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Differentiable invariant manifolds of nilpotent parabolic points
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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.
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CUFÍ CABRÉ, Clara and FONTICH, Ernest. Differentiable invariant manifolds of nilpotent parabolic points. Discrete and Continuous Dynamical Systems-Series A. 2021. Vol. 41, num. 10, pags. 4667- 4704. ISSN 1078-0947. [consulted: 7 of June of 2026]. Available at: https://hdl.handle.net/2445/183479