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2015, 9: 203-218. doi: 10.3934/jmd.2015.9.203

On the intersection of sectional-hyperbolic sets

Serafin Bautista 1, and  Carlos A. Morales 2,

1. 

Universidad Nacional de Colombia, Depto. de Matemáticas, Facultad de Ciencias, Bogota, Colombia
2. 

Instituto de Matemática, Universidade Federal do Rio de Janeiro,, P. O. Box 68530, 21945-970 Rio de Janeiro, Brazil

Received  October 2014 Revised  June 2015 Published  September 2015

We study the intersection of a positively sectional-hyperbolic set and a negatively sectional-hyperbolic set of a flow on a compact manifold. Indeed, we show that such an intersection is not a hyperbolic set in general. Next, we show that such an intersection is a hyperbolic set if the sets involved in the intersection are both transitive. In general, we prove that such an intersection is the disjoint union of a nonsingular hyperbolic set, a finite set of singularities, and a set of regular orbits joining these dynamical objects. Finally, we exhibit a three-dimensional star flow with a positively (but not negatively) sectional-hyperbolic homoclinic class and a negatively (but not positively) sectional-hyperbolic homoclinic class whose intersection is a periodic orbit. This provides a counterexample to a conjecture of Shi, Zhu, Gan and Wen ([25], [26]).
Keywords: Sectional-hyperbolic set, singularity, star flow.
Mathematics Subject Classification: Primary: 37D30; Secondary: 37C1.
Citation: Serafin Bautista, Carlos A. Morales. On the intersection of sectional-hyperbolic sets. Journal of Modern Dynamics, 2015, 9: 203-218. doi: 10.3934/jmd.2015.9.203
References:
[1]

V. S. Afraĭmovič, V. V. Bykov and L. P. Sil'nikov, The origin and structure of the Lorenz attractor, Dokl. Akad. Nauk SSSR, 234 (1977), 336-339.

[2]

V. Araújo and M. J. Pacifico, Three-Dimensional Flows, With a foreword by Marcelo Viana, Ergebnisse der Mathematik und ihrer Grenzgebiete, 3. Folge, A Series of Modern Surveys in Mathematics [Results in Mathematics and Related Areas, 3rd Series, A Series of Modern Surveys in Mathematics], 53, Springer, Heidelberg, 2010. doi: 10.1007/978-3-642-11414-4.

[3]

V. Araujo and L. Salgado, Infinitesimal Lyapunov functions for singular flows, Math. Z., 275 (2013), 863-897. doi: 10.1007/s00209-013-1163-8.

[4]

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.

[5]

R. Bamon, R. Labarca, R. Mañé and M. J. Pacifico, The explosion of singular cycles, Inst. Hautes Études Sci. Publ. Math., 78 (1993), 207-232.

[6]

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

[7]

S. Bautista and C. A. Morales, Lectures on Sectional-Anosov Flows,, preprint, (). 

[8]

S. Bautista, C. A. Morales and M. J. Pacifico, On the intersection of homoclinic classes on singular-hyperbolic sets, Discrete Contin. Dyn. Syst., 19 (2007), 761-775. doi: 10.3934/dcds.2007.19.761.

[9]

C. Bonatti, A. Pumariño and M. Viana, Lorenz attractors with arbitrary expanding dimension, C. R. Acad. Sci. Paris Sér. I Math., 325 (1997), 883-888. doi: 10.1016/S0764-4442(97)80131-0.

[10]

C. Diminnie, S. Gahler and A. White, $2$-inner product spaces, Collection of articles dedicated to Stanisław Gołąb on his 70th birthday, II, Demonstratio Math., 6 (1973), 525-536.

[11]

S. Gan and L. Wen, Nonsingular star flows satisfy Axiom A and the no-cycle condition, Invent. Math., 164 (2006), 279-315. doi: 10.1007/s00222-005-0479-3.

[12]

S. Gähler, Lineare 2-normierte Räume, Math. Nachr., 28 (1964), 1-43. doi: 10.1002/mana.19640280102.

[13]

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

[14]

M. Hirsch, C. Pugh and M. Shub, Invariant Manifolds, Lecture Notes in Math., Vol. 583, Springer-Verlag, Berlin-New York, 1977.

[15]

A. Kawaguchi, On areal spaces. I. Metric tensors in $n$-dimensional spaces based on the notion of two-dimensional area, Tensor N.S., 1 (1950), 14-45.

[16]

R. Labarca and M. J. Pacifico, Stability of singularity horseshoes, Topology, 25 (1986), 337-352. doi: 10.1016/0040-9383(86)90048-0.

[17]

A. M. López, Sectional Hyperbolic Sets in Higher Dimensions, Tese de Doutorado, Universidade Federal do Rio de Janeiro, Rio de Janeiro, Brazil, 2015.

[18]

A. M. López and H. Sánchez, Sectional Anosov flows: Existence of Venice masks with two singularities,, , (). 

[19]

R. Metzger and C. A. Morales, Sectional-hyperbolic systems, Ergodic Theory Dynam. Systems, 28 (2008), 1587-1597. doi: 10.1017/S0143385707000995.

[20]

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.

[21]

C. A. Morales and M. J. Pacifico, Sufficient conditions for robustness of attractors, Pacific J. Math., 216 (2004), 327-342. doi: 10.2140/pjm.2004.216.327.

[22]

C. A. Morales and M. J. Pacifico, A spectral decomposition for singular-hyperbolic sets, Pacific J. Math., 229 (2007), 223-232. doi: 10.2140/pjm.2007.229.223.

[23]

C. A. Morales and M. Vilches, On 2-Riemannian manifolds, SUT J. Math., 46 (2010), 119-153.

[24]

S. Newhouse, On simple arcs between structurally stable flows, in Dynamical Systems-Warwick 1974 (Proc. Sympos. Appl. Topology and Dynamical Systems, Univ. Warwick, Coventry, 1973/1974; presented to E. C. Zeeman on his fiftieth birthday), Lecture Notes in Math., Vol. 468, Springer, Berlin, 1975, 209-233.

[25]

Y. Shi, S. Gan and L. Wen, On the singular-hyperbolicity of star flows, J. Mod. Dyn., 8 (2014), 191-219. doi: 10.3934/jmd.2014.8.191.

[26]

S. Zhu, S. Gan and L. Wen, Indices of singularities of robustly transitive sets, Discrete Contin. Dyn. Syst., 21 (2008), 945-957. doi: 10.3934/dcds.2008.21.945.

show all references

References:
[1]

V. S. Afraĭmovič, V. V. Bykov and L. P. Sil'nikov, The origin and structure of the Lorenz attractor, Dokl. Akad. Nauk SSSR, 234 (1977), 336-339.

[2]

V. Araújo and M. J. Pacifico, Three-Dimensional Flows, With a foreword by Marcelo Viana, Ergebnisse der Mathematik und ihrer Grenzgebiete, 3. Folge, A Series of Modern Surveys in Mathematics [Results in Mathematics and Related Areas, 3rd Series, A Series of Modern Surveys in Mathematics], 53, Springer, Heidelberg, 2010. doi: 10.1007/978-3-642-11414-4.

[3]

V. Araujo and L. Salgado, Infinitesimal Lyapunov functions for singular flows, Math. Z., 275 (2013), 863-897. doi: 10.1007/s00209-013-1163-8.

[4]

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.

[5]

R. Bamon, R. Labarca, R. Mañé and M. J. Pacifico, The explosion of singular cycles, Inst. Hautes Études Sci. Publ. Math., 78 (1993), 207-232.

[6]

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

[7]

S. Bautista and C. A. Morales, Lectures on Sectional-Anosov Flows,, preprint, (). 

[8]

S. Bautista, C. A. Morales and M. J. Pacifico, On the intersection of homoclinic classes on singular-hyperbolic sets, Discrete Contin. Dyn. Syst., 19 (2007), 761-775. doi: 10.3934/dcds.2007.19.761.

[9]

C. Bonatti, A. Pumariño and M. Viana, Lorenz attractors with arbitrary expanding dimension, C. R. Acad. Sci. Paris Sér. I Math., 325 (1997), 883-888. doi: 10.1016/S0764-4442(97)80131-0.

[10]

C. Diminnie, S. Gahler and A. White, $2$-inner product spaces, Collection of articles dedicated to Stanisław Gołąb on his 70th birthday, II, Demonstratio Math., 6 (1973), 525-536.

[11]

S. Gan and L. Wen, Nonsingular star flows satisfy Axiom A and the no-cycle condition, Invent. Math., 164 (2006), 279-315. doi: 10.1007/s00222-005-0479-3.

[12]

S. Gähler, Lineare 2-normierte Räume, Math. Nachr., 28 (1964), 1-43. doi: 10.1002/mana.19640280102.

[13]

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

[14]

M. Hirsch, C. Pugh and M. Shub, Invariant Manifolds, Lecture Notes in Math., Vol. 583, Springer-Verlag, Berlin-New York, 1977.

[15]

A. Kawaguchi, On areal spaces. I. Metric tensors in $n$-dimensional spaces based on the notion of two-dimensional area, Tensor N.S., 1 (1950), 14-45.

[16]

R. Labarca and M. J. Pacifico, Stability of singularity horseshoes, Topology, 25 (1986), 337-352. doi: 10.1016/0040-9383(86)90048-0.

[17]

A. M. López, Sectional Hyperbolic Sets in Higher Dimensions, Tese de Doutorado, Universidade Federal do Rio de Janeiro, Rio de Janeiro, Brazil, 2015.

[18]

A. M. López and H. Sánchez, Sectional Anosov flows: Existence of Venice masks with two singularities,, , (). 

[19]

R. Metzger and C. A. Morales, Sectional-hyperbolic systems, Ergodic Theory Dynam. Systems, 28 (2008), 1587-1597. doi: 10.1017/S0143385707000995.

[20]

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.

[21]

C. A. Morales and M. J. Pacifico, Sufficient conditions for robustness of attractors, Pacific J. Math., 216 (2004), 327-342. doi: 10.2140/pjm.2004.216.327.

[22]

C. A. Morales and M. J. Pacifico, A spectral decomposition for singular-hyperbolic sets, Pacific J. Math., 229 (2007), 223-232. doi: 10.2140/pjm.2007.229.223.

[23]

C. A. Morales and M. Vilches, On 2-Riemannian manifolds, SUT J. Math., 46 (2010), 119-153.

[24]

S. Newhouse, On simple arcs between structurally stable flows, in Dynamical Systems-Warwick 1974 (Proc. Sympos. Appl. Topology and Dynamical Systems, Univ. Warwick, Coventry, 1973/1974; presented to E. C. Zeeman on his fiftieth birthday), Lecture Notes in Math., Vol. 468, Springer, Berlin, 1975, 209-233.

[25]

Y. Shi, S. Gan and L. Wen, On the singular-hyperbolicity of star flows, J. Mod. Dyn., 8 (2014), 191-219. doi: 10.3934/jmd.2014.8.191.

[26]

S. Zhu, S. Gan and L. Wen, Indices of singularities of robustly transitive sets, Discrete Contin. Dyn. Syst., 21 (2008), 945-957. doi: 10.3934/dcds.2008.21.945.

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