Non Resonant Limit for Sequences of Resonant Orbits: the Case of Holomorphic Maps

The approximation of a nonresonant orbit with a sequence of resonant orbits is considered for the holomorphic maps of the complex plane. The problem is motivated by hamiltonian dynamics (Greene's conjecture) and we consider a complexiied hamiltonian map in the region (far from the section of real dynamics) where it can be reduced to an holomorphic map of a single complex variable. For a sequence of maps in normal form with linear resonant frequencies, the limit to a linear map with nonresonant diophantine frequency has a simple interpretation: the ower-like resonant orbits become circles due to the increase of the number of petals and the freezing of radial motion. A similar nontrivial result is proved for small perturbation of the normal forms by investigating the behaviour of the conjugation functions.