There is only one KAM curve

TL;DR

This paper introduces a novel way to encode all KAM curves of area-preserving twist maps into a single complex function, revealing a form of quasianalyticity despite natural boundaries caused by resonances.

Contribution

It demonstrates that KAM curves can be represented by a monogenic function in a complex domain, unifying their description and establishing a quasianalyticity property.

Findings

Unified encoding of KAM curves via a monogenic function

Establishment of quasianalyticity for the encoding function

Natural boundary due to resonance density

Abstract

We consider the standard family of area-preserving twist maps of the annulus and the corresponding KAM curves. Addressing a question raised by Kolmogorov, we show that, instead of viewing these invariant curves as separate objects, each of which having its own Diophantine frequency, one can encode them in a single function of the frequency which is naturally defined in a complex domain containing the real Diophantine frequencies and which is monogenic in the sense of Borel; this implies a remarkable property of quasianalyticity, a form of uniqueness of the monogenic continuation, although real frequencies constitute a natural boundary for the analytic continuation from the Weierstrass point of view because of the density of the resonances.