Hyperbolic LowDimensional Invariant Toriand Summations of Divergent Series
Abstract
We consider a class of a priori stable quasiintegrable analytic Hamiltonian systems and study the regularity of lowdimensional 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.
 Publication:

Communications in Mathematical Physics
 Pub Date:
 2002
 DOI:
 10.1007/s002200200599
 arXiv:
 arXiv:mathph/0107012
 Bibcode:
 2002CMaPh.227..421G
 Keywords:

 Mathematical Physics;
 Mathematics  Dynamical Systems;
 Mathematics  Mathematical Physics
 EPrint:
 32 pages, 5 figures