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Partial hyperbolicity or dense elliptic periodic points for -generic symplectic diffeomorphisms
Authors:Radu Saghin  Zhihong Xia
Institution:Department of Mathematics, Northwestern University, Evanston, Illinois 60208 ; Department of Mathematics, Northwestern University, Evanston, Illinois 60208
Abstract:We prove that if a symplectic diffeomorphism is not partially hyperbolic, then with an arbitrarily small $ C^1$ perturbation we can create a totally elliptic periodic point inside any given open set. As a consequence, a $ 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.

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