## On the existence of attractors

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- by Christian Bonatti, Ming Li and Dawei Yang PDF
- Trans. Amer. Math. Soc.
**365**(2013), 1369-1391 Request permission

## 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

- 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), no. 2, 336–339 (Russian). MR**0462175** - A. Araujo, Existência de atratores hiperbólicos para difeomorfismos de superficies,
*Ph.D. Thesis, IMPA*, 1987. - Masayuki Asaoka,
*Hyperbolic sets exhibiting $C^1$-persistent homoclinic tangency for higher dimensions*, Proc. Amer. Math. Soc.**136**(2008), no. 2, 677–686. MR**2358509**, DOI 10.1090/S0002-9939-07-09115-0 - R. Bamon, J. Kiwi, and J. Rivera, Wild Lorenz like attractors,
*preprint*. - Christian Bonatti and Sylvain Crovisier,
*Récurrence et généricité*, Invent. Math.**158**(2004), no. 1, 33–104 (French, with English and French summaries). MR**2090361**, DOI 10.1007/s00222-004-0368-1 - Christian Bonatti and Lorenzo Díaz,
*On maximal transitive sets of generic diffeomorphisms*, Publ. Math. Inst. Hautes Études Sci.**96**(2002), 171–197 (2003). MR**1985032**, DOI 10.1007/s10240-003-0008-0 - C. Bonatti, L. J. Díaz, and E. R. Pujals,
*A $C^1$-generic dichotomy for diffeomorphisms: weak forms of hyperbolicity or infinitely many sinks or sources*, Ann. of Math. (2)**158**(2003), no. 2, 355–418 (English, with English and French summaries). MR**2018925**, DOI 10.4007/annals.2003.158.355 - Christian Bonatti, Lorenzo J. Díaz, and Marcelo Viana,
*Dynamics beyond uniform hyperbolicity*, Encyclopaedia of Mathematical Sciences, vol. 102, Springer-Verlag, Berlin, 2005. A global geometric and probabilistic perspective; Mathematical Physics, III. MR**2105774** - Christian Bonatti, Nikolas Gourmelon, and Thérèse Vivier,
*Perturbations of the derivative along periodic orbits*, Ergodic Theory Dynam. Systems**26**(2006), no. 5, 1307–1337. MR**2266363**, DOI 10.1017/S0143385706000253 - C. Bonatti, M. Li, and D. Yang, Robustly chain transitive attractor with singularities of different indices,
*preprint*, 2008. - Christian Bonatti and Marcelo Viana,
*SRB measures for partially hyperbolic systems whose central direction is mostly contracting*, Israel J. Math.**115**(2000), 157–193. MR**1749677**, DOI 10.1007/BF02810585 - Charles Conley,
*Isolated invariant sets and the Morse index*, CBMS Regional Conference Series in Mathematics, vol. 38, American Mathematical Society, Providence, R.I., 1978. MR**511133** - Shengzhi Zhu, Shaobo Gan, and Lan Wen,
*Indices of singularities of robustly transitive sets*, Discrete Contin. Dyn. Syst.**21**(2008), no. 3, 945–957. MR**2399444**, DOI 10.3934/dcds.2008.21.945 - Joel C. Gibbons,
*One-dimensional basic sets in the three-sphere*, Trans. Amer. Math. Soc.**164**(1972), 163–178. MR**292110**, DOI 10.1090/S0002-9947-1972-0292110-4 - J. Guckenheimer, A strange, strange attractor,
*The Hopf bifurcation theorems and its applications (Applied Mathematical Series,*, Springer-Verlag, 1976, pp. 368–381.**19**) - John Guckenheimer and R. F. Williams,
*Structural stability of Lorenz attractors*, Inst. Hautes Études Sci. Publ. Math.**50**(1979), 59–72. MR**556582** - Mike Hurley,
*Attractors: persistence, and density of their basins*, Trans. Amer. Math. Soc.**269**(1982), no. 1, 247–271. MR**637037**, DOI 10.1090/S0002-9947-1982-0637037-7 - Ming Li, Shaobo Gan, and Lan Wen,
*Robustly transitive singular sets via approach of an extended linear Poincaré flow*, Discrete Contin. Dyn. Syst.**13**(2005), no. 2, 239–269. MR**2152388**, DOI 10.3934/dcds.2005.13.239 - E. N. Lorenz, Deterministic nonperiodic flow,
*J. Atmosph. Sci.*,**20**(1963), 130–141. - Ricardo Mañé,
*An ergodic closing lemma*, Ann. of Math. (2)**116**(1982), no. 3, 503–540. MR**678479**, DOI 10.2307/2007021 - R. Metzger and C. Morales,
*Sectional-hyperbolic systems*, Ergodic Theory Dynam. Systems**28**(2008), no. 5, 1587–1597. MR**2449545**, DOI 10.1017/S0143385707000995 - John Milnor,
*On the concept of attractor*, Comm. Math. Phys.**99**(1985), no. 2, 177–195. MR**790735** - C. A. Morales and M. J. Pacifico,
*Lyapunov stability of $\omega$-limit sets*, Discrete Contin. Dyn. Syst.**8**(2002), no. 3, 671–674. MR**1897874**, DOI 10.3934/dcds.2002.8.671 - Carlos Arnoldo Morales, Maria José Pacífico, and Enrique Ramiro Pujals,
*On $C^1$ robust singular transitive sets for three-dimensional flows*, C. R. Acad. Sci. Paris Sér. I Math.**326**(1998), no. 1, 81–86 (English, with English and French summaries). MR**1649489**, DOI 10.1016/S0764-4442(97)82717-6 - C. A. Morales, M. J. Pacifico, and E. R. Pujals,
*Robust transitive singular sets for 3-flows are partially hyperbolic attractors or repellers*, Ann. of Math. (2)**160**(2004), no. 2, 375–432. MR**2123928**, DOI 10.4007/annals.2004.160.375 - Sheldon E. Newhouse,
*Nondensity of axiom $\textrm {A}(\textrm {a})$ on $S^{2}$*, Global Analysis (Proc. Sympos. Pure Math., Vol. XIV, Berkeley, Calif., 1968) Amer. Math. Soc., Providence, R.I., 1970, pp. 191–202. MR**0277005** - Sheldon E. Newhouse,
*Diffeomorphisms with infinitely many sinks*, Topology**13**(1974), 9–18. MR**339291**, DOI 10.1016/0040-9383(74)90034-2 - Sheldon E. 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** - Jacob Palis,
*A global view of dynamics and a conjecture on the denseness of finitude of attractors*, Astérisque**261**(2000), xiii–xiv, 335–347 (English, with English and French summaries). Géométrie complexe et systèmes dynamiques (Orsay, 1995). MR**1755446** - J. Palis,
*A global perspective for non-conservative dynamics*, Ann. Inst. H. Poincaré C Anal. Non Linéaire**22**(2005), no. 4, 485–507 (English, with English and French summaries). MR**2145722**, DOI 10.1016/j.anihpc.2005.01.001 - J. Palis,
*Open questions leading to a global perspective in dynamics*, Nonlinearity**21**(2008), no. 4, T37–T43. MR**2399817**, DOI 10.1088/0951-7715/21/4/T01 - J. Palis and C. C. Pugh (eds.),
*Fifty problems in dynamical systems*, 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, pp. 345–353. MR**0646829** - R. V. Plykin,
*Hyperbolic attractors of diffeomorphisms*, Uspekhi Mat. Nauk**35**(1980), no. 3(213), 94–104 (Russian). International Topology Conference (Moscow State Univ., Moscow, 1979). MR**580625** - Enrique R. Pujals and Martín Sambarino,
*Homoclinic tangencies and hyperbolicity for surface diffeomorphisms*, Ann. of Math. (2)**151**(2000), no. 3, 961–1023. MR**1779562**, DOI 10.2307/121127 - J. Palis and M. Viana,
*High dimension diffeomorphisms displaying infinitely many periodic attractors*, Ann. of Math. (2)**140**(1994), no. 1, 207–250. MR**1289496**, DOI 10.2307/2118546 - M. Shub,
*Topological transitive diffeomorphisms in $T^4$*, Lecture Notes in Math. Vol.**206**, Springer Verlag, 1971. - S. Smale,
*Differentiable dynamical systems*, Bull. Amer. Math. Soc.**73**(1967), 747–817. MR**228014**, DOI 10.1090/S0002-9904-1967-11798-1 - René Thom,
*Structural stability and morphogenesis*, W. A. Benjamin, Inc., Reading, Mass.-London-Amsterdam, 1976. An outline of a general theory of models; Translated from the French by D. H. Fowler; With a foreword by C. H. Waddington; Second printing. MR**0488156** - Lan Wen,
*Homoclinic tangencies and dominated splittings*, Nonlinearity**15**(2002), no. 5, 1445–1469. MR**1925423**, DOI 10.1088/0951-7715/15/5/306 - J. Yang, Lyapunov stable chain recurrent class,
*preprint*, 2007.

## Additional Information

**Christian Bonatti**- Affiliation: Institut de Mathématiques de Bourgogne, Université de Bourgogne, Dijon 21004, France
- Email: bonatti@u-bourgogne.fr
**Ming Li**- Affiliation: School of Mathematical Sciences, Nankai University, Tianjin 300071, People’s Republic of China
- MR Author ID: 763903
- Email: limingmath@nankai.edu.cn
**Dawei Yang**- Affiliation: School of Mathematics, Jilin University, Changchun 130000, People’s Republic of China
- Email: yangdw1981@gmail.com
- 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.
- © Copyright 2012
American Mathematical Society

The copyright for this article reverts to public domain 28 years after publication. - Journal: Trans. Amer. Math. Soc.
**365**(2013), 1369-1391 - MSC (2010): Primary 37B20, 37B25, 37C05, 37C10, 37C20, 37C29, 37C70, 37D05, 37D30, 37G25
- DOI: https://doi.org/10.1090/S0002-9947-2012-05644-3
- MathSciNet review: 3003268