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On the sensitivity of sectional-Anosov flows

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Abstract

We study sectional-Anosov flows on compact 3-manifolds for which the maximal invariant and nonwandering sets coincide. We prove that every vector field close to one of these flows is sensitive with respect to initial conditions.

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References

  1. Araujo, V., Pacifico, M., J.: Three dimensional flows. Publicações Matemáticas do IMPA. [IMPA Mathematical Publications] 26° Colquio Brasileiro de Matemática. [26th Brazilian Mathematics Colloquium] Instituto Nacional de Matemática Pura e Aplicada (IMPA), Rio de Janeiro (2007)

  2. Araujo V., Pacifico M.J., Pujals E.R., Viana M.: Singular-hyperbolic attractors are chaotic. Trans. Am. Math. Soc. 361(5), 2431–2485 (2009)

    Article  MATH  MathSciNet  Google Scholar 

  3. Bautista S., Morales C.: A sectional-Anosov connecting lemma. Ergodic Theory Dynam. Syst. 30, 339–359 (2010)

    Article  MATH  MathSciNet  Google Scholar 

  4. Bautista S., Morales C.: Existence of periodic orbits for singular-hyperbolic sets. Mosc. Math. J. 6(2), 265–297 (2006)

    MATH  MathSciNet  Google Scholar 

  5. Benedicks M., Viana M.: Solution of the basin problem for Hénon-like attractors. Invent. Math. 143(2), 375–434 (2001)

    Article  MATH  MathSciNet  Google Scholar 

  6. Carballo C.M., Morales C.A.: Omega-limit sets close to singular-hyperbolic attractors. Illinois J. Math. 48(2), 645–663 (2004)

    MATH  MathSciNet  Google Scholar 

  7. Fried D.: The geometry of cross sections to flows. Topology 21(4), 353–371 (1982)

    Article  MATH  MathSciNet  Google Scholar 

  8. Hirsch, M., Pugh, C., Shub, M.: Invariant manifolds. In: Lec. Not. in Math., vol. 583. Springer, Berlin (1977)

  9. Komuro M.: Expansive properties of Lorenz attractors, the theory of dynamical systems and its applications to nonlinear problems (Kyoto, 1984), pp. 4–26. World Sci. Publishing, Singapore (1982)

    Google Scholar 

  10. Metzger R., Morales C.: Sectional-hyperbolic systems. Ergodic Theory Dynam. Syst. 28(5), 1587–1597 (2008)

    Article  MATH  MathSciNet  Google Scholar 

  11. Morales, C.: An improved sectional-Anosov closing lemma. Math. Z. doi:10.1007/s00209-010-0673-x

  12. Morales C.: Sectional-Anosov systems. Monatsh. Math. 159, 253–260 (2010)

    Article  MATH  MathSciNet  Google Scholar 

  13. Morales C.: Strong stable manifolds for sectional-hyperbolic sets. Discrete Contin. Dyn. Syst. 17(3), 553–560 (2007)

    Article  MATH  MathSciNet  Google Scholar 

  14. Morales C.: The explosion of singular-hyperbolic attractors. Ergodic Theory Dynam. Syst. 24(2), 577–591 (2004)

    Article  MATH  Google Scholar 

  15. Morales C., Pacifico M.J., Pujals E.R.: Robust transitive singular sets for 3-flows are partially hyperbolic attractors or repellers. Ann. Math. (2) 160(2), 375–432 (2004)

    Article  MATH  MathSciNet  Google Scholar 

  16. Morales C., Pacifico M.J., Pujals E.R.: Singular-hyperbolic systems. Proc. Am. Math. Soc. 127, 3393–3401 (1999)

    Article  MATH  MathSciNet  Google Scholar 

  17. Morales C., Pujals E.R.: Singular strange attractors on the boundary of Morse-Smale systems. Ann. Sci. École Norm. Sup. (4) 30(6), 693–717 (1997)

    MATH  MathSciNet  Google Scholar 

  18. Shilnikov L.P., Turaev D.V.: An example of a wild strange attractor. (Russian) Mat. Sb. 189(2), 137–160 (1998)

    MathSciNet  Google Scholar 

  19. Shilnikov L.P., Turaev D.V.: An example of a wild strange attractor. translation in Sb. Math. 189(1–2), 291–314 (1998)

    MathSciNet  Google Scholar 

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Authors and Affiliations

  1. Instituto de Matemática, Universidade Federal do Rio de Janeiro, P. O. Box 68530, Rio de Janeiro, 21945-970, Brazil

    A. Arbieto, C. Morales & L. Senos

Authors
  1. A. Arbieto
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  2. C. Morales
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  3. L. Senos
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Corresponding author

Correspondence to C. Morales.

Additional information

Partially supported by CNPq, FAPERJ and PRONEX/DYN-SYS from Brazil.

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Arbieto, A., Morales, C. & Senos, L. On the sensitivity of sectional-Anosov flows. Math. Z. 270, 545–557 (2012). https://doi.org/10.1007/s00209-010-0811-5

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  • DOI: https://doi.org/10.1007/s00209-010-0811-5

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