## 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.

Original language | English (US) |
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Pages (from-to) | 5119-5136 |

Number of pages | 18 |

Journal | Transactions of the American Mathematical Society |

Volume | 358 |

Issue number | 11 |

DOIs | |

State | Published - Nov 2006 |

## ASJC Scopus subject areas

- Mathematics(all)
- Applied Mathematics

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