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)

 

 

Incompressibility of tori transverse to Axiom A flows


Author: C. A. Morales
Journal: Proc. Amer. Math. Soc. 136 (2008), 4349-4354
MSC (2000): Primary 37D20; Secondary 57M27
Published electronically: June 25, 2008
MathSciNet review: 2431049
Full-text PDF Free Access

Abstract | References | Similar Articles | Additional Information

Abstract: We prove that a torus transverse to an Axiom A vector field that does not exhibit sinks, sources or null homotopic periodic orbits on a closed irreducible $ 3$-manifold is incompressible. This strengthens the works of Brunella (1993), Fenley (1995), and Mosher (1992).


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

  • 1. Apaza., E., Soares, R., Axiom A flows without sinks nor sources on 3-manifolds, Discrete Contin. Dyn. Syst. 21 (2008), no. 2, 393-401.
  • 2. Marco Brunella, Separating the basic sets of a nontransitive Anosov flow, Bull. London Math. Soc. 25 (1993), no. 5, 487–490. MR 1233413, 10.1112/blms/25.5.487
  • 3. Sérgio R. Fenley, Quasigeodesic Anosov flows and homotopic properties of flow lines, J. Differential Geom. 41 (1995), no. 2, 479–514. MR 1331975
  • 4. John Franks and Bob Williams, Anomalous Anosov flows, Global theory of dynamical systems (Proc. Internat. Conf., Northwestern Univ., Evanston, Ill., 1979) Lecture Notes in Math., vol. 819, Springer, Berlin, 1980, pp. 158–174. MR 591182
  • 5. David Gabai, 3 lectures on foliations and laminations on 3-manifolds, Laminations and foliations in dynamics, geometry and topology (Stony Brook, NY, 1998) Contemp. Math., vol. 269, Amer. Math. Soc., Providence, RI, 2001, pp. 87–109. MR 1810537, 10.1090/conm/269/04330
  • 6. Anatole Katok and Boris Hasselblatt, Introduction to the modern theory of dynamical systems, Encyclopedia of Mathematics and its Applications, vol. 54, Cambridge University Press, Cambridge, 1995. With a supplementary chapter by Katok and Leonardo Mendoza. MR 1326374
  • 7. Hatcher, A., Notes on Basic 3-Manifold Topology, available at Hatcher's homepage.
  • 8. John Hempel, 3-Manifolds, Princeton University Press, Princeton, N. J.; University of Tokyo Press, Tokyo, 1976. Ann. of Math. Studies, No. 86. MR 0415619
  • 9. M. W. Hirsch, C. C. Pugh, and M. Shub, Invariant manifolds, Lecture Notes in Mathematics, Vol. 583, Springer-Verlag, Berlin-New York, 1977. MR 0501173
  • 10. William Jaco, Lectures on three-manifold topology, CBMS Regional Conference Series in Mathematics, vol. 43, American Mathematical Society, Providence, R.I., 1980. MR 565450
  • 11. Morales, C., Poincaré-Hopf index and partial hyperbolicity, Ann. Fac. Sci. Toulouse Math. XVII (2008), no. 1, 193-206.
  • 12. Morales, C., Examples of singular-hyperbolic attracting sets, Dyn. Syst. 22 (2007), no. 3, 339-349.
  • 13. C. A. Morales, Incompressibility of torus transverse to vector fields, Topology Proc. 28 (2004), no. 1, 219–228. Spring Topology and Dynamical Systems Conference. MR 2105459
  • 14. C. A. Morales, Axiom A flows with a transverse torus, Trans. Amer. Math. Soc. 355 (2003), no. 2, 735–745. MR 1932723, 10.1090/S0002-9947-02-03127-6
  • 15. Lee Mosher, Dynamical systems and the homology norm of a 3-manifold. I. Efficient intersection of surfaces and flows, Duke Math. J. 65 (1992), no. 3, 449–500. MR 1154179, 10.1215/S0012-7094-92-06518-5
  • 16. Jacob Palis Jr. and Welington de Melo, Geometric theory of dynamical systems, Springer-Verlag, New York-Berlin, 1982. An introduction; Translated from the Portuguese by A. K. Manning. MR 669541
  • 17. Clark Robinson, Dynamical systems, 2nd ed., Studies in Advanced Mathematics, CRC Press, Boca Raton, FL, 1999. Stability, symbolic dynamics, and chaos. MR 1792240

Similar Articles

Retrieve articles in Proceedings of the American Mathematical Society with MSC (2000): 37D20, 57M27

Retrieve articles in all journals with MSC (2000): 37D20, 57M27


Additional Information

C. A. Morales
Affiliation: Instituto de Matemática, Universidade Federal do Rio de Janeiro, P. O. Box 68530, 21945-970 Rio de Janeiro, Brazil
Email: morales@impa.br

DOI: https://doi.org/10.1090/S0002-9939-08-09409-4
Keywords: Sink, vector field, atoroidal, incompressible torus
Received by editor(s): May 22, 2007
Received by editor(s) in revised form: October 3, 2007, and October 24, 2007
Published electronically: June 25, 2008
Additional Notes: This work was supported in part by CNPq, FAPERJ and PRONEX-Brazil. The author thanks Professors E. Apaza, D. Carrasco-Olivera and B. San Martin for helpful conversations. He also thanks the Instituto de Matemáticas Puras e Aplicadas (IMPA) for its kind hospitality.
Communicated by: Jane M. Hawkins
Article copyright: © Copyright 2008 American Mathematical Society