Stability Domain and Invariant Manifolds of 2d Area-Preserving Diffeomorphisms

We study the stability domain of generic 2D area-preserving polynomialdiffeomorphisms. The starting point of our analysis is the study of thedistribution of stable and unstable fixed points. We show that the locationof fixed points and their stability type are linked to the degree of thepolynomial map. These results are based on a classification Theorem forplane automorphisms by Friedland and Milnor. Then we discuss the problem ofdetermining the domain in phase space where stable motion occurs. We showthat the boundary of the stability domain is given by the invariantmanifolds emanating from the outermost unstable fixed point of low period(one or two). This fact extends previous results obtained for reversiblearea-preserving polynomial maps of the plane. This analysis is based onanalytical arguments and is supported by the results of numericalsimulations.