In this paper, we consider the normal form problem of a commutative family of germs of diffeomorphisms at a fixed point, say the origin, of \(\mathbb {K}^n (\mathbb {K}={\mathbb {C}}\text {~or~} {\mathbb {R}})\). We define a notion of integrability of such a family. We give sufficient conditions which ensure that such an integrable family can be transformed into a normal form by an analytic (resp. a smooth) transformation if the initial diffeomorphisms are analytic (resp. smooth).

中文翻译：

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微差同构及其局部范式的完全可积性

在本文中，我们考虑了一个固定点上的同变易变种的交换族的正规形式问题，例如\（\ mathbb {K} ^ n（\ mathbb {K} = {\ mathbb {C} } \ text {〜or〜} {\ mathbb {R}}）\）。我们定义了这样一个家庭的可整合性概念。我们给出了足够的条件，以确保如果初始微分态是解析的（可平滑的），则可解析的（可平滑的）变换可将此类可积族转化为正常形式。