There is only one KAM curve

Carlo Carminati (1) , Stefano Marmi (2) , David Sauzin (2, 3)

1 Università di Pisa, Dipartimento di Matematica
2 SNS PISA - Gruppo di ricerca di sistemi dinamici
3 IMCCE - Institut de Mécanique Céleste et de Calcul des Ephémérides

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.

Mots clés: KAM theory; small divisors; Borel's monogenic functions; quasianalytic spaces

Domaines: Systèmes dynamiques [math.DS]

Carlo Carminati, Stefano Marmi, David Sauzin. There is only one KAM curve. Nonlinearity, 2014, 27 (9), pp.2035--2062. ⟨hal-00585973v2⟩