Quasiperiodic Poincare Persistence at High Degeneracy

TL;DR

This paper proves a quasiperiodic Poincare theorem demonstrating the persistence of resonant invariant tori in highly degenerate Hamiltonian systems, addressing a longstanding conjecture in the field.

Contribution

It establishes a general result on the persistence of resonant invariant tori at high degeneracy levels, extending classical theory to more complex Hamiltonian systems.

Findings

Proves quasiperiodic Poincare theorem at high degeneracy

Shows persistence of various types of resonant invariant tori

Answers a longstanding conjecture in Hamiltonian dynamics

Abstract

For Hamiltonian systems with degeneracy of any higher order, we study the persistence of resonant invariant tori, which as some lower-dimensional invariant tori might be elliptic, hyperbolic or of mixed types. Hence we prove a quasiperiodic Poincare theorem at high degeneracy. This answers a long standing conjecture on the persistence of resonant invariant tori in quite general situations.