American Institute of Mathematical Sciences

Advanced
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 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.

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).

[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.

[1]

Shengzhi Zhu, Shaobo Gan, Lan Wen. Indices of singularities of robustly transitive sets. Discrete and Continuous Dynamical Systems, 2008, 21 (3) : 945-957. doi: 10.3934/dcds.2008.21.945
[2]

Ming Li, Shaobo Gan, Lan Wen. Robustly transitive singular sets via approach of an extended linear Poincaré flow. Discrete and Continuous Dynamical Systems, 2005, 13 (2) : 239-269. doi: 10.3934/dcds.2005.13.239
[3]

Peter Müller, Gábor P. Nagy. On the non-existence of sharply transitive sets of permutations in certain finite permutation groups. Advances in Mathematics of Communications, 2011, 5 (2) : 303-308. doi: 10.3934/amc.2011.5.303
[4]

Jutamas Kerdkaew, Rabian Wangkeeree. Characterizing robust weak sharp solution sets of convex optimization problems with uncertainty. Journal of Industrial and Management Optimization, 2020, 16 (6) : 2651-2673. doi: 10.3934/jimo.2019074
[5]

D. Hilhorst, L. A. Peletier, A. I. Rotariu, G. Sivashinsky. Global attractor and inertial sets for a nonlocal Kuramoto-Sivashinsky equation. Discrete and Continuous Dynamical Systems, 2004, 10 (1&2) : 557-580. doi: 10.3934/dcds.2004.10.557
[6]

Eleonora Catsigeras, Xueting Tian. Dominated splitting, partial hyperbolicity and positive entropy. Discrete and Continuous Dynamical Systems, 2016, 36 (9) : 4739-4759. doi: 10.3934/dcds.2016006
[7]

Wenxiang Sun, Xueting Tian. Dominated splitting and Pesin's entropy formula. Discrete and Continuous Dynamical Systems, 2012, 32 (4) : 1421-1434. doi: 10.3934/dcds.2012.32.1421
[8]

Anh N. Le. Sublacunary sets and interpolation sets for nilsequences. Discrete and Continuous Dynamical Systems, 2022, 42 (4) : 1855-1871. doi: 10.3934/dcds.2021175
[9]

Nithirat Sisarat, Rabian Wangkeeree, Gue Myung Lee. Some characterizations of robust solution sets for uncertain convex optimization problems with locally Lipschitz inequality constraints. Journal of Industrial and Management Optimization, 2020, 16 (1) : 469-493. doi: 10.3934/jimo.2018163
[10]

Wan-Tong Li, Bin-Guo Wang. Attractor minimal sets for nonautonomous type-K competitive and semi-convex delay differential equations with applications. Discrete and Continuous Dynamical Systems, 2009, 24 (2) : 589-611. doi: 10.3934/dcds.2009.24.589
[11]

Xinsheng Wang, Lin Wang, Yujun Zhu. Formula of entropy along unstable foliations for $C^1$ diffeomorphisms with dominated splitting. Discrete and Continuous Dynamical Systems, 2018, 38 (4) : 2125-2140. doi: 10.3934/dcds.2018087
[12]

Pedro Duarte, Silvius Klein. Topological obstructions to dominated splitting for ergodic translations on the higher dimensional torus. Discrete and Continuous Dynamical Systems, 2018, 38 (11) : 5379-5387. doi: 10.3934/dcds.2018237
[13]

François Blanchard, Wen Huang. Entropy sets, weakly mixing sets and entropy capacity. Discrete and Continuous Dynamical Systems, 2008, 20 (2) : 275-311. doi: 10.3934/dcds.2008.20.275
[14]

Johannes Kellendonk, Lorenzo Sadun. Conjugacies of model sets. Discrete and Continuous Dynamical Systems, 2017, 37 (7) : 3805-3830. doi: 10.3934/dcds.2017161
[15]

S. Astels. Thickness measures for Cantor sets. Electronic Research Announcements, 1999, 5: 108-111.

[16]

Frank D. Grosshans, Jürgen Scheurle, Sebastian Walcher. Invariant sets forced by symmetry. Journal of Geometric Mechanics, 2012, 4 (3) : 271-296. doi: 10.3934/jgm.2012.4.271
[17]

Piotr Oprocha. Coherent lists and chaotic sets. Discrete and Continuous Dynamical Systems, 2011, 31 (3) : 797-825. doi: 10.3934/dcds.2011.31.797
[18]

Arya Mazumdar, Ron M. Roth, Pascal O. Vontobel. On linear balancing sets. Advances in Mathematics of Communications, 2010, 4 (3) : 345-361. doi: 10.3934/amc.2010.4.345
[19]

Rasul Shafikov, Christian Wolf. Stable sets, hyperbolicity and dimension. Discrete and Continuous Dynamical Systems, 2005, 12 (3) : 403-412. doi: 10.3934/dcds.2005.12.403
[20]

L. S. Grinblat. Theorems on sets not belonging to algebras. Electronic Research Announcements, 2004, 10: 51-57.

2020 Impact Factor: 1.392

Tools

Metrics

  • PDF downloads (46)
  • HTML views (0)
  • Cited by (0)

Other articles
by authors

[Back to Top]

Copyright © 2022 American Institute of Mathematical Sciences | Privacy Policy