TY - JOUR
T1 - Partial hyperbolicity or dense elliptic periodic points for C 1-generic symplectic diffeomorphisms
AU - Saghin, Radu
AU - Xia, Zhihong
PY - 2006/11
Y1 - 2006/11
N2 - 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.
AB - 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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U2 - 10.1090/S0002-9947-06-04171-7
DO - 10.1090/S0002-9947-06-04171-7
M3 - Article
AN - SCOPUS:33750183630
SN - 0002-9947
VL - 358
SP - 5119
EP - 5136
JO - Transactions of the American Mathematical Society
JF - Transactions of the American Mathematical Society
IS - 11
ER -