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Intransitive Sectional-Anosov Flows on 3-manifolds

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For each \(n\in \mathbb {Z}^+\), we show the existence of Venice masks (i.e. intransitive sectional-Anosov flows with dense periodic orbits, Bautista and Morales in Lectures on sectional-Anosov flows. http://preprint.impa.br/Shadows/SERIE_D/2011/86.html, 2011; Bautista and Morales in Discrete Contin Dyn Syst 19(4): 761–775, 2007; López Barragan and Sánchez in Bull Braz Math Soc N Ser 48(1): 1–18, 2017, Morales and Pacífico in Pac J Math 216(2): 327–342, 2004) containing n equilibria on certain compact 3-manifolds. These examples are characterized because of the maximal invariant set is a finite union of homoclinic classes. Here, the intersection between two different homoclinic classes is contained in the closure of the union of unstable manifolds of the singularities.

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References

  1. Afraimovich, V.S., Bykov, V.V., Shilnikov, L.P.: On structurally unstable attracting limit sets of lorenz attractor type. Trudy Moskov. Mat. Obshch. 44(2), 150–212 (1982)

    MathSciNet  Google Scholar 

  2. Arroyo, A., Pujals, E.: Dynamical properties of singular-hyperbolic attractors. Discrete Contin. Dyn. Syst. 19(1), 67–87 (2007)

    Article  MathSciNet  MATH  Google Scholar 

  3. Bautista, S.: The geometric lorenz attractor is a homoclinic class. Bol. Mat. (NS) 11(1), 69–78 (2004)

    MathSciNet  MATH  Google Scholar 

  4. Bautista, S., Morales, C.A.: Lectures on sectional-Anosov flows (2011). http://preprint.impa.br/Shadows/SERIE_D/2011/86.html. Accessed 2011

  5. Bautista, S., Morales, C.A.: On the intersection of sectional-hyperbolic sets. J. Mod. Dyn. 10(1), 1–16 (2016)

    MathSciNet  Google Scholar 

  6. Bautista, S., Morales, C.A., Pacifico, M.J.: On the intersection of homoclinic classes on singular-hyperbolic sets. Discrete Contin. Dyn. Syst. 19(4), 761–775 (2007)

    Article  MathSciNet  MATH  Google Scholar 

  7. Bonatti, C., Pumariño, A., Viana, M.: Lorenz attractors with arbitrary expanding dimension. C. R. Acad. Sci. Paris Sér. I Math. 325(8), 883–888 (1997)

    Article  MathSciNet  MATH  Google Scholar 

  8. Gähler, S.: Lineare 2-normierte räume. Math. Nachr. 28(1–2), 1–43 (1964)

    Article  MathSciNet  MATH  Google Scholar 

  9. Guckenheimer, J., Williams, R.F.: Structural stability of Lorenz attractors. Publ. Math. l’IHÉS 50(1), 59–72 (1979)

    Article  MathSciNet  MATH  Google Scholar 

  10. Hempel, J.: 3-manifolds. In: Annals of Mathematics Studies, No. 86. Princeton University Press, Princeton, N.J.; University of Tokyo Press, Tokyo (1976)

  11. Hirsch, M.W., Pugh, C.C., Shub, M.: Invariant Manifolds, vol. 583. Springer, Berlin (1977)

    Book  MATH  Google Scholar 

  12. Kawaguchi, A., Tandai, K.: On areal spaces i. Tensor NS 1, 14–45 (1950)

    Google Scholar 

  13. López Barragan, A.M., Sánchez, H.M.S.: Sectional anosov flows: existence of venice masks with two singularities. Bull. Braz. Math. Soc. N. Ser. 48(1), 1–18 (2017)

    Article  MathSciNet  MATH  Google Scholar 

  14. Metzger, R., Morales, C.A.: Sectional-hyperbolic systems. Ergod. Theory Dyn. Syst. 28(05), 1587–1597 (2008)

    Article  MathSciNet  MATH  Google Scholar 

  15. Morales, C.: Singular-hyperbolic attractors with handlebody basins. J. Dyn. Control Syst. 13(1), 15–24 (2007)

    Article  MathSciNet  MATH  Google Scholar 

  16. Morales, C.A.: Examples of singular-hyperbolic attracting sets. Dyn. Syst. 22(3), 339–349 (2007)

    Article  MathSciNet  MATH  Google Scholar 

  17. Morales, C.A.: Sectional-Anosov flows. Monatsh. Math. 159(3), 253–260 (2010)

    Article  MathSciNet  MATH  Google Scholar 

  18. Morales, C.A., Pacífico, M.J.: Sufficient conditions for robustness of attractors. Pac. J. Math. 216(2), 327–342 (2004)

    Article  MathSciNet  MATH  Google Scholar 

  19. Morales, C.A., Pacífico, M.J.: A spectral decomposition for singular-hyperbolic sets. Pac. J. Math. 229(1), 223–232 (2007)

    Article  MathSciNet  MATH  Google Scholar 

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

  21. Morales, C.A., Vilches, M.: On 2-riemannian manifolds. SUT J. Math. 46(1), 119–153 (2010)

    MathSciNet  MATH  Google Scholar 

  22. Reis, J.: Infinidade de órbitas periódicas para fluxos seccional anosov. Tese de Doutorado, UFRJ (2011)

  23. Smale, S.: Differentiable dynamical systems. Bull. Am. Math. Soc. 73(6), 747–817 (1967)

    Article  MathSciNet  MATH  Google Scholar 

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Authors and Affiliations

  1. Departamento de Matemáticas, Universidad Nacional de Colombia, Bogotá, Colombia

    S. Bautista

  2. Departamento de Matemática, Instituto de Ciências Exatas (ICE), Universidade Federal Rural do Rio de Janeiro, Seropédica, Brazil

    A. M. López

  3. Departamento de Matemáticas, Universidad Central, Bogotá, Colombia

    H. M. Sánchez

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  1. S. Bautista
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  2. A. M. López
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  3. H. M. Sánchez
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Correspondence to S. Bautista.

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Bautista, S., López, A.M. & Sánchez, H.M. Intransitive Sectional-Anosov Flows on 3-manifolds. Qual. Theory Dyn. Syst. 18, 615–630 (2019). https://doi.org/10.1007/s12346-018-0303-2

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