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

Saghin, Radu; Xia, Zhihong

doi:10.1090/S0002-9947-06-04171-7

Partial hyperbolicity or dense elliptic periodic points for $C^1$-generic symplectic diffeomorphisms
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by Radu Saghin and Zhihong Xia PDF

Trans. Amer. Math. Soc. 358 (2006), 5119-5138 Request permission

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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