Unique resonant normal forms for area preserving maps at an elliptic fixed point

TL;DR

This paper develops a unique resonant normal form for area-preserving maps near elliptic fixed points, providing a formal classification with infinite invariants, applicable to both weak and strong resonances.

Contribution

It introduces a novel method for constructing unique resonant normal forms for area-preserving maps at elliptic fixed points, including analytic families, using non-linear grading functions.

Findings

Normal forms are unique and provide formal local classification.

Constructed for both weak and strong resonances.

Applicable to analytic families of maps.

Abstract

We construct a resonant normal form for an area-preserving map near a generic resonant elliptic fixed point. The normal form is obtained by a simplification of a formal interpolating Hamiltonian. The resonant normal form is unique and therefore provides the formal local classification for area-preserving maps with the elliptic fixed point. The total number of formal invariants is infinite. We consider the cases of weak (of order $n\ge5$) and strong (of order $n=3,4$) resonances. We also construct unique normal forms for analytic families of area-preserving maps. We note that our constructions involve non-linear grading functions.