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 and Continuous Dynamical Systems, 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. 

[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.

[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.

[4]

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

[5]

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

[6]

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

[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.

[8]

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

[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.

[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. 

[11]

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

[12]

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

[13]

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

[14]

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

[15]

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

[16]

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

[17]

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

[18]

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

[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.

[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.

[21]

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

[22]

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

[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.

[24]

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

[25]

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

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. 

[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.

[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.

[4]

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

[5]

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

[6]

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

[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.

[8]

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

[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.

[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. 

[11]

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

[12]

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

[13]

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

[14]

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

[15]

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

[16]

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

[17]

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

[18]

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

[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.

[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.

[21]

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

[22]

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

[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.

[24]

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

[25]

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

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