# Gevrey estimates for one dimensional parabolic invariant manifolds of non-hyperbolic fixed points

## TL;DR

This paper investigates the Gevrey regularity of one-dimensional invariant manifolds near parabolic fixed points in analytic maps, providing explicit conditions, optimality examples, and applications to celestial mechanics.

## Contribution

It establishes Gevrey estimates for invariant manifolds at parabolic fixed points and links these estimates to explicit constants, advancing understanding of their regularity.

## Key findings

- Invariant manifolds are Gevrey with type related to explicit constants.
- Examples demonstrate the optimality of the Gevrey estimates.
- Applications include celestial mechanics problems like Sitnikov and three-body problems.

## Abstract

We study the Gevrey character of a natural parameterization of one dimensional invariant manifolds associated to a parabolic direction of fixed points of analytic maps, that is, a direction associated with an eigenvalue equal to $1$. We show that, under general hypotheses, these invariant manifolds are Gevrey with type related to some explicit constants. We provide examples of the optimality of our results as well as some applications to celestial mechanics, namely, the Sitnikov problem and the restricted planar three body problem.