NONRESONANT LIMIT FOR SEQUENCES OF RESONANT ORBITS - THE CASE OF HOLOMORPHIC MAPS

The approximation of a non-resonant orbit with a sequence of resonant orbits is considered for the holomorphic maps of the complex plane. Th e problem is motivated by Hamiltonian dynamics (Greene's conjecture) a nd we consider a complexified Hamiltonian map in the region (far from the section of real dynamics) where it can be reduced to a holomorphic map of a single complex variable. For a sequence of maps in normal fo rm with linear resonant frequencies, the limit to a linear map with no n-resonant diophantine frequency has a simple interpretation: the flow er-like resonant orbits become circles due to the increase of the numb er of petals and the freezing of radial motion. A similar non-trivial result is proved for small perturbations of the normal forms by invest igating the behaviour of the conjugation functions.