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

It is shown that every symplectic diffeomorphism of R 2 n 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 ∊ R n , y ∊ R n , and V is a polynomial R n → R and η∊ 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.