January  2017, 37(1): 337-354. doi: 10.3934/dcds.2017014

Finiteness and existence of attractors and repellers on sectional hyperbolic sets

Instituto de Ciências Exatas (ICE), Universidade Federal Rural do Rio de Janeiro, 23890-000 Seropédica, Rio de Janeiro, Brazil

* Corresponding author: A. M. López

Received  March 2015 Revised  September 2016 Published  November 2016

Fund Project: The author was supported by CAPES, Brazil

We study small perturbations of a sectional hyperbolic set of a vector field on a compact manifold. Indeed, we obtain an upper bound for the number of attractors and repellers that can arise from these perturbations. Moreover, no repeller can arise if the unperturbed set has singularities, is connected and consists of nonwandering points.

Citation: A. M. López. Finiteness and existence of attractors and repellers on sectional hyperbolic sets. Discrete & Continuous Dynamical Systems - A, 2017, 37 (1) : 337-354. doi: 10.3934/dcds.2017014
References:
[1]

V. AfraimovichV. Bykov and L. Shilnikov, On structurally unstable attracting limit sets of Lorenz attractor type, Trudy Moskov. Mat. Obshch., 44 (1982), 150-212.   Google Scholar

[2]

V. AraújoM. PacíficoE. Pujals and M. Viana, Singular-hyperbolic attractors are chaotic, Transactions of the American Mathematical Society, 361 (2009), 2431-2485.  doi: 10.1090/S0002-9947-08-04595-9.  Google Scholar

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A. ArbietoC. Morales and L. Senos, On the sensitivity of sectional-Anosov flows, Mathematische Zeitschrift, 270 (2012), 545-557.  doi: 10.1007/s00209-010-0811-5.  Google Scholar

[4]

S. Bautista and C. Morales, Lectures on sectional-Anosov flows, Available from: http://preprint.impa.br/Shadows/SERIE_D/2011/86.html. Google Scholar

[5]

S. Bautista and C. Morales, Existence of periodic orbits for singular-hyperbolic sets, Mosc. Math. J, 6 (2006), 265-297,406.   Google Scholar

[6]

C. Bonatti, The global dynamics of C1-generic diffeomorphisms or flows, In Second Latin American Congress of Mathematicians. Cancun, Mexico 2004. Google Scholar

[7]

D. Carrasco-Olivera and M. Chavez-Gordillo, An attracting singular-hyperbolic set containing a non trivial hyperbolic repeller, Lobachevskii Journal of Mathematics, 30 (2009), 12-16.  doi: 10.1134/S1995080209010028.  Google Scholar

[8]

C. Chicone, Ordinary Differential Equations with Applications Springer 1 1999. doi: 10.1007%2F0-387-35794-7.  Google Scholar

[9]

S. Crovisier and D. Yang, On the density of singular hyperbolic three-dimensional vector fields: A conjecture of Palis, Comptes Rendus Mathematique, 353 (2015), 85-88.  doi: 10.1016/j.crma.2014.10.015.  Google Scholar

[10]

C. Doering, Persistently transitive vector fields on three-dimensional manifolds, Dynamical Systems and Bifurcation Theory (Rio de Janeiro, 1985), Pitman Res. Notes Math. Ser., 160 (1987), 59-89.   Google Scholar

[11]

J. Franks and B. Williams, Anomalous Anosov flows, In Global theory of dynamical systems, Springer, 819 (1980), 158–174. Google Scholar

[12]

J. Guckenheimer and R. Williams, Structural stability of lorenz attractors, Publications Math{é}matiques de l'IHÉS, 50 (1979), 59-72.   Google Scholar

[13]

M. Hirsch, C. Pugh and M. Shub, Invariant Manifolds Springer Berlin, 583 1977. doi: 10.1007%2FBFb0092042.  Google Scholar

[14]

A. Katok and B. Hasselblatt, Introduction to the Modern Theory of Dynamical Systems Cambridge University Press, Cambridge, 1995. doi: 10.1017/CBO9780511809187.  Google Scholar

[15]

A. López, Existence of periodic orbits for sectional Anosov flows, arXiv: 1407.3471 (2014). Google Scholar

[16]

A. López, Sectional-Anosov flows in higher dimensions, Revista Colombiana de Matemáticas, 49 (2015), 39-55.  doi: 10.15446/recolma.v49n1.54162.  Google Scholar

[17]

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

[18]

C. Morales, The explosion of singular-hyperbolic attractors, Ergodic Theory and Dynamical Systems, 24 (2004), 577-591.  doi: 10.1017/S014338570300052X.  Google Scholar

[19]

C. Morales and M. Pacífico, A dichotomy for three-dimensional vector fields, Ergodic Theory and Dynamical Systems, 23 (2003), 1575-1600.  doi: 10.1017/S0143385702001621.  Google Scholar

[20]

C. MoralesM. Pacífico and E. Pujals, Singular hyperbolic systems, Proceedings of the American Mathematical Society, 127 (1999), 3393-3401.  doi: 10.1090/S0002-9939-99-04936-9.  Google Scholar

[21]

J. Palis, A global view of dynamics and a conjecture on the denseness of finitude of attractors, Astérisque, 261 (2000), 335-347.   Google Scholar

[22]

J. Palis and W. De Melo, Geometric Theory of Dynamical Systems, Springer, 1982. doi: 10.1007%2F978-1-4612-5703-5.  Google Scholar

[23]

J. Palis and F. Takens, Hyperbolicity and Sensitive Chaotic Dynamics at Homoclinic Bifurcations: Fractal Dimensions and Infinitely Many Attractors in Dynamics Cambridge University Press, 1993.  Google Scholar

[24]

E. Pujals and M. Sambarino, Homoclinic tangencies and hyperbolicity for surface diffeomorphisms, Annals of Mathematics, 151 (2000), 961-1023.  doi: 10.2307/121127.  Google Scholar

[25]

D. Ruelle and F. Takens, On the nature of turbulence, Communications in mathematical physics, 20 (1971), 167-192.  doi: 10.1007/BF01646553.  Google Scholar

show all references

References:
[1]

V. AfraimovichV. Bykov and L. Shilnikov, On structurally unstable attracting limit sets of Lorenz attractor type, Trudy Moskov. Mat. Obshch., 44 (1982), 150-212.   Google Scholar

[2]

V. AraújoM. PacíficoE. Pujals and M. Viana, Singular-hyperbolic attractors are chaotic, Transactions of the American Mathematical Society, 361 (2009), 2431-2485.  doi: 10.1090/S0002-9947-08-04595-9.  Google Scholar

[3]

A. ArbietoC. Morales and L. Senos, On the sensitivity of sectional-Anosov flows, Mathematische Zeitschrift, 270 (2012), 545-557.  doi: 10.1007/s00209-010-0811-5.  Google Scholar

[4]

S. Bautista and C. Morales, Lectures on sectional-Anosov flows, Available from: http://preprint.impa.br/Shadows/SERIE_D/2011/86.html. Google Scholar

[5]

S. Bautista and C. Morales, Existence of periodic orbits for singular-hyperbolic sets, Mosc. Math. J, 6 (2006), 265-297,406.   Google Scholar

[6]

C. Bonatti, The global dynamics of C1-generic diffeomorphisms or flows, In Second Latin American Congress of Mathematicians. Cancun, Mexico 2004. Google Scholar

[7]

D. Carrasco-Olivera and M. Chavez-Gordillo, An attracting singular-hyperbolic set containing a non trivial hyperbolic repeller, Lobachevskii Journal of Mathematics, 30 (2009), 12-16.  doi: 10.1134/S1995080209010028.  Google Scholar

[8]

C. Chicone, Ordinary Differential Equations with Applications Springer 1 1999. doi: 10.1007%2F0-387-35794-7.  Google Scholar

[9]

S. Crovisier and D. Yang, On the density of singular hyperbolic three-dimensional vector fields: A conjecture of Palis, Comptes Rendus Mathematique, 353 (2015), 85-88.  doi: 10.1016/j.crma.2014.10.015.  Google Scholar

[10]

C. Doering, Persistently transitive vector fields on three-dimensional manifolds, Dynamical Systems and Bifurcation Theory (Rio de Janeiro, 1985), Pitman Res. Notes Math. Ser., 160 (1987), 59-89.   Google Scholar

[11]

J. Franks and B. Williams, Anomalous Anosov flows, In Global theory of dynamical systems, Springer, 819 (1980), 158–174. Google Scholar

[12]

J. Guckenheimer and R. Williams, Structural stability of lorenz attractors, Publications Math{é}matiques de l'IHÉS, 50 (1979), 59-72.   Google Scholar

[13]

M. Hirsch, C. Pugh and M. Shub, Invariant Manifolds Springer Berlin, 583 1977. doi: 10.1007%2FBFb0092042.  Google Scholar

[14]

A. Katok and B. Hasselblatt, Introduction to the Modern Theory of Dynamical Systems Cambridge University Press, Cambridge, 1995. doi: 10.1017/CBO9780511809187.  Google Scholar

[15]

A. López, Existence of periodic orbits for sectional Anosov flows, arXiv: 1407.3471 (2014). Google Scholar

[16]

A. López, Sectional-Anosov flows in higher dimensions, Revista Colombiana de Matemáticas, 49 (2015), 39-55.  doi: 10.15446/recolma.v49n1.54162.  Google Scholar

[17]

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

[18]

C. Morales, The explosion of singular-hyperbolic attractors, Ergodic Theory and Dynamical Systems, 24 (2004), 577-591.  doi: 10.1017/S014338570300052X.  Google Scholar

[19]

C. Morales and M. Pacífico, A dichotomy for three-dimensional vector fields, Ergodic Theory and Dynamical Systems, 23 (2003), 1575-1600.  doi: 10.1017/S0143385702001621.  Google Scholar

[20]

C. MoralesM. Pacífico and E. Pujals, Singular hyperbolic systems, Proceedings of the American Mathematical Society, 127 (1999), 3393-3401.  doi: 10.1090/S0002-9939-99-04936-9.  Google Scholar

[21]

J. Palis, A global view of dynamics and a conjecture on the denseness of finitude of attractors, Astérisque, 261 (2000), 335-347.   Google Scholar

[22]

J. Palis and W. De Melo, Geometric Theory of Dynamical Systems, Springer, 1982. doi: 10.1007%2F978-1-4612-5703-5.  Google Scholar

[23]

J. Palis and F. Takens, Hyperbolicity and Sensitive Chaotic Dynamics at Homoclinic Bifurcations: Fractal Dimensions and Infinitely Many Attractors in Dynamics Cambridge University Press, 1993.  Google Scholar

[24]

E. Pujals and M. Sambarino, Homoclinic tangencies and hyperbolicity for surface diffeomorphisms, Annals of Mathematics, 151 (2000), 961-1023.  doi: 10.2307/121127.  Google Scholar

[25]

D. Ruelle and F. Takens, On the nature of turbulence, Communications in mathematical physics, 20 (1971), 167-192.  doi: 10.1007/BF01646553.  Google Scholar

Figure 1.  Three dimensional case. Cross-section
Figure 2.  Four dimensional case. Cross-sections
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