RESONANT NORMAL FORMS, INTERPOLATING HAMILTONIANS AND STABILITY ANALYSIS OF AREA PRESERVING-MAPS

The geometrical and dynamical properties of area preserving maps in the neighborhood of an elliptic fixed point are analyzed in the framework of resonant normal forms. The interpolating flow is not obtained from a map tangent to the identity, but from the normal form of the given map and a time independent interpolating Hamiltonian H is introduced. On this Hamiltonian the local stability properties of the fixed point and the geometric structure of the orbits are transparent. Numerical agreement between the level lines of H and the orbits of the map suggests that the perturbative expansion of H is asymptotic. This is confirmed by a rigorous error analysis, based on majorant series: the error for the normal form expansion grows as n! while the truncation error for H also has a factorial growth and in a disc of radius r can be made exponentially small with 1/r. The boundary of the global stability domain is considered; for the quadratic map the identification with the inner envelope of the homoclinic tangle of the hyperbolic fixed point is strongly suggested by numerical evidence.