Doi: http://dx.doi.org/10.15446/recolma.v49n1.54162

¹Universidade Federal do Rio de Janeiro, Rio de Janeiro, Brazil. Email: barragan@im.ufrj.br

A sectional-Anosov flow on a manifold is a C¹ vector field inwardly transverse to the boundary for which the maximal invariant is sectional hyperbolic [10]. We prove that every attractor of every vector field C¹ close to a transitive sectional-Anosov flow with singularities on a compact manifold has a singularity. This extends the three-dimensional result obtained in [9].

Key words: Transitive, Maximal invariant, Sectional-Anosov flow.

Un flujo seccional-Anosov sobre una variedad es un C¹ campo vectorial transversal a la frontera apuntando hacia el interior, para el cual su conjunto maximal invariante es un conjunto seccional hiperbólico [10]. Probamos que todo atractor de todo campo vectorial C¹ próximo a un flujo seccional-Anosov transitivo con singularidades sobre una variedad compacta tiene una singularidad. Este resultado extiende el resultado tres-dimensional obtenido en [9].

Palabras clave: Transitivo, maximal invariante, flujo seccional-Anosov.

Texto completo disponible en PDF

References

[1] V. S. Afraimovich, V. V. Bykov, and L. P. Shilnikov, `On Structurally Unstable Attracting Limit Sets of Lorenz Attractor Type´, Trudy Moskov. Mat. Obshch. 44, 2 (1982), 150-212.         [ Links ]

[2] V. Araújo and M. J. Pacífico, Ergebnisse der mathematik und ihrer grenzgebiete. 3. folge. a series of modern surveys in mathematics, `Three-dimensional flows.´, 2010, Vol. 53, Springer.         [ Links ]

[3] A. Arbieto, C. A. Morales, and L. Senos, `On the Sensitivity of Sectional-Anosov Flows´, Mathematische Zeitschrift 270, 1-2 (2012), 545-557.         [ Links ]

[4] S. Bautista and C. A. Morales, Lectures on Sectional-Anosov Flows, http://preprint.impa.br/Shadows/SERIE_D/2011/86.html, 0000.         [ Links ]

[5] C. Bonatti, A. Pumariño, and M. Viana, `Lorenz Attractors with Arbitrary Expanding Dimension´, C. R. Acad. Sci. Paris Sér. I Math. 325, 8 (1997), 883-888.         [ Links ]

[6] C. I. Doering, `Persistently Transitive Vector Fields on Three-Dimensional Manifolds´, Dynamical Systems and Bifurcation Theory (Rio de Janeiro, 1985), Pitman Res. Notes Math. Ser. 160, (1987), 59-89.         [ Links ]

[7] J. Guckenheimer and R. F. Williams, `Structural Stability of Lorenz Attractors´, Publications Mathématiques de l'IHÉS 50, 1 (1979), 59-72.         [ Links ]

[8] M. W. Hirsch, C. C. Pugh, and M. Shub, Invariant Manifolds, Vol. 583, Springer Berlin, 1977.         [ Links ]

[9] C. A. Morales, `The Explosion of Singular-Hyperbolic Attractors´, Ergodic Theory and Dynamical Systems 24, 2 (2004), 577-591.         [ Links ]

[10] C. A. Morales, `Sectional-Anosov Flows´, Monatshefte für Mathematik 159, 3 (2010), 253-260.         [ Links ]

[11] C. A. Morales, M. J. Pacífico, and E. R. Pujals, `Singular Hyperbolic Systems´, Proceedings of the American Mathematical Society 127, 11 (1999), 3393-3401.         [ Links ]

[12] J. Palis and W. De Melo, Geometric Theory of Dynamical Systems, Springer, 1982.         [ Links ]

[13] J. Palis and F. Takens, Hyperbolicity and Sensitive Chaotic Dynamics at Homoclinic Bifurcations: Fractal Dimensions and Infinitely Many Attractors in Dynamics, Cambridge University Press, 1993.         [ Links ]

Este artículo se puede citar en LaTeX utilizando la siguiente referencia bibliográfica de BibTeX:

Doi: http://dx.doi.org/ @ARTICLE{RCMv49n1a02,
    AUTHOR  = {López, Andrés Mauricio}, ]]>     TITLE   = {{Sectional-Anosov Flows in Higher Dimensions}},
    JOURNAL = {Revista Colombiana de Matemáticas},
    YEAR    = {2015},
    volume  = {49},
    number  = {1},
    pages   = {39--55}
}