Partial hyperbolicity or dense elliptic periodic points for 𝐶¹-generic symplectic diffeomorphisms

Abstract

We prove that if a symplectic diffeomorphism is not partially hyperbolic, then with an arbitrarily small C 1 C^1 perturbation we can create a totally elliptic periodic point inside any given open set. As a consequence, a C 1 C^1 -generic symplectic diffeomorphism is either partially hyperbolic or it has dense elliptic periodic points. This extends the similar results of S. Newhouse in dimension 2 and M.-C. Arnaud in dimension 4. Another interesting consequence is that stably ergodic symplectic diffeomorphisms must be partially hyperbolic, a converse to Shub-Pugh’s stable ergodicity conjecture for the symplectic case.