Differentiable invariant manifolds of nilpotent parabolic points

Clara Cufí-Cabré; Ernest Fontich

doi:10.3934/dcds.2021053

Article Contents

2021, Volume 41, Issue 10: 4667-4704. Doi: 10.3934/dcds.2021053

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Differentiable invariant manifolds of nilpotent parabolic points

Clara Cufí-Cabré^1, and
Ernest Fontich^2,

1.
Departament de Matemàtiques, Universitat Autònoma de Barcelona (UAB), Barcelona Graduate School of Mathematics (BGSMath), 08193 Bellaterra, Spain
2.
Departament de Matemàtiques i Informàtica, Universitat de Barcelona (UB), Barcelona Graduate School of Mathematics (BGSMath), Gran Via 585. 08007 Barcelona, Spain

Received: October 2020

Revised: February 2021

Early access: April 2021

Published: October 2021

The first author has been partially supported by the Spanish Government grants MTM2016-77278-P (MINECO/ FEDER, UE), PID2019-104658GB-I00 (MICINN/FEDER, UE) and BES-2017-081570, and by the Catalan Government grant 2017-SGR-1617. The second author has been partially supported by the Spanish Government grants MTM2016-80117-P (MINECO/ FEDER, UE) and PID2019-104851GB-I00 (MICINN/FEDER, UE) and by the Catalan Government grant 2017-SGR-1374

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Abstract

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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Mathematics Subject Classification: Primary: 37D10; Secondary: 37C25.

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