On Non-KAM Invariant Circles for Area-Preserving Twist Maps

TL;DR

This paper explores the existence and breakdown of invariant circles in area-preserving twist maps beyond classical KAM theory, using innovative methods to address longstanding open questions.

Contribution

It introduces new techniques to analyze non-KAM invariant circles, advancing understanding of their dynamics and providing evidence for open problems posed by Mather.

Findings

Established conditions for existence of non-KAM invariant circles

Proved breakdown scenarios for these invariant circles

Provided new insights supporting Mather's open questions

Abstract

In this note, we investigate the dynamics of invariant circles in area-preserving twist maps. The invariant circles under consideration lie beyond the applicability of classical KAM theory, as the perturbations involved exceed the scope of standard KAM methods. By integrating both constructive and non-constructive techniques, we establish several results on the existence and breakdown of such non-KAM invariant circles. These findings provide new evidence in support of affirmative answers to two open questions posed by Mather in 1998.