Remote Access Proceedings of the American Mathematical Society
Green Open Access

Proceedings of the American Mathematical Society

ISSN 1088-6826(online) ISSN 0002-9939(print)



Another dichotomy for surface diffeomorphisms

Author: C. Morales
Journal: Proc. Amer. Math. Soc. 137 (2009), 2639-2644
MSC (2000): Primary 37D30; Secondary 37E30
Published electronically: March 27, 2009
MathSciNet review: 2497476
Full-text PDF Free Access

Abstract | References | Similar Articles | Additional Information

Abstract: We prove that a $ C^1$ generic orientation-preserving diffeomorphism of a closed orientable surface either has infinitely many periodic points with complex (nonreal) eigenvalues or is Axiom A without cycles. This improves Mañé's dichotomy.

References [Enhancements On Off] (What's this?)

  • 1. Abdenur, F., Bonatti, C., Crovisier, S., Diaz, L. J.,
    Generic diffeomorphisms on compact surfaces.
    Fund. Math. 187 (2005), no. 2, 127-159. MR 2214876 (2006m:37024)
  • 2. Asaoka, M.,
    Markov covers and finiteness of periodic attractors for diffeomorphisms with a dominated splitting.
    Ergodic Theory Dynam. Systems 20 (2000), no. 1, 1-14. MR 1747033 (2001b:37040)
  • 3. Bonatti, C., Diaz, L. J., Pujals, E. R.,
    A $ C\sp 1$-generic dichotomy for diffeomorphisms: Weak forms of hyperbolicity or infinitely many sinks or sources.
    Ann. of Math. (2) 158 (2003), no. 2, 355-418. MR 2018925 (2007k:37032)
  • 4. Franks. F.,
    Necessary conditions for stability of diffeomorphisms.
    Trans. Amer. Math. Soc. 158 (1971), 301-308. MR 0283812 (44:1042)
  • 5. Mañé, R.,
    An ergodic closing lemma.
    Ann. of Math. (2) 116 (1982), 503-540. MR 678479 (84f:58070)
  • 6. Newhouse, S.,
    Lectures on Dynamical Systems.
    In Dynamical Systems, Progress in Mathematics (CIME Lectures 1978), 8, pages 1-114. Birkhäuser, Boston, 1980. MR 589590 (81m:58028)
  • 7. Pujals, E. R., Sambarino, M.,
    Homoclinic tangencies and hyperbolicity for surface diffeomorphisms.
    Ann. of Math. (2) 151 (2000), 961-1023. MR 1779562 (2001m:37057)

Similar Articles

Retrieve articles in Proceedings of the American Mathematical Society with MSC (2000): 37D30, 37E30

Retrieve articles in all journals with MSC (2000): 37D30, 37E30

Additional Information

C. Morales
Affiliation: Instituto de Matemática, Universidade Federal do Rio de Janeiro, C. P. 68.530, 21945-970, Rio de Janeiro, RJ, Brazil

Keywords: Axiom A diffeomorphism, homoclinic tangency, complex eigenvalues
Received by editor(s): August 18, 2008
Published electronically: March 27, 2009
Communicated by: Jane M. Hawkins
Article copyright: © Copyright 2009 American Mathematical Society

American Mathematical Society