Partial hyperbolicity or dense elliptic periodic points for C 1-generic symplectic diffeomorphisms

RADU SAGHIN , Zhihong Xia

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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 languageEnglish
Pages (from-to)5119-5136
Number of pages18
JournalTransactions of the American Mathematical Society
Issue number11
StatePublished - 1 Nov 2006

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