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July  2019, 39(7): 4057-4071. doi: 10.3934/dcds.2019163

Asymptotically sectional-hyperbolic attractors

B. San Martín and  Kendry J. Vivas ,

Departamento de Matemáticas, Universidad Católica del Norte, Antofagasta, Chile

*Corresponding author.

Received  August 2018 Revised  November 2018 Published  April 2019

Fund Project: The first author was partially supported by Proyecto FONDECYT 1181183, Chile. The second author was supported by CONICYT grant 21180163, Chile.

The notion of asymptotically sectional-hyperbolic set was recently introduced. The main feature is that any point outside of the stable manifolds of its singularities has arbitrarily large hyperbolic times. In this paper we prove the existence, on any three-dimensional Riemannian manifold, of attractors with Rovella-like singularities satisfying this kind of hyperbolicity. Furthermore, we prove that asymptotically sectional-hyperbolic Lyapunov-stable sets, under certain conditions, have positive topological entropy.

Keywords: Flow, sectional-hyperbolic, asymptotically sectional-hyperbolic, attractors, topological entropy.
Mathematics Subject Classification: Primary: 54H20, 37B25; Secondary: 37B40.
Citation: B. San Martín, Kendry J. Vivas. Asymptotically sectional-hyperbolic attractors. Discrete & Continuous Dynamical Systems - A, 2019, 39 (7) : 4057-4071. doi: 10.3934/dcds.2019163
References:
[1]

V. S. Afraimovic, V. V. Bykov and L. P. Shilnikov, The origin and structure of the Lorenz attractor, Dokl. Akad. Nauk SSSR, 234 (1977), 336-339.   Google Scholar

[2]

A. Arbieto, A. M. Lopez and C. A. Morales, Homoclinic classes for sectional-hyperbolic sets, Kyoto J. Math, 56 (2016), 531-538.  doi: 10.1215/21562261-3600157.  Google Scholar

[3]

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

[4]

S. Bautista and C. A. Morales, On the intersection of sectional-hyperbolic sets, J. Mod. Dyn., 9 (2015), 203-218.  doi: 10.3934/jmd.2015.9.203.  Google Scholar

[5]

C. Bonatti, A. Pumariño and M. Viana, Lorenz attractors with arbitrary expanding dimension, International Conference on Differential Equations, Vol. 1, 2 (Berlin, 1999), 39–44, World Sci. Publ., River Edge, NJ, 2000. doi: 10.1016/S0764-4442(97)80131-0.  Google Scholar

[6]

J. Carmona, D. Carrasco-Olivera and B. San Martín, On the $C^1$ robust transitivity of the geometric Lorenz attractor, J. Differential Equations, 262 (2017), 5928-5938.   Google Scholar

[7]

E. Catsigeras, M. Cerminara and H. Enrich, The Pesin entropy formula for $C^1$ diffeomorphisms with dominated splitting, Ergodic Theory Dynam. Systems, 35 (2015), 737-761.  doi: 10.1017/etds.2013.93.  Google Scholar

[8]

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

[9]

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

[10]

J. Guckenheimer, A strange, strange attractor, in The Hopf Bifurcation and Its Applications, Applied Mathematical Sciences, (1976), 368–381. doi: 10.1007/978-1-4612-6374-6_25.  Google Scholar

[11]

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

[12]

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

[13]

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

[14]

E. N. Lorenz, Deterministic nonperiodic flow, J. Atmospheric Sci., 20 (1963), 130-141.   Google Scholar

[15]

R. J. Metzger, Stochastic stability for contracting Lorenz maps and flows, Comm. Math. Phys., 212 (2000), 277-296.  doi: 10.1007/s002200000220.  Google Scholar

[16]

R. J. Metzger, Sinai-Ruelle-Bowen measures for contracting Lorenz maps and flows, Ann. Inst. H. Poincaré Anal. Non Linéaire, 17 (2000), 247-276.  doi: 10.1016/S0294-1449(00)00111-6.  Google Scholar

[17]

R. J. Metzger and C. A. Morales, The Rovella attractor is a homoclinic class, Bull. Braz. Math. Soc., 37 (2006), 89-101.  doi: 10.1007/s00574-006-0005-2.  Google Scholar

[18]

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

[19]

C. A. Morales, M. J. Pacifico and and E. R. Pujals, Singular hyperbolic systems, Proc. Amer. Math. Soc., 127 (1999), 3393-3401.  doi: 10.1090/S0002-9939-99-04936-9.  Google Scholar

[20]

C. A. Morales, M. J. Pacifico and B. San Martín, Contracting Lorenz attractors through resonant double homoclinic loops, SIAM J. Math. Anal., 38 (2006), 309-332.  doi: 10.1137/S0036141004443907.  Google Scholar

[21]

C. A. Morales and B. San Martín, Contracting singular horseshoe, Nonlinearity, 30 (2017), 4208-4219.  doi: 10.1088/1361-6544/aa864e.  Google Scholar

[22]

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

[23]

E. M. Muñoz, B. San Martín and J. A. Vera, Nonhyperbolic persistent attractors near the Morse-Smale boundary, Ann. Inst. H. Poincaré Anal. Non Linéaire, 20 (2003), 867-888.  doi: 10.1016/S0294-1449(03)00015-5.  Google Scholar

[24]

M. J. Pacifico and M. Todd, Thermodynamic formalism for contracting Lorenz flows, J. Stat. Phys., 139 (2010), 159-176.  doi: 10.1007/s10955-010-9939-2.  Google Scholar

[25]

R. C. Robinson, An Introduction to Dynamical Systems: Continuous and Discrete, 2$^nd$ edition, American Mathematical Society, Providence, 2012.  Google Scholar

[26]

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

[27]

D. V. Turaev and L. P. Shilnikov, An example of a wild strange attractor, Sb. Math., 189 (1998), 291-314.  doi: 10.1070/SM1998v189n02ABEH000300.  Google Scholar

[28]

R. F. Williams, The structure of Lorenz attractors, Inst. Hautes Études Sci. Publ. Math., 50 (1979), 73-99.   Google Scholar

show all references

References:
[1]

V. S. Afraimovic, V. V. Bykov and L. P. Shilnikov, The origin and structure of the Lorenz attractor, Dokl. Akad. Nauk SSSR, 234 (1977), 336-339.   Google Scholar

[2]

A. Arbieto, A. M. Lopez and C. A. Morales, Homoclinic classes for sectional-hyperbolic sets, Kyoto J. Math, 56 (2016), 531-538.  doi: 10.1215/21562261-3600157.  Google Scholar

[3]

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

[4]

S. Bautista and C. A. Morales, On the intersection of sectional-hyperbolic sets, J. Mod. Dyn., 9 (2015), 203-218.  doi: 10.3934/jmd.2015.9.203.  Google Scholar

[5]

C. Bonatti, A. Pumariño and M. Viana, Lorenz attractors with arbitrary expanding dimension, International Conference on Differential Equations, Vol. 1, 2 (Berlin, 1999), 39–44, World Sci. Publ., River Edge, NJ, 2000. doi: 10.1016/S0764-4442(97)80131-0.  Google Scholar

[6]

J. Carmona, D. Carrasco-Olivera and B. San Martín, On the $C^1$ robust transitivity of the geometric Lorenz attractor, J. Differential Equations, 262 (2017), 5928-5938.   Google Scholar

[7]

E. Catsigeras, M. Cerminara and H. Enrich, The Pesin entropy formula for $C^1$ diffeomorphisms with dominated splitting, Ergodic Theory Dynam. Systems, 35 (2015), 737-761.  doi: 10.1017/etds.2013.93.  Google Scholar

[8]

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

[9]

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

[10]

J. Guckenheimer, A strange, strange attractor, in The Hopf Bifurcation and Its Applications, Applied Mathematical Sciences, (1976), 368–381. doi: 10.1007/978-1-4612-6374-6_25.  Google Scholar

[11]

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

[12]

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

[13]

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

[14]

E. N. Lorenz, Deterministic nonperiodic flow, J. Atmospheric Sci., 20 (1963), 130-141.   Google Scholar

[15]

R. J. Metzger, Stochastic stability for contracting Lorenz maps and flows, Comm. Math. Phys., 212 (2000), 277-296.  doi: 10.1007/s002200000220.  Google Scholar

[16]

R. J. Metzger, Sinai-Ruelle-Bowen measures for contracting Lorenz maps and flows, Ann. Inst. H. Poincaré Anal. Non Linéaire, 17 (2000), 247-276.  doi: 10.1016/S0294-1449(00)00111-6.  Google Scholar

[17]

R. J. Metzger and C. A. Morales, The Rovella attractor is a homoclinic class, Bull. Braz. Math. Soc., 37 (2006), 89-101.  doi: 10.1007/s00574-006-0005-2.  Google Scholar

[18]

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

[19]

C. A. Morales, M. J. Pacifico and and E. R. Pujals, Singular hyperbolic systems, Proc. Amer. Math. Soc., 127 (1999), 3393-3401.  doi: 10.1090/S0002-9939-99-04936-9.  Google Scholar

[20]

C. A. Morales, M. J. Pacifico and B. San Martín, Contracting Lorenz attractors through resonant double homoclinic loops, SIAM J. Math. Anal., 38 (2006), 309-332.  doi: 10.1137/S0036141004443907.  Google Scholar

[21]

C. A. Morales and B. San Martín, Contracting singular horseshoe, Nonlinearity, 30 (2017), 4208-4219.  doi: 10.1088/1361-6544/aa864e.  Google Scholar

[22]

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

[23]

E. M. Muñoz, B. San Martín and J. A. Vera, Nonhyperbolic persistent attractors near the Morse-Smale boundary, Ann. Inst. H. Poincaré Anal. Non Linéaire, 20 (2003), 867-888.  doi: 10.1016/S0294-1449(03)00015-5.  Google Scholar

[24]

M. J. Pacifico and M. Todd, Thermodynamic formalism for contracting Lorenz flows, J. Stat. Phys., 139 (2010), 159-176.  doi: 10.1007/s10955-010-9939-2.  Google Scholar

[25]

R. C. Robinson, An Introduction to Dynamical Systems: Continuous and Discrete, 2$^nd$ edition, American Mathematical Society, Providence, 2012.  Google Scholar

[26]

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

[27]

D. V. Turaev and L. P. Shilnikov, An example of a wild strange attractor, Sb. Math., 189 (1998), 291-314.  doi: 10.1070/SM1998v189n02ABEH000300.  Google Scholar

[28]

R. F. Williams, The structure of Lorenz attractors, Inst. Hautes Études Sci. Publ. Math., 50 (1979), 73-99.   Google Scholar

Figure 1.  A vector field $ X $ without non-atomic SRB-like measures
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Figure 2.  The cube $ Q $ and the cells $ T_l $ and $ T_r $
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Figure 3.  The flow in the resulting $ 3 $-cell
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Figure 4.  The flow from the cusp triangles to $ S^+ $ and $ S^- $
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Figure 6.  The trapping region $ U $
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