Right Type Departmental Bulletin Paper

We shall show simultaneous normal forms of a system of vector fields and local diffeomorphisms under Brjuno condition. These results are proved by a new scheme of a generalized impl.icit function theorem with high loss of derivatives such that for some $\epsilon, $ $0<\epsilon<1 $, $\exp(e^{\frac{1}{(\sigma-\sigma)^{\epsilon}})}, $ $0<\sigma’<\sigma$. The nonlinear equations (homological equation) are overdetermined system of equa-tions. 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} $, (1) $X^{\mu} = \langle X^{\mu}, \partial_{x}\rangle=\sum_{1j=}^{n}X_{j}^{\mu}(x)\partial xj $ $1\leq\mu\leq d$ $\partial_{x}=(\partial_{x_{1}}, \cdots, \partial_{Xn}) $ , $\partial_{x_{\dot{r}}}=\frac{\partial}{\mathrm{o}_{---}}$ $\partial_{x_{j}}=\overline{\partial_{X_{j}}^{-}}$ We assume (2) $X^{\mu}(1\leq\mu\leq d) $ are singular i.e. $X^{\mu}(\mathrm{O})=0 $ for $1\leq\mu\leq d$. The linear parts of $X^{\mu}(1\leq\mu\leq d) $ are semi-simple i.e., (3) $X^{\mu}(x)=(X_{1}^{\mu}(x), \cdots, X^{\mu}(nx))=\Lambda^{\mu}x+R^{\mu}(x)$, $1\leq\mu\leq d $, 1Supported by the Volkswagen-Stiftung ( $\mathrm{R}\mathrm{i}\mathrm{P}$-program at Oberwolfach) and partially supported b