Non-resonant tori in symplectic twist maps without conjugate points

TL;DR

This paper investigates symplectic twist maps lacking conjugate points, demonstrating the existence of many invariant tori near periodic Lagrangian manifolds where the dynamics resemble non-resonant translations.

Contribution

It establishes the presence of a large family of invariant Lagrangian tori in neighborhoods of periodic manifolds for such twist maps, extending understanding of their structure.

Findings

Existence of invariant Lagrangian tori near periodic manifolds

Conjugation of the twist map to non-resonant translations on these tori

Insights into the dynamics of symplectic twist maps without conjugate points

Abstract

We study the dynamics of a symplectic twist map without conjugate points. We show that in a neighborhood of a totally periodic Lagrangian manifold, there exists a large family of invariant Lagrangian tori on which the twist map is conjugated to a translation of non-resonant vector.