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

On the intersection of sectional-hyperbolic sets

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]).
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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[6]

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

[7]

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

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[12]

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

[13]

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

[14]

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

[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.  Google Scholar

[16]

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

[17]

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

[18]

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

[19]

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

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[23]

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

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[6]

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

[7]

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

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[12]

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

[13]

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

[14]

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

[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.  Google Scholar

[16]

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

[17]

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

[18]

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

[19]

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

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[23]

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

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

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