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Transactions of the American Mathematical Society

ISSN 1088-6850(online) ISSN 0002-9947(print)

 

 

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


Authors: Radu Saghin and Zhihong Xia
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
Published electronically: June 19, 2006
MathSciNet review: 2231887
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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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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
Email: xia@math.northwestern.edu

DOI: https://doi.org/10.1090/S0002-9947-06-04171-7
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.
Article copyright: © Copyright 2006 American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication.