April  2020, 40(4): 2011-2016. doi: 10.3934/dcds.2020103

Sectional-hyperbolic Lyapunov stable sets

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

* Corresponding author: sbautistad@unal.edu.co

Received  April 2018 Revised  August 2019 Published  January 2020

Fund Project: The first author is supported by UNAL from Colombia. The second author is supported by COLCIENCIAS and UNAL from Colombia.

In hyperbolic dynamics, a well-known result is: every hyperbolic Lyapunov stable set, is attracting; it's natural to wonder if this result is maintained in the sectional-hyperbolic dynamics. This question is still open, although some partial results have been presented. We will prove that all sectional-hyperbolic transitive Lyapunov stable set of codimension one of a vector field $ X $ over a compact manifold, with unique singularity Lorenz-like, which is of boundary-type, is an attractor of $ X $.

Citation: Serafin Bautista, Yeison Sánchez. Sectional-hyperbolic Lyapunov stable sets. Discrete & Continuous Dynamical Systems - A, 2020, 40 (4) : 2011-2016. doi: 10.3934/dcds.2020103
References:
[1]

V. Araujo and M. J. Pacifico, Three-Dimensional Flows, A Series of Modern Surveys in Mathematics, 53. Springer, Heidelberg, 2010. doi: 10.1007/978-3-642-11414-4.  Google Scholar

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S. Bautista and C. Morales, Recent progress on sectional-hyperbolic systems, Synamical Systems: An international Journal., 30 (2015), 369-382.  doi: 10.1080/14689367.2015.1056093.  Google Scholar

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S. Bautista, V. Sales and Y. Sánchez, Sectional connecting lema, preprint, arXiv: 1804.00646. Google Scholar

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M. W. Hirsch, C. C. Pugh and M. Shub, Invariant Manifolds, Lecture Notes in Mathematics, Vol. 583. Springer-Verlag, Berlin-New York, 1977.  Google Scholar

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C. A. Morales and M. J. Pacifico, A dichotomy for three-dimensional vector fields, Ergodic Theory Dynam. Systems, 23 (2003), 1575-1600.  doi: 10.1017/S0143385702001621.  Google Scholar

show all references

References:
[1]

V. Araujo and M. J. Pacifico, Three-Dimensional Flows, A Series of Modern Surveys in Mathematics, 53. Springer, Heidelberg, 2010. doi: 10.1007/978-3-642-11414-4.  Google Scholar

[2]

A. Arbieto and C. A. Morales, A dichotomy for higher-dimensional flows, Proc. Amer. Math. Soc., 141 (2013), 2817-2827.  doi: 10.1090/S0002-9939-2013-11536-4.  Google Scholar

[3]

A. ArbietoA. M. Lopez Barragán and C. Morales, Homoclinic classes for sectional-hyperbolic sets, Kyoto Journal of Mathematics, 56 (2016), 531-538.  doi: 10.1215/21562261-3600157.  Google Scholar

[4]

S. Bautista and C. Morales, Lectures on Sectional-Anosov Flows, Preprint IMPA Serie D 84, 2011. Google Scholar

[5]

S. Bautista and C. Morales, A sectional-Anosov connecting lemma, Ergodic Theory Dynam. Systems, 30 (2010), 339-359.  doi: 10.1017/S0143385709000157.  Google Scholar

[6]

S. Bautista and C. Morales, Characterizing omega-limit sets which are closed orbits, J. Differential Equations, 245 (2008), 637-652.  doi: 10.1016/j.jde.2007.11.007.  Google Scholar

[7]

S. Bautista and C. Morales, Recent progress on sectional-hyperbolic systems, Synamical Systems: An international Journal., 30 (2015), 369-382.  doi: 10.1080/14689367.2015.1056093.  Google Scholar

[8]

S. Bautista, V. Sales and Y. Sánchez, Sectional connecting lema, preprint, arXiv: 1804.00646. Google Scholar

[9]

M. W. Hirsch, C. C. Pugh and M. Shub, Invariant Manifolds, Lecture Notes in Mathematics, Vol. 583. Springer-Verlag, Berlin-New York, 1977.  Google Scholar

[10] A. Katok and B. Hasselblatt, Introduction to the Modern Theory of Dynamical Systems, Encyclopedia of Mathematics and its Applications, 54. Cambridge University Press, Cambridge, 1995.  doi: 10.1017/CBO9780511809187.  Google Scholar
[11]

C. A. Morales and M. J. Pacifico, A dichotomy for three-dimensional vector fields, Ergodic Theory Dynam. Systems, 23 (2003), 1575-1600.  doi: 10.1017/S0143385702001621.  Google Scholar

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