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

It is shown that every symplectic diffeomorphism of R²ⁿ can be approximated, in the C^∞-topology, on any compact set, by some iteration of some map of the form (x, y) → (y + η, -x + ∇V(y)) where x ∈ Rⁿ, y ∈ Rⁿ, and V is a polynomial Rⁿ → R and η ∈ Rⁿ 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.