One dimensional invariant manifolds of Gevrey type in real-analytic maps

I. Baldomá; Àlex Haro

doi:10.3934/dcdsb.2008.10.295

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2008, Volume 10, Issue 2&3: 295-322. Doi: 10.3934/dcdsb.2008.10.295

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One dimensional invariant manifolds of Gevrey type in real-analytic maps

I. Baldomá^1, and
Àlex Haro^2,

1.
Departament de Matemàtica Aplicada I, Universitat Politècnica de Catalunya, Diagonal 647, 08028 Barcelona
2.
Departament de Matemàtica Aplicada i Anàlisi, Universitat de Barcelona, Gran Via de les Corts Catalanes 585, 08007 Barcelona

Received: October 31, 2006

Revised: June 30, 2007

Published: August 2008

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Abstract

In this paper we study the basic questions of existence, uniqueness, differentiability, analyticity and computability of one dimensional parabolic manifolds of degenerate fixed points, i.e. invariant manifolds tangent to the eigenspace of 1, which is assumed to be a simple eigenvalue. We use the parameterization method, reducing the dynamics on the parabolic manifold to a polynomial. We prove that the asymptotic expansions of the parabolic manifold are of Gevrey type. Moreover, under suitable hypothesis, we also prove that the asymptotic expansions correspond to a real-analytic parameterization of an invariant curve that goes to the fixed point. The parameterization is Gevrey at the fixed point, hence $C^\infty$.

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Mathematics Subject Classification: Primary: 37D10.

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