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Isentropes and Lyapunov exponents
Sectional-hyperbolic Lyapunov stable sets
Departamento de Matemáticas, Universidad Nacional de Colombia, Bogotá, 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 $.
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. |
| [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. |
| [3] |
A. Arbieto, A. 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. |
| [4] |
S. Bautista and C. Morales, Lectures on Sectional-Anosov Flows, Preprint IMPA Serie D 84, 2011. |
| [5] |
S. Bautista and C. Morales,
A sectional-Anosov connecting lemma, Ergodic Theory Dynam. Systems, 30 (2010), 339-359.
doi: 10.1017/S0143385709000157. |
| [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. |
| [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. |
| [8] |
S. Bautista, V. Sales and Y. Sánchez, Sectional connecting lema, preprint, arXiv: 1804.00646. |
| [9] |
M. W. Hirsch, C. C. Pugh and M. Shub, Invariant Manifolds, Lecture Notes in Mathematics, Vol. 583. Springer-Verlag, Berlin-New York, 1977. |
| [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.
|
| [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. |
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. |
| [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. |
| [3] |
A. Arbieto, A. 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. |
| [4] |
S. Bautista and C. Morales, Lectures on Sectional-Anosov Flows, Preprint IMPA Serie D 84, 2011. |
| [5] |
S. Bautista and C. Morales,
A sectional-Anosov connecting lemma, Ergodic Theory Dynam. Systems, 30 (2010), 339-359.
doi: 10.1017/S0143385709000157. |
| [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. |
| [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. |
| [8] |
S. Bautista, V. Sales and Y. Sánchez, Sectional connecting lema, preprint, arXiv: 1804.00646. |
| [9] |
M. W. Hirsch, C. C. Pugh and M. Shub, Invariant Manifolds, Lecture Notes in Mathematics, Vol. 583. Springer-Verlag, Berlin-New York, 1977. |
| [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.
|
| [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. |
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