数理解析研究所講究録983 巻 1997 年 200-208

200

Normal Forms of Vector Fields andDiffeomorphisms
By
Masafumi Yoshinol
Abstract
We shall show simultaneous normal forms of a system of vector fields anddiffeomorphisms under Brjuno condition. These results are proved by a newscheme of a rapidly convergent iteration with high loss of derivatives such that$0<\sigma’<\sigma$
for some $0<\epsilon<1,$.
We solve an overdetermined system of equations arising in the study ofnormal forms and diffeomorphisms by this method.
$\exp(\exp((\sigma-\sigma’)^{-\mathcal{E}})),$$\epsilon,$

1

Normal forms of vector fields

Let us consider a system of analytic vector fields $X^{\mu}(\mu=1, \cdots , d)$ in some neighborhood of the origin of $x=(x_{1}, \cdots, x_{n})\in R^{n}$ ,$X^{\mu}= \langle X^{\mu}, \partial_{x}\rangle=\sum_{j=1}^{n}x_{j}\mu(X)\partial_{x}j$with the convention that$\partial_{x}=(\partial_{x_{1}’ x_{n}}\ldots, \partial),$$X^{\mu}(1\leq\mu\leq d)$’$1\leq\mu\leq d$$\partial_{x_{j}}=\partial/\partial x_{j}$are singular i.e.

(1.1)

. We assumefor$X^{\mu}(\mathrm{O})=0$,$1\leq\mu\leq d$

.

(1.2)

The linear parts of $X^{\mu}(1\leq\mu\leq d)$ are semi-simple i.e.,$X^{\mu}(x)=(X_{1}^{\mu}(x), \cdots, X^{\mu}n(x))=\Lambda^{\mu}x+R^{\mu}(X)$,$1\leq\mu\leq d$

,

(1.3)

where

$\Lambda^{\mu}=$ ,and where$R^{\mu}(x)$$\lambda_{j}^{\mu}\in C$

are analytic at the origin and satisfy$R^{\mu}(\mathrm{O})=\partial xR^{\mu}(\mathrm{o})=0$

1Supported by the Volkswagen-Stiftung (,$1\leq\mu\leq d$.

-program at Oberwolfach) and partially supported
by Grant-in-Aid for Scientific Research (No. 07640250), Ministry of Education, Science and Culture,
Japan and by Chuo University special research fund, Tokyo, Japan.
This is a joint work with T.Gramchev in Dipartimento di Matematica, Universit\‘a di Cagliari via
Ospedale 72, 09124 Cagliari, Italia.
$\mathrm{R}\mathrm{i}\mathrm{P}$201. (1.4)are pairwise commuting, . . $[X^{\mu}, X^{\nu}]=0$ ,, $(1 \leq\mu\leq d)$ . We are interested in reduction of vector fields
Set
$d=1$
(single case), a normal form was obtained by Poincare’ under
to normal forms. If$X^{\mu}(1\leq\mu\leq d)$$1\leq\nu,$ $\mu\leq d$$\mathrm{i}$$\mathrm{e}$$\lambda^{\mu}=(\lambda_{1}^{\mu}, \cdots, \lambda_{n}^{\mu})$the condition$(*)$$|\lambda\alpha|\geq c_{0}|\alpha|$for$\alpha\in Z_{+}^{n},$$|\alpha|>>1$

Roughly speaking, in order to find a change of variables which reduces a vector
field to its normal form we must solve a nonlinear equation, a so-called homologicalimplies the existence of the bounded inverse of theequation. The condition
linearized operator. The solvability of certain nonlinear equations under Poincar\’econdition was proved by Kaplan for more general equation.([6]).
The solvability of these nonlinear equations with unbounded inverse was proved bySiegel in case $d=1([12])$ under a famous Siegel condition:$(^{*})$$\exists c>0,$$\exists\gamma>0;|\lambda\alpha-\lambda_{k}|\geq c|\alpha|^{-\gamma}$

for $1\leq k\leq n,$$\alpha\in Z_{+}^{n}$

.

(1.5)

R\"ussman ([10]) generalized his idea and proved

ppose (1.2), (1.3) and (1.5). Then the vector field (1.1) can be
Assume $d=1$ .
transformed to a normal form by a holomorphic change of variables.
$S\mathrm{u}$By the studies of normal forms of mappings by Yoccoz ([13]) and M. Perez ([9]), it isnatural to weaken the condition (1.5) to the following simultaneous Brjuno condition:$\exists c>0,$ $\exists\gamma>0$$(Br)$

such that$\max_{1\leq\mu\leq d}|\lambda\mu\alpha-\lambda_{k}^{\mu}|\geq c\exp(-\frac{|\alpha|}{\log(2+|\alpha|)^{1+\gamma}})$$\forall\alpha\in Z_{+}^{n},$$1\leq\forall k\leq n$.

We note that our condition is weaker because the bound from the below is exponen, and there is a maximum in in the left-hand side. Hencetially small wheneach vectors could be resonant and may not satisfy a Brjuno condition as a single
equation, while they simultaneously satisp (Br).
$|\alpha|arrow\infty$$\mu$

We note that (Br) implies that$\lambda_{1}^{\mu},$$\lambda_{n}^{\mu}$$\ldots,$,

are non simultaneous resonant, namely$\forall\alpha\in \mathrm{Z}_{+}^{n},$$\max_{1\leq\mu\leq d}|\lambda^{\mu}\cdot\alpha-\lambda_{k}^{\mu}|\neq 0$$1\leq k\leq n$

.

(1.6)

Then we havevector, $X^{d}(x)$ be pairwise commuting holomo
Theorem 1. 1 Let $X^{1}(x),$Brjunovenfythe(1.4).
and(1.3)(1.2),
If
fields satisfying the conditions
condition $(Br)$ we can find a neighborhood of the origin and a holomo rphic change$x^{d}(x)$which transforms simultaneously $X^{1}(x),$of the variables $x=y+u(y),$, respectively. Moreover, is a solution of the following equation
into$7ph\dot{i}C$$\ldots$$\lambda^{d}$$\lambda^{1},$$\ldots,$$\Omega$$y\in\Omega$$\ldots,$$u$$\lambda^{d}y\partial_{y}$$\lambda^{1}y\partial_{y},$$\ldots,$$\mathcal{L}_{\lambda^{\mu}}u-R^{\mu}(y+u)=0$,$1\leq\mu\leq d$

.

(1.7)

202

1.1

Approximate solution to a homological equation

First we need to introduce some Banach spaces of holomorphic functions. Let bean open ball containing the origin in
and letbe the set of holomorphic
functions on . Following [4] we define for $0<T<d_{\dot{i}am}(\Omega)/2$$\Omega$$\mathrm{C}^{n}$$\mathcal{O}(\Omega)$$\Omega$$H(T)= \{u(x)=\sum_{\mathrm{Z}\alpha\in n}u_{\alpha}x^{\alpha}\in O(\Omega) :|v\mathrm{b}= \sum_{n,\alpha\in Z}|u_{\alpha}|T^{||}\alpha<\infty\}$(1.8)

Theorem 1. 2 The following estimate is $tme$(1.9)$|D^{\beta}u \mathrm{b}_{1}\leq\frac{C}{(T-T_{1})^{1\beta}1}\mathfrak{p}\ \cdot$

for all $0<T_{1}<T$ .
We define$Mf= \sum_{=\mu 1}^{d}\mathcal{L}\lambda-\mu \mathcal{L}_{\lambda\mu}f$,

If we expand $f(x)$ into Taylor series
we can see that
$M(\alpha)=\mathrm{d}\mathrm{i}\mathrm{a}\mathrm{g}(M_{1}(\alpha), \ldots, M_{n}(\alpha))$$f\in(H(T))n:=H(\tau).\cross\cdots \mathrm{X}H(T)$$f(x)= \sum_{\alpha}f_{\alpha}x^{\alpha}$,.

(1.10)and if we set $Mf= \sum M(\alpha)f_{\alpha}x\alpha$$M_{j}( \alpha)=\sum_{1\mu=}^{d}|\lambda^{\mu}\cdot\alpha-\lambda\mu|^{2}j$’$1\leq j\leq n$. (1.11)

Then we have
Lemma 1. 3 Let $T_{0}>0$ be given. Suppose that $(Br)$ is satisfied. Then, for any$0<T’<T<T_{0}$ there exists an inverse: $(H(T’))^{n}arrow(H(T))^{n}$ and a constant$c_{0}>0$ such that the following estimate holds$M^{-1}$$||M^{-1}||_{T}\prime_{arrow}\tau\leq\exp(2\exp(C_{0}(\tau-T’)^{-1/(+}1\mathcal{T})))$

Proof. Set $\delta=T’/T$ . We have, for$||M^{-1}f||_{T}$,$=$$\leq$,$T_{0}<\forall T’<\forall T<2T_{0}$.

(1.12)$f= \sum f_{\alpha}x^{\alpha}\in(H(T))^{n}$(1.13)$\sum_{\alpha}T^{;|\alpha|}|M^{-1}(\alpha)f_{\alpha}|\leq$.

$\sum_{\alpha}\delta^{|\alpha|}\exp(2|\alpha|(ln(|\alpha|+2))^{-\mathcal{T}-1})|f_{\alpha}|T^{|\alpha|}$Since$\sup_{|\alpha|\geq 1}(\delta^{|\alpha|}\exp(2|\alpha|(ln(|\alpha|+2))^{-\mathcal{T}-1}))\leq\exp(\exp(c(ln\delta^{-1})-1/(1+\tau)))$(1.14)

for some $c>0$ and since $ln(T/T’)=ln(1+(T-T’)/T’)$ is bounded by the constanttimes of $T-T’$ from the above and the below we have (1.12). .
$\square$203

For the later use we define the approximate inverse to a homological operator as
follows:$\vec{f}=(f1, \ldots, f_{d})\in(H(T))nd$ .
(1.15),$P(D) \vec{f}=\sum^{d}\mathcal{L}M-1f_{\mu}\mu=1\overline{\lambda^{\mu}}$

We observe that
$1\leq\forall\mu\leq d$$\mathcal{L}_{\lambda^{\mu}}P(D)\vec{f}=f\mu+M^{-1}\sum_{\nu=1,\nu\neq\mu}^{d}\mathcal{L}\overline{\lambda^{\nu}}(\mathcal{L}_{\lambda^{\mu}}f_{\nu}-\mathcal{L}\lambda\nu f_{\mu})$

1.2.

(1.16)

Rapidly convergent iteration scheme

Now we will prove Theorem 2.1. We shall find $u(x)$ such that(1.17)$(1+ \frac{\partial u}{\partial x})^{-1}X^{\mu}(x+u(x))=\Lambda^{\mu}x=t(\lambda_{1}^{\mu}X_{1}, \ldots, \lambda^{\mu}X_{n})n$for $1\leq\mu\leq d$. The equation (1.17) is equivalent to solving the following overdetermined system of equations$\mathcal{L}_{\lambda^{\mu}}u(X)=R_{\mu}(x+u(x))$,$1\leq\mu\leq d$.

(1.18)We set$v_{0}(x)= \nu 1\sum_{=}^{d}\mathcal{L}M^{-}1R_{\nu}^{0}(X)\overline{\lambda^{\nu}}$,$R_{\nu}^{0}(x)=R_{\nu}(x)$.

(1.19). Then we consider the
By a scale change of variables we may assume thatchange of the variables $x+v_{0}(x)$ and obtain the new system of vector fields$|v_{0}|_{T}\ll 1$$X^{\mu,1}(X)=(1+ \frac{\partial v_{0}}{\partial x})^{-1}X^{\mu}(x+v_{0}(X))\equiv\Lambda^{\mu}x+R_{\mu}^{1}(x)$,$1\leq\mu\leq d$(1.20).

from the left andStraightforward calculations show (multiplying by$X^{\mu}(x+v_{0}(X))=\Lambda^{\mu}x+\Lambda^{\mu}v_{0}(X)+R_{\mu}^{0}(x+v_{0}(x)))$recalling that$(1+\partial v_{0}/\partial x)$,$\Lambda^{\mu}x+\Lambda\mu v_{0}(x)+R^{0}(\mu(_{X}x+v_{0}))=(1+\frac{\partial v_{0}}{\partial x}\mathrm{I}^{\Lambda^{\mu}}x+(1+\frac{\partial v_{0}}{\partial x})R^{1}\mu(_{X})$i.e.,$(1+ \frac{\partial v_{0}}{\partial x})R_{\mu}^{1}(x)$$=$$- \frac{\partial v_{0}}{\partial x}\Lambda^{\mu}x+\Lambda^{\mu}v_{0}(X)+R_{\mu}^{0}(x+v_{0}(x))$$=$$-\mathcal{L}_{\lambda^{\mu}}v_{0}+R_{\mu}^{0}(x+v_{0}(x))$(1.21)

204$=$$-R_{\mu}^{0}(_{X)+\sum_{\nu 1}}=d\mathcal{L}M\overline{\lambda^{\nu}}-1(\mathcal{L}\lambda^{\nu}R0_{-\mathcal{L}\lambda\mu}R_{\nu}^{0})\mu+R^{0}\mu(X+v\mathrm{o}(X))$$=$$v_{0}(x) \int_{0}1d)R^{0}’(\mu x+tv\mathrm{o}(x))dt+\nu=\sum_{1}\mathcal{L}M-1(\overline{\lambda^{\nu}}x-R^{0_{\partial}0}xR_{\mu}R_{\mu\nu}0_{\partial R}0\nu$’

where we have used$\mathcal{L}_{\lambda}\mu R^{\nu}(x)-\mathcal{L}\lambda\nu R^{\mu}(X)=\partial R\mu(X)R^{\nu}(X)-\partial R^{\nu}(X)R^{\mu}(_{X)},$$1\leq\nu,$ $\mu\leq d$.

This is equivalent to $[X^{\mu}, X^{\nu}]=0$ . Therefore,
is estimated quadratically.
We conditnue this process. Suppose that we have constructed$v_{k1}-(x)$such that after a change of variables$R_{\mu}^{1}$$v_{0}(x),$$\ldots,$$(1+v_{0})\circ(1+v1)0\cdots \mathrm{O}(1+v_{k-}1)(_{X)}$

we have obtained$X^{\mu,k}(x)=\Lambda^{\mu}x+R_{\mu}^{k}(x)$,$1\leq\mu\leq n$.

(1.22)

Next we define(1.23)$v_{k}(X)= \sum_{1\nu=}^{d}\mathcal{L}M-1Rk(X)\overline{\lambda^{\nu}}\nu$’and$X^{\mu,k+1}(X)$(1.24)$=$$(1+\partial_{xk}v)-1X^{\mu,k}(x+v_{k}(x))$$=$$(1+\partial_{xk}v)-1(1+\partial v-1)^{-1}xk\ldots(1+\partial_{x}v_{0})-1$$\cross$$=$$X^{\mu}((1+vk)\mathrm{o}(1+v_{k}-1)0\cdots \mathrm{O}(1+v0)(_{X)})$$\Lambda^{\mu}x+R^{k1}+(_{X)}\mu$.

As before we get$(1+\partial_{xk}v)R^{k+1}(\mu)X=-\mathcal{L}_{\lambda^{\mu}}vk+R^{k}(\mu x+vk(_{X))}$$=$$-R_{\mu}^{k}(x)+ \nu 1\sum_{=}^{d}\mathcal{L}_{\overline{\lambda^{\nu}}}M-1(\mathcal{L}_{\lambda^{\nu}}R^{k}-\mu \mathcal{L}\lambda\mu R_{\nu}k)+Rk(\mu x+vk(x))$$=$$v_{k}(x) \int_{0}1)R_{\mu}^{k/}(x+tv_{k}(X)dt+\sum_{\nu=1}^{d}\mathcal{L}_{\overline{\lambda^{\nu}}}M^{-1}(R_{\mu}^{k}\partial xR_{\nu}^{k}-R_{\nu}k\partial_{x}R_{\mu}^{k})$Hence, there exist $c>0$ and$|R_{\mu}^{k+1}|_{T}$$\leq$$+$$c_{1}>\mathrm{s}\mathrm{u}\mathrm{c}\mathrm{h}$. (1.25)that$c(|(1+ \partial vxk)-1|_{T}|v_{k}|_{\tau}\frac{c_{1}}{T-T},$ $|DR^{k}\mu|T’+$(1.26)$|(1+\partial_{xk}v)-1|_{T}\exp(2\exp(c(\tau-T^{J})-1/(1+\tau)))|Rk|\tau’|DR_{\mu}^{k}|_{T)}’$.

By using the estimate for composition of maps we see that$(1+v_{0})\circ(1+v_{1})\circ\cdots\circ(1+v_{k})(x)arrow 1+u$and$|R^{k}|_{T}arrow 0$

in $H(T)$ as$rarrow\infty$(1.27), as $karrow\infty$ . It follows that$(1+\partial u_{k})^{-1}X\mu(X+uk(x))arrow(1+\partial u)^{-1\mu}X(x+u(x))=\Lambda^{\mu}x$.

(1.28)

205

1.3

Normal forms in the case of Jordan blocks

Now we shall remove the restriction (1.2). Namely,$X^{\mu}(x)={}^{t}(x_{1}^{\mu}(X), \ldots, X^{\mu}(nx))=J^{\mu}x+R^{\mu}(x)$,$1\leq\mu\leq d$,

are not necessarily semi-simple and we use the same notations
where the matricesbyas before. We define the homological operator$J^{\mu}$$\mathcal{L}_{\lambda^{\mu}}$$\mathcal{L}_{\lambda^{\nu}}u=\partial uJ^{\mu}x-J^{\mu}u$,$(\mu=1, \ldots d)$,$u\in(H(T))^{n}$.

(1.29)

imply that the matrices
commute each other, namely
The commutativity of$[J^{\mu}, J^{\nu}]=0$ for everyup
to their Jordan blocks if weand . This determines
fix one Jordan normal form of some . In the following, we may assume that all
are diagonalized up to their Jordan blocks.
We assume the following simultaneous Poincar\’e condition; there exists $c>0$ suchthat.
(1.30)$X^{\mu}(x)$$J^{\mu}$$J^{\mu}$

$\nu$

$\mu$

$J^{\mu}$$J^{\mu}$$\max_{1\leq\mu\leq d}|\lambda^{\mu}\alpha-\lambda_{k}^{\mu}|\geq c|\alpha|$$\forall\alpha\in \mathrm{Z}_{+}^{n}1\leq\forall k\leq n$

We note that this condition is stronger than the usual Poincar\’e condition if $d=1$
because we have a nonresonance condition. Then we have the following, $X^{d}(x)$ be pairwise commuting holomorphic vector
Theorem 1. 4 Let $X^{1}(x),$(1.30) we can find a neighborhood of the origin
verifyabove.
If
fields as
and a holomorphic change of the variables $x=y+u(y),$ $y\in\Omega$ which transforms$x^{d}(x)$ into their linear parts, respectively.
simultaneously $X^{1}(x),$Moreover, is a solution of (1.7).
$\ldots$$\lambda^{d}$$\lambda^{1},$$\Omega$$\ldots,$$J^{d}y\partial_{y}$$J^{1}y\partial_{y},$$\ldots,$$\ldots,$$u$Remark. The novelity of the theorem lies in the case $d\geq 2$ under natural extensionof a usual Poincar\’e condition for a single equation.
In order to prove Theorem 2.4 we define the operator $M$ by (1.10). Then we have
Lemma 1. 5 Suppose that (1.30) is satisfied. Then the followings holds;$\tau_{0>}^{\dot{i})}\mathrm{o}su\tau he_{C}re_{h}eXi_{S}tSaSthat_{\dot{i}nthe}CaleneCwhangcoOerd_{\dot{i}na}teofvar\dot{i}ablesX_{j\rho_{j}}=y,theinverseM(yj\rho_{j}>0,j:-1(H(T))^{n}=1,$$\ldots,n)andarrow(H(T))^{n}$exists as a continuous linear operator for any $0<T<T_{0}$ .
We have$i\dot{i})$$[M^{-1}, \mathcal{L}_{\lambda^{\mu}}]=[M^{-1}, \mathcal{L}_{\overline{\lambda^{\mu}}}]=0$

Especially, the operator $P(D),$Proof. We write$(\mathit{1}.\mathit{1}\mathit{5})$for every$1\leq\mu\leq d$.

(1.31)

satisfies (1.16).

correspond to semi-simple, whereandchangeof variables in the lemmaand nilpotent part of , respectively. Because the’can beintotransforms’ it follows that$M$
has themade arbitrarily small by an appropriate choice of . It follows that$\mathcal{L}_{\lambda^{\mu}}=\mathcal{L}_{\lambda^{\mu}}’+\mathcal{L}_{\lambda\mu}^{\prime J}$$\mathcal{L}_{\lambda^{\mu}}’$$\mathcal{L}_{\lambda^{\mu}}’’$$J^{\mu}$$x_{\nu+l}\partial_{x_{\nu}}$$\rho_{\nu+}\ell\rho_{\nu}^{-1}y_{\nu}+\ell\partial y\nu$$( \sum_{\mu=1}^{d\prime}\mathcal{L}_{\frac{\prime}{\lambda^{\mu}}\mathcal{L}_{\lambda^{\mu}}})-1\mathcal{L}_{\lambda\mu}’$$\rho_{j}$206representation, where has small norm while the first term isinvertible by (1.30). This proves i).
In order to prove ii) it is sufficient to show thatfor every $1\leq\mu\leq d$ . In
$M$view of the definition of
we shall show the commutativity ofand. Notingthat $[J^{\mu}, J^{\nu}]=0$ and the symmetry of
we have, for $u\in(H(T))^{n}$$M= \Sigma_{\mu=1\lambda\mu}^{d}\mathcal{L}\prime \mathcal{L}\frac{\prime}{\lambda^{\mu}}+\mathit{6}$$\epsilon$$[M, \mathcal{L}_{\lambda^{\mu}}]=0$$\mathcal{L}_{\lambda^{\mu}}$$\mathcal{L}_{\lambda^{\nu}}$$\partial^{2}u$$[\mathcal{L}_{\lambda^{\mu}}, \mathcal{L}_{\lambda^{\nu}}]u$$-\partial_{x}(\partial_{x}uJ\mu x-J\mu u)J\nu x$$=$$\partial_{x}(\partial_{x}uJ^{\nu}x-J^{\nu}u)J^{\mu}X-J^{\mu}(\partial_{x}uJ^{\nu}X-J\nu u)$$+$$J^{\nu}(\partial_{x}uJ^{\mu_{X}}-J\mu u)=J^{\mu_{X}}\partial 2J^{\nu}x-tJtu\nu_{X\partial u}2J\mu_{X}=0$.

This ends the proof.
The proof of Theorem 2.4 can be proved by the same argument as in Theorem 2.1.

2

Normal forms of commuting holomorphic diffeomorphisms

We consider pairwise commuting local biholomorphic maps
: $C^{n}arrow C^{n},$ $\mu=$in a neighbourhood of the fixed point . Hence we can write$d$$\Phi_{\mu}$

$1,$

$d$

$\Omega$$0$$\ldots,$$\Phi_{\mu}(x)=\Lambda_{\mu}x+\varphi_{\mu}(x)$where $\Lambda_{\mu}\in GL(n:C)$ and$\varphi_{\mu}\in(C_{1}\{x\})^{n}$$\varphi_{\mu}(x)=o(|x|^{2}),$

,

(2.1)

$\mu=1,$ $\ldots,$ $d$, i.e.

$|x|arrow 0$

,

(2.2)

$\mu=1,$ $\ldots,$ $d$

The commuting relation
implies
for all$\nu=1,$. Without loss of generality (after a linear change of thebles)
we may assume that all matrices are in the Jordan normal forms with identical block
structures. We set
to be the vector consisting of the eigenvalues$\mu=1,$of the matrix.
Our result for (simultaneous) analytic equvalence of the maps to their linear parts
will be proved under the additional requirement that all matrices are semisimple.
Therefore we can set$\Phi_{\mu}\circ\Phi_{\nu}=\Phi_{\nu}\circ\Phi_{\mu}$$\mu,$$\Lambda_{\mu}\circ\Lambda_{\nu}=\Lambda_{\nu}0\Lambda_{\mu}$$d$$\mathrm{v}\mathrm{a}\mathrm{r}\mathrm{i}\dot{\mathrm{a}}$$\ldots,$$\lambda_{\mu}=$$(\lambda_{\mu 1}, \ldots , \lambda_{\mu n})$$d$$\Lambda_{\mu},$$\ldots,$$\Lambda_{\mu}=d_{\dot{i}}ag\{\lambda 1, \ldots, \lambda_{\mu 1}\}\mu$’

We suppose that the vectors,$\lambda_{\mu}^{\alpha}=\lambda^{\alpha_{1}}\mu 1\ldots\lambda_{\mu}^{\alpha_{n}}n\neq\lambda_{\mu j}$$\lambda_{\mu},$$j=1,$$\lambda_{\mu j}\in C,$$\mu=1,$ $\ldots,$ $d$$\alpha\in \mathbb{Z}_{+}^{n},$$\ldots,$$n,$$\mu=1,$ $\ldots,$ $d$

.

(2.3)

are nonresonant, namely$|\alpha|\geq 2,$$j=1,$$\ldots,$$n,$$\mu=1,$ $\ldots,$ $d$

.

(2.4)

In fact, we will impose a simultaneous Brjuno type condition: there exist two
positive constants and such that$c_{0}$$\tau$$\min_{1\leq j\leq n\leq}\max_{1\mu\leq d}|\lambda\alpha-\lambda_{\mu}j|\mu c_{0}\geq\exp(-\frac{|\alpha|}{(\ln|\alpha|)1+\tau})$,$\alpha\in \mathbb{Z}_{+}^{n},$$|\alpha|\geq 2$

.

(2.5)

207

Furthermore, in the case of more than one diffeomorphism (i.e. $d\geq 2$ ) we impose

the following restriction: there exist a constant $C_{0}>0$ such that$\frac{\Sigma_{\mu^{--}1}^{d}|\lambda^{\mu}\gamma|}{1+\Sigma_{\mu=}^{d}1|\lambda\mu\alpha|}\leq C_{0}$,$\forall\alpha,$$\gamma\in \mathrm{Z}_{+}^{n},$$|\gamma|\geq 2,$$\gamma\leq\alpha$

.

(2.6)

belongs to the
We note that in the case of a single map $(d=1)$ the vector(cf. [1, p. 311]). In thatorPoincar\’e domain if eithercase the condition (2.6) is always fulfilled. One checks easily that it is also true when
the space dimension is one $(n=1)$ while is arbitrary positive integer. Then we have$\lambda_{1}$$j^{\max_{=1}},\ldots,d|\lambda_{1j}|<1$$\min_{j=1,\ldots,d}|\lambda_{1j}|>1$$d$be pairwise commuting local biholomorphic maps preTheorem 2. 1 Let
seming the origin and satisfying the conditions (2.5) and (2.6). Then we can find aneighborhood $B$ of the origin and a holomorphic change of the variables $yarrow x=u(y)$reinto their linear parts
which transforms simultaneously
spectively.
$\Phi_{1},$$\Phi_{d}$$\ldots,$$\Phi_{d}$$\Phi_{1},$$\Lambda_{1^{X..\Lambda}d^{X}},.,,$$\ldots,$

References[1] V. I. Arnold, Geometrical Methods in the Theory of Ordinary Differential Equations, Springer, New York- Heidelberg-Berlin, 1983.
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$Ast\acute{e}r\dot{i}sque$,[3] A.D. Brjuno, Local Methods in Nonlinear Differential Equations, Springer,
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[8] J. Moser, On commuting circle mappings and simultaneous Diophantine approximations, Math. Z., 205 (1990), 105-121.
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Department of Mathematics, Chuo University
Higashinakano Hachioji, Tokyo 192-03, Japan-mail: yoshinom@tamacc.chuo-u.ac.jp$\mathrm{e}$