Polynomial approximations of symplectic dynamics and richness of chaos in non-hyperbolic area-preserving maps

It is shown that every symplectic diffeomorphism of R-2n, can be approximated, in the C-infinity-topology, on any compact set, by some iteration of some map of the form (x, y) bar right arrow (y + eta, -x + delV(y)) where x is an element of R-n, y is an element of R-n, and V is a polynomial R-n --> R and eta is an element of R-n is a constant vector. For the case of area-preserving maps (i.e. n = 1), it is shown how this result can be applied to prove that C-r-universal maps (a map is universal if its iterations approximate dynamics of all C-r-smooth area-preserving maps altogether) are dense in the C-r-topology in the Newhouse regions.