FM: 01-07; mp_arc ??; xxx ?????

Autori : G. Gallavotti, G.Gentile

Titolo: Hyperbolic low-dimensional invariant tori and summations of divergent series

Riassunto: We consider a class of {\sl a priori} stable quasi-integrable analytic Hamiltonian systems and study the regularity of low-dimensional hyperbolic invariant tori as functions of the perturbation parameter. We show that, under natural nonresonance conditions, such tori exist and can be identified through the maxima or minima of a suitable potential. They are analytic inside a disc centered at the origin and deprived of a region around the positive or negative real axis with a quadratic cusp at the origin. The invariant tori admit an asymptotic series at the origin with Taylor coefficients that grow at most as a power of a factorial and a remainder that to any order $N$ is bounded by the $(N+1)$-st power of the argument times a power of $N!$. We show the existence of a summation criterion of the (generically divergent) series, in powers of the perturbation size, that represent the parametric equations of the tori by following the renormalization group methods for the resummations of perturbative series in quantum field theory.

  Keywords: Mechanics, Stability, Divergent Series, Lindstedt Series,Resummations, Renormalization Group, Quantum Field Theory

Giovanni Gallavotti
Fisica, Universita' di Roma 1
P.le Moro 2
00185 Roma, Italia
tel +39-06-4991-4370
fax +39-06-4957697

Guido Gentile
Matematica, Universita' di Roma 3
L. S. Leonardo Murialdo
00100 Roma, Italia
tel +39-06-54-888-226