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

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- by Radu Saghin and Zhihong Xia PDF
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**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.## References

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## Additional Information

**Radu Saghin**- Affiliation: Department of Mathematics, Northwestern University, Evanston, Illinois 60208
- Address at time of publication: Department of Mathematics, University of Toronto, Toronto, Ontario, Canada M5S 2E4
- Email: rsaghin@math.northwestern.edu, rsaghin@fields.utoronto.ca
**Zhihong Xia**- Affiliation: Department of Mathematics, Northwestern University, Evanston, Illinois 60208
- MR Author ID: 271126
- Email: xia@math.northwestern.edu
- Received by editor(s): December 2, 2004
- Published electronically: June 19, 2006
- Additional Notes: This research was supported in part by the National Science Foundation.
- © Copyright 2006
American Mathematical Society

The copyright for this article reverts to public domain 28 years after publication. - Journal: Trans. Amer. Math. Soc.
**358**(2006), 5119-5138 - MSC (2000): Primary 37C25, 37D30
- DOI: https://doi.org/10.1090/S0002-9947-06-04171-7
- MathSciNet review: 2231887