Quasi-analytic properties of the KAM curve

TL;DR

This paper investigates the quasi-analytic properties of KAM curves in near-integrable systems, demonstrating that these curves uniquely determine the systems and exploring their dynamical implications.

Contribution

It establishes strong quasi-analyticity results for KAM curves under analytic regularity, showing they fully characterize the underlying dynamical systems.

Findings

KAM curves uniquely determine the systems they originate from

Shared features of KAM curves imply specific dynamical behaviors

Strong quasi-analyticity properties are proven for KAM curves in analytic settings

Abstract

Classical KAM theory guarantees the existence of a positive measure set of invariant tori for sufficiently smooth non-degenerate near-integrable systems. When seen as a function of the frequency this invariant collection of tori is called the KAM curve of the system. Restricted to analytic regularity, we obtain strong quasi-analyticity properties for these objects. In particular, we prove that KAM curves completely characterize the underlying systems. We also show some of the dynamical implications on systems whose KAM curves share certain common features.