Remote Access Transactions of the American Mathematical Society
Green Open Access

Transactions of the American Mathematical Society

ISSN 1088-6850(online) ISSN 0002-9947(print)

Request Permissions   Purchase Content 


On the existence of attractors

Authors: Christian Bonatti, Ming Li and Dawei Yang
Journal: Trans. Amer. Math. Soc. 365 (2013), 1369-1391
MSC (2010): Primary 37B20, 37B25, 37C05, 37C10, 37C20, 37C29, 37C70, 37D05, 37D30, 37G25
Published electronically: August 22, 2012
MathSciNet review: 3003268
Full-text PDF

Abstract | References | Similar Articles | Additional Information

Abstract: On every compact $ 3$-manifold, we build a non-empty open set $ \mathcal U$ of $ \operatorname {Diff}^1(M)$ such that, for every $ r\geq 1$, every $ C^r$-generic diffeomorphism $ f\in \mathcal U\cap \operatorname {Diff}^r(M)$ has no topological attractors. On higher-dimensional manifolds, one may require that $ f$ has neither topological attractors nor topological repellers. Our examples have finitely many quasi-attractors. For flows, we may require that these quasi-attractors contain singular points. Finally we discuss alternative definitions of attractors which may be better adapted to generic dynamics.

References [Enhancements On Off] (What's this?)

  • [ABS] V. Afraĭmovič, V. Bykov, and L. Sil'nikov, The origin and structure of the Lorenz attractor, Dokl. Akad. Nauk SSSR, 234 (1977), 336-339. MR 0462175 (57:2150)
  • [Ar] A. Araujo, Existência de atratores hiperbólicos para difeomorfismos de superficies, Ph.D. Thesis, IMPA, 1987.
  • [As] M. Asaoka, Hyperbolic sets exhibiting $ C\sp 1$-persistent homoclinic tangency for higher dimensions, Proc. Amer. Math. Soc., 136 (2008), no. 2, 677-686. MR 2358509 (2008k:37049)
  • [BKR] R. Bamon, J. Kiwi, and J. Rivera, Wild Lorenz like attractors, preprint.
  • [BC] C. Bonatti and S. Crovisier, Récurrence et généricité (French), Invent. Math., 158 (2004), 33-104. MR 2090361 (2007b:37036)
  • [BD] C. Bonatti and L. Díaz, On maximal transitive sets of generic diffeomorphisms, Inst. Hautes études Sci. Publ. Math., 96 (2002), 171-197. MR 1985032 (2007d:37017)
  • [BDP] C. Bonatti, L. Diaz, and E. Pujals, A $ C^1$-generic dichotomy for diffeomorphisms: Weak forms of hyperbolicity or infinitely many sinks or sources, Ann. of Math., 158 (2003), 355-418. MR 2018925 (2007k:37032)
  • [BDV] C. Bonatti, L. Diaz, and and M. Viana, Dynamics beyond uniform hyperbolicity. A global geometric and probabilistic perspective, Encyclopaedia of Mathematical Sciences, 102. Mathematical Physics, III. Springer-Verlag, Berlin, 2005, xviii+384 pp. MR 2105774 (2005g:37001)
  • [BGV] C. Bonatti, N. Gourmelon and T. Vivier, Perturbations of the derivative along periodic orbits, Ergodic Theory Dynam. Systems, 26 (2006), 1307-1337. MR 2266363 (2007i:37062)
  • [BLY] C. Bonatti, M. Li, and D. Yang, Robustly chain transitive attractor with singularities of different indices, preprint, 2008.
  • [BV] C. Bonatti and M. Viana, SRB measures for partially hyperbolic systems whose central direction is mostly contracting, Israel J. Math., 115 (2002), 157-193. MR 1749677 (2001j:37063a)
  • [Co] C. Conley, Isolated invariant sets and Morse index, CBMS Regional Conference Series in Mathematics, 38, AMS Providence, R.I., 1978. MR 511133 (80c:58009)
  • [GWZ] S. Gan, L. Wen, and S. Zhu, Indices of singularities of robustly transitive sets, Disc. Cont. Dynam. Syst., 21 (2008), 945-957. MR 2399444 (2009a:37065)
  • [Gi] J. C. Gibbons, One-dimensional basic sets in the three-sphere, Transactions of the American Mathematical Society, 164 (1972), 163-178. MR 0292110 (45:1197)
  • [Gu] J. Guckenheimer, A strange, strange attractor, The Hopf bifurcation theorems and its applications (Applied Mathematical Series, 19), Springer-Verlag, 1976, pp. 368-381.
  • [GuW] J. Guckenheimer and R. Williams, Structural stability of Lorenz attractors, Inst. Hautes Études Sci. Publ. Math., 50 (1979), 59-72. MR 556582 (82b:58055a)
  • [H] M. Hurley, Attractors: Persistence, and density of their basins, Trans. Am. Math. Soc., 269 (1982), 247-271. MR 637037 (83c:58049)
  • [LGW] M. Li, S. Gan, and L. Wen, Robustly transitive singular sets via approach of an extended linear Poincaré flow, Discrete Contin. Dyn. Syst., 13 (2005), 239-269. MR 2152388 (2006b:37056)
  • [Lo] E. N. Lorenz, Deterministic nonperiodic flow, J. Atmosph. Sci., 20 (1963), 130-141.
  • [Ma] R. Mañé, An ergodic closing lemma, Ann. Math., 116 (1982), 503-540. MR 678479 (84f:58070)
  • [MM] R. Metzger and C. Morales, Sectional-hyperbolic systems, Ergodic Theory Dynam. Systems, 28 (2008), no. 5, 1587-1597. MR 2449545 (2010g:37045)
  • [Mi] J. Milnor, On the concept of attractor, Commum. Math. Phys., 99 (1985), 177-195. MR 790735 (87i:58109a)
  • [MP] C. Morales and M. Pacifico, Lyapunov stability of $ \omega $-limit sets, Disc. Cont. Dyn. Sys., 8 (2002), 671-674. MR 1897874 (2003b:37024)
  • [MPP1] C. Morales, M. Pacifico, and E. Pujals, On $ C\sp 1$ robust singular transitive sets for three-dimensional flows, C. R. Acad. Sci. Paris, 326 (1998), 81-86. MR 1649489 (99j:58183)
  • [MPP2] C. Morales, M. Pacifico, and E. Pujals, Robust transitive singular sets for 3-flows are partially hyperbolic attractors or repellers, Ann. of Math., 160 (2004), 375-432. MR 2123928 (2005k:37054)
  • [N1] S. Newhouse, Nondensity of axiom $ {\rm A}({\rm a})$ on $ S^2$, Global Analysis (Proc. Sympos. Pure Math., Vol. XIV, Berkeley, Calif., 1968), Amer. Math. Soc., Providence, R.I., 1970, 191-202. MR 0277005 (43:2742)
  • [N2] S. Newhouse, Diffeomorphisms with infinitely many sinks, Topology, 13 (1974), 9-18. MR 0339291 (49:4051)
  • [N3] S. Newhouse, The abundance of wild hyperbolic sets and nonsmooth stable sets for diffeomorphisms, Inst. Hautes Études Sci. Publ. Math., 50 (1979), 101-151. MR 556584 (82e:58067)
  • [P1] J. Palis, A global view of dynamics and a conjecture of the denseness of finitude of attractors, Astérisque, 261 (2000), 335-347. MR 1755446 (2001d:37025)
  • [P2] J. Palis, A global perspective for non-conservative dynamics, Ann. Inst. H. Poincaré Anal. Non Linéaire, 22 (2005), 485-507. MR 2145722 (2006b:37037)
  • [P3] J. Palis, Open questions leading to a global perspective in dynamics, Nonlinearity, 21 (2008), 37-43. MR 2399817 (2009i:37003)
  • [PP] J. Palis and C. Pugh, Fifty problems in dynamical systems, Lect. Notes Math., 468 (1975), 345-353, Springer-Verlag. MR 0646829 (58:31134)
  • [Pl] R. V. Plykin, Hyperbolic attractors of diffeomorphisms, Usp. Math. Nauk, 35 (1980), no. 3, 94-104. [English Transl.: Russ. Math. Survey, 35 (1980), no. 3, 109-121.] MR 580625 (82k:58081)
  • [PS] E. Pujals and M. Sambarino, Homoclinic tangencies and hyperbolicity for surface diffeomorphisms, Ann. Math., 151(2000), 961-1023. MR 1779562 (2001m:37057)
  • [PV] J. Palis and M. Viana, High dimension diffeomorphisms displaying infinitely many periodic attractors, Ann. of Math., 140 (1994), 207-250. MR 1289496 (95g:58140)
  • [Sh] M. Shub, Topological transitive diffeomorphisms in $ T^4$, Lecture Notes in Math. Vol. 206, Springer Verlag, 1971.
  • [Sm] S. Smale, Differentiable dynamical systems, Bull. Amer. Math.Soc., 73 (1967), 747-817. MR 0228014 (37:3598)
  • [T] R. Thom, Structually stability and morphogenesis, Benjamin, Reading, Mass., 1976. MR 0488156 (58:7722b)
  • [W] L. Wen, Homoclinic tangencies and dominated splittings, Nonlinearity, 15 (2002), 1445-1469. MR 1925423 (2003f:37055)
  • [Y] J. Yang, Lyapunov stable chain recurrent class, preprint, 2007.

Similar Articles

Retrieve articles in Transactions of the American Mathematical Society with MSC (2010): 37B20, 37B25, 37C05, 37C10, 37C20, 37C29, 37C70, 37D05, 37D30, 37G25

Retrieve articles in all journals with MSC (2010): 37B20, 37B25, 37C05, 37C10, 37C20, 37C29, 37C70, 37D05, 37D30, 37G25

Additional Information

Christian Bonatti
Affiliation: Institut de Mathématiques de Bourgogne, Université de Bourgogne, Dijon 21004, France

Ming Li
Affiliation: School of Mathematical Sciences, Nankai University, Tianjin 300071, People’s Republic of China

Dawei Yang
Affiliation: School of Mathematics, Jilin University, Changchun 130000, People’s Republic of China

Received by editor(s): March 20, 2010
Received by editor(s) in revised form: April 15, 2011
Published electronically: August 22, 2012
Additional Notes: This work was done during the stays of the second and third authors at the IMB, Université de Bourgogne, and they thank the IMB for its warm hospitality. The second author was supported by a postdoctoral grant of the Région Bourgogne, and the third author was supported by CSC of Chinese Education Ministry. This is a part of the third author’s Ph.D. thesis at Peking University.
Article copyright: © Copyright 2012 American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication.

American Mathematical Society