Robust attractors without dominated splitting on manifolds with boundary

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November 2014, 34(11): 4555-4563. doi: 10.3934/dcds.2014.34.4555

Robust attractors without dominated splitting on manifolds with boundary

Dante Carrasco-Olivera ^1, and Bernardo San Martín ^2,

1.	Departamento de Matemática, Universidad del Bío-Bío, Av. Collao 1202, Casilla 5-C, Concepción, Chile
2.	Departamento de Matemáticas, Universidad Católica del Norte, Av. Angamos 0610, Casilla 1280, Antofagasta, Chile

Received August 2013 Revised February 2014 Published May 2014

Abstract
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In this paper we prove that there exists a positive integer $k$ with the following property: Every compact $3$-manifold with boundary carries a $C^\infty$ vector field exhibiting a $C^k$-robust attractor without dominated splitting in a robust sense.

Keywords: attractor sets, dominated splitting., Robust transitive sets.

Mathematics Subject Classification: Primary: 34D45, 37D30; Secondary: 37C10, 37F15, 37D9.

Citation: Dante Carrasco-Olivera, Bernardo San Martín. Robust attractors without dominated splitting on manifolds with boundary. Discrete and Continuous Dynamical Systems, 2014, 34 (11) : 4555-4563. doi: 10.3934/dcds.2014.34.4555

References:

[1]	V. S. Afraimovich, V. V. Bykov and L. P. Shilnikov, On attracting structurally unstable limit sets of Lorenz attractor type, Trudy Moskov. Mat. Obshch., 44 (1982), 150-212 (Russian).
[2]	V. Araújo and M. J. Pacifico, Three-dimensional Flows, Ergebnisse der Mathematik und Ihrer Grenzgebiete. 3. Folge. A Series of Modern Surveys in Mathematics [Results in Mathematics and Related Areas. 3rd Series], 53. Springer, Heidelberg, 2010. doi: 10.1007/978-3-642-11414-4.
[3]	R. Bamón, R. Labarca, R. Mańé and M.J. Pacífico, The explosion of singulay cycles, Inst. Hautes Études Sci. Publ. Math., 78 (1993), 207-232.
[4]	S. Bautista, The geometric Lorenz attractor is a homoclinic class, Bol. Mat. (N.S.), 11 (2004), 69-78.
[5]	C. Bonatti, L. J. Díaz and M. Viana, Dynamics Beyond Uniform Hyperbolicity. A Global Geometric and Probabilistic Perspective, Encyclopaedia of Mathematical Sciences,102, Mathematical Physics III, Springer-Verlag, Berlin, 2005.
[6]	D. Carrasco-Olivera, C. A. Morales and B. San Martín, Singular cycles and $C^k$-robust transitive set on manifold with boundary, Communications in Contemporary Mathematics, 13 (2011), 191-211. doi: 10.1142/S0219199711004233.
[7]	C. I. Doering, Persistently transitive vector fields on three-dimensional manifolds, In Procs. on Dynamical Systems and Bifurcations Theory, 160 (1987), 59-89.
[8]	J. Guckenheimer and R. F. Williams, Structural stability of Lorenz attractors, Inst. Hautes Études Sci. Publ. Math., 50 (1979), 59-72.
[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]	M. Milnor, On the concept of attractor, Comm. Math. Phys., 99 (1985), 177-195. doi: 10.1007/BF01212280.
[11]	C. A. Morales, Sufficient conditions for a partially hyperbolic attractor to be a homoclinic class, J. Differential Equations, 249 (2010), 2005-2020. doi: 10.1016/j.jde.2010.05.014.
[12]	C. A. Morales, M. J. Pacifico and E. R. Pujals, Robust transitive singular sets for 3-flows are partially hyperbolic attractors or repellers, Ann. of Math., 160 (2004), 375-432. doi: 10.4007/annals.2004.160.375.
[13]	C. A. Morales, M. J. Pacifico and E. R. Pujals, On $C^1$ robust singular transitive sets for three-dimensional flows, C. R. Acad. Sci. Paris, 326 (1998), 81-86. doi: 10.1016/S0764-4442(97)82717-6.
[14]	C. A. Morales, M. J. Pacifico and E. R. Pujals, Singular hyperbolic systems, Proc. Am. Math. Soc., 127 (1999), 3393-3401. doi: 10.1090/S0002-9939-99-04936-9.
[15]	A. Rovella, The dynamics of perturbations of the contracting Lorenz attractors, Bol. Soc. Bras. Math., 24 (1993), 233-259. doi: 10.1007/BF01237679.
[16]	S. Sternberg, On the structure of local homeomorphism of eucliden $n$-spane II, Am. Journal Math., 80 (1958), 623-631. doi: 10.2307/2372774.
[17]	T. Vivier, Flots robustement transitifs sur les variétés compactes (French) [Robustly transitive flows on compact manifolds], Comptes Rendus Acad. Sci. Paris, 337 (2003), 791-796. doi: 10.1016/j.crma.2003.10.001.

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References:

[1]	V. S. Afraimovich, V. V. Bykov and L. P. Shilnikov, On attracting structurally unstable limit sets of Lorenz attractor type, Trudy Moskov. Mat. Obshch., 44 (1982), 150-212 (Russian).
[2]	V. Araújo and M. J. Pacifico, Three-dimensional Flows, Ergebnisse der Mathematik und Ihrer Grenzgebiete. 3. Folge. A Series of Modern Surveys in Mathematics [Results in Mathematics and Related Areas. 3rd Series], 53. Springer, Heidelberg, 2010. doi: 10.1007/978-3-642-11414-4.
[3]	R. Bamón, R. Labarca, R. Mańé and M.J. Pacífico, The explosion of singulay cycles, Inst. Hautes Études Sci. Publ. Math., 78 (1993), 207-232.
[4]	S. Bautista, The geometric Lorenz attractor is a homoclinic class, Bol. Mat. (N.S.), 11 (2004), 69-78.
[5]	C. Bonatti, L. J. Díaz and M. Viana, Dynamics Beyond Uniform Hyperbolicity. A Global Geometric and Probabilistic Perspective, Encyclopaedia of Mathematical Sciences,102, Mathematical Physics III, Springer-Verlag, Berlin, 2005.
[6]	D. Carrasco-Olivera, C. A. Morales and B. San Martín, Singular cycles and $C^k$-robust transitive set on manifold with boundary, Communications in Contemporary Mathematics, 13 (2011), 191-211. doi: 10.1142/S0219199711004233.
[7]	C. I. Doering, Persistently transitive vector fields on three-dimensional manifolds, In Procs. on Dynamical Systems and Bifurcations Theory, 160 (1987), 59-89.
[8]	J. Guckenheimer and R. F. Williams, Structural stability of Lorenz attractors, Inst. Hautes Études Sci. Publ. Math., 50 (1979), 59-72.
[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]	M. Milnor, On the concept of attractor, Comm. Math. Phys., 99 (1985), 177-195. doi: 10.1007/BF01212280.
[11]	C. A. Morales, Sufficient conditions for a partially hyperbolic attractor to be a homoclinic class, J. Differential Equations, 249 (2010), 2005-2020. doi: 10.1016/j.jde.2010.05.014.
[12]	C. A. Morales, M. J. Pacifico and E. R. Pujals, Robust transitive singular sets for 3-flows are partially hyperbolic attractors or repellers, Ann. of Math., 160 (2004), 375-432. doi: 10.4007/annals.2004.160.375.
[13]	C. A. Morales, M. J. Pacifico and E. R. Pujals, On $C^1$ robust singular transitive sets for three-dimensional flows, C. R. Acad. Sci. Paris, 326 (1998), 81-86. doi: 10.1016/S0764-4442(97)82717-6.
[14]	C. A. Morales, M. J. Pacifico and E. R. Pujals, Singular hyperbolic systems, Proc. Am. Math. Soc., 127 (1999), 3393-3401. doi: 10.1090/S0002-9939-99-04936-9.
[15]	A. Rovella, The dynamics of perturbations of the contracting Lorenz attractors, Bol. Soc. Bras. Math., 24 (1993), 233-259. doi: 10.1007/BF01237679.
[16]	S. Sternberg, On the structure of local homeomorphism of eucliden $n$-spane II, Am. Journal Math., 80 (1958), 623-631. doi: 10.2307/2372774.
[17]	T. Vivier, Flots robustement transitifs sur les variétés compactes (French) [Robustly transitive flows on compact manifolds], Comptes Rendus Acad. Sci. Paris, 337 (2003), 791-796. doi: 10.1016/j.crma.2003.10.001.

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