Differentiable invariant manifolds of nilpotent parabolic points

TL;DR

This paper investigates the existence, regularity, and computation of invariant manifolds near parabolic fixed points of maps with non-diagonalizable derivatives, using the parameterization method and fiber contraction theorem.

Contribution

It establishes conditions for the existence and smoothness of invariant curves and provides an algorithm for their approximation and normal form computation.

Findings

Invariant curves of class C^r exist under certain conditions.

Invariant curves are analytic if the map is analytic.

An algorithm for approximating invariant curves and normal forms is proposed.

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.