数理解析研究所講究録
第 1586 巻 2008 年 109-117

109

Existence of invariant manifolds at
an indeterminate point
Tomoko Shinohara 1
Tokyo Metropolitan College of Industrial Technology
E-mail address: sinohara@tokyo-tmct.ac.jp
篠原知子
東京都立産業技術高等専門学校

Abstract
In this note, consider dynamics of a rational mapping $F$ on 2-dimensional
complex projective space
which has a periodic indeterminate point . By
whichusing a symbol sequence $j\in\{1,2\}^{N}$ , we will define some family
consists of locally invariant holomorphic curves at by $F$ , algebraically.
$P^{2}$$p$$\{V_{j}\}_{j\in J}$$p$

1. Introduction.
In this note, we consider a local dynamical structure of a rational mapping $F$ ofnear a periodic indeterminate point . Using a blow up, we construct a family
which consists of locally invariant curves at by , where is a subset of$P^{2}$$p$$p$$\{V_{j}\}_{j\in J}$

$F$

$J$

.
the Cantor set
Here, prepare some notation and terminology. Let $f_{1}(x, y,t)(i=0,1,2)$ be homogeneous polynomials with degree $F:[x:y:t]rightarrow[f_{0} : f_{1} : f_{2}]$ a rational mapping$\{1, 2\}^{N}$$d,$: $(x, y,t)rightarrow(f_{0}, f_{1}, f_{2})$ a polynomial mapping on . Then, we have
except some analytic sets, where :onis the canonical projection. A point $p\in P^{2}$ is said to be an indetenninate point. In general, if is an indeterfor some pointof if
is not a single point, where the intersection isminate point, thenon$P^{2}$

and$\pi\circ\tilde{F}=F\circ\pi$$F$$C^{3}$$\tilde{F}$$C^{3}$$\pi$$\tilde{F}(\tilde{p})=(0,0,0)$$\tilde{p}\in\pi^{-1}(p)$$C^{3}\backslash \{(0,0, O)\}arrow P^{2}$$p$$\bigcap_{U_{p}}\overline{F(U_{p}\backslash \{p\})}$

of . So, no definition of the image $F(p)$taken over all open neighborhoods, it is called amakes the mapping be continuous. Moreover, if
periodic indeterminate point. It can be seen from the definition that a periodic indeterminate point has a certain recurrent property, hence we expect a local dynamical$U_{p}$$F$$p$$p \in\bigcap_{U_{p}}\overline{F(U_{p}\backslash \{p\})}$

structure at this point.
1 Supported by the Ministry of Education, Culture, Sports, Science and Technology, Grants-in-

Aid for Young Scientists (B) No. 18740094

110
In this note, we assume that $F$ : $P^{2}arrow P^{2}$ is a rational mapping with anwith the affine chart ofindeterminate point $p=[0:0:1]$ . We often identified$C^{2}$$P^{2}$

which is defined by $\{[x:y:t]\in P^{2}|t\neq 0\}$ , and put $p=(0,0)$ . Let$X$$:=\{(x,y)x[u:v]\in C^{2}xP^{1}|xv-yu=0\}$

be a subset of $C^{2}xP^{1}$ . Then, $X$ is a subvariety of $C^{2}xP^{1}$ and covered by the,following two coordinate neighborhoods$\{(U^{j},\mu^{j})\}_{j=0,1}$$U^{0}$$U^{1}$$:= \{(x, y)x[u :v]\in X|y=\frac{v}{u}x\},$$:= \{(x, y)x[u :v]\in X|x=\frac{u}{v}y\},$$\mu^{0}$$\mu^{1}$: $U^{0}\ni(x,y)x[u : v]\mapsto(x,$: $U^{1}\ni(x,y)x[u : v]-\rangle$$\frac{v}{u})\in C^{2}$$( \frac{u}{v},y)\in C^{2}$,.

Definition 1 (see [4]). The mapping : $Xarrow C^{2}$ defined by restricting the firstcentered at $p=(0,0)$ andprojection $C^{2}xP^{1}arrow C^{2}$ is called the blow up of$E:=\pi^{-1}(0,0)=(0,0)xP^{1}$ is called the exceptional curve.
$\pi$$C^{2}$

is a biholomorphic mapping,
It is remarked here thatwith the affine chart $\{[x:y:t]\in P^{2}|t\neq 0\}$ of , it can be
and by replacing
centered at $p=[0$ : : 1 . To simplify thenaturally extended as the blow up ofcentered at , too.
notation, we call it blow up of
The study of local dynamics of a periodic indeterminate point was started by Y.
$\pi:X\backslash Earrow C^{2}\backslash \{(0,0)\}$$P^{2}$$C^{2}$$P^{2}$$P^{2}$

$0$

$]$

$p$

Yamagishi (see [8] and [9]). Here, introduce his results. Set
Assume that $F$ satisfies the following assumption.

$\tilde{F}$$:=F\circ\pi$: $Xarrow P^{2}$ .

$(A.0)\{\begin{array}{l}\tilde{F}E\tilde{F}^{-1}(p)\cap E=\{p_{1},p_{2}\}N_{1}p_{1}\tilde{F}|_{N}i=1,2\end{array}$is a periodic indeterminate point of $F$ . Moreover, heassumed that is contracting in the horizontal direction on . Then, it has been
proved that there exists a family of local stable manifolds of which is indexed byHere, we remark that$p$$N_{i}$$\tilde{F}$$p$

the Cantor set$\{1, 2\}^{N}$ $:=${

$j=(j_{1},j_{2},$ $\ldots)|j_{n}=1,2$

for $n\in N$ }.

111
It ls called the Cantor bouquet (for detail, see [8] and [9]).

In this note, we consider the following family of curves which is a generalizationof the Cantor bouquet.

Definition 2. A familyof curves is locally invari ant at(1) everyis given by a graph of some continuous function$\{W_{\lambda}\}_{\lambda\in\Lambda}$$p$

by $F$ if$W_{\lambda}$$\phi_{\lambda}$

with$\phi_{\lambda}(0)=0$(2) for any, where$\Delta_{\rho_{\lambda}}$:$\Delta_{\rho_{\lambda}}\ni xrightarrow y=\phi_{\lambda}(x)\in C$$:=\{x\in C||x|<\rho_{\lambda}\}$there is a$\lim_{xarrow 0}F(x, \phi_{\lambda}(x))=p$ and$\lambda’\in\Lambda$$W_{\lambda}$

and some open neighborhood$F(x, \phi_{\lambda}(x))\cap N_{\lambda’}\subset W_{\lambda’}$Especially, if every$\phi_{\lambda}$

, and

for

$x\in\Delta_{\rho_{\lambda}}\backslash \{0\}$is a holomorphic function, then$\{W_{\lambda}\}$$N_{\lambda’}$of such that$p$.

is called a family ofholomorphic curves.
Remark. Let be a mapping$\Phi_{\lambda}$:$\Delta_{\rho_{\lambda}}arrow C^{2}$

by$\Phi_{\lambda}(x)=(x, \phi_{\lambda}(x))$. Assumethat, even if
is a holomorphic mapping. Then,is well-defined on
is an indeterminate point of , that is, there is a unique holomorphic mapping:(for detail, see [1]).
such that $g(z)=Fo\Phi_{\lambda}(z)$ for$Fo\Phi_{\lambda}$$\Phi_{\lambda}$

$F$

$p$

$g$

$\Delta_{\rho_{\lambda}}$$\Delta_{\rho_{\lambda}}arrow C^{2}$$z\in\Delta_{\rho\lambda}\backslash \{0\}$Now, we state our Main theorems. In the reminder of this note, denote $j_{n}=1,2$

for every $n\in N$ . Assume that$F$

satisfies the condition (A.O). Then, the following

claim (A.1) holds.

(A.1)$\{\begin{array}{l}F_{0}:=\pi^{-1}0\tilde{F}N(E)\{p_{1},p_{2}\}F_{0}N(E)E\pi_{j_{1}}X_{j_{1}}arrow XXp_{j_{1}}\tilde{F}_{j_{1}}:=F_{0}\circ\pi_{j_{1}}X_{j\iota}arrow X\tilde{F}_{j\iota}|_{Eg_{1}}E_{j_{1}}arrow Ep_{j_{1}j_{2}}:=\tilde{F}_{j_{1}}^{-1}(p_{j_{2}})\in E_{j\iota}N_{j_{1}j_{2}}p_{j_{1}j_{2}}\tilde{F}_{j_{1}}|_{N_{j_{1}f_{2}}}\end{array}$

Theorem 1 (see [5]). We can repeat this process inductively for all $n\in N$ and$8ymbol$ sequences $j=(j_{1}, \ldots)\in\{1,2\}^{N}$ , and succeed with infinitely many times of

112
blow ups:$\pi_{j_{1}\ldots j_{n+1}}$$p_{j_{1}\ldots j_{n+1}}\in X_{j_{1}\ldots j_{n}}$$X_{j_{1}\ldots j_{n+1}}arrow X_{j_{1}\ldots j_{n}}$. In particular, there exist sequences of points.

In addition to the condition (A.0), we suppose the following condition (B).
$(B)p_{j_{1}\ldots j_{n}}\in U_{j_{1}\ldots j_{\mathfrak{n}-1}}^{0}$for every $n\in N$ ,analogue to that definedis the coordinate neighborhood of
by using the local coordinates systemfor $X$ . Then, we can set. Finally, for all symbol sequences $j\in\{1,2\}^{N}$ with $j=(j_{1},j_{2}, \ldots)$ , define
of

where

$X_{j_{1}\ldots j_{n-1}}$$U_{j\iota\cdots j_{n-1}}^{0}$$p_{j_{1}\ldots j_{n}}$$:=(0, \alpha_{j_{1}\ldots j_{n}})$$U_{j\iota\cdots j_{n-1}}^{0}$a formal power series$y=\phi_{j}(x)$$J:=\{j\in\{1,2\}^{N}|\phi_{j}(x)$$V_{j}$$:=\alpha_{j_{1}}x+\alpha_{j_{1}j_{2}}x^{2}+\cdots$,

has a positive convergent radius$:=\{(x, y)\in N_{j}|y=\phi_{j}(x)$on$\rho_{j}>0\}$,for all $j\in J$.

$\Delta_{\rho_{f}}\}$

Then, we have the following Theorem 2.

is a family of locally invari ant holomo rphic curves
Theorem 2 (see [5]).
of locally invariant holomorphic
at by F. In particular, every family.
curves at by must be a subfamily of$\{V_{j}\}_{j\in J}$$\{W_{\lambda}\}_{\lambda\in\Lambda}$

$p$

$p$

$F$

$\{V_{j}\}_{j\in J}$

As applications, consider the following rational mappings of$(*1)$$(*2)$$F(x,y)=(ax,$$\frac{y(y-x)}{x^{2}})$$C^{2}$.

$|a|>4$ ,,,$F(x,y)=(x+ax^{2},$ $\frac{y(2y-x)}{x^{2}}$$|a|\neq 0$.

ノ

Theorem 3 (see [6]). Suppose that is the rational mapping in $(*1)$ . For all
symbol sequences $j=(j_{1},j_{2}, \ldots)\in\{1,2\}^{N}$ , one of the following claims holds.
and, thensuch that $j_{n}=1$ for any
there exists an integer
(1)
$F$

If

$n\geq n_{0}$$n_{0}$$V_{j}\subset F^{-n0}(V_{11}\ldots)=F^{-n0}(\{y=0\})$. Especially,(2) If there exist infinitely many $n_{0}\in N$ with

For the rational mapping$F$$V_{j}$$V_{j}\neq\emptyset$

are unstable manifolds of .
.
, then$j_{\mathfrak{n}0}=2$$p$$V_{j}=\emptyset$

in $(*2)$ , the following theorems 4 and 5 hold.

113
Theorem 4. For every symbol sequence $j\in\{1,2\}^{N}$ there exists a continuousfunction$y=\psi_{j}(x)$on$\Delta_{\delta}$$W_{j}$

In particular,. Put$:=\{(x, y)\in C^{2}|y=\psi_{j}(x)$$\{W_{j}\}_{j\in\{1,2\}^{N}}$on$\Delta_{\delta}\}$.

is a family of curves which is locally invariant at$p$

by $F$ .

Theorem 5. For any symbol sequence $j\in\{1,2\}^{N}$ , there enists $j’=(j_{1}’,j_{2}’, \ldots)\in$
such that the fomal power series
is the asymptotic$\{1,2\}^{N}$$\phi_{j’}(x)=\sum\alpha_{j_{1}’\ldots j_{n}’}x^{n}$

expansion of$\psi_{j}(x)$. That is, for all $n\in N$ , there exist positive constants$\delta_{n}$and$M_{n}$

such that,$|\psi_{j}(x)-\alpha_{j_{1}’}x-\cdots-\alpha_{j_{1}’\ldots j_{n-1}’}x^{n-1}|\leq M_{n}|x|^{n}$for any$x\in\Delta_{\delta_{n}}$.

may not be a convergent power series, for any fixed
Remark. Although$n\in N,$is approximated by the polynomialwith theorder $O(|x|^{n})$ by taking the limit as $xarrow 0$ .
$\phi_{j’}$$\psi_{j}(0)$$\alpha_{j_{1}’}x+\cdots+\alpha_{j_{n-1}’}x^{n-1}$

Theorems 1, 2 and 3 have been obtained by [5] and [6]. In this note, we will givean outline of proof of Theorems 4 and 5.

2. Proof of Theorem 4.
Put $q(x)$ $:=x+ax^{2}$ . This is the first component of in $(*2)$ . We begin withbasic facts on dynamics of the polynomial $q(x)$ at $x=0$ (for detail, see [7]). For
the polynomial $q(x),$ $x=0$ is a rationally indifferent fixed point and there exist an
attracting petal and a repelling petal such that
$F$

$P$

$R$

,

(1)

$q( \overline{P})\subset P\cup\{0\},\bigcap_{n=1}^{\infty}(F)=\{0\}$(2)$(q|_{R})^{-1}( \overline{R})\subset R\cup\{0\},\bigcap_{n=1}^{\infty}(q|_{R})^{-n}(\overline{R})=\{0\}$(3) $\{0\}\cup P\cup R$ is an open neighborhood of

$0$

,

.

Now, let us start the proof of Theorem 4. In the following part, we shall give a114
proof which is based on an argument by Hadamard-Perron Theorem in [3] and theconstruction of the Cantor bouquet in [8].

For$\alpha_{j}\in C,$$p=(O,O)\in C^{2}$

and$p_{j}$,$:=\{x\in C||x-\alpha_{j}|<r\}$$\Delta_{r}(\alpha_{j})$$\Delta_{r}^{2}(p)$$:=\Delta_{r}x\Delta_{r}$, define the following sets;$:=(0, \alpha_{j})\in C^{2}$,$\Delta_{r}^{2}(p_{j})$$\Delta_{f}$$:=\Delta_{r}(0)$$:=\Delta_{r}x\Delta_{r}(\alpha_{j})$,.

From some easy calculation, one can check that our $F$ satisfies the conditions(A.0) and $(B)$ . Hence, Theorems 1 and 2 hold and for any infinite symbol sequence$j\in\{1,2\}^{N}$ , there exists the sequence of points.
In the reminder of this note, denote $l=1,2$ . IFMrom (A.0), is alocally biholomorphic mapping on some neighborhoods of , and there are positive constants
and and branches :of , Let $\rho:C^{2}arrow[0,1]$ be a -function
such that$\{\alpha_{j_{1}\ldots j_{n}}\}_{n\geq 1}$$k,$$\tilde{F}$$r$$p_{l}$$r’$$G_{l}$$\tilde{F}$$\Delta_{r}^{2}(p)arrow\Delta_{r}^{2},(p_{l})$$C^{1}$$\rho(x,y)=\{\begin{array}{ll}1 on \Delta_{f}^{2}(p_{k})0 on \Delta_{2r}^{2}(p_{k})^{c}.\end{array}$By using this$C^{1}$-function , define a$C^{1}$$\rho$-mapping$f_{kl}$:$C^{2}arrow C^{2}$such that$f_{kl}=\{\begin{array}{ll}G_{l}o\pi on \Delta_{r}^{2}(p_{k})J(G_{l}\circ\pi)_{p} on \Delta_{2r}^{2}(p_{k})^{c},\end{array}$

where$J(G_{l}o\pi)_{p_{k}}$$C_{\gamma^{k}}^{p}$$:=$is the Jacobian matrix of$C_{\gamma}$

at the point$p_{k}${ $\phi:Carrow C$ , Lipshitz ft. with Lipshitz constant and$\gamma$$C_{\gamma}$

Then,$G_{l}\circ\pi$$:=C_{\gamma}^{p\iota}\cup C_{\gamma}^{P2}$$\phi(0)=\alpha_{k}$$d$defined as follows;$:=\{\begin{array}{ll}\sup_{x\in}^{\phi x-x}c_{\backslash \{0\}}^{\ovalbox{\tt\small REJECT}_{|x|}} if \phi, \psi\in C_{\gamma^{h}}^{p}3if \phi\in C_{\gamma}^{Ph} a d \psi\in C_{\gamma^{l}}^{p}(k\neq l).\end{array}$

It can be seen that for any$\phi\in C_{\gamma^{k}}^{p}$$f_{kl}$there exists$\psi\in C_{\gamma}^{Pl}$such that(graph ) $=graph\psi$ .
$\phi$

By using this fact, one can define the action of},.

is a complete metric space with respect to the metric$d(\phi, \psi)$. Set$g_{l}$$g_{l}(graph\phi):=graph((f_{kl}).\phi)$on$C_{\gamma}$

, if

by

$\phi\in C_{\gamma}^{Pk}$115:
is a contraction mapping.
Let be the space of non-empty compact subsets of . Then, is a complete
metric space with respect to the Hausdorff metric. Setting a mappingand know that$g_{l}$$C_{\gamma}arrow C_{\gamma^{l}}^{p}$

$S$

$S$

$C_{\gamma}$. $G:Sarrow S$, by $A\mapsto G(A)$ $:=g_{1}(A)\cup g_{2}(A)$we can show that is contraction on , since is a contraction mapping.
has theThus, it follows from Banach’s contraction mapping theorem thatfor any $A\in S$ . Here, we
unique fixed point $E\in S$ , and $G^{n}(A)$ converges tofor $l=1,2$ . Thenchoose a subset of satisfying
$S$

$G$

$g_{l}$

$G$

$E$

$A$

$S$

$g_{l}(A)\subset A$$\bigcap_{n=0}^{\infty}G^{n}(A)=E$.

Consequently, since $g_{1}(A)\cap g_{2}(A)=\emptyset$ , there exists the unique point
for every symbol sequenoe $j\in\{1,2\}^{N}$ .
that$\tilde{\phi}_{j}\in C_{\gamma}$such$g_{j_{1}}\circ\cdots\circ g_{j_{n}}(A)arrow\tilde{\phi}_{j}(narrow\infty)$

By using$\tilde{\phi}_{j}$, let us set$\tilde{W}_{j}$Then, it implies that$:=\{(x,y)\in C^{2}|y=\tilde{\phi}_{j}(x)\}$, where is the shift mapping onwith $0<\delta<r$ , and put$g_{l}(\tilde{W}_{j})=\tilde{W}_{\sigma(j)}$a small positive constant$\tilde{W}_{j}^{\delta}$$\delta$.

$:=\tilde{W}_{j}\cap\Delta_{\delta}xC$$\sigma$and$W_{j}:=\pi(\tilde{W}_{j}^{\delta})$$\{1, 2\}^{N}$. Ihke.

Finarlly, we can prove that
$W_{j}=\{(x, y)\in C^{2}|y=x\tilde{\phi}_{j}(x)\}$and$\{W_{j}\}_{j\in\langle 1,2\}^{N}}$is a family of curves which is locally invariant at$p$

by $F$ . This isrequired.

Remark. Unfortunatelly, depends on the construction of an extension mapis determined uniquely forping
and does not have uniqueness. However,
any $x\in P$ , where is an attracting petal of $q(x)$ at , and$\tilde{\phi}_{j}$$\tilde{\phi}_{j}(x)$$f_{kl}$$0$$P$$F^{n}(x,y)arrow p$

as$narrow\infty$

for any $(x,y)\in W_{j}\cap\{P\cross C\}$ with $x\neq 0$ .

116
3. Proof of Theorem 5.
To prove Theorem 5, we need the following Lemmas 1 and 2.

Lemma 1. For every symbol sequence $j\in\{1,2\}^{N}$ the following claims hold;(1) there exists a pointsuch that, and put$p_{j_{1}’}\in\{p_{1},p_{2}\}$$(W_{j})_{j_{1}’}$$\overline{\pi^{-1}(W_{j}\backslash \{p\})}\cap E=\{p_{j_{1}’}\}$$:=\overline{\pi^{-1}(W_{j}\backslash \{p\})}$(2) there exists a continuous function$\phi_{j_{1}’}$

on$\Delta_{\delta}$,

such that$(W_{j})_{j_{1}’}=\{(x, y)\in\Delta_{\delta}xC|y=\phi_{j_{1}’}(x)$Sincefor everyon$\Delta_{\delta}\}$.

is a family of curves which is locally invariant at by $F$ ,there exists a symbol sequence $i=(i_{1}, i_{2}, \ldots)\in\{1,2\}^{N}$ and an open
neighborhoodof such that
. IFlrom Lemma 1 (1), there
is a pointsuch that. Put$\{W_{j}\}_{j\in\{1,2\}^{N}}$$p$$W_{j}$$N_{i}$$F(W_{j}\backslash \{p\})\cap N_{i}\subset W_{i}$$p$$p_{1_{1}’}\in\{p_{1},p_{2}\}$$\pi^{-1}(VV_{i}^{r}\backslash \{p\})\cap E=\{p_{i_{1}’}\}$$(W_{i})_{i_{1}’}$$:=\overline{\pi^{-1}(VV\cdot\backslash \{p\})}$

and$F_{0}$$:=\pi^{-1}\circ\tilde{F}$.

Then, we have the following lemma.

Lemma 2.
(1) There exists an open neighborhood$\lim_{xarrow 0}F_{0}(x, \phi_{j_{1}’}(x))=p_{i_{1}’}$and(2) There exist positive constants$\delta_{j_{1}}$$(N_{1})_{i_{1}’}$of$p_{i_{1}’}$such that$F_{0}((W_{j})_{j_{1}’}\backslash \{p_{j_{1}’}\})\cap(N_{i})_{i_{1}’}\subset(W_{i})_{i_{1}’}$

and$M_{j_{1}}$$|\psi_{j}(x)-\alpha_{j_{1}’}x|\leq M_{j_{1}}$such thatfor$x\in\Delta_{\delta_{j_{1}}}$.

We can repeat this process inductively for every $n\in N$ and prove Theorem 5.

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721-772.

$P^{2}$, Indiana Univ. Math. J. 45 (1996),[2] T. C. Dinh, R. Dujardin and N. Sibony, On the dynamics near infinity of somepolynomial mapping in , Math. Ann. 333 (2005), 703-739.
$C^{2}$

117[3] A. Katok and B. Hasselblatt, Introduction to the modern theory of dynamicalsystems, Cambridge University Press, (1995).
[4] I. R. Shafarevic, Basic Algebraic Geometry Vols I andBerlin, (1994).

$\Pi$

Springer-Verlag,[5] T. Shinohara, A const ruction of invariant curves at a periodic indeterminate
point, Suurikaiseki kenkyuusho Koukyuroku, 1494 (2006), 13-23.
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kenkyuusho Koukyuroku, 1537 (2007).
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[8] Y. Yamagishi, Cantor bouquet of holomo$rp$ hic stable manifolds for a periodic
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[9] Y. Yamagishi, On the local convergence of Newton’s method, Journal of theMathematical Society of Japan, 55 (2003), 897-908.