Unique normal forms for area preserving maps near a fixed point with neutral multipliers

TL;DR

This paper develops unique normal forms for area-preserving maps near fixed points with neutral multipliers, providing a comprehensive set of invariants and simplifying Takens normal forms under certain conditions.

Contribution

It introduces a new approach to normal forms for area-preserving maps with neutral multipliers, establishing their uniqueness and formal invariants.

Findings

Takens normal form can be significantly simplified.

Normal forms are unique under non-degeneracy conditions.

Provides a full system of formal invariants.

Abstract

We study normal forms for families of area-preserving maps which have a fixed point with neutral multipliers -1 or +1 at epsilon=0. Our study covers both the orientation-preserving and orientation-reversing cases. In these cases Birkhoff normal forms do not provide a substantial simplification of the system. In the paper we prove that the Takens normal form vector field can be substantially simplified. We also show that if certain non-degeneracy conditions are satisfied no further simplification is generically possible since the constructed normal forms are unique. In particular, we provide a full system of formal invariants with respect to formal coordinate changes.