We prove that for each characteristic direction $[v]$ of a tangent to the identity diffeomorphism of order $k+1$ in $\mathbb{C}^2$ there exist either an analytic curve of fixed points tangent to $[v]$ or $k$ parabolic manifolds where all the orbits are tangent to $[v]$, and that at least one of these parabolic manifolds is or contains a parabolic curve.

中文翻译：

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二维双全同胚的特征方向

我们证明，对于 $\mathbb{C}^2$ 中 $k+1$ 阶恒等微分同构的切线的每个特征方向 $[v]$，存在一条与 $[v 相切的不动点的解析曲线]$ 或 $k$ 抛物线流形，其中所有轨道都与 $[v]$ 相切，并且这些抛物线形流形中的至少一个是或包含抛物线。