $m=n/2$ and $R_{jj}$ is a block of (2, 2) type defined by

$R_{jj}=\left(\begin{array}[]{ccc}\cos\frac{2\pi j}{n+1}&&-\sin\frac{2\pi
j}{n+1}\\\ \sin\frac{2\pi j}{n+1}&&\cos\frac{2\pi j}{n+1}\\\
\end{array}\right),\ \mbox{for}\ \ 1\leq j\leq m.$

If $n$ is odd, $m=(n+1)/2$ and $R_{jj}$ is the same as above for $1\leq
j\leq(n-1)/2$ and $R_{mm}=(-1)$ is a block of (1, 1) type. The generator
$C\mid_{R_{n}}$ is represented in the form:

$C\mid_{R_{n}}=\left(\begin{array}[]{cccccc}C_{11}&&&&&\\\
&C_{22}&&&$\mbox{\Huge 0}$&\\\ &&&&&\\\ &&\ddots&&\\\ &&&&&\\\ &$\mbox{\Huge
0}$&&&&\\\ &&&&&C_{mm}\end{array}\right).$

If $n$ is even, $m=n/2$ and $C_{jj}$ is a block of (2,2) type defined by

$C_{jj}=\left(\begin{array}[]{cc}1&0\\\ 0&-1\\\ \end{array}\right),\
\mbox{for}\ \ 1\leq j\leq m.$

If $n$ is odd, $m=(n+1)/2$ and $C_{jj}$ is the same as above for $1\leq
j\leq(n-1)/2$ and $C_{mm}=(1)$ is a block of (1, 1) type.

###  2.2. Morphisms on orbit varieties

From (2.6) and (2,7) we can define an induced map of $f_{d}$ on
${\mathbb{R}}^{n}/D_{n+1}$, where $D_{n+1}$ is the finite matrix group in
$GL(n,{\mathbb{R}})$. Let $k$ be the field ${\mathbb{R}}$ or the field
${\mathbb{C}}$ . The $D_{n+1}$-orbit of ${\bf a}\in k^{n}$ is a set

$D_{n+1}\cdot{\bf a}=\\{A\cdot{\bf a}:A\in D_{n+1}\\}.$

The set of all $D_{n+1}$-orbits in $k^{n}$ is denoted by $k^{n}/D_{n+1}$ and
is called the orbit space.

We begin with the construction of the maps on ${\mathbb{R}}^{n}/D_{n+1}$. We
use invariant theory. See [3], [5] and [20].

Let $k[x_{1},\cdots,x_{n}]^{D_{n+1}}$ is the set of invariant polynomials
$f({\bf x})\in k[x_{1},\cdots,x_{n}]$ such that $f({\bf x})=f(A\cdot{\bf x})$,
for all $A\in D_{n+1}$. By Emmy Noether’s theorem (Theorem 5 in [3, Chapter 7,
§3]), we know that $k[x_{1},\dots,x_{n}]^{D_{n+1}}$ is generated by finitely
many homogeneous invariants in $k[x_{1},\dots,x_{n}]$. We denote these
homogeneous invariants by $p_{1}({\bf x}),\dots,p_{m}({\bf x})$ . Since
$D_{n+1}\subset GL(n,{\mathbb{R}})$, by the proof of Theorem 5 in [3, Chapter
7, §3], we can select a basis $\\{p_{1}({\bf x}),\dots,p_{m}({\bf x})\\}$ such
that

(2.9)
$\begin{split}{\mathbb{R}}[x_{1},\dots,x_{n}]^{D_{n+1}}={\mathbb{R}}[p_{1},\dots,p_{m}],\\\
\mbox{ and}\hskip 142.26378pt\\\
{\mathbb{C}}[x_{1},\dots,x_{n}]^{D_{n+1}}={\mathbb{C}}[p_{1},\dots,p_{m}].\end{split}$

We need some more results on invariant theory. See [3, Chapter 7, §4]. Using a
basis $\\{p_{1},...,p_{m}\\}$ satisfying (2.9), we have that
$k[x_{1},\dots,x_{n}]^{D_{n+1}}=k[p_{1},\dots,p_{m}]$. We let the ideal
$F=(p_{1},\dots,p_{m})$. Then we define the syzygy ideal $I_{F}$ for $F$ :

$I_{F}:=\\{h\in k[y_{1},\dots,y_{m}]:h(p_{1},\dots,p_{m})=0\ \mbox{in}\
k[x_{1},\dots,x_{n}]\\}.$

Then we have the affine variety $V_{F}:=V(I_{F})$. Let $k[V_{F}]$ be the
coordinate ring of $V_{F}$. Then there is a ring isomorphism $k[V_{F}]\simeq
k[x_{1},\dots,x_{n}]^{D_{n+1}}$.

######  Theorem 2.3.

([3, Chapter 7, §4, Theorem10])

We assume that $k={\mathbb{C}}$. Then :
(1) The polynomial map $\tilde{F}:{\mathbb{C}}^{n}\to V_{F}$ defined by
$\tilde{F}({\bf a})=(p_{1}({\bf a}),\dots,p_{m}({\bf a}))$ is subjective.
(2) The map sending the $D_{n+1}$-orbit $D_{n+1}\cdot{\bf
a}\subset{\mathbb{C}}^{n}$ to a point $\tilde{F}({\bf a})\in V_{F}$ includes a
one-to-one correspondence ${\mathbb{C}}^{n}/D_{n+1}\simeq V_{F}$.

We recall that the map $f_{d}({\bf x})$ is the map $T_{d}$ restricted to
$R_{n}\simeq{\mathbb{R}}^{n}$. Then $f_{d}$ is the map from ${\mathbb{R}}^{n}$
to ${\mathbb{R}}^{n}$. We know from (2.9) that $p_{j}({\bf x})$ is a
polynomial over ${\mathbb{R}}$. Let

$P_{j}({\bf x}):=p_{j}(f_{d}({\bf x})),\ j=1,\dots,m.$

Then we define a map $\tilde{f}_{d}$ from $\tilde{F}({\mathbb{R}}^{n})$ to
${\mathbb{R}}^{m}$ by

$\tilde{f}_{d}(p_{1}({\bf x}),\dots,p_{m}({\bf x}))=(P_{1}({\bf
x}),\dots,P_{m}({\bf x})).$

We will verify that $\tilde{f}_{d}$ is well-defined. We suppose
$\tilde{F}({\bf x})=\tilde{F}({\bf x}^{\prime})$ for ${\bf x},{\bf
x}^{\prime}\in{\mathbb{R}}^{n}$. From Theorem 2.3(2), we see that ${\bf
x}^{\prime}$ belongs to the $D_{n+1}$-orbit $D_{n+1}\cdot{\bf x}$. Then there
exists an element $A\in D_{n+1}$ such ${\bf x}^{\prime}=A\cdot{\bf x}$. Then
by (2.6) and (2.7), there exists an element $B\in D_{n+1}$ such that
$f_{d}(A\cdot{\bf x})=B\cdot f_{d}({\bf x})$. Hence

$P_{j}(A\cdot{\bf x})=p_{j}(f_{d}(A\cdot{\bf x}))=p_{j}(B\cdot f_{d}({\bf
x}))=p_{j}(f_{d}({\bf x}))=P_{j}({\bf x}).$

Then $\tilde{f}_{d}$ is well-defined. And we also show that

$P_{j}(A\cdot{\bf x})=P_{j}({\bf x}),\ \mbox{for any}\ A\in D_{n+1}.$

Then

$P_{j}({\bf
x})\in{\mathbb{R}}[x_{1},\dots,x_{n}]^{D_{n+1}}={\mathbb{R}}[(p_{1}({\bf
x}),\dots,p_{m}({\bf x})].$

Hence $P_{j}({\bf x})$ is a polynomial $P_{j}(p_{1}({\bf x}),\dots,p_{m}({\bf
x}))$ in $p_{1}({\bf x}),\dots,p_{m}({\bf x})$ over ${\mathbb{R}}$. Here we
use the same symbol $P_{j}$ for a polynomial of $p_{1},\dots,p_{m}$. We set

$f({\bf x}):=P_{j}(p_{1}({\bf x}),\dots,p_{m}({\bf x}))-(p_{j}(f_{d}({\bf
x})).$

Then $f\in{\mathbb{R}}[x_{1},\dots,x_{n}]\ \mbox{and}\
f:{\mathbb{R}}^{n}\to{\mathbb{R}}$ is the zero function. Hence by Proposition
5 in [3, Chapter 1, §1], we have $f=0$ in ${\mathbb{R}}[x_{1},\dots,x_{n}]$.
That is,

$P_{j}(p_{1}({\bf x}),\dots,p_{m}({\bf x}))\equiv p_{j}(f_{d}({\bf x})).$

Then

(2.10) $\begin{split}\tilde{f}_{d}(p_{1}({\bf x}),\dots,p_{m}({\bf
x}))=(P_{1}(p_{1},\dots,p_{m})({\bf x}),\dots,P_{m}(p_{1},\dots,p_{m})({\bf
x})),\\\ \mbox{ where}\ P_{j}(p_{1},\dots,p_{m})({\bf x}):=P_{j}(p_{1}({\bf
x}),\dots,p_{m}({\bf x})),j=1,\dots,m.\end{split}$

And so, for ${\bf x}\in{\mathbb{R}}^{n}$, we have

(2.11) $\tilde{f}_{d}\circ\tilde{F}({\bf x})=\tilde{F}\circ f_{d}({\bf x}).$

We have the following commutative diagram.

(2.12) $\begin{array}[]{ccc}\\{t_{1},\dots,t_{n+1}\\}&\smash{\mathop{\hbox
to36.98866pt{\rightarrowfill}}\limits}&\\{t_{1}^{d},\dots,t_{n+1}^{d}\\}\\\
\Big{\updownarrow}\hbox
to0.0pt{\hss$\vbox{\hbox{$\scriptstyle\,$}}$}&&\Big{\updownarrow}\hbox
to0.0pt{\hss$\vbox{\hbox{$\scriptstyle\,$}}$}\\\
(z_{1},\dots,z_{n})&\smash{\mathop{\hbox
to36.98866pt{\rightarrowfill}}\limits^{T_{d}}}&(z_{1}^{(d)},\dots,z_{n}^{(d)})\\\
\Big{\uparrow}\hbox to0.0pt{$\vbox{\hbox{$\scriptstyle
M_{n}\,$}}$\hss}&&\Big{\uparrow}\hbox to0.0pt{$\vbox{\hbox{$\scriptstyle
M_{n}\,$}}$\hss}\\\ (x_{1},\dots,x_{n})&\smash{\mathop{\hbox
to36.98866pt{\rightarrowfill}}\limits^{f_{d}}}&(X_{1},\dots,X_{n})\\\
\Big{\downarrow}\hbox
to0.0pt{\hss$\vbox{\hbox{$\scriptstyle\,$}}$}&&\Big{\downarrow}\hbox
to0.0pt{\hss$\vbox{\hbox{$\scriptstyle\,$}}$}\\\
(p_{1},\dots,p_{m})&\smash{\mathop{\hbox
to36.98866pt{\rightarrowfill}}\limits}&(P_{1},\dots,P_{m}),\end{array}$

where $T_{d}:{\mathbb{C}}^{n}\to{\mathbb{C}}^{n}$ and
$f_{d}:{\mathbb{R}}^{n}\to{\mathbb{R}}^{n}$.

To use the correspondence ${\mathbb{C}}^{n}/D_{n+1}\simeq V_{F}$ in Theorem
2.3(2), we consider complexification. We regard $x_{1},\dots,x_{n}$ as complex
variables and $f_{d}$ as a polynomial map from ${\mathbb{C}}^{n}$ to
${\mathbb{C}}^{n}$. The map $M_{n}$ in (2.8) is an invertible linear map from
${\mathbb{C}}^{n}$ to ${\mathbb{C}}^{n}$. We consider also complexification of
$\tilde{f_{d}}$. Recall that the bijection caused by $\tilde{F}$ gives
${\mathbb{C}}^{n}/D_{n+1}$ the structure of the affine algebraic variety
$V_{F}$. Then from (2.10), we can define a map $g_{d}$ from $V_{F}$ to
$V_{F}$.

Then we have the following diagram.

(2.13) $\begin{array}[]{ccc}\\{t_{1},\dots,t_{n+1}\\}&\smash{\mathop{\hbox
to36.98866pt{\rightarrowfill}}\limits}&\\{t_{1}^{d},\dots,t_{n+1}^{d}\\}\\\
\Big{\updownarrow}\hbox
to0.0pt{\hss$\vbox{\hbox{$\scriptstyle\,$}}$}&&\Big{\updownarrow}\hbox
to0.0pt{\hss$\vbox{\hbox{$\scriptstyle\,$}}$}\\\
(z_{1},\dots,z_{n})&\smash{\mathop{\hbox
to36.98866pt{\rightarrowfill}}\limits^{T_{d}}}&(z_{1}^{(d)},\dots,z_{n}^{(d)})\\\
\Big{\downarrow}\hbox to0.0pt{$\vbox{\hbox{$\scriptstyle
M_{n}^{-1}\,$}}$\hss}&&\Big{\downarrow}\hbox
to0.0pt{$\vbox{\hbox{$\scriptstyle M_{n}^{-1}\,$}}$\hss}\\\
(x_{1},\dots,x_{n})&\smash{\mathop{\hbox
to36.98866pt{\rightarrowfill}}\limits^{f_{d}}}&(X_{1},\dots,X_{n})\\\
\Big{\downarrow}\hbox
to0.0pt{$\vbox{\hbox{$\scriptstyle\tilde{F}\,$}}$\hss}&&\Big{\downarrow}\hbox
to0.0pt{$\vbox{\hbox{$\scriptstyle\tilde{F}\,$}}$\hss}\\\
(p_{1},\dots,p_{m})&\smash{\mathop{\hbox
to36.98866pt{\rightarrowfill}}\limits^{g_{d}}}&(P_{1},\dots,P_{m}),\end{array}$

where $t_{j}\in{\mathbb{C}}^{*}$, $T_{d}:{\mathbb{C}}^{n}\to{\mathbb{C}}^{n},$
$f_{d}:{\mathbb{C}}^{n}\to{\mathbb{C}}^{n}$ and $g_{d}:V_{F}\to V_{F}$.

Then the element $(t_{1},\dots,t_{n+1})$ may take any value of
$({\mathbb{C}}^{*})^{n+1}$ with $t_{1}\cdots t_{n+1}=1.$ Note that $T_{d}$ and
$f_{d}$ are polynomial maps and the diagram (2.13) is commutable if
$(x_{1},\dots,x_{n})\in{\mathbb{R}}^{n}.$ Then the diagram (2.13) is also
commutable for any $(x_{1},\dots,x_{n})\in{\mathbb{C}}^{n}.$

######  Definition 2.4.

We define a map $g_{d}$ from $V_{F}$ to $V_{F}$ by

$g_{d}(p_{1},\dots,p_{m})=(P_{1}(p_{1},\dots,p_{m}),\dots,P_{m}(p_{1},\dots,p_{m})).$

Note that $P_{j}$ is a polynomial in $p_{1},\dots,p_{m}$ over $\mathbb{R}$.
Then we have the following.

######  Proposition 2.5.

Let $g_{d}$ be a map in Definition 2.4. The map $g_{d}$ is a morphism from
$V_{F}$ to $V_{F}$ defined over $\mathbb{R}$ and is surjective.

###### Proof.

It is enough to prove that $g_{d}$ is surjective. Note that the vertical
arrows in (2.13) are surjective and that the morphism
$\\{t_{1},\dots,t_{n+1}\\}\mapsto\\{t_{1}^{d},\dots,t_{n+1}^{d}\\}$ is
surjective. Then the morphism $g_{d}$ is surjective. ∎

Let $\\{p_{1}({\bf x}),\dots,p_{m}({\bf x})\\}$ be a basis of
${\mathbb{C}}[x_{1},\dots,x_{n}]^{D_{n+1}}$ satisfying (2.9). Then by (2.10)
and (2.11) we have

(2.14) $\begin{split}g_{d}(p_{1}({\bf x}),\dots,p_{m}({\bf
x}))=(P_{1}(p_{1},\dots,p_{m})({\bf x}),\dots,P_{m}(p_{1},\dots,p_{m})({\bf
x}))\\\ ={\hat{f}}_{d}(p_{1}({\bf x}),\dots,p_{m}({\bf
x})),\qquad\qquad\qquad\qquad\end{split}$

for any ${\bf x}\in{\mathbb{C}}^{n}$.

In Sections 3 and 4, we study the maps $f_{d}$ on ${\mathbb{C}}^{2}$ or
${\mathbb{C}}^{3}$ and their induced maps $g_{d}$. In the cases, we can select
a basis $\\{p_{1}({\bf x}),\dots,p_{m}({\bf x})\\}$ satisfying the following
properties :
${\mbox{(}1)}\ p_{j}({\bf x})\in{\mathbb{Z}}[x_{1},\dots,x_{n}],\
(j=1,\dots,m),\ (n=2\ {\mbox{o}r}\ 3),$
(2) $g_{d}$ is a morphism defined over ${\mathbb{Z}}$.

We consider the set $\\{g_{d}:d\in{\mathbb{Z}}\setminus\\{0\\}\\}$ of
morphisms.

Since $f_{1}({\bf x})={\bf x},\ {\tilde{f}}_{1}(p_{1}({\bf x}),\ \dots\ ,\
p_{m}({\bf x}))=(p_{1}({\bf x}),\ \dots\ ,\ p_{m}({\bf x}))$. Then $g_{1}$ is
an identity map. We show the set
$\\{g_{d}:d\in{\mathbb{Z}}\setminus\\{0\\}\\}$ is a commutative monoid.

######  Proposition 2.6.

We have $g_{e}\circ g_{d}=g_{de},$ for any
$d,e\in{\mathbb{Z}}\setminus\\{0\\}.$

###### Proof.

By (2.3), we have $T_{e}\circ T_{d}=T_{de}.$ If $f_{d}$ is the map $T_{d}$
restricted to $R_{n}$, then $f_{e}\circ f_{d}=f_{de}$. If we view $f_{d}$ as a
map from ${\mathbb{C}}^{n}\to{\mathbb{C}}^{n}$, then the same formula holds.
We set

(2.15) $g_{e}\circ
g_{d}(p_{1},\dots,p_{m})-g_{de}(p_{1},\dots,p_{m})=h(p_{1},\dots,p_{m}).$

Note that $h(p_{1},\dots,p_{m})\in{\mathbb{R}}[p_{1},\dots,p_{m}].$

We substitute $p_{j}({\bf x})$ for $p_{j},\ (j=1,\dots,m)$ in (2.15). By
(2.10), (2.11) and (2.14), we deduce

$g_{d}(p_{1}({\bf x}),\dots,p_{m}({\bf x}))=(P_{1}(p_{1},\dots,p_{m})({\bf
x}),\dots,P_{m}(p_{1},\dots,p_{m})({\bf x}))$ $={\hat{f}}_{d}(p_{1}({\bf
x}),\dots,p_{m}({\bf x}))=\tilde{f}_{d}\circ\tilde{F}({\bf x})=\tilde{F}\circ
f_{d}({\bf x}),$ $\mbox{ where}\ P_{j}(p_{1},\dots,p_{m})({\bf
x}):=P_{j}(p_{1}({\bf x}),\dots,p_{m}({\bf x})),j=1,\dots,m.$

And we have

$g_{e}(P_{1}(p_{1},\dots,p_{m})({\bf x}),\dots,P_{m}(p_{1},\dots,p_{m})({\bf
x}))={\tilde{f}}_{e}(P_{1}(p_{1},\dots,p_{m})({\bf
x}),\dots,P_{m}(p_{1},\dots,p_{m}({\bf x}))$
$={\tilde{f}}_{e}\circ\tilde{F}\circ f_{d}({\bf x})=\tilde{F}\circ f_{e}\circ
f_{d}({\bf x})=\tilde{F}\circ f_{ed}({\bf x}).$

Hence $g_{e}\circ g_{d}(p_{1}({\bf x}),\dots,p_{m}({\bf x}))=g_{de}(p_{1}({\bf
x}),\dots,p_{m}({\bf x})).$
Then $h(p_{1}({\bf x}),\dots,p_{m}({\bf x}))=0$, for any ${\bf
x}\in{\mathbb{C}}^{n}$. Therefore $h(y_{1},\dots,y_{m})$ belongs to the syzygy
ideal $I_{F}$ for $F$. Thus, for any $(p_{1},\dots,p_{m})\in V_{F}=V(I_{F})$,
we have $g_{e}\circ g_{d}(p_{1},\dots,p_{m})=g_{de}(p_{1},\dots,p_{m}).$ ∎

######  Corollary 2.7.

For any $d\in{\mathbb{N}}$, we have $g_{-d}=g_{d}$.

###### Proof.

It suffices to show $g_{-1}=g_{1}$. We see in Section 2.1 that

$T_{-1}(z_{1},\dots,z_{n})=T_{1}(\bar{z}_{1},\ \dots,\ \bar{z}_{n})\
{\mbox{i}n}\ R_{n}.$

Then
$f_{-1}(x_{1},x_{2},\dots,x_{n})=(x_{1},-x_{2},x_{3},\dots,(-1)^{n-1}x_{n})$.

Recall that $f_{1}(x_{1},\dots,x_{n})=(x_{1},\dots,x_{n})$. Then
$(C\mid_{R_{n}})(f_{1}(x_{1},\dots,x_{n})=f_{-1}(x_{1},\dots,x_{n})$. Hence
$p_{j}(f_{-1}({\bf x}))=p_{j}({\bf x}),\ j=1,\dots,m$. Therefore

$\tilde{f}_{-1}(p_{1}({\bf x}),\dots,p_{m}({\bf x}))=(p_{1}({f}_{-1}({\bf
x}),\dots,p_{m}({f}_{-1}({\bf x}))=(p_{1}({\bf x}),\dots,p_{m}({\bf x})).$

Thus we have $g_{-1}=g_{1}$. ∎

From now on we consider only morphisms $g_{d}$ with $d\in{\mathbb{N}}$.

We consider the properties of the morphisms $g_{d}$. We use the following
theorem.

######  Theorem 2.8.

([19, II, §5, 3, Theorems 6 and 7 ])

If a morphism $f:X\to Y$ is finite, dominant and separable, the varieties $X$
and $Y$ are irreducible of equal dimension, and $Y$ is normal, then any point
of some non-empty subsets $U\subset Y$ has deg $f$ distinct inverse images,
and any point of the complement has fewer inverse images.

We have the following proposition.

######  Proposition 2.9.

(1) The affine algebraic variety $V_{F}\simeq{\mathbb{C}}^{n}/D_{n+1}$ is
normal.
(2) We assume that $g_{d}$ is a finite morphism. Then deg $g_{d}=d^{n}$.

###### Proof.

(1) Clearly ${\mathbb{C}}[x_{1},\dots,x_{n}]$ is integrally closed. Then by
Propsition 2.3.11 in [5] we know that
${\mathbb{C}}[x_{1},\dots,x_{n}]^{D_{n+1}}$ is also integrally closed. Recall
that ${\mathbb{C}}[V_{F}]\simeq{\mathbb{C}}[x_{1},\dots,x_{n}]^{D_{n+1}}$.
Then by [19, II, §5 ], the affine variety $V_{F}$ is normal.
(2) Note that the vertical arrrows in the diagram (2.13) are at most finite
surjective morphisms. Then it is easy to see that the degree of $g_{d}$ is the
same as the morphism
$\\{t_{1},\dots,t_{n+1}\\}\mapsto\\{t_{1}^{d},\dots,t_{n+1}^{d}\\}$. Then deg
$g_{d}=d^{n}$. ∎

##  3\. Morphisms on ${\mathbb{C}}^{2}/D_{3}$

###  3.1. Morphisms and branch locus

We begin to study the orbit variety ${\mathbb{C}}^{2}/D_{3}\simeq V_{F}$. We
set $x:=x_{1}$ and $y:=x_{2}$. By Algorithm 2.5.14 in [20], we know that
${\mathbb{C}}[x,y]^{D_{3}}={\mathbb{C}}[x^{2}+y^{2},x^{3}-3xy^{2}]$. See also
Exercises(3) in [20, §2.2]. Clearly $x^{2}+y^{2}$ and $x^{3}-3xy^{2}$ are
algebraically independent. Set $p=x^{2}+y^{2},q=x^{3}-3xy^{2}$ and $F=(p,q)$.
Then the syzygy ideal $I_{F}$ for $F$ is equal to the zero ideal. Hence
$V_{F}=V(I_{F})\simeq{\mathbb{C}}^{2}$. We use a coordinate $(p,q)$ of
${\mathbb{C}}^{2}$.

We consider another fundamental system $\\{p^{\prime},q^{\prime}\\}$ of
invariants of ${\mathbb{C}}[x,y]^{D_{3}}$. By Molien’ theorem ([20, Theorem
2.2.1]) , the Hilbert series $\Phi_{D_{3}}(t)$ of the invariant ring
${\mathbb{C}}[x,y]^{D_{3}}$ is written as

$\Phi_{D_{3}}(t)=1+t^{2}+t^{3}+t^{4}+t^{5}+2t^{6}+t^{7}+2t^{8}+2t^{9}+\it{O}(t^{10}).$

Then
$dim({\mathbb{C}}[x,y]_{2}^{D_{3}})=dim({\mathbb{C}}[x,y]_{3}^{D_{3}})=1$.
Since $x^{2}+y^{2}$ and $x^{3}-3xy^{2}\in{\mathbb{C}}[x,y]^{D_{3}}$, then it
follows that $p^{\prime}=a(x^{2}+y^{2})$ and $q^{\prime}=b(x^{3}-3xy^{2})$,
where $a$ and $b$ are non-zero complex numbers. Hence the fundamental system
$\\{p,q\\}$ of invariants of ${\mathbb{C}}[x,y]^{D_{3}}$ is unique up to
constants.

By [22, p.996], we have $f_{2}(x,y)=(x^{2}-y^{2}-2x,\ 2xy+2y)$,
$f_{3}(x,y)=(x^{3}-3xy^{2}-3x^{2}-3y^{2}+3,\ 3x^{2}y-y^{3}).$ We compute two
morphisms $g_{2}$ and $g_{3}$ on $V_{F}$ :

(3.1) $g_{2}(p,q)=(4p+p^{2}-4q,12p^{2}-8q-6pq+2q^{2}-p^{3}),$ (3.2)
$\begin{split}g_{3}(p,q)=(9-18p+9p^{2}+p^{3}+6q-6pq,\qquad\qquad\qquad\qquad\qquad\qquad\\\
9p^{4}-3p^{3}q-36p^{3}+27p^{2}q+81p^{2}-18pq^{2}-54pq-81p+4q^{3}+18q^{2}+27q+27).\end{split}$

These are morphisms defined over ${\mathbb{Z}}$. This fact holds for general
morphisms $g_{d}$ on ${\mathbb{C}}^{2}/D_{3}$.

######  Theorem 3.1.

(1) Let $p=x^{2}+y^{2}\ {\mbox{a}nd}\ q=x^{3}-3xy^{2}$. Then we have
${\mathbb{R}}[x,y]^{D_{3}}\cap{\mathbb{Z}}[x,y]={\mathbb{Z}}[p,q].$
(2) Let $g_{d}$ be a morphism on ${\mathbb{C}}^{2}/D_{3}$ in Definition 2.4.
Then $g_{d}$ is defined over ${\mathbb{Z}}$ for any $d\in{\mathbb{N}}.$

###### Proof.

(1) Suppose $P(p,q)$ be a polynomial such that

(3.3) $P(p({\bf x}),q({\bf x}))\in{\mathbb{R}}[p({\bf x}),q({\bf
x})]\cap{\mathbb{Z}}[x,y].$

Then we will show that

$P(p,q)\in{\mathbb{Z}}[p,q].$

We define the xy-degree of $p^{i}q^{j}$ to be $2i+3j$. Let $h_{k}(p,q)$ denote
the sum of terms $a_{ij}p^{i}q^{j}$ in $P(p,q)$ with the xy-degree $k$. Then

$P(p,q)=\sum_{k}h_{k}(p,q).$

It is enough to prove that $h_{k}(p,q)\in{\mathbb{Z}}[p,q]$ for any $k$.
Clearly

(3.4) $h_{k}(p({\bf x}),q({\bf x}))\in{\mathbb{R}}[p({\bf x}),q({\bf
x})]\cap{\mathbb{Z}}[x,y].$

Let

$h_{k}(p,q)=\alpha p^{j}+qH(p,q),\ {\mbox{w}here}\ \alpha\in{\mathbb{R}}\
{\mbox{a}nd}\ H(p,q)\in{\mathbb{R}}[p,q],$

and the xy-degree of any term in $H(p,q)$ is $k-3$.
If $\alpha\neq 0$, then $k=2j$. We consider the monomial $y^{2j}$ in $\alpha
p({\bf x})^{j}+q({\bf x})H(p({\bf x}),q({\bf x})).$ The monomial $y^{2j}$
occurs only in the term of $\alpha p({\bf x})^{j}$. Since $h_{k}(p({\bf
x}),q({\bf x}))\in{\mathbb{Z}}[x,y],$ we have $\alpha\in{\mathbb{Z}}$ and so

(3.5) $q({\bf x})H(p({\bf x}),q({\bf x}))\in{\mathbb{Z}}[x,y].$

We consider the case $k$ is odd. Since $p(x,y),\
q(x,y)\in{\mathbb{Z}}[x,y^{2}],$ we may set

$H(p({\bf x}),q({\bf x}))=\sum_{j=0}^{(k-3)/2}a_{j}x^{k-3-2j}y^{2j},\
a_{j}\in{\mathbb{R}}.$

Then

$q({\bf x})H(p({\bf x}),q({\bf
x}))=a_{0}x^{k}+(\sum_{j=1}^{(k-3)/2}(a_{j}-3a_{j-1})x^{k-2j}y^{2j})-3a_{(k-3)/2}xy^{k-1}.$

By (3.5), we see that

$a_{0},a_{1}-3a_{0},a_{2}-3a_{1},\dots,a_{(k-3)/2}-3a_{(k-5)/2}\in{\mathbb{Z}}.$

Hence

$a_{0},a_{1},\dots,a_{(k-3)/2}\in{\mathbb{Z}}.$

Therefore

(3.6) $H(p({\bf x}),q({\bf x}))\in{\mathbb{R}}[p({\bf x}),q({\bf
x})]\cap{\mathbb{Z}}[x,y].$

For the case $k$ is even, we can prove (3.6) similarly.

We get (3.6) from (3.4). Repeating this procedure, eventually we have a
polynomial $H_{*}(p({\bf x}),q({\bf x}))$ satisfying
(i) the xy-degree of any monomial in $H_{*}(p,q)$ is 2 or 0,
(ii) $H_{*}(p({\bf x}),q({\bf x}))\in{\mathbb{R}}[p({\bf x}),q({\bf
x})]\cap{\mathbb{Z}}[x,y].$

Then $H_{*}(p,q)=bp$ or $b,\ b\in{\mathbb{Z}}.$ The assertions (1) follows.
(2) We see in (2.10) that

$\tilde{f}_{d}(p({\bf x}),q({\bf x}))=(P_{1}({\bf x}),P_{2}({\bf x})),\mbox{
with}\ {\bf x}=(x,y),$ $\mbox{and}\ P_{j}({\bf x})=P_{j}(p({\bf x}),q({\bf
x}))\in{\mathbb{R}}[p({\bf x}),q({\bf x})],\ j=1,2.$

Since $T_{d}(z_{1},z_{2})$ is a polynomial map over ${\mathbb{Z}}$,
$f_{d}({\bf x})$ is a polynomial map defined over ${\mathbb{Z}}$. Then
$P_{j}(p({\bf x}),q({\bf x})),(j=1,2),$ satisfies (3.3). Then
$P_{j}(p,q)\in{\mathbb{Z}}[p,q]$.

∎

We note that since $I_{F}=0,$ the expression of $P_{j}(p,q)$ is unique.

We study the dynamics of $g_{d}$ on ${\mathbb{C}}^{2}/D_{3}$. We have seen
that ${\mathbb{C}}^{2}/D_{3}\simeq{\mathbb{C}}^{2}$. To start, we introduce a
new parametrization of ${\mathbb{C}}^{2}$ which is convenient for
understanding the dynamics of $g_{d}$. From (2.13), we have

(3.7) $\begin{array}[]{ccc}\\{t_{1},t_{2},t_{3}\\}&\smash{\mathop{\hbox
to36.98866pt{\rightarrowfill}}\limits}&\\{t_{1}^{d},t_{2}^{d},t_{3}^{d}\\}\\\
\Big{\updownarrow}\hbox
to0.0pt{\hss$\vbox{\hbox{$\scriptstyle\,$}}$}&&\Big{\updownarrow}\hbox
to0.0pt{\hss$\vbox{\hbox{$\scriptstyle\,$}}$}\\\
(z_{1},z_{2})&\smash{\mathop{\hbox
to36.98866pt{\rightarrowfill}}\limits^{T_{d}}}&(z_{1}^{(d)},z_{2}^{(d)})\\\
\Big{\downarrow}\hbox to0.0pt{$\vbox{\hbox{$\scriptstyle
M_{2}^{-1}\,$}}$\hss}&&\Big{\downarrow}\hbox
to0.0pt{$\vbox{\hbox{$\scriptstyle M_{2}^{-1}\,$}}$\hss}\\\
(x,y)&\smash{\mathop{\hbox
to36.98866pt{\rightarrowfill}}\limits^{f_{d}}}&(X,Y)\\\ \Big{\downarrow}\hbox
to0.0pt{$\vbox{\hbox{$\scriptstyle\tilde{F}\,$}}$\hss}&&\Big{\downarrow}\hbox
to0.0pt{$\vbox{\hbox{$\scriptstyle\tilde{F}\,$}}$\hss}\\\
(p,q)&\smash{\mathop{\hbox
to36.98866pt{\rightarrowfill}}\limits^{g_{d}}}&(P_{d},Q_{d}).\end{array}$

Set $t_{1}=e^{i(\alpha+\beta)},\ t_{2}=e^{i(\alpha-\beta)},\
t_{3}=e^{-2i\alpha},$ for $\alpha,\beta\in{\mathbb{C}}.$ Then

(3.8) $z_{1}=e^{i(\alpha+\beta)}+e^{i(\alpha-\beta)}+e^{-2i\alpha}\
\mbox{and}\ z_{2}=e^{-i(\alpha+\beta)}+e^{-i(\alpha-\beta)}+e^{2i\alpha}.$

Hence by (2.8) we have

$x=\cos(\alpha+\beta)+\cos(\alpha-\beta)+\cos 2\alpha,\quad
y=\sin(\alpha+\beta)+\sin(\alpha-\beta)-\sin 2\alpha.$

Since $p(x,y)=x^{2}+y^{2}$ and $q(x,y)=x^{3}-3xy^{2}$ , it follows that

$p(x,y)=3+2\cos(3\alpha-\beta)+2\cos 2\beta+2\cos(3\alpha+\beta),$
$q(x,y)=\cos 6\alpha+2\cos\beta(5\cos
3\alpha+\cos(3\alpha-2\beta)+6\cos\beta+\cos(3\alpha+2\beta)).$

Using trigonometrical identities, we can rewrite $p(x,y)$ and $q(x,y)$ as

(3.9) $\begin{split}p=1+4\cos 3\alpha\cos\beta+4(\cos\beta)^{2}\qquad\qquad\\\
q=\frac{1}{2}(-2+4(\cos 3\alpha)^{2}+24(\cos\beta)^{2}+4\cos
3\alpha\cos\beta(3+4(\cos\beta)^{2}).\end{split}$

Note that all the identities which we use to get (3.9) are valid for complex
variables $\alpha$ and $\beta$. Then setting $u=2\cos 3\alpha$ and
$v=2\cos\beta$, we have

(3.10) $\begin{split}p(u,v)=1+uv+v^{2},\\\
q(u,v)=\frac{1}{2}(-2+u^{2}+6v^{2}+uv(3+v^{2})).\end{split}$

We can think of this as the mapping
$G_{{\mathbb{C}}^{2}}:{\mathbb{C}}^{2}\to{\mathbb{C}}^{2}$ defined by

$G_{{\mathbb{C}}^{2}}(u,v)=(1+uv+v^{2},\
\frac{1}{2}(-2+u^{2}+6v^{2}+uv(3+v^{2})).$

######  Proposition 3.2.

Under the above notations, we have
$G_{{\mathbb{C}}^{2}}({\mathbb{C}}^{2})={\mathbb{C}}^{2}$.

###### Proof.

It suffices to prove that the polynomial parametrization
$G_{{\mathbb{C}}^{2}}({\mathbb{C}}^{2})$ covers all points of
${\mathbb{C}}^{2}$. Since $\tilde{F}$ is surjective, for any
$(p,q)\in{\mathbb{C}}^{2}$, there exists an element $(x,y)\in{\mathbb{C}}^{2}$
satisfying $\tilde{F}(x,y)=(p,q)$. Then there exists an element
${}^{t}(z_{1},z_{2})=M_{2}\ ^{t}(x,y)$. Since $\alpha,\beta\in{\mathbb{C}}$,
the parametrization $t_{1}=e^{i(\alpha+\beta)},\ t_{2}=e^{i(\alpha-\beta)},\
t_{3}=e^{-2\alpha i}$ covers all points of $\\{t_{1},t_{2},t_{3}\\}$ with
$t_{1}t_{2}t_{3}=1$. Hence we have an element
$(\alpha,\beta)\in{\mathbb{C}}^{2}$ such that the two elements $(x,y)$ and
$(\alpha,\beta)$ satisfy

$x=\cos(\alpha+\beta)+\cos(\alpha-\beta)+\cos 2\alpha,\quad
y=\sin(\alpha+\beta)+\sin(\alpha-\beta)-\sin 2\alpha.$

Then $G_{{\mathbb{C}}^{2}}(2\cos 3\alpha,2\cos\beta)=(p,q).$ ∎

Through we have a simpler proof in this case $n=2$, we note this proof because
this proof is valid for $n=2,3.$ See Section 4.

For any invariant affine algebraic variety $V$ in ${\mathbb{C}}^{m}$ under
$g_{d}$, we define the set of bounded orbits of $g_{d}\mid_{V}$ by

$K(g_{d}\mid_{V})=\\{w\in V:\\{g_{d}^{k}(w):k=1,2,\cdots\\}\ \mbox{is bounded
in }\ {\mathbb{C}}^{m}\\}.$

Let $T_{d}(z)$ be the $d^{th}$ Chebyshev polynomial (See [17, §6.2])
satisfying $T_{d}(s+s^{-1})=s^{d}+s^{-d}$, with $z=s+s^{-1}$.

######  Theorem 3.3.

Let $g_{d}$ be a morphism on ${\mathbb{C}}^{2}/D_{3}$ in Definition 2.4.

$(1)\
g_{d}(G_{{\mathbb{C}}^{2}}(u,v))=G_{{\mathbb{C}}^{2}}(T_{d}(u),T_{d}(v)),\quad\mbox{for
any}\quad(u,v)\in{\mathbb{C}}^{2}.\qquad\qquad\qquad\qquad\qquad$ $(2)\
K(g_{d}\mid_{\mathbb{C}^{2}})=\\{G_{{\mathbb{C}}^{2}}(2\cos 3\alpha,\
2\cos\beta):0\leq 3\alpha,\beta<2\pi\\},\ \mbox{for}\ d\geq
2.\qquad\qquad\qquad$

###### Proof.

(1) We consider the parametrization $p(u,v)$ and $q(u,v)$ in (3.10). Set

$g_{d}(p(u,v),\ q(u,v))=(P_{d}(u,v),\ Q_{d}(u,v)).$

Setting $s=e^{i3\alpha}$, we have $s+s^{-1}=2\cos 3\alpha.$ Hence $T_{d}(2\cos
3\alpha)=2\cos 3d\alpha$. Also we have $T_{d}(2\cos\beta)=2\cos d\beta$.

Note that $T_{d}(z_{1},z_{2})=(z_{1}^{(d)},z_{2}^{(d)})$ in (3.7). The map
$T_{d}$ gives rise to a map :
$(t_{1},t_{2},t_{3})\mapsto(t_{1}^{d},t_{2}^{d},t_{3}^{d})$ such that
$\Phi(t_{1}^{d},t_{2}^{d},t_{3}^{d})=(z_{1}^{(d)},z_{2}^{(d)}).$ Since $z_{1}$
and $z_{2}$ are written as (3.8), we have

$z_{1}^{(d)}=e^{id(\alpha+\beta)}+e^{id(\alpha-\beta)}+e^{-2di\alpha},$
$z_{2}^{(d)}=e^{-id(\alpha+\beta)}+e^{-id(\alpha-\beta)}+e^{2di\alpha}.$

Set $f_{d}(x,y)=(X,Y).$ So, ${}^{t}(X,Y)=M_{2}^{-1}\
{}^{t}(z_{1}^{(d)},z_{2}^{(d)}).$ Then we get $(X,Y)$ by substituting
$d\alpha$ for $\alpha$ and $d\beta$ for $\beta$ in $(x,y)$.

And also we get $P_{d}$ and $Q_{d}$ by substituting $d\alpha$ for $\alpha$ and
$d\beta$ for $\beta$ in $(p,q)$ in (3.9). Therefore we have

(3.11) $\begin{split}P_{d}(u,v)=1+T_{d}(u)T_{d}(v)+T_{d}(v)^{2},\\\
Q_{d}(u,v)=\frac{1}{2}(-2+T_{d}(u)^{2}+6T_{d}(v)^{2}+T_{d}(u)T_{d}(v)(3+T_{d}(v)^{2})),\end{split}$

for every $(u,v)\in{\mathbb{C}}^{2}.$
(2) Let $u=t_{1}+t_{1}^{-1}$ and $v=t_{2}+t_{2}^{-1}$, with
$t_{j}=r_{j}e^{i\theta_{j}},\ r_{j}\geq 1,\ \theta_{j}\in{\mathbb{R}},$ for
$j=1,2.$ Then $T_{d}(u)=t_{1}^{d}+t_{1}^{-d}$. Let $T_{d}^{n}(u)$ denote the
$n$-th iterate of $T_{d}$. Then $T_{d}^{n}(u)=t_{1}^{d^{n}}+t_{1}^{{-d}^{n}}$.
Set

$g_{d}^{n}(p(u,v),\ q(u,v))=(p_{n}(u,v),\ q_{n}(u,v)).$

We assume that $\\{p_{n}(u,v):n=1,2,\dots\\}$ is bounded. We suppose that
$r_{1}\neq r_{2}$. We assume $r_{1}>r_{2}\geq 1$. Then the dominant term of
$p_{n}(u,v)$ is $T_{d}^{n}(u)T_{d}^{n}(v)$ and
$\lim_{n\to\infty}T_{d}^{n}(u)T_{d}^{n}(v)=\infty.$ A contradiction. Also from
the assumption $r_{2}>r_{1}\geq 1$, we have a contradiction. Then
$r_{1}=r_{2}\geq 1$. We suppose that $r_{1}=r_{2}>1$. Then
$\lim_{n\to\infty}q_{n}(u,v)=\infty.$ Hence if $g_{d}^{n}(p(u,v),\
q(u,v)):n=1,2,\dots\\}$ is bounded, then $r_{1}=r_{2}=1$. ∎

The set $K(g_{d}\mid_{{\mathbb{C}}^{2}})$ is a closed region lying on the real
slice ${\mathbb{R}}^{2}$ in ${\mathbb{C}}^{2}$. It is invariant under $g_{d}$.
It is depicted in Figure 1.

Figure 1.  An invariant closed domain.

It is known that Chebyshev endomorphism $P_{A_{n}}^{d}$ on ${\mathbb{C}}^{n}$
is postcritically finite. More precisely, the set of critical values of
$P_{A_{n}}^{d}$ is invariant under $P_{A_{n}}^{d}$ (See [22] and [23]).

Inspired by the above result, we consider the branch points of the morphisms
$g_{d}$.

We recall Theorem 2.8. We use the notations in the theorem. A point $y\in Y$
is called a branch point of $f$ if the number of inverse images of $y$ under
$f$ is less than the degree of $f$. In this paper we use ”a branch point” in
place of ”a ramification point” in [19, II, §5].

We consider the inverse images of $(a,b)$ under the morphism $g_{2}$ in (3.1).
Let $(P,Q)$ denote the right hand side of (3.1). We consider the resultants

$\mbox{ Res}(P-a,Q-b,q)\ \mbox{ and \ Res}(P-a,Q-b,p)\ \mbox{of}\ P-a\ \mbox{
and }\ Q-b$

with respect to $q$ or $p$.
By direct computations, we have

$\mbox{
Res}(P-a,Q-b,q)=32a+2a^{2}-16b-128p+8ap+96p^{2}-4ap^{2}-24p^{3}+2p^{4}$

and

$\mbox{ Res}(P-a,Q-b,p)=192a^{2}-a^{3}-256b-36ab+b^{2}-2048q+864aq-24a^{2}q$
$-152bq+768q^{2}-108aq^{2}-4bq^{2}-96q^{3}+4q^{4}.$

Then $g_{2}$ is finite (See [19, p.48]). By Proposition 2.9(2), we have deg
$g_{2}=4.$ To get the branch points of $g_{2}$, we compute the discriminant of
Res$(P-a,Q-b,q)$ with respect to $p$. The discriminant is written in the form
: $-1048576(-27+18a+a^{2}-8b)(a^{3}-b^{2})$. We have two affine varieties
$V(-27+18a+a^{2}-8b)$ and $V(a^{3}-b^{2})$. We denote the cuspidal cubic
$V(p^{3}-q^{2})$ by $C_{C}$ and denote the parabola $V(-27+18p+p^{2}-8q)$ by
$C_{D}$. We will show that two curves $C_{C}$ and $C_{D}$ are branch locus of
$g_{2}$. Let $O$ denote the origin $(0,0)$ and $S$ denote the points of
intersection $C_{C}\cap C_{D}$ in ${\mathbb{C}}^{2}$. That is,
$S=\\{(9,27),(1,-1)\\}$.

######  Proposition 3.4.

Under the above notations we have the followings :
(1) If $(a,b)\in C_{C}\setminus\\{O\cup S\\},\ \mbox{then }\ \sharp\
g_{2}^{-1}(a,b)=3.$ Two points lie on $C_{C}$ and one point with multiplicity
2 lies on $V(3p-q-4)$.
(2) If $(a,b)\in C_{D}\setminus S,\ \mbox{then }\ \sharp\ g_{2}^{-1}(a,b)=3.$
Two points lie on $C_{D}$ and one point with multiplicity 2 lie on $V(p-1)$.
(3) $\sharp\ g_{2}^{-1}(p,q)=2,$ for any point $(p,q)$ in $O\cup S$.

###### Proof.

By direct computations, we can prove this proposition. ∎

Next we consider the inverse images of $(a,b)$ under the morphism $g_{3}$ in
(3.2). Let $(P,Q)$ denote the right hand side of (3.2). The resultant $\mbox{
Res}(P-a,Q-b,q)$ is a polynomial in $p$ of degree 9 over ${\mathbb{Z}}[x,y].$
Its leading term is $-4p^{9}$. The resultant $\mbox{Res}(P-a,Q-b,p)$ is also a
polynomial in $q$ whose leading term is $64q^{9}$. Then $g_{3}$ is a finite
map. By Proposition 2.9(2), we have deg $g_{3}=9.$ The discriminant of
$\mbox{Res}(P-a,Q-b,q)$ is written in the form
$n(a-1)^{6}(-27+18a+a^{2}-8b)^{3}(a^{3}-b^{2})^{3}$, where $n$ is a negative
integer.

We consider the case $a=1$. If $a=1$, the resultant $\mbox{ Res}(P-a,Q-b,q)$
has a zero $p=1$ with multiplicity 3. But if $a=1$ and $b\neq\pm 1$, then
$\sharp\ g_{3}^{-1}(1,b)=9.$ We compute $\sharp\ g_{3}^{-1}(1,1)=6$ and
$\sharp\ g_{3}^{-1}(1,-1)=4.$ Clearly, the points $(1,\pm 1)$ belong to
$V(a^{3}-b^{2})$.

######  Proposition 3.5.

We have the followings :
(1) If $(a,b)\in C_{C}\setminus\\{O\cup S\\},$ then $\sharp\
g_{3}^{-1}(a,b)=6$. Three points lie on $C_{C}$ and other three points with
multiplicity 2 lie on $V(-27+54p-27p^{2}+p^{3}-18q+18pq-4q^{2})$.
(2) If $(a,b)\in C_{D}\setminus S$ , then $\sharp\ g_{3}^{-1}(a,b)=6$. Three
points lie on $C_{D}$ and other three points with multiplicity 2 lie on
$V(p^{2}-2q)$.
(3) $\sharp\ g_{3}^{-1}(0,0)=3,$ $\sharp\ g_{3}^{-1}(9,27)=3,$ and $\sharp\
g_{3}^{-1}(1,-1)=4.$

###### Proof.

By direct computations, we can prove this proposition. ∎

######  Proposition 3.6.

(1) The branch locus of $g_{d}:{\mathbb{C}}^{2}\to{\mathbb{C}}^{2}$ consists
of $C_{C}$ and $C_{D}$, for $d=2,3.$
(2) The branch locus of $\tilde{F}:{\mathbb{C}}^{2}\to V_{F}$ is $C_{C}$.

###### Proof.

(1) This assertion follows from the above arguments.
(2) Recall that $p=x^{2}+y^{2}$ and $q=x^{3}-3xy^{2}.$ Then we have a equation
$4x^{3}-3px-q=0.$ Its discriminant with respect to $x$ is $432(p^{3}-q^{2}).$
Then $C_{C}$ is the branch divisor of $\tilde{F}$. ∎

In the next subsections we study the dynamics of $g_{d}$ on these branch
divisors.

###  3.2. The dynamics of $g_{d}$ on the cuspidal cubic $C_{C}$.

In the previous subsection, we see the relation between $g_{d}^{-1}(C_{C})$
and $C_{C}$ for $d=2,3$. We will prove that $g_{d}(C_{C})=C_{C}$ for any
$d\in{\mathbb{N}}$ and show the dynamics of $g_{d}$ on $C_{C}$. A polynomial
parametric representation of $C_{C}=V(p^{3}-q^{2})$ consists of $p=t^{2}$ and
$q=t^{3}$. We consider a mapping $G_{C}:{\mathbb{C}}\to{\mathbb{C}}^{2}$
defined by

$G_{C}(z)=((1+z)^{2},(1+z)^{3}).$

######  Theorem 3.7.

Let $g_{d}$ be a morphism on ${\mathbb{C}}^{2}/D_{3}$ in Definition 2.4.
(1) $\ g_{d}(G_{C}(z))=G_{C}(T_{d}(z)),$ for any $z\in{\mathbb{C}}\quad
and\quad d\in{\mathbb{N}}.$
(2) $g_{d}(C_{C})=C_{C},$ for any $d\in{\mathbb{N}}$.

###### Proof.

(1)The proof is similar to the proof of Theorem 3.3.

Using (3.9) we compute

$p(x,y)^{3}-q(x,y)^{2}=4(\cos 3\alpha-\cos 3\beta)^{2}(\sin 3\alpha)^{2}.$

Since $p^{3}=q^{2}$, we assume that $\alpha=0.$ Clearly $d\alpha=0.$ Hence we
put $u=2$ in (3.10). Then from $G_{{\mathbb{C}}^{2}}$ we get $G_{C}$. Then we
have the assertion (1).
(2) We consider a polynomial parametrization of $C_{C}$ given by $p=t^{2}$ and
$q=t^{3}$. Equivalently, we consider the mapping
$G:{\mathbb{C}}\to{\mathbb{C}}^{2}$ defined by $G(t)=(t^{2},t^{3})$. We know
that the polynomial parametrization $G({\mathbb{C}})$ fills up all of
$V(p^{3}-q^{2}).$
Then the assertion (2) follows from (1). ∎

We remark that if we assume that $\cos 3\alpha=\cos 3\beta$, then
$p(x,y)=(1+2\cos 2\beta)^{2},\ q(x,y)=(1+2\cos 2\beta)^{3}$. Then the similar
result holds.

We may view $V(p^{3}-q^{2})$ as an affine curve in ${\mathbb{C}}^{2}$. Then by
the argument similar to that in the proof of Theorem 3.3 (2), we have

$K(g_{d}\mid_{C_{C}})=\\{((1+2\cos\theta)^{2},\
(1+2\cos\theta)^{3}):0\leq\theta<2\pi\\}.$

The set $K(g_{d}\mid_{C_{C}})$ is a part of the cuspidal cubic
$V(p^{3}-q^{2})$ in ${\mathbb{R}}^{2}$ containing the cusp point $(0,0)$. It
is a Jordan arc, denoted by $\gamma_{1}$. Its end points are $(1,-1)$ and
$(9,27)$. Recall that the set of these two points coincides with the
intersection $C_{C}\cap C_{P}$ in ${\mathbb{C}}^{2}$.

The dynamics of $g_{d}\mid_{C_{C}}$ can be deduced by the dynamics of the
Chebyshev map $T_{d}:{\mathbb{C}}\to{\mathbb{C}}$. It is known (e. g. in [12,
§7]) that $K(T_{d})=[-2,2],\ T_{d}([-2,2])=[-2,2]$, and Julia set of $T_{d}$
is also $[-2,2]$. The Jordan arc $\gamma_{1}$ corresponds to the interval
$[-2,2]$. Then all preperiodic points of $g_{d}\mid_{C_{C}}$ lie on
$\gamma_{1}$. It is know that $T_{2}$ restricted to $[-2,2]$ is conjugate to a
typical chaotic map $L(x)=4x(1-x)$ on $[0,1]$ which is studied by Ulam and von
Newman [25]. In this meaning, we say that $g_{d}\mid_{C_{C}}$ is chaotic on
$\gamma_{1}$.

######  Corollary 3.8.

Under the above notations we have the following :
(1) $K(g_{d}\mid_{C_{C}})=\gamma_{1},\ g_{d}(\gamma_{1})=\gamma_{1}$ and
$g_{d}\mid_{C_{C}}$ is chaotic on $\gamma_{1}$.
(2) If $3\not|\ d,$ then the cusp point $O$ in $\gamma_{1}$ is a fixed point
of $g_{d}$.
If $3\mid d$, then $g_{d}(O)=(9,27)$. And $g_{d}(9,27)=(9,27)$ for any $d\geq
1$.

###### Proof.

(1) By the above argument, we have this assertion.
(2) Note that the point $O=(0,0).$ We set $z=s+s^{-1}$. If $1+s+s^{-1}=0,$
then $s$ is a primitive cubic root of unity. Then the assertion (2) is
obvious. ∎

Next we will show that $g_{d}\mid_{C_{C}}$ extends to a morphism on
${\mathbb{P}}^{2}$. Let $g_{d}(p,q)=(P_{d}(p,q),Q_{d}(p,q))$. Then
$g_{d}\mid_{C_{C}}(p,q)=([P_{d}(p,q)],\ [Q_{d}(p,q)])$ where $[P_{d}(p,q)]$
and $[Q_{d}(p,q)]$ are equivalent classes in the coordinate ring
${\mathbb{C}}[p,q]/(p^{3}-q^{2})$.

######  Lemma 3.9.

We can represent the equivalent classes in the form :
(1) $P_{d}(p,q)=p^{d}+H_{1}(p,q)$,
(2) $Q_{d}(p,q)\equiv q^{d}+pH_{2}(p,q)+H_{3}(p,q)\quad mod\ (p^{3}-p^{2})$,
where $H_{j}(p,q)\in{\mathbb{Z}}[p,q],\ j=1,2,3,$ and the degrees of
$H_{1}(p,q)$ and $H_{3}(p,q)$ are less than or equal to $d-1$ and $H_{2}(p,q)$
is a homogeneous polynomial of degree $d-1$.

###### Proof.

(1) We consider the diagram (3.7) in the case ${z_{2}}=\overline{z_{1}}$. Then
$T_{d}(z_{1},\overline{z_{1}})=(z_{1}^{(d)},\overline{z_{1}^{(d)}})$. Let
$z_{1}=x+iy$, where $x$ and $y$ are real variables. Then $X(x,y)=Re\
z_{1}^{(d)}$ and $Y(x,y)=Im\ z_{1}^{(d)}$. Then $f_{d}(x,y)=(X(x,y),\
Y(x,y)).$ We recall that $T_{d}(z_{1},z_{2})=(z_{1}^{(d)},z_{2}^{(d)})$. By
[11] and [22, p.996], we have a recurrence equation

$z_{1}^{(d)}=z_{1}z_{1}^{(d-1)}-z_{2}z_{1}^{(d-2)}+z_{1}^{(d-3)},\
z_{1}^{(1)}=z_{1},\ z_{1}^{(2)}=z_{1}^{2}-2z_{2},\
z_{1}^{(3)}=z_{1}^{3}-3z_{1}z_{2}+3.$

Then $z_{1}^{(d)}(z_{1},z_{2})$ is a polynomial of degree $d$ and its leading
term is $z_{1}^{d}$. Then the leading term of
$z_{1}^{(d)}(z_{1},\overline{z_{1}})$ is $z_{1}^{d}$. Set
$(x+iy)^{d}=X_{1}+iY_{1}$. Set $X=X_{1}+X_{2}$ and $Y=Y_{1}+Y_{2}$, where
degrees of $X_{2}$ and $Y_{2}$ are less than or equal to $d-1$.

If $d$ is even, the term $y^{d}$ occurs in $X_{1}$. If $d$ is odd, the term
$y^{d}$ occurs in $Y_{1}$. Then the term $y^{2d}$ occurs in $X^{2}+Y^{2}$.
Since $P_{d}(p({\bf x}),q({\bf x}))=X^{2}+Y^{2}$, it follows that the term
$(x^{2}+y^{2})^{d}$ must occur in $P_{d}(p({\bf x}),q({\bf x}))$. Hence the
term $p^{d}$ occurs in $P_{d}(p,q)$. The xy-degree of any term in $P_{d}(p,q)$
is less than or equal to $2d$. If there is a term $p^{i}q^{j}$ in $P_{d}(p,q)$
with $2i+3j=2d$ and $j>0$, then $i+j<d$.
(2) We consider a monomial $p^{3n+j}q^{k}$, where
$j,k,n\in{\mathbb{N}}\cup\\{0\\}$ and $0\leq j\leq 2$. We define a degree
minimization function $m$ by $m(p^{3n+j}q^{k})=p^{j}q^{2n+k}$. Clearly,
$m(p^{3n+j}q^{k})\equiv p^{3n+j}q^{k}\ mod\ (p^{3}-q^{2})$. Let
$Q_{d}(p,q)=\sum a_{ij}p^{i}q^{j}$. We replace any term $a_{ij}p^{i}q^{j}$
with $a_{ij}m(p^{i}q^{j})$. Since the leading term of $z_{1}^{(d)}$ is
$z_{1}^{d}$, the xy-degree of any term in $Q_{d}(p,q)$ is less than or equal
to $3d$.
(i) The monomials with the xy-degree $3d$ :
$q^{d},q^{d-2}p^{3},\dots\dots,q^{d-2k}p^{3k},$ where $k=d/2$ or $(d-1)/2$
according to whether $d$ is even or odd.
The images of these monomials under the function $m$ are $q^{d}$.
(ii) The monomials with the xy-degree $3d-1$ :
$q^{d-1}p,q^{d-3}p^{4},\dots\dots,p^{3(d-1)/2}\ \mbox{or}\ p^{3d/2-2}q.$ The
images of these monomials under the function $m$ are $q^{d-1}p$.
(iii) The monomials with the xy-degree $3d-2$ :
$q^{d-2}p^{2},q^{d-4}p^{5},\dots\dots,p^{3d/2-1}\ \mbox{or}\ p^{3d/2-5/2}q.$
The images of these monomials under the function $m$ are $q^{d-2}p^{2}$.

Then to prove that the polynomial $\sum a_{ij}m(p^{i}q^{j})$ satisfies the
conditions of the right hand side of (2), it suffices to prove the following.
Let $\sum_{h=0}^{k}a_{h}q^{d-2h}p^{3h}$ be the sum of terms in $Q_{d}(p,q)$
with the xy-degree $3d$. Then it is enough to prove $\sum_{h=0}^{k}a_{h}=1$.
Note that

$X_{1}^{3}-3X_{1}Y_{1}^{2}=\sum_{h=0}^{k}a_{h}q({\bf x})^{d-2h}p({\bf
x})^{3h}.$

Since $X_{1}^{3}-3X_{1}Y_{1}^{2}=Re(X_{1}+iY_{1})^{3}=Re(x+iy)^{3d},$ the term
$x^{3d}$ occurs in $X_{1}^{3}-3X_{1}Y_{1}^{2}$ and so it occurs in
$\sum_{h=0}^{k}a_{h}q({\bf x})^{d-2h}p({\bf x})^{3h}$. Hence
$\sum_{h=0}^{k}a_{h}=1$. ∎

We will show that the morphism $g_{d}$ restricted to $V(p^{3}-q^{2})$ can
extend to a morphism $\bar{g}_{d}$ on ${\mathbb{P}}^{2}({\mathbb{C}})$. Let
$(p:q:r)$ be homogeneous coordinates of ${\mathbb{P}}^{2}({\mathbb{C}})$. The
projective closure of $V(p^{3}-q^{2})$ is the variety $V(p^{3}-q^{2}r)$. Based
on Lemma 3.9, we define morphisms on ${\mathbb{P}}^{2}({\mathbb{C}})$. We set

$\bar{P}_{d}(p,q,r)=p^{d}+r^{d}H_{1}(p/r,q/r)=p^{d}+r\bar{H}_{1}(p,q,r),$

and

$\bar{Q}_{d}(p,q,r)=q^{d}+pH_{2}(p,q)+r^{d}H_{3}(p/r,q/r)=q^{d}+pH_{2}(p,q)+r\bar{H}_{3}(p,q,r).$

Here $\bar{H}_{1}(p,q,r)$ and $\bar{H}_{3}(p,q,r)$ are homogeneous
polynomials. Then we define mappings $\bar{g}_{d}$ on
${\mathbb{P}}^{2}({\mathbb{C}})$ by

(3.12) $\bar{g}_{d}(p:q:r)=(\bar{P}_{d}:\bar{Q}_{d}:r^{d}).$

######  Proposition 3.10.

Let $\bar{g}_{d}$ be a mapping in (3.12).
(1) The mapping
$\bar{g}_{d}:{\mathbb{P}}^{2}({\mathbb{C}})-\to{\mathbb{P}}^{2}({\mathbb{C}})$
is a morphism.
(2) $\bar{g}_{d}(V(p^{3}-q^{2}r))=V(p^{3}-q^{2}r)$.

###### Proof.

(1) We suppose that $\bar{P}_{d}=0,\ \bar{Q}_{d}=0$ and $r^{d}=0$. Then we
have $p=q=r=0.$
(2) If $r=1$, then by Theorem 3.7(2), we have
$\bar{g}_{d}(V(p^{3}-q^{2}))=V(p^{3}-q^{2})$. Clearly
$V(p^{2}-q^{2}r)\cap\\{(p:q:0)\\}=(0:1:0),$ and
$(\bar{g}_{d})^{-1}(0:1:0)=\\{(0:1:0)\\}$. ∎

By [18, II2], we know that $\bar{g}_{d}:V(p^{3}-q^{2}r)\to V(p^{3}-q^{2}r)$ is
finite. Diller [4] and Uehara [24] studied surface automorphisms preserving
the cuspidal cubic.

###  3.3. The dynamics of $g_{d}$ on the affine curve $C_{D}$ and $K$ sets.

We first show the relation between the curve $C_{D}$ and a relative orbit
variety of a deltoid which has a connection with the set of critical values of
$f_{d}$ on $R_{2}$. The critical set of $P_{A_{2}}^{d}\ (d\geq 2)$ on
${\mathbb{C}}^{2}$ is described in Lemma 2.1 in [22] :

$z_{1}=t_{0}+\varepsilon t_{0}+\frac{1}{\varepsilon t_{0}^{2}},\quad
z_{2}=\frac{1}{t}_{0}+\frac{1}{\varepsilon t_{0}}+{\varepsilon t_{0}^{2}},\
where\ \varepsilon=e^{2\pi\sqrt{-1}/d},\quad t_{0}\in{\mathbb{C}}^{*},\
j\in\mathbb{N}.$

We see that $z_{1}=\bar{z}_{2}$ if and only if $\mid t_{0}\mid=1.$ The
critical values are written in the form
$z_{1}^{(d)}=t_{0}^{d}+t_{0}^{d}+t_{0}^{-2d}.$ Note that the set of critical
values of $f_{d}$ is the same set for any $d\geq 2$. We set $t=t_{0}^{d}$.
Then $z_{1}^{(d)}=t+t+t^{-2}.$ We set $t=e^{i\alpha}$. Then the set of
critical values of $f_{d}$ on $R_{2}$ can be parametrized as

(3.13) $x=2\cos\alpha+\cos 2\alpha,\quad y=2\sin\alpha-\sin 2\alpha.$

The real variety parametrized by the above equations for $0\leq\alpha<2\pi$ is
a deltoid. Its implicit representation is
$h(x,y):=(x^{2}+y^{2}+9)^{2}+8(-x^{3}+3xy^{2})-108=0$. (See [22, p. 998]).

We regard $\alpha$ as a complex variable in (3.13). Hence we view $h(x,y)$ as
a polynomial in ${\mathbb{C}}[x,y]$. Since $h(x,y)$ is invariant under the
action of $D_{3}$, we can construct the relative orbit variety
$V(h(x,y))/D_{3}$ whose points are $D_{3}$-orbits of zeros of $h(x,y)$. See
[20, §2.6]. By Algorithm 2.6.2 in [20], we see that
$V(h(x,y))/D_{3}=V(-27+18p+p^{2}-8q)$. Hence the $C_{D}=V(-27+18p+p^{2}-8q)$
is the relative orbit variety $V(h(x,y))/D_{3}$.

Note that by setting $\beta=0$ in (3.8), we have (3.13). Hence, we put $v=2$
in (3.10). Then we have

$p(u)=5+2u,\quad q(u)=11+7u+\frac{u^{2}}{2}.$

Hence from $G_{{\mathbb{C}}^{2}}$ we get a mapping from ${\mathbb{C}}$ to
${\mathbb{C}}^{2}$ defined by $G_{D}(u)=(5+2u,11+7u+u^{2}/2).$ We prove in the
next theorem that the mapping $G_{D}$ parametrizes the curve $C_{D}$. The
morphism $g_{d}$ restricted to $C_{D}$ is represented in the following form.

######  Theorem 3.11.

Let $g_{d}$ be a morphism on ${\mathbb{C}}^{2}/D_{3}$ in Definition 2.4.
(1) $g_{d}(G_{D}(u))=G_{D}(T_{d}(u)),$ for any $u\in{\mathbb{C}}\ and\
d\in{\mathbb{N}}.$
(2) $g_{d}(C_{D})=C_{D},$ for any $d\in{\mathbb{N}}$.

###### Proof.

(1) By the argument similar to that used in the proof of Theorem 3.3(1), we
can prove this assertion.
(2) We can show that by Theorem 1 in [3, Chapter 3, §3], $V(p^{2}+18p-8q-27)$
is the smallest algebraic set containing $G_{D}({\mathbb{C}})$. We know from
[3, Chapter 3 §3, p.131] that $G_{D}({\mathbb{C}})$ covers all points of
$V(p^{2}+18p-8q-27)$. Then the assertion (2) follows. ∎

By the similar way to the case of $g_{d}\mid_{C_{C}}$, we have

$K(g_{d}\mid_{C_{D}})=\\{(5+4\cos\alpha,11+14\cos\alpha+2\cos^{2}\alpha):0\leq\alpha\leq
2\pi\\},\ \mbox{for}\ d\geq 2.$

Set $\gamma_{2}=K(g_{d}\mid_{C_{D}})$. The Jordan arc $\gamma_{2}$ corresponds
to the interval $[-2,2]$ for $T_{d}$. Then $g_{d}(\gamma_{2})=\gamma_{2}$ and
$g_{d}\mid_{C_{D}}$ is chaotic on $\gamma_{2}$. The end points of $\gamma_{2}$
are $(1,-1)$ and (9, 27). Then we can connect two Jordan arcs $\gamma_{1}$ and
$\gamma_{2}$ and get a Jordan curve

(3.14) $\gamma=\gamma_{1}\cup\gamma_{2}.$

The Jordan curve $\gamma$ lies on the real slice ${\mathbb{R}}^{2}$ in
${\mathbb{C}}^{2}$. See Figure 2. We will show that it is the boundary of
$K(g_{d}\mid_{{\mathbb{C}}^{2}})$ in ${\mathbb{R}}^{2}$. See Figure 1. Clearly
$g_{d}$ maps ${\mathbb{R}}^{2}$ to ${\mathbb{R}}^{2}$ and
$g_{d}(\gamma)=\gamma$ and $g_{d}\mid_{{\mathbb{R}}^{2}}$ is chaotic on
$\gamma$ for $d\geq 2$.

Figure 2. The Jordan curve $\gamma$. Figure 3. The zeros of det[J]. Figure 4.
A partition in the ($\alpha^{\prime},\beta$) space.
---

######  Theorem 3.12.

Let $\gamma$ be the the Jordan curve defined in (3.14). Then $\gamma$ is the
boundary of $K(g_{d}\mid_{{\mathbb{C}}^{2}})$ in ${\mathbb{R}}^{2}$.

###### Proof.

Set $I^{2}=[-2,2]\times[-2,2]$. Let $h(u,v)$ denote the map
$G_{{\mathbb{C}}^{2}}\mid_{I^{2}},\ i.e.,$

$h(u,v)=(1+uv+v^{2},\ \frac{1}{2}(-2+u^{2}+6v^{2}+uv(3+v^{2})),\ -2\leq
u,v\leq 2.$

By Theorem 3.3(2), we have $h(I^{2})=K(g_{d}\mid_{{\mathbb{C}}^{2}})$. We
consider $p(\alpha,\beta)$ and $q(\alpha,\beta)$ in (3.9) for
$0\leq\alpha,\beta<2\pi$. Then

(3.15) $\begin{split}p^{3}-q^{2}=4(\sin 3\alpha)^{2}(\cos 3\alpha-\cos
3\beta)^{2}\geq 0,\\\
\mbox{and}\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\\\
27+8q-18p-p^{2}=16(\sin\beta)^{2}(\cos 3\alpha-\cos\beta)^{2}\geq
0.\end{split}$

Let $D_{1}$ denote the closed domain in ${\mathbb{R}}^{2}$ bounded by
$\gamma$. Then $h(I^{2})\subset D_{1}$. We will show that $h$ maps $I^{2}$
onto $D_{1}$.

We compute the Jacobian matrix $J=\partial(p,q)/\partial(u,v)$ of $h(u,v)$ and
its determinant. Then

$\det[J]=-(u-v)(u+3v-v^{3}).$

The set of zeros of $\det[J]$ in the $(u,v)$ space is depicted in Figure 3.
The square $I^{2}$ is partitioned into six domains. Set

$D_{2}=\\{(u,v):u\geq v^{3}-3v,\ v\geq u,\ -2\leq u,v\leq 2\\}.$

We will show $D_{2}$ is mapped onto $D_{1}$ under the map $h$. Let $\Gamma$ be
the boundary of $D_{2}$. Then $h(\Gamma)=\gamma.$ As a point on $\Gamma$ moves
anticlockwise round $\Gamma$, its image under $h$ moves round $\gamma$ once
anticlockwise. Note that $J>0$ in the interior of $D_{2}$. We use a
topological argument principle ([13, Chapter 7, IV, 4]). It states that
$\nu(h(\Gamma),w)=\Sigma\nu(a_{j}),$ where $\nu(h(\Gamma),w)$ is the winding
number of a closed loop $h(\Gamma)=\gamma$ about a point $w$ in the interior
of $D_{1}$ and the sum is over $w$-points $a_{j}$ lying inside $\Gamma$ and
$\nu(a_{j})=$ the sign of det$[J[a_{j}]]$. Then we have that the mapping $h$
maps the interior of $\Gamma$ onto the interior of $\gamma$ in one-to-one
fashion. Then $h(D_{2})=D_{1}.$ Any of other five domains in $I^{2}$ is also
mapped onto $D_{1}$ under $h$. ∎

Set $\alpha^{\prime}=3\alpha$. Set
$\tilde{I}^{2}:=\\{(\alpha^{\prime},\beta):-\pi\leq\alpha^{\prime},\beta\leq
0\\}.$ The partion of the square $\tilde{I}^{2}$ depicted in Figure 4
corresponds to the partition of $I^{2}$ depicted in Figure 3.

Next we will show that $g_{d}\mid_{C_{D}}$ extends to a morphism on
${\mathbb{P}}^{2}$ as in the case of $C_{C}$. We use the same notations used
in Lemma 3.9.

######  Lemma 3.13.

Let $g_{d}(p,q)=(P_{d}(p,q),Q_{d}(p,q))$.
(1) $P_{d}(p,q)=p^{d}+H_{1}(p,q)$,
(2) $Q_{d}(p,q)\equiv 2^{d-1}q^{d}+pH_{2}(p,q)+H_{3}(p,q)\quad mod\
(p^{2}+18p-8q-27)$,
where $H_{j}(p,q)\in{\mathbb{Z}}[p,q],\ j=1,2,3,$ and the degrees of
$H_{1}(p,q)$ and $H_{3}(p,q)$ are less than or equal to $d-1$ and $H_{2}(p,q)$
is a homogeneous polynomial of degree $d-1$.

###### Proof.

(1) The proof is given in Lemma 3.9(1). (2) We consider a monomial
$p^{2n+j}q^{k}$, where $j,k,n\in{\mathbb{N}}\cup\\{0\\}$ and $0\leq j\leq 1$.
We define a function $m$ by $m(p^{2n+j}q^{k})=(27-18q+8q)^{n}p^{j}q^{k}$. We
replace any term $a_{ij}p^{i}q^{j}$ in $Q_{d}(p,q)$ with
$a_{ij}m(p^{i}q^{j})$.
(i) The monomials with the xy-degree $3d$ :
$q^{d},q^{d-2}p^{3},\dots,q^{d-2k}p^{3k}.$ Then

$m(q^{d-2}p^{3})=(27-18p+8q)pq^{d-2},\ \dots\ .$

(ii) The monomials with the xy-degree $3d-1$ :
$q^{d-1}p,q^{d-3}p^{4},q^{d-5}p^{7},\dots.$ Then

$m(q^{d-3}p^{4})=(27-18p+8q)^{2}q^{d-3},\ \dots\ .$

Then to prove that the replaced polynomial satisfies the properties of the
right hand side of (2), it suffices to prove the term $2^{d-1}q^{d}$ occurs in
$Q_{d}(p,q)$. We see in the proof of Lemma 3.9 that

$X_{1}^{3}-3X_{1}Y_{1}^{2}=\sum_{h=0}^{k}a_{h}q({\bf x})^{d-2h}p({\bf
x})^{3h}.$

We substitute $x^{2}$ for $-y^{2}$ in the both sides. Let $u$ and $v^{2}$
denote the polynomials obtained by substituting $x^{2}$ for $(iy)^{2}$ in
$X_{1}$ and $Y_{1}^{2}$, respectively. Then $u=mx^{d}$ and
$v^{2}=-n^{2}x^{2d}$, where

$m=\sum_{j=0}^{[d/2]}{}_{d}C_{2j},\quad
n=\sum_{j=0}^{[(d-1)/2]}{}_{d}C_{2j+1}.$

By the identity ${}_{d}C_{k}={}_{d-1}C_{k}+{}_{d-1}C_{k-1},$ we know
$m=n=2^{d}/2=2^{d-1}$. Hence $a_{0}=(m^{3}+3mn^{2})/4^{d}=2^{d-1}.$ ∎

The projective closure of $V(p^{2}+18p-8q-27)$ is $V(p^{2}+18pr-8qr-27r^{2})$.
Hence we define a morphism $\bar{g}_{d}$ on ${\mathbb{P}}^{2}({\mathbb{C}})$
by the similar way to (3.12).

######  Proposition 3.14.

There is a morphism
$\bar{g}_{d}:{\mathbb{P}}^{2}({\mathbb{C}})\to{\mathbb{P}}^{2}({\mathbb{C}})$
such that

$\bar{g}_{d}(V(p^{2}+18pr-8qr-27r^{2}))=V(p^{2}+18pr-8qr-27r^{2}).$

###### Proof.

The proof is similar to the argument given in Proposition 3.10. ∎

§3.4. An infinite number of invariant curves.

In Sections 3.2 and 3.3, we see two invariant curves $C_{C}$ and $C_{D}$ under
$g_{d}$. The invariability of $C_{C}$ is based on the fact that $C_{C}$
corresponds to $\alpha=0$ and $g_{d}(C_{C})$ corresponds to $d\alpha=0$. Also
the invariability of $C_{D}$ is based on the fact that $C_{D}$ corresponds to
$\beta=0$ and $g_{d}(C_{D})$ corresponds to $d\beta=0$. These properties can
be generalized. Let $C$ be the curve corresponding to
$\beta-\frac{n}{m}\alpha=0,\ (m,n\in{\mathbb{N})}$. Then $g_{d}(C)$
corresponds to $d\beta-\frac{n}{m}d\alpha=0$. Then we will show that there is
an infinite number of invariant affine algebraic curves under $g_{d}$ for any
$\alpha\in{\mathbb{N}}$.

To prove this we need some preparations. We begin to show the following lemma.

######  Lemma 3.15.

Assume that an affine algebraic set $C$ in ${\mathbb{C}}^{2}$ has a polynomial
parametric representation $x=p(t),\ y=q(t),$ where $p(t),\
q(t)\in{\mathbb{C}}[t]$. Then the polynomial parametric representation covers
all points of $C$.

We remark that by Proposition 5 in [3, Chapter 4 §5], we know that $C$ is
irreducible. Then if $p(t)$ and $q(t)$ are non-constant polynomials, then $C$
is an affine algebraic curve.

###### Proof.

Let $f_{1}=x-p(t)$ and $f_{2}=y-q(t)$. Set $I=(f_{1},f_{2})$.

Let $I_{1}=I\cap{\mathbb{C}}[x,y]$. Then $C=V(I_{1})$.

Let $p(t)=a_{n}t^{n}+a_{n-1}t^{n-1}+\cdot\cdot\cdot+a_{0},\enskip a_{n}\neq 0$

$q(t)=b_{m}t^{m}+b_{m-1}t^{m-1}+\cdot\cdot\cdot+b_{0},\enskip b_{n}\neq 0$.

We assume the lex order with $t>x>y$. Then by Corollary 4 in [3, Chapter 3
§1], we see that if $(x_{0},y_{0})\in V(I_{1})$, then there is a
$t_{0}\in{\mathbb{C}}$ such that $(t_{0},x_{0},y_{0})\in V(I)$. ∎

We consider the case that $\beta=\frac{n}{m}\alpha,\ (m,n\in{\mathbb{N})}$ in
(3.9). Set $t=2\cos\frac{\beta}{n}$. Then $2\cos 3\alpha=T_{3m}(t)$ and
$2\cos\beta=T_{n}(t)$. Then we define an affine algebraic curve $C_{m,n}$ by
the following polynomial parametric representation :

(3.16)
$\begin{split}p(t)=1+T_{3m}(t)T_{n}(t)+T_{n}(t)^{2},\qquad\qquad\qquad\qquad\\\
q(t)=\frac{1}{2}(-2+T_{3m}(t)^{2}+6T_{n}(t)^{2}+T_{3m}(t)T_{n}(t)(3+T_{n}(t)^{2}).\end{split}$

######  Lemma 3.16.

Let $C_{m,n}$ be the affine algebraic curve defined in (3.16). Then
$g_{d}(C_{m,n})=C_{m,n}$.

###### Proof.

By Lemma 3.15, we may assume that any point of $C_{m,n}$ is written as
$(p(t),\ q(t))$ in (3.16). Set $g_{d}(p(t),\ q(t))=(P(t),\ Q(t))$. Then by
Theorem 3.3(1), $(P(t),\ Q(t))=(p(T_{d}(t),\ q(T_{d}(t))$. Then this lemma
follows. ∎

######  Proposition 3.17.

There is an infinite number of invariant affine algebraic curves under $g_{d}$
for any $d\in{\mathbb{N}}$.

###### Proof.

We consider a family of curves $\\{C_{2k,3}:k=1,2,\dots\\}$. By Lemma 3.16,
each curve $C_{2k,3}$ is invariant under $g_{d}$. Then, to prove this
proposition it suffices to show that all curves $C_{2k,3}$ are different from
each other. To prove this, we consider continuous arcs
$\gamma_{2k}:=C_{2k,3}\cap K(g_{d}\mid_{{\mathbb{C}}^{2}})$. Then we will show
that the point $P=(1,-1)$ is an end point of each $\gamma_{2k}$ and that the
slopes of the tangent lines of $\gamma_{2k}$ at $P$ in
$K(g_{d}\mid_{{\mathbb{C}}^{2}})$ are different from each other. The point
$P=(1,-1)$ is a fixed point of $g_{d}$.

To prove the above facts we need some preparations. Let $T_{d}(z)$ be the
$d^{th}$ Chebyshev polynomial in one variable. Then

(3.17) $T_{d+2}(z)=zT_{d+1}(z)-T_{d}(z),\enskip T_{0}(z)=2,\enskip
T_{1}(z)=z.$

By induction, we have

(3.18) $\begin{split}T_{2k}(-2)=2,\enskip T_{2k+1}(-2)=-2,\enskip
T_{2k}(0)=(-1)^{k}\times 2,\\\ T_{2k+1}(0)=0,\enskip
T_{2k}(2)=T_{2k+1}(2)=2,\enskip\mbox{for}\enskip k\in{\mathbb{N}}.\end{split}$

Let $T^{\prime}_{d}(z)$ denote the derivative of $T_{d}(z)$. Then by (3.17)
and (3.18), we have

(3.19) $T_{2k}^{\prime}(-2)=-(2k)^{2},\ \mbox{and}\enskip
T_{2k+1}^{\prime}(-2)=(2k+1)^{2},\enskip\mbox{for}\enskip k\in{\mathbb{N}}.$

We consider a polynomial parametric representation of $C_{2k,3}$. We set
$v=2\cos\beta$. Then $2\cos 3\alpha=T_{2k}(v)$. Then we have a polynomial
parametric representation :

(3.20) $\begin{split}p(v)=1+vT_{2k}(v)+v^{2},\qquad\qquad\qquad\qquad\\\
q(v)=\frac{1}{2}(-2+T_{2k}(v)^{2}+6v^{2}+T_{2k}(v)(3v+v^{3})).\end{split}$

By Theorem 3.3,

$C_{2k,3}\cap K(g_{d}\mid_{{\mathbb{C}}^{2}})=\\{(p(v),\ q(v)):-2\leq v\leq
2\\}.$

Then we set

$\gamma_{2k}(v)=\\{(p(v),\ q(v)):-2\leq v\leq 2\\}.$

We claim that if $p(v)=1$ and $q(v)=-1,$ in $C_{2k,3}$ for $v\in{\mathbb{C}}$,
then $v=-2$.
Indeed. Suppose $p(v)=1$ in (3.20). Then $v=0$ or $T_{2k}(v)=-v$. If $v=0$,
then by (3.18), $q(0)=1\neq-1$. If $T_{2k}(v)=-v$, then
$q(v)=-1+2v^{2}-\frac{1}{2}v^{4}$. We assume that $T_{2k}(v)=-v$, $q(v)=-1$
and $v\neq 0$. Then $v=\pm 2.$ If $v=2,\enskip T_{2k}(2)=2\neq-2$. If $v=-2$,
then we have $p(-2)=1$ and $q(-2)=-1.$ This completes the proof the claim.

Hence if $\gamma_{2k}(v)=(1,-1),$ then $v=-2$ and an end point of any
$\gamma_{2k}(v)$ is the point $P=(1,-1)$.

We compute the slope of the tangent line of $\gamma_{2k}(v)$ at the point $P$.
Clearly,

$\frac{dp}{dv}=vT_{2k}^{\prime}(v)+T_{2k}(v)+2v.$

Then by (3.18) and (3.19),

$\left.\frac{dp}{dv}\right|_{v=-2}=2(2k)^{2}-2>0.$

Hence

$\left.\frac{dp}{dv}\right|_{v}>0,\ \mbox{for}\ v\in(-2-\varepsilon,\
-2+\varepsilon),\ \mbox{with a small positive number}\ \varepsilon.$

Similarly, we have that $dg/dv$ is a polynomial in $v$ and

$\left.\frac{dq}{dv}\right|_{v=-2}=5(2k)^{2}+3.$

Then

$\left.\frac{dq}{dp}\right|_{(p,q)=(1,-1)}=\
\frac{\left.\frac{dq}{dv}\right|_{v=-2}}{\left.\frac{dp}{dv}\right|_{v=-2}}\
=\frac{5}{2}+\frac{4}{4k^{2}-1}.$

Hence

$\lim_{v\to-2+0}\left.\frac{dq}{dp}\right|_{\gamma_{2k}(v)}=\frac{5}{2}+\frac{4}{4k^{2}-1}.$

The slopes of the tangent lines of the continuous arcs ${\gamma_{2k}(v)},\
(k\in{\mathbb{N}})$, at $v=-2+0$ are different from each other. ∎

##  4\. Morphisms on ${\mathbb{C}}^{3}/D_{4}$

###  4.1. The orbit variety ${\mathbb{C}}^{3}/D_{4}$ and morphisms defined
over ${\mathbb{Z}}$

We begin with the study of the orbit variety ${\mathbb{C}}^{3}/D_{4}$. We set
$x:=x_{1},\quad y:=x_{2}$ $z:=x_{3}$. By Algorithm 2.5.14 in [20], we know
that $\\{x^{2}+y^{2},\ z^{2},\ z(x^{2}-y^{2}),\ x^{2}y^{2}\\}$ is a
fundamental system of invariants of ${\mathbb{C}}[x,y,z]^{D_{4}}$. But, to
simplify the syzygy ideal we select another fundamental system of invariants
$\\{x^{2}+y^{2},\ z^{2},\ z(x^{2}-y^{2}),\ (x^{2}-y^{2})^{2}\\}$. Set

(4.1) $p=x^{2}+y^{2},\ q=z^{2},\ r=z(x^{2}-y^{2}),\ s=(x^{2}-y^{2})^{2},$

and $F=(p,q,r,s).$

Then by Proposition 3 in [3, Chapter 7 §4], we see that the syzygy ideal
$I_{F}$ for $F$ is the ideal $(r^{2}-qs)$. By Theorem 2.3, we have
${\mathbb{C}}^{3}/D_{4}\simeq V_{F}=V(I_{F})$. Then the orbit variety $V_{F}$
can be embedded into ${\mathbb{C}}^{4}$ as an affine subvariety $V(r^{2}-qs)$.
The subvariety $V(r^{2}-qs)$ is a quadric hypersurface in ${\mathbb{C}}^{4}$,
denoted by $X_{Q}$. The singular locus of $X_{Q}$ is the set
$\\{(p,0,0,0):p\in{\mathbb{C}}\\}$.

Now we consider a morphism $g_{2}$ on $X_{Q}$. By [23, p.198], we have

$f_{2}(x,y,z)=(x^{2}-y^{2}-2z,\ 2xy,\ z^{2}-2x^{2}-2y^{2}+2).$

Using the generators $\\{p,q,r,s\\}$ in (4.1), we can compute $g_{2}$ :

(4.2)
$g_{2}(p,q,r,s)=(p^{2}+4q-4r,(q-2p+2)^{2},(q-2p+2)(2s-p^{2}-4r+4q),(2s-p^{2}-4r+4q)^{2}).$

The morphism $g_{2}$ is defined over ${\mathbb{Z}}$. This fact holds for any
morphisms $g_{d}$ on $X_{Q}$.

######  Theorem 4.1.

Let $g_{d}$ be a morphism on $X_{Q}$ as in Definition 2.4. Then any morphism
$g_{d}$ from $X_{Q}$ to $X_{Q}$ is defined over ${\mathbb{Z}}$.

###### Proof.

We use recurrence formulas of $z_{1}^{(d)}$ and $z_{2}^{(d)}$ (See [11] and
[23, p.201]) :

(4.3) $\begin{split}z_{1}^{(k)}=z_{1}\cdot z_{1}^{(k-1)}-z_{2}\cdot
z_{1}^{(k-2)}+z_{3}\cdot z_{1}^{(k-3)}-z_{1}^{(k-4)},\\\ z_{1}^{(0)}=4,\
z_{1}^{(1)}=z_{1},\ z_{1}^{(2)}=z_{1}^{2}-2z_{2},\
z_{1}^{(3)}=z_{1}^{3}-3z_{1}z_{2}+z_{3},\end{split}$

(4.4) $\begin{split}z_{2}^{(k)}=z_{2}\cdot z_{2}^{(k-1)}-(z_{1}z_{3}-1)\cdot
z_{2}^{(k-2)}+(z_{1}^{2}-2z_{2}+z_{3}^{2})\cdot z_{2}^{(k-3)}\\\
-(z_{1}z_{3}-1)\cdot z_{2}^{(k-4)}+z_{2}\cdot z_{2}^{(k-5)}-z_{2}^{(k-6)},\\\
z_{2}^{(-2)}=z_{2}^{2}-2z_{1}z_{2}+2,\ z_{2}^{(-1)}=z_{2},\ z_{2}^{(0)}=6,\
z_{2}^{(1)}=z_{2},\\\ z_{2}^{(2)}=z_{2}^{(-2)},\
z_{2}^{(3)}=z_{2}^{3}-3z_{1}z_{2}z_{3}+3z_{3}^{2}+3z_{1}^{2}-3z_{2}.\end{split}$

We assume that $z_{1}=\bar{z}_{3}$ and $z_{2}\in{\mathbb{R}}$. Let
$z_{1}=x+iy$ and $\ z_{2}=z$ where $x,y,z$ are real variables. We set

(4.5) $\begin{split}z_{1}^{(d)}(x+iy,\ z,\ x-iy)=X_{d}(x,\ y,\ z)+iY_{d}(x,\
y,\ z),\\\ \quad z_{2}^{(d)}(x+iy,\ z,\ x-iy)=Z_{d}(x,\ y,\ z),\\\
\mbox{where}\ X_{d},\ Y_{d},\ Z_{d}\in{\mathbb{R}}[x,\ y,\ z].\end{split}$

By Proposition 2.6 and (4.2), to prove this theorem it suffices to consider
only the case $d$ is odd. We assume $d$ is a positive old integer. Then it is
enough to prove the following lemma. Set $R={\mathbb{Z}}[x^{2}+y^{2},\ z^{2},\
z(x^{2}-y^{2}),\ (x^{2}-y^{2})^{2}]$.

######  Lemma 4.2.

Let $d$ be a positive odd integer.
(a) $X_{d}^{2}+Y_{d}^{2}\in R$. (b) $Z_{d}^{2}\in R$. (c)
$Z_{d}(X_{d}^{2}-Y_{d}^{2})\in R$. (d) $(X_{d}^{2}-Y_{d}^{2})^{2}\in R$.

###### Proof of (b).

Using (4.4), we can prove the following properties (4.6) by induction:

(4.6) $\begin{split}\mbox{if}\ k\ \mbox{is odd},\quad
Z_{k}=c_{1}(x^{2}-y^{2})+c_{2}z,\\\ \mbox{if}\ k\ \mbox{is even},\quad
Z_{k}=c_{3}z(x^{2}-y^{2})+c_{4},\\\ \mbox{where}\
c_{1},c_{2},c_{3},c_{4}\in{\mathbb{Z}}[x^{2}+y^{2},z^{2},(x^{2}-y^{2})^{2}].\end{split}$

Then the assertion (b) follows. ∎

To prove the assertions (a), (c) and (d), it suffices to show the following
lemma.

######  Lemma 4.3.

Let $d$ be a positive odd integer.
(a) $X_{d}^{2}+Y_{d}^{2}\in R$. (e) $X_{d}^{2}-Y_{d}^{2}\in(x^{2}-y^{2})R+zR$.

The assertion (c) follows from (4.6) and (e). The assertion (d) follows from
(e). The proof of Lemma 4.3 needs some preparations and will be given after
Lemma 4.7.

We begin to show the following lemma.

######  Lemma 4.4.

Let $d$ be a positive odd integer. (1) Let
$\alpha_{abc}z_{1}^{a}z_{2}^{b}z_{3}^{c}$ be a term in
$z_{1}^{(d)}(z_{1},z_{2},z_{3})$. Then $a+2b+3c\equiv d$ (mod 4) and
$\alpha_{abc}\in{\mathbb{Z}}$.
(2) The term $z_{1}^{d}$ occurs in $z_{1}^{(d)}$ and it is the only term with
degree $d$.

###### Proof.

Using (4.3) , we can prove this lemma by induction. ∎

We define the weighted degree of $z_{1}^{a}z_{2}^{b}z_{3}^{c}$ to be
$a+2b+3c$. Set $R_{0}={\mathbb{Z}}[x^{2}+y^{2},z^{2}]$. Since
$z_{1}z_{3}=x^{2}+y^{2}$ and $z_{2}^{2}=z^{2}$, we pick out monomials
$z_{1}z_{3}$ and $z_{2}^{2}$ in $z_{1}^{(d)}$ as coefficients belonging to
$R_{0}$. Then we may view the polynomial $z_{1}^{(d)}$ as a polynomial
$\tilde{z}_{1}^{(d)}$ whose terms are written as

(4.7) $\alpha z_{1}^{n_{1}}\ \mbox{or}\ \beta z_{1}^{n_{2}}z_{2}\ \mbox{or}\
\gamma z_{3}^{n_{3}}\ \mbox{or}\ \delta z_{3}^{n_{4}}z_{2},\ \mbox{where}\
\alpha,\beta,\gamma,\delta\ \in R_{0},\ n_{j}\in{\mathbb{N}},\ (j=1,2,3,4).$

The weighted degrees of $z_{1}z_{3}$ and $z_{2}^{2}$ are 4. Then by Lemma
4.4(1), we see that if $n_{j}^{,}$s are the degrees of monomials in (4.7),
then

(4.8) $n_{1}\equiv n_{2}+2\equiv 3{n_{3}}\equiv 3{n_{4}}+2\equiv d\
(\mbox{mod}\ 4).$ $\mbox{Let}\quad\tilde{z}_{1}^{(d)}=\sum
a(1,n_{1})z_{1}^{n_{1}}+\sum a(2,n_{2})z_{1}^{n_{2}}z_{2}+\sum
a(3,n_{3})z_{3}^{n_{3}}+\sum a(4,n_{4})z_{1}^{n_{4}}z_{2},$

where the sums are over $n_{j}\in I_{j}\quad j=1,2,3,4.$ Here $a(j,n_{j})\in
R_{0},\ j=1,2,3,4.$

Now we assume that $z_{1}=x+iy,\ z_{2}=z,\ z_{3}=x-iy$. Let
$z_{1}^{m}=U_{m}+iV_{m},\ U_{m},\ V_{m}\in{\mathbb{Z}}[x,y]$. Then
$z_{3}^{m}=U_{m}-iV_{m}$. Hence

$\tilde{z}_{1}^{(d)}(x+iy,z,x-iy)=\sum a(1,n_{1})(U_{n_{1}}+iV_{n_{1}})+\sum
a(2,n_{2})(U_{n_{2}}+iV_{n_{2}})z$ $+\sum
a(3,n_{3})(U_{n_{3}}-iV_{n_{3}})+\sum a(4,n_{4})(U_{n_{4}}-iV_{n_{4}})z.$

Then by (4.5),

$X_{d}=\sum a(1,n_{1})U_{n_{1}}+\sum a(2,n_{2})U_{n_{2}}z+\sum
a(3,n_{3})U_{n_{3}}+\sum a(4,n_{4})U_{n_{4}}z,$ $Y_{d}=\sum
a(1,n_{1})V_{n_{1}}+\sum a(2,n_{2})V_{n_{2}}z-\sum a(3,n_{3})V_{n_{3}}-\sum
a(4,n_{4})V_{n_{4}}z.$

Hence

$X_{d}^{2}=\sum_{1\leq j,k\leq 4}(\sum a(j,m_{j})U_{m_{j}}z^{\lambda(j)})(\sum
a(k,n_{k})U_{n_{k}}z^{\lambda(k)}),$ $Y_{d}^{2}=\sum_{1\leq j,k\leq
4}\varepsilon_{jk}(\sum a(j,m_{j})V_{m_{j}}z^{\lambda(j)})(\sum
a(k,n_{k})V_{n_{k}}z^{\lambda(k)}),$

where if $j=2$ or $4$, then $\lambda(j)=1$, otherwise $\lambda(j)=0$, and if
$\\{j,k\\}=\\{1,3\\}$ or $\\{1,4\\}$ or $\\{2,3\\}$ or $\\{2,4\\}$, then
$\varepsilon_{jk}=-1$, otherwise $\varepsilon_{jk}=1$. Set

(4.9)
$s(\pm,j,k,m_{j},n_{k})=U_{m_{j}}U_{n_{k}}\pm\varepsilon_{jk}V_{m_{j}}V_{n_{k}},$

and

(4.10) $s(\pm,j,k)=\sum
a(j,m_{j})a(k,n_{k})s(\pm,j,k,m_{j},n_{k})z^{\lambda(j)+\lambda(k)},$

Then

(4.11) $X_{d}^{2}\pm Y_{d}^{2}=\sum_{1\leq j,k\leq 4}s(\pm,j,k).$

First, we compute $s(\pm,j,k,m_{j},n_{k})$. Note that

(4.12) $Re(z_{1}^{m}z_{3}^{n})=U_{m}U_{n}+V_{m}V_{n},\quad
Re(z_{1}^{m+n})=U_{m}U_{n}-V_{m}V_{n}.$

By (4.9) and (4.12), we have

(4.13)
$s(+,j,k,m_{j},n_{k})=\left\\{\begin{array}[]{ll}\displaystyle{Re(z_{1}^{m_{j}}z_{3}^{n_{k}})}&\quad\
\mbox{if}\ \varepsilon_{jk}=1\\\ \displaystyle{Re(z_{1}^{m_{j}+n_{k}})}&\quad\
\mbox{if}\ \varepsilon_{jk}=-1\end{array}\right.$ (4.14)
$s(-,j,k,m_{j},n_{k})=\left\\{\begin{array}[]{ll}\displaystyle{Re(z_{1}^{m_{j}+n_{k}})}&\quad\
\mbox{if}\ \varepsilon_{jk}=1\\\
\displaystyle{Re(z_{1}^{m_{j}}z_{3}^{n_{k}})}&\quad\ \mbox{if}\
\varepsilon_{jk}=-1\end{array}\right.$

Note that $Re(z_{1}^{m}z_{3}^{n})=Re(z_{1}^{n}z_{3}^{m}).$ By (4.13) and
(4.14), we have

$s(+,j,k,m_{j},n_{k})=c_{1}Re(z_{1}^{m_{j}-\varepsilon_{j_{k}}n_{k}})\ \
\mbox{or}\ c_{2}Re(z_{1}^{n_{k}-\varepsilon_{j_{k}}m_{k}}),$
$s(-,j,k,m_{j},n_{k})=c_{3}Re(z_{1}^{m_{j}+\varepsilon_{j_{k}}n_{k}})\ \
\mbox{or}\ c_{4}Re(z_{1}^{n_{k}+\varepsilon_{j_{k}}m_{k}}),$

where $c_{j}\in{\mathbb{Z}}[x^{2}+y^{2}],\ j=1,2,3,4.$

######  Lemma 4.5.

Let $d$ be a positive odd integer. Then :
(1) If $\mid j-k\mid$ is even, then $m_{j}+\varepsilon_{jk}n_{k}\equiv 2$,
$m_{j}-\varepsilon_{jk}n_{k}\equiv 0$ (mod 4),
$n_{k}+\varepsilon_{jk}m_{j}\equiv 2$, $n_{k}-\varepsilon_{jk}m_{j}\equiv 0$
(mod 4).
(2) If $\mid j-k\mid$ is odd, then $m_{j}+\varepsilon_{jk}n_{k}\equiv 0$,
$m_{j}-\varepsilon_{jk}n_{k}\equiv 2$ (mod 4),
$n_{k}+\varepsilon_{jk}m_{j}\equiv 0$, $n_{k}-\varepsilon_{jk}m_{j}\equiv 2$
(mod 4).

###### Proof.

From (4.8) and the definition of $\varepsilon_{jk}$ we have this lemma. ∎

######  Lemma 4.6.

Let $R_{1}={\mathbb{Z}}[x^{2}+y^{2},(x^{2}-y^{2})^{2}]$.
(1) If $m\equiv 0$ (mod 4), then $Re(z_{1}^{m})\in R_{1}$.
(2) If $m\equiv 2$ (mod 4), then $Re(z_{1}^{m})\in(x^{2}-y^{2})R_{1}$.

###### Proof.

(1) Clearly, $z_{1}^{4h}=(z_{1}^{2})^{2h}=(x^{2}-y^{2}+2xyi)^{2h},\
h\in{\mathbb{N}}$. Hence

$Re(z_{1}^{4h})=\sum_{j=0}^{h}{}_{2h}C_{2j}(x^{2}-y^{2})^{2h-2j}(2xyi)^{2j}$
$=\sum_{j=0}^{h}{}_{2h}C_{2j}((x^{2}-y^{2})^{2})^{h-j}(-(x^{2}+y^{2})^{2}+(x^{2}-y^{2})^{2})^{j}\in
R_{1}.$

(2) Clearly, $z_{1}^{4h+2}=(x^{2}-y^{2}+2xyi)^{2h+1}$. Hence

$Re(z_{1}^{4h+2})=\sum_{j=0}^{h}{}_{2h+1}C_{2j}(x^{2}-y^{2})^{2h+1-2j}(2xyi)^{2j}$
$=\sum_{j=0}^{h}{}_{2h+1}C_{2j}(x^{2}-y^{2})(x^{2}-y^{2})^{2(h-j)}(-4x^{2}y^{2})^{j}\in(x^{2}-y^{2})R_{1}.$

∎

Hence from Lemmas 4.5 and 4.6, we have the following.

######  Lemma 4.7.

(1) If $\mid j-k\mid$ is even, then

$s(+,j,k,m_{j},n_{k})\in R_{1}\ \mbox{and}\
s(-,j,k,m_{j},n_{k})\in(x^{2}-y^{2})R_{1}.$

(2) If $\mid j-k\mid$ is odd, then

$s(+,j,k,m_{j},n_{k})\in(x^{2}-y^{2})R_{1}\ \mbox{and}\
s(-,j,k,m_{j},n_{k})\in R_{1}.$

Now we can prove Lemma 4.3.

###### proof of Lemma 4.3..

Note that if $\mid j-k\mid$ is even, then $\lambda(j)+\lambda(k)=0\ \mbox{or}\
2,$ and if $\mid j-k\mid$ is odd, then $\lambda(j)+\lambda(k)=1$. Then by
(4.10) and Lemma 4.7, we have

$\begin{split}s(+,j,k)\in R,\hskip 163.60333pt\\\
s(-,j,k)\in\left\\{\begin{array}[]{ll}\displaystyle{(x^{2}-y^{2})R}&\quad\
\mbox{if}\ \mid j-k\mid\ \mbox{is even},\\\ \displaystyle{\quad zR}&\quad\
\mbox{if}\ \mid j-k\mid\ \mbox{is odd}.\end{array}\right.\end{split}$
$\mbox{Hence}\quad X_{d}^{2}+Y_{d}^{2}\in R\ \mbox{and}\
X_{d}^{2}-Y_{d}^{2}\in zR+(x^{2}-y^{2})R.\qquad\qquad\qquad$

∎

Then Theorem 4.1 is proved. ∎

We remark that the original basis $\\{p,q,r,s^{\prime}\\}$ with
$s^{\prime}=x^{2}y^{2}$ satisfies

${\mathbb{R}}[x,y,z]^{D_{4}}\cap{\mathbb{Z}}[x,y,z]={\mathbb{Z}}[p,q,r,s^{\prime}],$

while $\\{p,q,r,s\\}$ does not.

Apart from the fundamental system $\\{p,q,r,s\\}$, there are many fundamental
systems of invariants of ${\mathbb{C}}[x,y,z]^{D_{4}}$. Let $F^{\prime}$ be
the ideal generated by such a fundamental system. Then there are the
corresponding affine algebraic varieties $V_{F^{\prime}}$ and morphisms
$g^{\prime}_{d}$ on $V_{F^{\prime}}$. We will show in Section 4.7 that there
is an isomorphism $\varphi:X_{Q}\to V_{F^{\prime}}\ \mbox{such that}\
\varphi\circ g_{d}=g^{\prime}_{d}\circ\varphi.$

We consider the dynamics of $g_{d}$ on $X_{Q}$. As in Section 3.1, we
introduce a parametrization of $X_{Q}$ which is convenient to understand the
dynamics of $g_{d}$. From (2.13), we have

(4.15) $\begin{array}[]{ccc}\\{t_{1},t_{2},t_{3},t_{4}\\}&\smash{\mathop{\hbox
to36.98866pt{\rightarrowfill}}\limits}&\\{t_{1}^{d},t_{2}^{d},t_{3}^{d},t_{4}^{d}\\}\\\
\Big{\updownarrow}\hbox
to0.0pt{\hss$\vbox{\hbox{$\scriptstyle\,$}}$}&&\Big{\updownarrow}\hbox
to0.0pt{\hss$\vbox{\hbox{$\scriptstyle\,$}}$}\\\
(z_{1},z_{2},z_{3})&\smash{\mathop{\hbox
to36.98866pt{\rightarrowfill}}\limits^{T_{d}}}&(z_{1}^{(d)},z_{2}^{(d)},z_{3}^{(d)})\\\
\Big{\downarrow}\hbox to0.0pt{$\vbox{\hbox{$\scriptstyle
M_{3}^{-1}\,$}}$\hss}&&\Big{\downarrow}\hbox
to0.0pt{$\vbox{\hbox{$\scriptstyle M_{3}^{-1}\,$}}$\hss}\\\
(x,y,z)&\smash{\mathop{\hbox
to36.98866pt{\rightarrowfill}}\limits^{f_{d}}}&(X,Y,Z)\\\
\Big{\downarrow}\hbox
to0.0pt{$\vbox{\hbox{$\scriptstyle\tilde{F}\,$}}$\hss}&&\Big{\downarrow}\hbox
to0.0pt{$\vbox{\hbox{$\scriptstyle\tilde{F}\,$}}$\hss}\\\
(p,q,r,s)&\smash{\mathop{\hbox
to36.98866pt{\rightarrowfill}}\limits^{g_{d}}}&(P_{d},Q_{d},R_{d},S_{d}).\end{array}$

Set

(4.16) $t_{1}=e^{i(\alpha+\gamma)},\ t_{2}=e^{i(\alpha-\gamma)},\
t_{3}=e^{i\beta},\ t_{4}=e^{-i(2\alpha+\beta)},\quad\mbox{for}\
\alpha,\beta,\gamma\in{\mathbb{C}}.$

Then

$z_{1}=e^{i(\alpha+\gamma)}+e^{i(\alpha-\gamma)}+e^{i\beta}+e^{-i(2\alpha+\beta)},\
z_{2}=2(\cos 2\alpha+\cos(\alpha+\beta+\gamma)+\cos(\alpha+\beta-\gamma)),$
$z_{3}=e^{-i(\alpha+\gamma)}+e^{-i(\alpha-\gamma)}+e^{-i\beta}+e^{i(2\alpha+\beta)}.$

Hence by (2.8) we have

(4.17)
$\begin{split}x=\cos(\alpha+\gamma)+\cos(\alpha-\gamma)+\cos\beta+\cos(2\alpha+\beta),\qquad\qquad\\\
y=\sin(\alpha+\gamma)+\sin(\alpha-\gamma)+\sin\beta-\sin(2\alpha+\beta),\qquad\qquad\\\
z=2\cos
2\alpha+(2\cos(\alpha+\beta))(2\cos\gamma).\qquad\qquad\qquad\qquad\qquad\end{split}$

Therefore, using trigonometrical identities we have

(4.18)
$\begin{split}p(\alpha,\beta,\gamma)=(2\cos\gamma)^{2}+(2\cos\gamma)(2\cos
2\alpha)(2\cos(\alpha+\beta))+(2\cos(\alpha+\beta))^{2},\qquad\\\
q(\alpha,\beta,\gamma)=((2\cos(\alpha+\beta))(2\cos\gamma)+(2\cos
2\alpha))^{2},\qquad\qquad\qquad\qquad\qquad\qquad\\\
r(\alpha,\beta,\gamma)=((2\cos(\alpha+\beta))(2\cos\gamma)+(2\cos
2\alpha))\times\quad\qquad\qquad\qquad\qquad\qquad\quad\\\
(2(2\cos(\alpha+\beta))(2\cos\gamma)+\frac{1}{2}(2\cos
2\alpha)((2\cos(\alpha+\beta))^{2}+(2\cos\gamma)^{2})),\\\
s(\alpha,\beta,\gamma)=(2(2\cos(\alpha+\beta))(2\cos\gamma)+\frac{1}{2}(2\cos
2\alpha)((2\cos(\alpha+\beta))^{2}+(2\cos\gamma)^{2}))^{2}.\end{split}$

Note that all the identities which we use to get (4.18) are valid for complex
variables $\alpha,\beta$ and $\gamma$. Then setting $u=2\cos 2\alpha,\
v=2\cos(\alpha+\beta)$ and $w=2\cos\gamma$, we have

(4.19) $\begin{split}p(u,v,w)=v^{2}+uvw+w^{2},\qquad\qquad\\\
q(u,v,w)=(u+vw)^{2},\qquad\qquad\\\
r(u,v,w)=(u+vw)(2vw+\frac{1}{2}u(v^{2}+w^{2})),\\\
s(u,v,w)=(2vw+\frac{1}{2}u(v^{2}+w^{2}))^{2}.\qquad\end{split}$

Let $G_{X}$ be a mapping from ${\mathbb{C}}^{3}$ to ${\mathbb{C}}^{4}$ defined
by

$G_{X}(u,v,w)=(v^{2}+uvw+w^{2},\ (u+vw)^{2},\
(u+vw)(2vw+\frac{1}{2}u(v^{2}+w^{2})),\ (2vw+\frac{1}{2}u(v^{2}+w^{2}))^{2}).$

We show that the mapping $G_{X}(u,v,w)$ is a polynomial parametric
representation of $X_{Q}$. Indeed. By the definition of $G_{X}$ and (4.19), we
have $r(u,v,w)^{2}=q(u,v,w)s(u,v,w).$ Then for any
$(u,v,w)\in{\mathbb{C}}^{3},\ G_{X}(u,v,w)\in V(r^{2}-qs)=X_{Q}.$ By an
argument similar to that use in the proof of Proposition 3.2, then we can show
that $G_{X}({\mathbb{C}}^{3})$ covers all points of $X_{Q}$.

Using this parametrization we describe the dynamics of $g_{d}$ on $X_{Q}$.

######  Theorem 4.8.

Let $g_{d}$ be a morphism on $X_{Q}$ in Definition 2.4.

$(1)\ g_{d}(G_{X}(u,v,w))=G_{X}(T_{d}(u),T_{d}(v),T_{d}(w))\ \mbox{for
any}\quad u,v,w\in{\mathbb{C}}.$ $(2)\ K(g_{d}\mid_{X_{Q}})=\\{G_{X}(2\cos
2\alpha,2\cos(\alpha+\beta),2\cos\gamma):\ 0\leq\alpha,\beta,\gamma<2\pi\\}\
\mbox{for}\quad d\geq 2.$

###### Proof.

(1) The proof of the assertion (1) is similar to the proof of Theorem 3.3 (1).
Then it is omitted.
(2) The proof of the assertion (2) is similar to the proof of Theorem 3.3 (2).
Let $u=t_{1}+t_{1}^{-1}$, $v=t_{2}+t_{2}^{-1}$ and $w=t_{3}+t_{3}^{-1}$, with
$t_{j}=r_{j}e^{i\theta_{j}},\ r_{j}\geq 1,\ \theta_{j}\in{\mathbb{R}},$ for
$j=1,2,3.$ Then $T_{d}(u)=t_{1}^{d}+t_{1}^{{-d}}$ and
$T_{d}^{n}(u)=t_{1}^{d^{n}}+t_{1}^{{-d}^{n}}$. Let $G_{X}(u,v,w)=(p(u,v,w),\
q(u,v,w),\ r(u,v,w),\ s(u,v,w))$. Set

$g_{d}^{n}(p(u,v,w),\ q(u,v,w),\ r(u,v,w),\ s(u,v,w))=(p_{n}(u,v,w),\
q_{n}(u,v,w),\ r_{n}(u,v,w),\ s_{n}(u,v,w)).$

We assume that $\\{q_{n}(u,v,w):n=1,2,\dots\\}$ is bounded. Then we have

(4.20) $r_{1}=r_{2}r_{3}.$

Besides, we assume that $\\{p_{n}(u,v,w):n=1,2,\dots\\}$ is bounded. We
consider the set $Dom(p)=max\\{r_{2}^{2},\ r_{1}r_{2}r_{3},\ r_{3}^{2}\\}.$ If
$Dom(p)$ is a singleton, then $\lim_{n\to\infty}p_{n}=\infty.$ If
$r_{2}=r_{3}$, then we have $r_{1}=r_{3}^{2}$ by (4.20). If $r_{2}=r_{3}>1$,
then $r_{1}r_{2}r_{3}>r_{2}^{2}=r_{3}^{2}$. We suppose that
$r_{1}r_{2}r_{3}=r_{2}^{2}$. Then by (4.20) we have $r_{1}=r_{2}$ and
$r_{3}=1$. If $r_{1}=r_{2}>1$, then $\lim_{n\to\infty}s_{n}=\infty.$ Hence if
$\\{g_{d}^{n}(G_{X}(u,v,w)):n=1,2,\dots\\}$ is bounded, then
$r_{1}=r_{2}=r_{3}=1$. In the case $r_{1}r_{2}r_{3}=r_{3}^{2}$, we get also
$r_{1}=r_{2}=r_{3}=1$. ∎

###  4.2. The branch loci of $g_{2}(p,q,r,s)$ and $\tilde{F}(p,q,r,s)$.

As in Section 3.1, we consider the branch locus of $g_{2}(p,q,r,s)$. We will
show that the branch locus of $g_{2}$ consists of three affine algebraic
varieties $L_{1}$, $S_{P}$ and $S_{A}$;

(4.21) $\begin{split}L_{1}:=V(q,r,s),\ S_{P}:=V(p^{2}-s,qs-r^{2}),\
S_{A}:=V(qs-r^{2},A_{h}),\ \mbox{where}\hskip 99.58464pt\\\
A_{h}(p,q,r,s):=256-192p+48p^{2}-4p^{3}-128q-80pq+p^{2}q+16q^{2}+288r+36pr-8qr-108s.\end{split}$

The algebraic sets $S_{P}$ and $S_{A}$ will be proved to be affine algebraic
surfaces.

Recall that $g_{2}$ is a morphism from $X_{Q}$ to $X_{Q}$. We may view it as a
morphism from $X$ to $Y$, where the coordinate rings ${\mathbb{C}}[X]$ and
${\mathbb{C}}[Y]$ are given by ${\mathbb{C}}[p,q,r,s]/(qs-r^{2})$ and
${\mathbb{C}}[a,b,c,d]/(bd-c^{2})$, respectively. We consider the inverse
images of $(a,b,c,d)\in Y$ under $g_{2}$. We suppose that
$g_{2}(p,q,r,s)=(a,b,c,d)$. A polynomial parametric representation of
$V(bd-c^{2})$ is given by $b=u^{2},\ c=uv$ and $d=v^{2}$. So by (4.2), we
consider equations :

(4.22) $\left\\{\begin{split}p^{2}+4q-4r-a=0,\qquad\qquad\\\
q-2p+2-u=0,\qquad\qquad\\\ (q-2p+2)(2s-p^{2}+4q-4r)-uv=0,\\\
2s-p^{2}+4q-4r-v=0,\qquad\end{split}\right.$ (4.23)
$\left\\{\begin{split}p^{2}+4q-4r-a=0,\qquad\qquad\\\
q-2p+2+u=0,\qquad\qquad\\\ (q-2p+2)(2s-p^{2}+4q-4r)-uv=0,\\\
2s-p^{2}+4q-4r+v=0.\qquad\end{split}\right.$

Let $I_{+}$ be the ideal generated by the four polynomials in the left hand
sides of (4.22). Substituting -u and -v for u and v in $I_{+}$, we have
another ideal $I_{-}$ for (4.23). We compute Gr$\ddot{\mbox{o}}$bner basis
$\mathcal{G}_{\pm}$ for the ideals $I_{\pm}+(qs-r^{2})$, using lex order with
$q>r>s>p>a>u>v:$

$\mathcal{G}_{+}=\\{64+a^{2}-128p+80p^{2}-2ap^{2}-16p^{3}+p^{4}-64u+64pu-8p^{2}u+16u^{2}+16v-16pv-8uv,$
$a-2p^{2}+2s-v,2-2p+q-u,8+a-8p-p^{2}+4r-4u\\},$
$\mathcal{G}_{-}=\\{64+a^{2}-128p+80p^{2}-2ap^{2}-16p^{3}+p^{4}+64u-64pu+8p^{2}u+16u^{2}-16v+16pv-8uv,$
$a-2p^{2}+2s+v,2-2p+q+u,8+a-8p-p^{2}+4r+4u\\}.$

Let

(4.24) $\left\\{\begin{split}a-2p^{2}+2s-v=0,\qquad\qquad\\\
2-2p+q-u=0,\qquad\qquad\\\ 8+a-8p-p^{2}+4r-4u=0,\qquad\end{split}\right.$
(4.25) $\left\\{\begin{split}a-2p^{2}+2s+v=0,\qquad\qquad\\\
2-2p+q+u=0,\qquad\qquad\\\ 8+a-8p-p^{2}+4r+4u=0.\qquad\end{split}\right.$

We set

$h_{\pm}=\mathcal{G}_{\pm}\cap{\mathbb{C}}[a,p,u,v].$

Then

$h_{+}=64+a^{2}-128p+80p^{2}-2ap^{2}-16p^{3}+p^{4}-64u+64pu-8p^{2}u+16u^{2}+16v-16pv-8uv,$
$h_{-}=64+a^{2}-128p+80p^{2}-2ap^{2}-16p^{3}+p^{4}+64u-64pu+8p^{2}u+16u^{2}-16v+16pv-8uv.$

The product $h_{+}h_{-}$ is invariant under the action $\sigma:u\mapsto-u,\
v\mapsto-v$. Hence, if a monomial $u^{i}v^{j}$ occurs in $h_{+}h_{-}$, then
$i+j$ is even. Then $h_{+}h_{-}$ is reduced to a polynomial in
${\mathbb{Z}}[a,b,c,d,p]$. Thus $h_{+}h_{-}$ is a monic polynomial in $p$ of
degree 8 with coefficients in ${\mathbb{C}}(Y)$, denoted by $H(p)$.

######  Proposition 4.9.

The morphism $g_{2}$ is a finite morphism.

###### Proof.

Since $g_{2}:X\to Y$ is a surjective morphism, we may regard ${\mathbb{C}}[Y]$
as a subring of ${\mathbb{C}}[X]$ . Then it is enough to verify that $p,q,r$
and $s$ are integral over ${\mathbb{C}}[Y]$. Considering $H(p)$, we know that
$p$ is integral over ${\mathbb{C}}[Y]$. The function $q$ is integral over
${\mathbb{C}}[Y][p]$ and $r$ is integral over ${\mathbb{C}}[Y][p,q]$ and $s$
is integral over ${\mathbb{C}}[Y][p,q,r]$. Thus $p,q,r$ and $s$ are integral
over ${\mathbb{C}}[Y]$ by the transitivity property of integral dependence
(See [1, Corollary 5.4]). ∎

Then by Proposition 2.9, we have the following.

######  Proposition 4.10.

Let $g_{2}:X_{Q}\to X_{Q}$ be the morphism in (4.2). Then deg $g_{2}=8$.

We consider the ramification points of $g_{2}$. We study multiple points of
the inverse images $(p,q,r,s)$ of $(a,b,c,d)$ under $g_{2}$. We assume that
the values of $a,b,c,d$ and $u=\sqrt{b}$, $v=\sqrt{d}$ are given. Let $p_{+}$
and $p_{-}$ are zeros of $h_{+}$ and $h_{-}$, respectively. Then for $p_{+}$,
there is the unique solution $(p_{+},q_{+},r_{+},s_{+})$ of (4.24) and for
$p_{-}$, there is the unique solution $(p_{-},q_{-},r_{-},s_{-})$ of (4.25).
If there are multiple points of the inverse image $(p,q,r,s)$, then $H(p)$
must have a multiple zero. Then there are three cases :(i) $h_{+}$ has a
multiple zero, (ii) $h_{-}$ has a multiple zero, (iii) $h_{+}$ and $h_{-}$
have a common zero.

Cases (i) and (ii). We compute the discriminants of $h_{+}$ and $h_{-}$ with
respect to $p$ :

$disc(h_{+})=-16384(a+v)^{2}(-256+192a-48a^{2}+4a^{3}+128u^{2}$
$\qquad+80au^{2}-a^{2}u^{2}-16u^{4}-288uv-36auv+8u^{3}v+108v^{2}),$

$disc(h_{-})=-16384(a-v)^{2}(-256+192a-48a^{2}+4a^{3}+128u^{2}$
$\qquad+80au^{2}-a^{2}u^{2}-16u^{4}-288uv-36auv+8u^{3}v+108v^{2}).$

Clearly,

$V(v+a)\cup V(v-a)=V(a^{2}-d),$

and

$V(-256+192a-48a^{2}+4a^{3}+128u^{2}+80au^{2}-a^{2}u^{2}-16u^{4}-288uv-36auv+8u^{3}v+108v^{2})$
$=V(256-192a+48a^{2}-4a^{3}-128b-80ab+a^{2}b+16b^{2}+288c+36ac-8bc-108d).$

Then we have two algebraic sets $V(a^{2}-d,bd-c^{2})$ and

$V(256-192a+48a^{2}-4a^{3}-128b-80ab+a^{2}b+16b^{2}+288c+36ac-8bc-108d,bd-c^{2}).$

These are $S_{P}$ and $S_{A}$ in (4.21). These will be proved to be
irreducible and 2-dimensional varieties in Sections 4.4 and 4.5.

Case (iii). We assume that
$(p_{+},q_{+},r_{+},s_{+})=(p_{-},q_{-},r_{-},s_{-})$. Then from (4.24) and
(4.25), we have $u=v=0$. The equality $u=v=0$ is equivalent to $b=c=d=0$. If
$u=v=0$, then $h_{+}=h_{-}$. Then $H(p)$ has four multiple zeros. Then we have
a variety $L_{1}=V(b,c,d)$. This variety is the singular locus of $X_{Q}$.

As in Section 3.1, we consider the ramification points of the varieties
$L_{1},\ S_{P}$ and $S_{A}$. In Sections 4.4 and 4.6, we will show that the
singular loci of $S_{P}$ and $S_{A}$ are $V(a,c,d)$ and
$V(-12+3a-b,c^{2}-bd,-54d+108c+bc,108b+b^{2}-54c)$, which are denoted by
$L_{2}$ and $C_{A}$, respectively.

By direct computations we have the following.

######  Proposition 4.11.

Under the above notations we have the followings :

$\mbox{(1)}\ g_{2}^{-1}(L_{1})=V(r^{2}+s-2ps,\ -8+8p-p^{2}-4r+2s,\ 2-2p+q).$

If $y\in L_{1}$, then $g_{2}^{-1}(y)$ consists of four points with
multiplicity 2.

$\mbox{(2)}\ g_{2}^{-1}(S_{P}\setminus L_{2})=V(qs-r^{2},\ 4q-4r+s)\cup
S_{P}.\qquad\qquad\qquad\qquad\qquad\qquad\qquad$

If $y\in S_{P}\setminus L_{2}$, then $\sharp g_{2}^{-1}(y)=6$. Two points with
multiplicity 2 lie on $V(qs-r^{2},\ 4q-4r+s)$ and four points with
multiplicity 1 lie on $S_{P}$.

$\mbox{(3)}\ g_{2}^{-1}(L_{2})=V(2r-s,\ 4q-s,\
p^{2}-s).\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad$

If $y\in L_{2}$, then $g_{2}^{-1}(y)$ consists of four points with
multiplicity 2.

$\mbox{(4)}\ g_{2}^{-1}(S_{A})=V(qs-r^{2},\ p^{2}q-4pr+4s)\cup
S_{A}.\qquad\qquad\qquad\qquad\qquad\qquad\qquad$

If $y\in S_{A}\setminus C_{A}$, then $\sharp g_{2}^{-1}(y)=6$. Two points with
multiplicity 2 lie on $V(qs-r^{2},\ p^{2}q-4pr+4s)$ and four points with
multiplicity 1 lie on $S_{A}$.

If $y\in C_{A}$, then $\sharp g_{2}^{-1}(y)=4$. Two points with multiplicity 3
lie on $V(qs-r^{2},\ p^{2}q-4pr+4s)\cap S_{A}$ and other two points with
multiplicity 1 lie on $S_{A}$.

######  Proposition 4.12.

(1) The branch locus of $g_{2}$: $X_{Q}\to X_{Q}$ consists of the three affine
algebraic sets $L_{1},\ S_{P}$ and $S_{A}$.
(2) The branch locus of $\tilde{F}:{\mathbb{C}}^{3}\to X_{Q}$ consists of
$L_{1}$ and $S_{P}$.

###### Proof.

(1) This assertion follows from the above arguments.
(2) Let $\tilde{F}(x,y,z)=(p,q,r,s)$. Clearly $\tilde{F}$ is finite. By direct
computations we have the following. If $q=0$ and $s=0$, then
$\sharp\tilde{F}^{-1}(p,0,0,0)\leq 4,$ else if $p^{2}=s$, then
$\sharp\tilde{F}^{-1}(p,q,r,s)\leq 4,$ else $\sharp\tilde{F}^{-1}(p,q,r,s)=8.$
Then the assertion (2) follows. ∎

In the following subsections, we consider the behavior of $g_{d}$ on the sets
$L_{1},S_{P},S_{A}$. The line $L_{1}$ is preperiodic under $g_{d}$ and the
branch divisors $S_{P}$ and $S_{A}$ are invariant under $g_{d}$ for any
$d\in{\mathbb{N}}$.

###  4.3. The line $L_{1}$ and the morphism $g_{d}$

In this section we will show the behavior of $g_{d}$ on $L_{1}$. We prepare
two lemmas.

######  Lemma 4.13.

Let $T_{d}(w)$ be the $d^{th}$ Chebyshev polynomial in one variable. Then

$T_{d}(w)=\left\\{\begin{array}[]{ll}\displaystyle{p_{d/2}(w^{2})}&\quad\
\mbox{if}\ d\ \mbox{is even},\\\ \displaystyle{wq_{(d-1)/2}(w^{2})}&\quad\
\mbox{if}\ d\ \mbox{is odd},\end{array}\right.$

where $p_{n}(x)$ and $q_{n}(x)$ are monic polynomials of degree $n$.

###### Proof.

Chebyshev polynomials $T_{d}(x)$ satisfies the recurrence (See [17, 6.2]) :
$T_{0}(w)=2,\ T_{1}(w)=w$, and $T_{d+2}(w)=wT_{d+1}(w)-T_{d}(w)$ for $d\geq
0$. Then this lemma follows by induction. ∎

By (4.2), we see that the morphism $g_{2}$ maps $L_{1}=\\{(t,0,0,0)\\}$ to
$\\{(t^{2},\ (2t-2)^{2},\ (2t-2)t^{2},\ t^{4})\\}$.

Then we consider a polynomial parametric representation given by

$p=t^{2},\ q=(2t-2)^{2},\ r=(2t-2)t^{2},\ s=t^{4}.$

Let $G_{1}(t)=(t^{2},\ (2t-2)^{2},\ (2t-2)t^{2},\ t^{4})$. Then by the
implicitization algorithm for polynomial parametrization [3, Chapter 3, §3,
Theorem 1], we compute the Gr$\ddot{o}$bner basis for $I=(t^{2}-p,\
(2-2t)^{2}-q,\ (2t-2)t^{2}-r,\ s-p^{2}),$ using graded reverse lex order with
$t>p>q>r>s$. The Gr$\ddot{o}$bner basis for $I$ is given by

$\\{-4-4p+q+8t,\ r^{2}-qs,-16p-12r+qr+20s-4ps+qs,\ 4pr+4s-4ps+qs,$
$16-8q+q^{2}+32r-16s,\ 4p+pq+4r-4s,p^{2}-s\\}.$

Set

(4.26) $\begin{split}C_{1}=V(r^{2}-qs,-16p-12r+qr+20s-4ps+qs,\\\
4pr+4s-4ps+qs,\ 16-8q+q^{2}+32r-16s,\ 4p+pq+4r-4s,p^{2}-s).\end{split}$

By [3, Chapter 3, §3, p.131], we know that $G_{1}({\mathbb{C}})$ covers all
points of $C_{1}$. By Proposition 5 in [3, Chapter 4, §5], we have that
$C_{1}$ is irreducible. Clearly, $dim\ C_{1}=1$. Then $C_{1}$ is an affine
algebraic curve.

######  Lemma 4.14.

Let $C_{1}$ be the algebraic set defined in (4.26). Then $C_{1}$ is an affine
algebraic curve.

######  Proposition 4.15.

Under the above notations, we have :
(1) If $d$ is odd, then $g_{d}(L_{1})=L_{1}.$ In particular,
$g_{d}(t,0,0,0)=(T_{d}(\sqrt{t})^{2},0,0,0).$
(2) $g_{2}(L_{1})=C_{1}$.
(3) $g_{d}(C_{1})=C_{1}$, for any $d\geq 1$. In particular,

$g_{d}(G_{1}(w+2))=G_{1}(T_{d}(w)+2),\ \mbox{for any}\ w\in{\mathbb{C}}.$

###### Proof.

(1) We note that by Lemma 4.13, $T_{d}(\sqrt{t})^{2}$ is a polynomial in $t$.
Set

$g_{d}(p,q,r,s)=(P_{d}(p,q,r,s),\ Q_{d}(p,q,r,s),\ R_{d}(p,q,r,s),\
S_{d}(p,q,r,s)).$

We must show that

(4.27) $\begin{split}P_{d}(t,0,0,0)=T_{d}(\sqrt{t})^{2},\hskip 85.35826pt\\\
Q_{d}(t,0,0,0)=R_{d}(t,0,0,0)=S_{d}(t,0,0,0)=0.\end{split}$

Since $T_{d}(\sqrt{t})^{2},\ P_{d}(t),\ Q_{d}(t),\ R_{d}(t)$ and $S_{d}(t)$
are polynomials, it is enough to prove (4.27) for infinitely many values $t$.
Then it suffices to show (4.27) for $t=4\cos^{2}\theta,\ 0\leq\theta\leq
2\pi$. See [10, Chapter V, §4]. We consider the diagram (4.15) in the case
${\bar{z}}_{3}=z_{1}$ and $z_{2}\in{\mathbb{R}}$.

We consider a correspondence between the top part and the bottom part in the
diagram. The operators $R$ and $C$ defined in Section 2.1 act on the space

$U_{4}=\\{\\{t_{1},t_{2},t_{3},t_{4}\\}:t_{j}\in{\mathbb{C}}^{*},\ \mid
t_{j}\mid=1,\ t_{1}t_{2}t_{3}t_{4}=1\\}.$

Then we consider the quotient space $U_{4}/<R,C>$. We claim that

(4.28) $\begin{split}(4\cos^{2}\theta,0,0,0)\in X_{Q}\ \mbox{ if and only
if}\hskip 85.35826pt\\\ [\\{\xi,\ -\xi,\ \xi e^{i\theta},\ \xi
e^{-i\theta}\\}]\in U_{4}/<R,C>,\ \mbox{where}\ \xi=e^{2\pi i/8},\
0\leq\theta\leq 2\pi.\end{split}$

Proof of the ”if ” part. Recall that $R(t_{j})=\xi^{2}t_{j},j=1,2,3,4.$ Then

$\\{\xi,\ R(\xi),\ R^{2}(\xi),\ R^{3}(\xi)\\}=\\{\xi,\ \xi^{3},\ \xi^{5},\
\xi^{7}\\}=\\{(\pm 1\pm i)/\sqrt{2}\\}.$

Since $C(\xi)=\xi^{7},$ we may consider the sets $\\{\xi^{k},\ -\xi^{k},\
\xi^{k}e^{i\theta},\ \xi^{k}e^{-i\theta}\\}$ for $k=1,3,5,7$. Hence
$z_{1}=\xi^{k}\cdot 2\cos\theta,\ z_{2}=0,$ and so $p=4\cos^{2}\theta,\ q=0,\
r=0,\ s=0$.
Proof of the ”only if ” part. By the assumption $q=r=s=0$, we have $x=\pm y,$
and $z=0.$ Hence $x=\pm\sqrt{2}\cos\theta$ and $y=\pm\sqrt{2}\cos\theta$. Then
${t_{j}}^{\prime}s$ satisfy one of the equations

$t^{4}-(2\cos\theta)\xi^{k}t^{3}-(2\cos\theta)\bar{\xi}^{k}t+1=0,\ k=1,3,5,7.$

Then the roots satisfy the property that

$\\{t_{1},t_{2},t_{3},t_{4}\\}\in\\{\\{\xi^{k},\ -\xi^{k},\
\xi^{k}e^{i\theta},\ \xi^{k}e^{-i\theta}\\}:k=1,3,5,7\\}.$

This completes the proof of the claim (4.28).

We consider the morphism $g_{d}$ with $d$ odd. Hence by (4.15) and (4.28),
$[\\{\xi,\ -\xi,\ \xi e^{i\theta},\ \xi e^{-i\theta}\\}]$ is mapped to
$[\\{\xi^{d},\ -\xi^{d},\ \xi^{d}e^{id\theta},\ \xi^{d}e^{-id\theta}\\}]$.
Note that if $\xi$ is a primitive eighth root of unity, then $\xi^{d}$ is also
a primitive eighth root of unity. Thus

$g_{d}(4\cos^{2}\theta,0,0,0)=(4(\cos d\theta)^{2},0,0,0).$

Setting $t=(2\cos\theta)^{2}$, we have (4.27), for all $0\leq\theta\leq 2\pi$.
(2) Since $g_{2}(t,0,0,0)=G_{1}(t),$ the assertion (2) is proved by the
argument before Lemma 4.14.
(3) Clearly, $t=2+2\cos 2\theta$. Then the parametric representation $G_{1}$
implies

$p=(2+2\cos 2\theta)^{2},\ q=(2+4\cos 2\theta)^{2},\ r=(2+4\cos
2\theta)(2+2\cos 2\theta)^{2},\ s=(2+2\cos 2\theta)^{4}.$

To the element $(p,q,r,s)$ we consider the corresponding set
$\\{t_{1},t_{2},t_{3},t_{4}\\}$. Then we claim the following.

(4.29) $\begin{split}((2+2\cos 2\theta)^{2},\ (2+4\cos 2\theta)^{2},\ (2+4\cos
2\theta)(2+2\cos 2\theta)^{2},\ (2+2\cos 2\theta)^{4})\in X_{Q}\\\ \mbox{ if
and only if}\quad[\\{1,\ 1,\ e^{i2\theta},\ e^{-i2\theta}\\}]\in
U_{4}/<R,C>.\end{split}$

We can prove (4.29) by the similar argument given in the proof of (4.28). Then
we can prove the assertion (3) similarly. ∎

The line $L_{1}$ is the singular locus of $X_{Q}$. For any invariant
subvariety $V$ under $g_{d}$ of $X_{Q}$, we define the set $K(g_{d}\mid_{V})$
of bounded orbits of $V$ under $g_{d}$ by

$K(g_{d}\mid_{V})=\\{w\in V:\\{g_{d}^{k}(w):k=1,2,\cdots\\}\ \mbox{is bounded
in }\ {\mathbb{C}}^{4}\\}.$

######  Corollary 4.16.

(1) If $d$ is odd and greater than 2, then

$K(g_{d}\mid_{L_{1}})=\\{((2\cos\theta)^{2},\ 0,0,0):\ 0\leq\theta\leq
2\pi\\}.\qquad\qquad\qquad\qquad\qquad\qquad$
$(2)K(g_{d}\mid_{C_{1}})=\\{G_{1}(2\cos\theta+2):0\leq\theta\leq 2\pi\\},\
\mbox{for}\ d\geq 2.\qquad\qquad\qquad\qquad\qquad\qquad$

$(3)\ C_{1}\subset S_{A}\cap S_{P}$.

###### Proof.

(1) and (2) The proof is similar to the proof of Theorem 4.8 (2).
(3) If we put $p=t^{2},\ q=(2-2t)^{2},\ r=(2t-2)t^{2},s=t^{4}$ in the
polynomial $A_{h}$ in (4.21), then the polynomial vanishes. Note that
$S_{A}=V(A_{h},\ qs-r^{2})$. Hence $C_{1}\subset S_{A}$. Then from (4.26), we
have this assertion. ∎

We will show that two surfaces $S_{A}$ and $S_{P}$ are invariant under
$g_{d}$.

###  4.4. The dynamics of $g_{d}$ on the surface $S_{P}$

We will show that $S_{P}$ is an affine algebraic surface and is invariant
under $g_{d}$. Recall that $S_{P}=V(p^{2}-s,qs-r^{2})$.

######  Proposition 4.17.

The algebraic set $S_{P}$ is an affine algebraic surface and is birationally
equivalent to ${\mathbb{C}}^{2}$.

###### Proof.

We consider a polynomial parametric representation of $S_{P}$ given by
$\phi(u,v)=(u,v^{2},uv,u^{2})$. By the similar arguments given in the argument
above Lemma 4.14, we can verify that $\phi({\mathbb{C}}^{2})$ fills up all of
$S_{P}$. Then $S_{P}$ is irreducible. Clearly, dim $S_{P}=2$. We consider a
rational map from $S_{P}$ to ${\mathbb{C}}^{2}$ given by
$\psi(p,q,r,s)=(p,r/p)$. Then $\psi\circ\phi=id_{{\mathbb{C}}^{2}}$ and
$\phi\circ\psi=id_{S_{P}}$. ∎

Next we consider the map $g_{d}$ on $S_{P}$. We consider a mapping $\rho$ from
${\mathbb{C}}^{2}$ to ${\mathbb{C}}^{2}$ defined by
$\rho(w,t)=((w+t)^{2},(wt+2))$. Then we consider a mapping
$G_{P}:{\mathbb{C}}^{2}\to{\mathbb{C}}^{4}$ defined by $G_{P}=\phi\circ\rho$.

######  Theorem 4.18.

Let $g_{d}$ be a morphism in Theorem 4.8.
(1) $g_{d}(G_{P}(w,t))=G_{P}(T_{d}(w),T_{d}(t)),$ for any $w,\
t\in{\mathbb{C}}$ and $d\in{\mathbb{N}}$.
(2) $g_{d}(S_{P})=S_{P}$ , for any $d\in{\mathbb{N}}$.
(3) $K(g_{d}\mid_{S_{P}})=\\{G_{P}(2\cos\alpha,2\cos\beta):\
0\leq\alpha,\beta\leq 2\pi\\},\ \mbox{for}\ d\geq 2.$

###### Proof.

(1) We consider the condition $p^{2}=s$ in $V(p^{2}-s,\ qs-r^{2})$. Since
$p=x^{2}+y^{2}$ and $s=(x^{2}-y^{2})^{2},$ then the condition $p^{2}=s$ is
equivalent to the condition $xy=0.$ We consider this equation $xy=0$. In
(4.16), we set

$t_{1}=e^{i(\alpha+\gamma)},\ t_{2}=e^{i(\alpha-\gamma)},\ t_{3}=e^{i\beta},\
t_{4}=e^{-i(2\alpha+\beta)}.$

Then by (4.17), we have

$\begin{split}x=\cos(\alpha+\gamma)+\cos(\alpha-\gamma)+\cos\beta+\cos(2\alpha+\beta),\qquad\qquad\\\
y=\sin(\alpha+\gamma)+\sin(\alpha-\gamma)+\sin\beta-\sin(2\alpha+\beta),\qquad\qquad\\\
=4\sin\alpha\sin(\frac{1}{2}(\alpha+\beta+\gamma))\sin(\frac{1}{2}(\alpha+\beta-\gamma)),\qquad\qquad\\\
z=2\cos
2\alpha+(2\cos(\alpha+\beta))(2\cos\gamma).\qquad\qquad\qquad\qquad\qquad\end{split}$

We assume that $\alpha=0$. Then $y=0$ and so $xy=0$. If $\alpha=0$, then
$x=(2\cos\beta+2\cos\gamma),\ y=0$ and $z=(2\cos\beta)(2\cos\gamma)+2$. Hence
putting $\alpha=0$ in (4.18), we have $p,q,r$ and $s$, and so we put $u=2$ in
(4.19). Then from the parametrization $G_{X}$ we get $G_{P}$.

To prove the assertion (1) we use an argument similar to that used in the
proof of Theorem 3.3(1).
(2) Clearly $\rho$ is a surjection. Set $\rho(w,t)=(u,v)$ and consider a
polynomial parametric representation $\phi(u,v)$ of $S_{P}$. Then we have the
assertion (2).
(3) The proof is similar to the proof of Theorem 4.8 (2).
∎

We remark that if we assume that $\alpha+\beta=\pm\gamma$, then we have the
same result.

By computing the rank of the Jacobian matrix
$\partial(p^{2}-s,qs-r^{2})/\partial(p,q,r,s)$, we can prove that the singular
locus of $S_{P}$ is $V(p,r,s)$. We consider the singular locus
$L_{2}=\\{(0,q,0,0):q\in{\mathbb{C}}\\}$ of $S_{P}$. The inverse images of
$L_{2}$ under $g_{2}$ is described in Proposition 4.11(3). We have the
following proposition that is similar to Proposition 4.15.

######  Proposition 4.19.

Let $g_{d}$ be a morphism in Theorem 4.8.
(1) If $d$ is odd, $g_{d}(L_{2})=L_{2}$. In particular,
$g_{d}(0,t^{2},0,0)=(0,T_{d}(t)^{2},0,0).$
(2) $g_{2}(L_{2})=C_{2}$, where

$C_{2}=V(-64+16q-8r+s,-8p+4r-s,16r^{2}-64s-8rs+s^{2}).$

$C_{2}$ is an affine algebraic curve. The curve $C_{2}$ has a polynomial
parametric representation $G_{2}(T)=(4T,\ (T+2)^{2},\ 4T(T+2),\ 16T^{2}).$
(3) $g_{d}(C_{2})=C_{2}$, for any $d\geq 1$. In particular,

$g_{d}(G_{2}(t^{2}))=G_{2}(T_{d}(t)^{2}),\ \mbox{for any}\ t\in{\mathbb{C}}.$

###### Proof.

(1) The proof is similar to the argument given in Proposition 4.15. Then it
suffices to prove the following claim :

(4.30) $\begin{split}(0,4\cos^{2}\theta,0,0)\in X_{Q}\ \mbox{ if and only
if}\\\ [\\{e^{i\theta/2},\ -e^{i\theta/2},\ e^{-i\theta/2},\
-e^{-i\theta/2}\\}]\in U_{4}/<R,C>\ \mbox{where}\ 0\leq\theta\leq
2\pi.\end{split}$

Proof of the ”if ” part. Under the condition, we have $z_{1}=0,\ z_{2}=\pm
2\cos\theta$ and so $q=4\cos^{2}\theta$.
Proof of the ”only if ” part. Then $[\\{e^{i\theta/2},\ -e^{i\theta/2},\
e^{-i\theta/2},\ -e^{-i\theta/2}\\}]$ is the set of roots of the equation :
$t^{4}\pm 2\cos\theta t^{2}+1=0.$ This completes the proof of the claim
(4.30).
(2) By (4.2), we have $g_{2}(0,t^{2},0,0)=(4t^{2},\ (t^{2}+2)^{2},\
4t^{2}(t^{2}+2),\ 16t^{4})$. We consider the polynomial parametric
representation

$G_{2}(T)=(4T,\ (T+2)^{2},\ 4T(T+2),\ 16T^{2}).$

Then we see that $G_{2}({\mathbb{C}})$ fills up all of $C_{2}$. Hence $C_{2}$
is an affine algebraic curve.
(3) We consider the case $t=2\cos\theta$. We see from (2) and (4.30) that

$(4(2\cos\theta)^{2},\ ((2\cos\theta)^{2}+2)^{2},\
4(2\cos\theta)^{2}((2\cos\theta)^{2}+2),\ 16(2\cos\theta)^{4})\in X_{Q}.$

if and only if

$[\\{e^{i\theta},\ e^{i\theta},\ e^{-i\theta},\ e^{-i\theta}\\}]\in
U_{4}/<R,C>.$

Then the assertion (3) follows. ∎

######  Corollary 4.20.

(1) If $d$ is odd and greater than 2, then
$K(g_{d}\mid_{L_{2}})=\\{(0,(2\cos\theta)^{2},0,0):0\leq\theta\leq 2\pi\\}.$
$(2)K(g_{d}\mid_{C_{2}})=\\{G_{2}((2\cos\theta)^{2}):0\leq\theta\leq 2\pi\\},\
\mbox{for}\ d\geq 2.$
$(3)\ C_{2}\subset S_{A}\cap S_{P}$.

###### Proof.

(1) and (2) are obvious.
(3) We use the polynomial parametric representation $G_{2}(T)$. Then
$C_{2}\subset S_{P}$. If we put $p=4T,\ q=(T+2)^{2},\ r=4T(T+2),$ and
$s=16T^{2}$ in the polynomial $A_{h}$ in (4.21) of $S_{A}$, then the
polynomial vanishes. ∎

Then combining Corollary 4.16 with Corollary 4.20, we have $C_{1}\cup
C_{2}\subset S_{A}\cap S_{P}$. In the next subsection we will show that

$C_{1}\cup C_{2}=S_{A}\cap S_{P}\ \mbox{and }\ K(g_{d}\mid_{C_{1}})\cup
K(g_{d}\mid_{C_{2}})=K(g_{d}\mid_{S_{A}})\cap K(g_{d}\mid_{S_{P}}).$

###  4.5. The dynamics of $g_{d}$ on the surface $S_{A}$

We begin to show that $S_{A}$ is an affine algebraic surface and is invariant
under $g_{d}$. The surface $S_{A}$ is related to the astroidalhedron
$\mathcal{A}$ that is studied in [23, p.205]. The astroidalhedron
$\mathcal{A}$ has connection with catastrophe theory (See [15, 16]). It is a
ruled surface in ${\mathbb{R}}^{3}$. In particular, it is the tangent
developable of the astroid in space (See [23, Figures 3 and 4]). The
astroidalhedron $\mathcal{A}$ lies in $R_{3}$ and it is the boundary of
$K(P_{A_{3}}^{d})$ . We will show that $\mathcal{A}$ is invariant under the
action of $D_{4}$ and that the relative orbit variety $\mathcal{A}/D_{4}$ is
related to the surface $S_{A}$.

The implicit representation of $\mathcal{A}$ is given in [23, p.205] by the
following ;

$256-27(z_{1}^{4}+{\bar{z}}_{1}^{4})+(z_{1}^{2}+{\bar{z}}_{1}^{2})(144z_{2}-4z_{2}^{3}+18z_{1}{\bar{z}}_{1}z_{2})$
$-80z_{1}{\bar{z}}_{1}z_{2}^{2}+z_{1}^{2}{\bar{z}}_{1}^{2}z_{2}^{2}-192z_{1}{\bar{z}}_{1}-4z_{1}^{3}{\bar{z}}_{1}^{3}-6z_{1}^{2}{\bar{z}}_{1}^{2}-128z_{2}^{2}+16z_{2}^{4}=0.$

Setting $z_{1}=x+iy$ and $z_{2}=z$, we have the equation ;

$A(x,y,z)=256-192x^{2}-60x^{4}-4x^{6}-192y^{2}+312x^{2}y^{2}-12x^{4}y^{2}-60y^{4}-12x^{2}y^{4}-4y^{6}+288x^{2}z$
$+36x^{4}z-288y^{2}z-36y^{4}z-128z^{2}-80x^{2}z^{2}+x^{4}z^{2}-80y^{2}z^{2}+2x^{2}y^{2}z^{2}+y^{4}z^{2}-8x^{2}z^{3}+8y^{2}z^{3}+16z^{4}=0.$

We can write $A(x,y,z)$ in the terms $p,q,r,s$ in (4.1). Then we have

$A_{h}=256-192p+48p^{2}-4p^{3}-128q-80pq+p^{2}q+16q^{2}+288r+36pr-8qr-108s.$

That is,

$A_{h}(p({\bf x}),\ q({\bf x}),\ r({\bf x}),\ s({\bf x}))=A({\bf x}),\
\mbox{where}\ {\bf x}=(x,y,z).$

Then $\mathcal{A}$ is invariant under $D_{4}$. Hence we can construct the
relative orbit variety $\mathcal{A}/D_{4}$ by Algorithm 2.6.2 in [20, §2.6].
By computing a Gr$\ddot{o}$bner basis for the ideal

$(A(x,y,z),\ x^{2}+y^{2}-p,\ z^{2}-q,\ z(x^{2}-y^{2})-r,\
(x^{2}-y^{2})^{2}-s),$

we know that the ideal of $\mathcal{A}/D_{4}$ is $(A_{h}(p,q,r,s),\
r^{2}-qs)$. We view the ideal $(A_{h}(p,q,r,s),\ r^{2}-qs)$ as an ideal in
${\mathbb{C}}[(p,q,r,s].$ Then we have the affine algebraic surface
$S_{A}=V(A_{h}(p,q,r,s),\ r^{2}-qs)$.

Next, using parametric representation of $\mathcal{A}$, we construct a
polynomial parametric representation of the affine algebraic surface $S_{A}$.
The astroidalhedron $\mathcal{A}$ is defined in $R_{3}$ using the coordinates
$(t_{1},t_{2},t_{3},t_{4})$ satisfying

$t_{1}=e^{i\alpha},\ t_{2}=e^{i\alpha},\ t_{3}=e^{i\beta},\
t_{4}=e^{-i(2\alpha+\beta)},\ (0\leq\alpha,\beta<2\pi),$

(See [23, pp. 204-205]). Note that this is the case $\gamma=0$ in (4.16). Then
putting $w=2$ in (4.19), we have

(4.31) $\begin{split}p=4+2uv+v^{2},\quad q=(u+2v)^{2},\\\
r=(u+2v)(2u+4v+uv^{2}/2),\quad s=(2u+4v+uv^{2}/2)^{2}.\end{split}$

We define a mapping $G_{A}$ from ${\mathbb{C}}^{2}$ to ${\mathbb{C}}^{4}$ by

$G_{A}(u,v)=(4+2uv+v^{2},\ (u+2v)^{2},\ (u+2v)(2u+4v+uv^{2}/2),\
(2u+4v+uv^{2}/2)^{2}).$

Then we can show that $G_{A}(u,v)$ is the polynomial parametrization of
$S_{A}$. Indeed. Using lex order with $u>v>p>q>r>s,$ we compute the
Gr$\ddot{o}$bner basis for the ideal

$(p-(4+2uv+v^{2}),\ q-(u+2v)^{2},\ r-(u+2v)(2u+4v+uv^{2}/2),\
s-(2u+4v+uv^{2}/2)^{2}).$

The Gr$\ddot{o}$bner basis has 12 elements altogether. The following two
elements contain only $p,q,r,s$ terms ;

$-r^{2}+qs,-256+192p-48p^{2}+4p^{3}+128q+80pq-p^{2}q-16q^{2}-288r-36pr+8qr+108s.$

Then $G_{A}(u,v)$ is a polynomial parametrization of
$S_{A}=V(-r^{2}+qs,A_{h})$. The Gr$\ddot{o}$bner basis contains also the
following two polynomials ; $-16+8p-p^{2}-24q+12r-2qv^{2}+9v^{4},\
-8+2p-q+u^{2}+2v^{2}.$ Then we know from [3, Chapter 3, §3, p.131] that
$G_{A}({\mathbb{C}}^{2})$ fills up all of $V(A_{h},r^{2}-qs)$. By Proposition
5 in [3, Chapter 4, §5], $S_{A}$ is irreducible. Also we have dim$S_{A}=2$ by
the argument after Theorem 8 in [3, Chapter 9, §3].

We remark that if we assume $\alpha+\beta=0$ in (4.16) in stead of $\gamma=0$,
then we get also the same variety $S_{A}$.

######  Proposition 4.21.

The algebraic set $S_{A}$ is an affine algebraic surface.

We call $S_{A}$ an astroid surface.

Next we consider the morphisms $g_{d}$ on the astroid surface $S_{A}$.

######  Theorem 4.22.

Let $g_{d}$ be a morphism in Theorem 4.8.
(1) $g_{d}(G_{A}(u,v))=G_{A}(T_{d}(u),T_{d}(v)).$
(2) $g_{d}(S_{A})=S_{A}$ .

###### Proof.

(1) In this case we consider the case $\gamma=0$ in (4.16). Hence by the
similar argument given in Theorems 3.3(1) and 4.18, we have the assertion (1).
(2) This assertion follows from the fact that the polynomial parametrization
$G_{A}({\mathbb{C}}^{2})$ covers all points of $S_{A}$. ∎

######  Proposition 4.23.

$(1)K(g_{d}\mid_{S_{A}})=\\{G_{A}(2\cos
2\alpha,2\cos(\alpha+\beta)):0\leq\alpha,\beta<2\pi\\},\ \mbox{for}\ d\geq 2.$
$(2)\ S_{A}\cap S_{P}=C_{1}\cup C_{2}$.
$(3)\ K(g_{d}\mid_{S_{A}})\cap K(g_{d}\mid_{S_{P}})=K(g_{d}\mid_{C_{1}})\cup
K(g_{d}\mid_{C_{2}}),\ \mbox{for}\ d\geq 2.$
$(4)\ K(g_{d}\mid_{S_{A}})\cup K(g_{d}\mid_{S_{P}})$ is a real 2-dimensional
surface.
The surface $K(g_{d}\mid_{S_{A}})\cup K(g_{d}\mid_{S_{P}})$ is invariant under
$g_{d}$ for any $d\in{\mathbb{N}}$.

###### Proof.

(1) The proof is similar to the proof of Theorem 4.8 (2).
(2) By Corollaries 4.16 and 4.20, We have $C_{1}\cup C_{2}\subset S_{A}\cap
S_{P}$. To prove the opposite inclusion, we consider the parametrization
(4.31) of $S_{A}$. Suppose

$(p(u,v),\ q(u,v),\ r(u,v),\ s(u,v))\in S_{A}\cap S_{P}.$

Hence if $p^{2}(u,v)-s(u,v)=0,$ then

$0=p^{2}(u,v)-s(u,v)=\frac{1}{4}(2-u)(2+u)(2-v)^{2}(2+v)^{2}.$

Then the two cases occur : $u=\pm 2$ and $v=\pm 2$.
Case 1: $u=\pm 2$. Then

$G_{A}(\pm 2,\ v)=((2\pm v)^{2},\ 4(1\pm v)^{2},\ 2(1\pm v)(2\pm v)^{2},\
(2\pm v)^{4}).$

By the parametrization $G_{1}$, we have

$C_{1}=\\{(t^{2},\ (2t-2)^{2},\ (2t-2)t^{2},\ t^{4}):t\in{\mathbb{C}}\\}.$

Then $G_{A}(\pm 2,v)\in C_{1}.$

Case 2: $v=\pm 2$. Then

$G_{A}(u,\ \pm 2)=(4(2\pm u),\ (4\pm u)^{2},\ 4(2\pm u)(4\pm u),\ 16(2\pm
u)^{2}).$

By the parametrization $G_{2}$ in Proposition 4.19(2), we have

$C_{2}=\\{(4T,\ (T+2)^{2},\ 4T(T+2),\ 16T^{2}):T\in{\mathbb{C}}\\}.$

Then $G_{A}(u,\ \pm 2)\in C_{2}.$
(3) From (2), we have $K(g_{d}\mid_{C_{1}})\cup K(g_{d}\mid_{C_{2}})\subset
K(g_{d}\mid_{S_{A}})\cap K(g_{d}\mid_{S_{P}})$. To prove the opposite
inclusion we note that the points $G_{A}(\pm 2,2\cos(\alpha+\beta))$ lie on
$K(g_{d}\mid_{C_{1}})$ by Corollary 4.16(2). Also the points $G_{A}(2\cos
2\alpha,\pm 2)$ lie on $K(g_{d}\mid_{C_{2}})$.
(4) $K(g_{d}\mid_{S_{A}})$ is described in (1) and $K(g_{d}\mid_{S_{P}})$ is
described in Theorem 4.18(3). These are connected by $K(g_{d}\mid_{C_{1}})\cup
K(g_{d}\mid_{C_{2}})$. Clearly $K(g_{d}\mid_{S_{A}})$ is invariant under
$g_{d}$ and $K(g_{d}\mid_{S_{P}})$ is invariant under $g_{d}$. ∎

We have the inequalities (3.15) for $K(g_{d}\mid_{{\mathbb{C}}^{2}})$ in §3.3.
Similarly we have the following ; for $0\leq\alpha,\beta,\gamma\leq 2\pi$,

(4.32)
$p^{2}-s=64(\cos\alpha)^{2}(\sin\alpha)^{2}(\cos(\alpha+\beta)-\cos\gamma)^{2}(\cos(\alpha+\beta)+\cos\gamma)^{2}\geq
0,\\\ $ $256-192p+48p^{2}-4p^{3}-128q-80pq+p^{2}q+16q^{2}+288r+36pr-8qr-108s=$
$64(1+\cos 4\alpha+\cos 2(\alpha+\beta)-4\cos
2\alpha\cos(\alpha+\beta)\cos\gamma+\cos
2\gamma)^{2}(\sin(\alpha+\beta))^{2}(\sin\gamma)^{2}\geq 0.$

Then we conjecture that $K(g_{d}\mid_{S_{A}})\cup K(g_{d}\mid_{S_{P}})$ is the
boundary of $K(g_{d}\mid_{X_{Q}})$ in ${\mathbb{R}}^{4}$.

We have seen that the astroid surface $S_{A}$ has relation to the
astroidalhedron $\mathcal{A}$. We know that the astroidalhedron $\mathcal{A}$
is a ruled surface in the sense of differential geometry. It is natural to ask
whether the affine algebraic surface $S_{A}$ is related to a ruled surface in
the sense of algebraic geometry. The projective closure of $S_{A}$ is a
birationally ruled surface. Indeed. We will show that $S_{A}$ is birationally
equivalent to an affine quadric cone $S_{C}$ in ${\mathbb{C}}^{3}$. It is
known that the blowing-up of the projective closure of $S_{C}$ at the vertex
$P_{0}=(1:0:0:0)$ is isomorphic to the Hirzebruch surface
${\mathbb{P}}({\mathcal{O}}\oplus{\mathcal{O}}(2))$ which is a geometrically
ruled surface. (See [6].) We assume that the coordinate ring
${\mathbb{C}}[S_{C}]$ is ${\mathbb{C}}[a,b,c]/(ac-b^{2})$. The cone
$V(ac-b^{2})$ has a polynomial parametric representation given by $a=u^{2},\
b=uv$ and $c=v^{2}$.

######  Proposition 4.24.

Under the above notation, the astroid surface $S_{A}$ is birationally
equivalent to the affine quadric cone $S_{C}$.

###### Proof.

We construct rational mappings $\phi:S_{A}-\to S_{C}$ and $\psi:S_{C}-\to
S_{A}$. We define the mapping $\psi$ by

$\psi(a,b,c)=(4+2b+c,\ a+4b+4c,\ 2a+8b+8c+\frac{1}{2}ac+bc,\
4a+16b+16c+2ac+4bc+\frac{1}{4}ac^{2}).$

And we define the mapping $\phi$ by

$\phi(p,q,r,s)=(\frac{-64+32p-4p^{2}+4q+5pq-q^{2}-12r}{-12+3p-q},\
\frac{64-32p+4p^{2}+16q-pq-6r}{2(-12+3p-q)},$
$\frac{-16+8p-p^{2}-12q+6r}{-12+3p-q}).$

We use the polynomial parametrizations of $S_{C}$ and $S_{A}$. By the
definition of $\psi$, we have

(4.33) $\psi(u^{2},uv,u^{2})=(p(u,v),\ q(u,v),\ r(u,v),\ s(u,v)),$

where $p(u,v),\ q(u,v),\ r(u,v)$ and $s(u,v)$ are the polynomials in (4.31).
We set

$\phi(p(u,v),\ q(u,v),\ r(u,v),\ s(u,v))=(a(u,v),\ b(u,v),\ c(u,v)).$

We compute $a(u,v),\ b(u,v)$ and $c(u,v)$. Then the numerator of
$a(u,v)=-u^{2}(u-v)^{2}$ and the denominator of $a(u,v)=-(u-v)^{2}$ . Then if
$u\neq v$, the function $a(u,v)$ is reduced to $u^{2}$. Similarly, if $u\neq
v$,

$b(u,v)=\frac{-2uv(u-v)^{2}}{-2(u-v)^{2}}=uv,\ \mbox{and}\
c(u,v)=\frac{-v^{2}(u-v)^{2}}{-(u-v)^{2}}=v^{2}.$

Hence if $u\neq v$, then $\phi\circ\psi(u^{2},uv,v^{2})=(u^{2},uv,v^{2})$.

Next we compute $\psi\circ\phi$. Let $(p(u,v),\ q(u,v),\ r(u,v),\ s(u,v))$ be
the parametrization of $S_{A}$ in (4.31). Hence if $u\neq v$, then

(4.34) $\phi(p(u,v),\ q(u,v),\ r(u,v),\ s(u,v))=(u^{2},\ uv,\ v^{2}).$

Then by (4.33), we have

$\psi\circ\phi(p(u,v),\ q(u,v),\ r(u,v),\ s(u,v))=(p(u,v),\ q(u,v),\ r(u,v),\
s(u,v)).$

We consider a proper subvariety $V_{S_{C}}(I)$ of $S_{C}$ satisfying $u=v$.
Then $V_{S_{C}}(I)=V(a-b,b-c).$ We denote a proper subvariety $V_{S_{A}}(J)$
of $S_{A}$ satisfying $u=v$ by $C_{A}$. Then $\phi\circ\psi$ is the identity
on $S_{C}\setminus V(a-b,b-c)$ , and $\psi\circ\phi$ is the identity on
$S_{A}\setminus C_{A}$. Hence $\phi\circ\psi=id_{S_{C}}$ and
$\psi\circ\phi=id_{S_{A}}$. ∎

The subvariety $C_{A}$ will be studied in the next subsection. It is an affine
algebraic curve.

We consider the mapping $\phi\circ g_{d}\circ\psi(a,b,c)$ from $S_{C}$ to
$S_{C}$. By (4.33), (4.34) and Theorem 4.22, it follows that if $u\neq v$,
then

(4.35) $\phi\circ g_{d}\circ\psi(u^{2},uv,u^{2})=(T_{d}(u)^{2},\
T_{d}(u)T_{d}(v),\ T_{d}(v)^{2}).$

We use lemma 4.13 to study Chebyshev polynomials. Set $j=d/2$ if $d$ is even
and $j=(d-1)/2$ if $d$ is odd. Then

$T_{d}(w)=\left\\{\begin{array}[]{ll}\displaystyle{p_{j}(w^{2})}&\quad\
\mbox{if}\ d\ \mbox{is even},\\\ \displaystyle{wq_{j}(w^{2})}&\quad\
\mbox{if}\ d\ \mbox{is odd},\end{array}\right.$

where $p_{j}(x)$ and $q_{j}(x)$ are monic polynomials of degree $j$. First, we
assume that $d$ is odd. Recall that $a=u^{2},\ b=uv$ and $c=v^{2}$. Then by
(4.35), we have

$\phi\circ g_{d}\circ\psi(a,b,c)=(aq_{j}(a)^{2},\ bq_{j}(a)q_{j}(c),\
cq_{j}(c)^{2}).$

Note that $bq_{j}(a)q_{j}(c)\equiv b^{d}+b_{d-1}(a,b,c)\quad mod\ (b^{2}-ac),$
where $b_{d-1}(a,b,c)=bq_{j}(a)q_{j}(c)-ba^{j}c^{j}.$ Next we assume that $d$
is even. Then by (4.35), we have

$\phi\circ g_{d}\circ\psi(a,b,c)=(p_{j}(a)^{2},\ p_{j}(a)p_{j}(c),\
p_{j}(c)^{2}).$

Note that $p_{j}(a)p_{j}(c)\equiv b^{d}+b^{\prime}_{d-1}(a,b,c)\quad mod\
(b^{2}-ac),$ where $b^{\prime}_{d-1}(a,b,c)=p_{j}(a)p_{j}(c)-a^{j}c^{j}.$

Then we can define a morphism $h_{d}:S_{C}\to S_{C}$ by the following:
if $d$ is odd, then

$h_{d}(a,b,c)=(aq_{j}(a)^{2},\ b^{d}+b_{d-1}(a,b,c),\ cq_{j}(c)^{2}),$

if $d$ is even, then

$h_{d}(a,b,c)=(p_{j}(a)^{2},\ b^{d}+b^{\prime}_{d-1}(a,b,c),\ p_{j}(c)^{2}).$

Hence

$h_{d}(a,b,c)=(a^{d}+A_{d-1}(a),\ b^{d}+B_{d-1}(a,b,c),\ c^{d}+C_{d-1}(c)),$

where $A_{d-1},\ B_{d-1},\ C_{d-1}$ are polynomials of degree $\leq d-1$. The
morphism $h_{d}$ can be extended to a morphism
$\bar{h}_{d}:{\mathbb{P}}^{3}\to{\mathbb{P}}^{3}$.

Note that the projective closure $\bar{S}_{c}$ of ${S}_{c}$ is given by
$V(x_{2}^{2}-x_{1}x_{3})$, where $(x_{0}:x_{1}:x_{2}:x_{3})$ are homogeneous
coordinates on ${\mathbb{P}}^{3}$. Then by the argument similar to that in the
proof of Proposition 3.10, we get the following proposition.

######  Proposition 4.25.

Under the above notation, we have $\bar{h}_{d}(\bar{S}_{c})=\bar{S}_{c}$.

###  4.6. The dynamics of $g_{d}$ on an astroid curve $C_{A}$

From the previous subsection, the parametrization of $C_{A}$ is obtained by
setting $v=u$ in $G_{A}$ ;

$G_{A}(u,u)=(4+3u^{2},\ (3u)^{2},\ 3u(6u+u^{3}/2),\ (6u+u^{3}/2)^{2}).$

Then we define a polynomial parametrization by

$G_{CA}(t)=(4+3t,\ 9t,\ 3t(6+t/2),\ t(6+t/2)^{2}).$

Hence by polynomial implicitization algorithm in Theorem 1 in [3, Chapter 3,
§3], we compute implicit representation of the variety. We compute the
Gr$\ddot{\mbox{o}}$bner basis for the ideal

$(4+3t-p,\ 9t-q,\ 3t(6+t/2)-r,\ t(6+t/2)^{2}-s),$

using graded reverse lex order with $t>p>q>r>s.$ The Gr$\ddot{\mbox{o}}$bner
basis is

$(-12+3p-q,\ -q+9t,\ r^{2}-qs,\ 108r+qr-54s,\ 108q+q^{2}-54r).$

Set

$I=(-12+3p-q,\ r^{2}-qs,\ 108r+qr-54s,\ 108q+q^{2}-54r).$

Then the smallest variety containing $G_{CA}({\mathbb{C}})$ is $V(I)$. And
$G_{CA}({\mathbb{C}})$ fills up all of $V(I)$. Then $C_{A}=V(I)$ and $C_{A}$
is an affine algebraic curve. The curve $C_{A}$ is related to the astroid in
space that is studied in [23, p. 205].

The astroid in space is the curve $\\{(4\cos^{3}\theta,\ 4\sin^{3}\theta,\
6\cos 2\theta):0\leq\theta<2\pi\\}$ in ${\mathbb{R}}^{3}$. It is a part of the
edge of the astroidalhedron $\mathcal{A}$. Setting
$\cos\theta=(1-t^{2})/(1+t^{2}),$ and $\sin\theta=2t/(1+t^{2}),$ we have a
rational parametric representation of the astroid in space. We define a
mapping $\rho:{\mathbb{R}}\to{\mathbb{R}}^{3}$, $t\mapsto(x,y,z)$ defined by