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

2025/10/24 by Guo, Jiashen, Liu, Yi, Wang, Lin · 1 citation
#Dynamical Systems (math.DS) #FOS: Mathematics

paper · doi:10.48550/arxiv.2510.21098

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.

Citations

• Quantitative destruction of invariant circles
• Non-differentiable irrational curves for C¹ twist map
• Some questions around quasi-periodic dynamics
• Boundaries of instability zones for symplectic twist maps
• Regularity properties of critical invariant circles of twist maps, and their universality

Cited by

• Herman's converse KAM mechanism revisited

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