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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

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 & 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).  Google Scholar

[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.  Google Scholar

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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.  Google Scholar

[4]

S. Bautista, The geometric Lorenz attractor is a homoclinic class, Bol. Mat. (N.S.), 11 (2004), 69-78.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[7]

C. I. Doering, Persistently transitive vector fields on three-dimensional manifolds, In Procs. on Dynamical Systems and Bifurcations Theory, 160 (1987), 59-89.  Google Scholar

[8]

J. Guckenheimer and R. F. Williams, Structural stability of Lorenz attractors, Inst. Hautes Études Sci. Publ. Math., 50 (1979), 59-72.  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]

M. Milnor, On the concept of attractor, Comm. Math. Phys., 99 (1985), 177-195. doi: 10.1007/BF01212280.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[15]

A. Rovella, The dynamics of perturbations of the contracting Lorenz attractors, Bol. Soc. Bras. Math., 24 (1993), 233-259. doi: 10.1007/BF01237679.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

show all references

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).  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[4]

S. Bautista, The geometric Lorenz attractor is a homoclinic class, Bol. Mat. (N.S.), 11 (2004), 69-78.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[7]

C. I. Doering, Persistently transitive vector fields on three-dimensional manifolds, In Procs. on Dynamical Systems and Bifurcations Theory, 160 (1987), 59-89.  Google Scholar

[8]

J. Guckenheimer and R. F. Williams, Structural stability of Lorenz attractors, Inst. Hautes Études Sci. Publ. Math., 50 (1979), 59-72.  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]

M. Milnor, On the concept of attractor, Comm. Math. Phys., 99 (1985), 177-195. doi: 10.1007/BF01212280.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[15]

A. Rovella, The dynamics of perturbations of the contracting Lorenz attractors, Bol. Soc. Bras. Math., 24 (1993), 233-259. doi: 10.1007/BF01237679.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

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