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Transactions of the American Mathematical Society

Published by the American Mathematical Society, the Transactions of the American Mathematical Society (TRAN) is devoted to research articles of the highest quality in all areas of pure and applied mathematics.

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

The 2020 MCQ for Transactions of the American Mathematical Society is 1.43.

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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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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