We consider a map $ F $ of class $ C^r $ with a fixed point of parabolic type whose differential is not diagonalizable, and we study the existence and regularity of the invariant manifolds associated with the fixed point using the parameterization method. Concretely, we show that under suitable conditions on the coefficients of $ F $, there exist invariant curves of class $ C^r $ away from the fixed point, and that they are analytic when $ F $ is analytic. The differentiability result is obtained as an application of the fiber contraction theorem. We also provide an algorithm to compute an approximation of a parameterization of the invariant curves and a normal form of the restricted dynamics of $ F $ on them.

中文翻译：

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幂零抛物线点的可微不变流形

我们考虑一个类$C^r$的映射$F$，其微分是不可对角化的抛物型不动点，我们使用参数化方法研究了与该不动点相关的不变流形的存在性和规律性。具体地，我们证明在$F$的系数合适的条件下，存在远离不动点的$C^r$类不变曲线，并且当$F$是解析的时，它们是解析的。可微性结果是作为纤维收缩定理的应用而获得的。我们还提供了一种算法来计算不变曲线的参数化的近似值和对它们的限制动力学的标准形式。