Giovanni Gallavotti and Guido Gentile
Hyperbolic low-dimensional invariant tori
and summations of divergent series
(611K, PostScript)
ABSTRACT. We consider a class of 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.