FM 05-17; mp_arc 05-337; math-ph/0509056

Author: Giovanni Gallavotti, Guido Gentile, Alessandro Giuliani

TitleFractional Lindstedt series: (version 0.1: comments welcome; )

Abstract: The parametric equations of the surfaces on which highly resonant quasi-periodic motions develop (lower-dimensional tori) cannot be analytically continued, in general, in the perturbation parameter $\e$, i.e. they are not analytic functions of $\e$. However rather generally quasi-periodic motions whose frequencies satisfy only one rational relation (``resonances of order $1$'') admit formal perturbation expansions in terms of a fractional power of $\e$ depending on the degeneration of the resonance. We find conditions for this to happen, and in such a case we prove that the formal expansion is convergent after suitable resummation.

Key words:  Resonances, KAM theory, Divergent Series, Lindstedt Series
Giovanni Gallavotti
Dipartimento di Fisica, INFN
Universita' di Roma "La Sapienza"
P.le A. Moro 2
I 00185 Roma, Italia

Guido Gentile
Dipartimento di Matematica
Universita` di Roma 3
Largo S. Leonardo Murialdo

Alessandro Giuliani
Mathematics Department
Princeton University
Princeton, NJ 08540