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SURVEY Towards a global view of dynamical systems, for the C1-topology

Published online by Cambridge University Press:  31 May 2011

C. BONATTI
C. BONATTI*
Affiliation:
Institut de Mathématiques de Bourgogne, UMR 5584 du CNRS, Université de Bourgogne, Dijon 21004, France (email: bonatti@u-bourgogne.fr)
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Abstract

This paper suggests a program for getting a global view of the dynamics of diffeomorphisms, from the point of view of the C1-topology. More precisely, given any compact manifold M, one splits Diff1(M) into disjoint C1-open regions whose union is C1-dense, and conjectures state that each of these open sets and their complements is characterized by the presence of:

  • • either a robust local phenomenon;

  • • or a global structure forbidding this local phenomenon.

Other conjectures state that some of these regions are empty. This set of conjectures draws a global view of the dynamics, putting in evidence the coherence of the numerous recent results on C1-generic dynamics.

Information

Type
Research Article
Information
Ergodic Theory and Dynamical Systems , Volume 31 , Issue 4 , August 2011 , pp. 959 - 993
Copyright
Copyright © Cambridge University Press 2011

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References

[Ab]Abdenur, F.. Generic robustness of spectral decompositions. Ann. Sci. École Norm. Sup. 36 (2003), 213–224.Google Scholar
[ABCD]Abdenur, F., Bonatti, Ch., Crovisier, S. and Díaz, L. J.. Generic diffeomorphisms on compact surfaces. Fund. Math. 187(2) (2005), 127–159.Google Scholar
[ABC]Abdenur, F., Bonatti, Ch. and Crovisier, S.. Nonuniform hyperbolicity for C 1-generic diffeomorphisms. Israel J. Math. to appear.Google Scholar
[AS]Abraham, R. and Smale, S.. Nongenericity of Ω-stability. Global Analysis I (Proceedings of Symposia in Pure Mathematics, 14). American Mathematical Society, Providence, RI, 1968, pp. 5–8.Google Scholar
[Ar1]Arnaud, M.-C.. Création de connexions en topologie C 1. Ergod. Th. & Dynam. Sys. 21 (2001), 339–381.Google Scholar
[ABV]Alves, J., Bonatti, Ch. and Viana, M.. SRB measures for partially hyperbolic systems whose central direction is mostly expanding. Invent. Math. 140(2) (2000), 351–398.Google Scholar
[AHK]Akin, E., Hurley, M. and Kennedy, J.. Dynamics of topologically generic homeomorphisms. Mem. Amer. Math. Soc. 164(783) (2003), viii+130.Google Scholar
[As]Asaoka, M.. Hyperbolic sets exhibiting C 1-persistent homoclinic tangency for higher dimensions. Proc. Amer. Math. Soc. 136(2) (2008), 677–686.Google Scholar
[BKR]Bamon, R., Kiwi, J. and Rivera, J.. Wild Lorenz like attractors. Preprint, 2006.Google Scholar
[BoBo]Bochi, J. and Bonatti, Ch.. Perturbation of the Lyapunov spectrum of periodic orbits. Preprint, 2010, arXiv:1004.5029.Google Scholar
[B]Bonatti, Ch.. Robust tangencies. Note of the mini-course, avaliable at: http://www.math.pku.edu.cn/teachers/gansb/conference09/BeijingAout2009.pdf.Google Scholar
[BC]Bonatti, Ch. and Crovisier, S.. Récurrence et généricité. Invent. Math. 158(1) (2004), 33–104.Google Scholar
[BC2]Bonatti, Ch. and Crovisier, S.. Recurrence and genericity. C. R. Math. Acad. Sci. Paris 336 (2003), 839–844.CrossRefGoogle Scholar
[BCDG]Bonatti, Ch., Crovisier, S., Díaz, L. J. and Gourmelon, N.. Internal perturbations of homoclinic classes: non-domination, cycles, and self-replication. Preprint, arXiv:1011.2935.Google Scholar
[BD1]Bonatti, Ch. and Díaz, L. J.. Persistent nonhyperbolic transitive diffeomorphisms. Ann. of Math. (2) 143 (1996), 357–396.Google Scholar
[BD2]Bonatti, Ch. and Díaz, L. J.. Connexions hétéroclines et généricité d’une infinité de puits ou de sources. Ann. Sci. École Norm. Sup. 32 (1999), 135–150.Google Scholar
[BD3]Bonatti, Ch. and Díaz, L. J.. On maximal transitive sets of generic diffeomorphisms. Publ. Math. Inst. Hautes Études Sci. 96 (2003), 171–197.Google Scholar
[BD4]Bonatti, Ch. and Díaz, L. J.. Aboundance of robust tangencies. Trans. Amer. Math. Soc. to appear.Google Scholar
[BD5]Bonatti, Ch. and Díaz, L. J.. Fragile cycles. Preprint, arXiv:1103.3255.Google Scholar
[BDK]Bonatti, Ch., Díaz, L. J. and Kiriki, S.. Robust heterodimensional cycles for hyperbolic continuations. Work in progress.Google Scholar
[BDP]Bonatti, Ch., Díaz, L. J. and Pujals, E. R.. A 𝒞1-generic dichotomy for diffeomorphisms: weak forms of hyperbolicity or infinitely many sinks or sources. Ann. of Math. (2) 158 (2003), 355–418.Google Scholar
[BDV]Bonatti, Ch., Díaz, L. J. and Viana, M.. Dynamics Beyond Uniform Hyperbolicity (Encyclopaedia of Mathematical Sciences, 102). Springer, Berlin, 2005, p. xviii+384 (Mathematical Physics, III).Google Scholar
[BGW]Bonatti, Ch., Gan, Sh. and Wen, L.. On the existence of non-trivial homoclinic classes. Ergod. Th. & Dynam. Sys. 27 (2007), 1473–1508.Google Scholar
[BGLY]Bonatti, Ch., Gan, Sh., Li, M. and Yang, D.. Lyapunov stable classes and essential attractors. Work in progress.Google Scholar
[BLY2]Bonatti, Ch., Li, M. and Yang, D.. Robustly chain transitive singular attractors with singularities of different indices. Preprint, 2008.Google Scholar
[BLY]Bonatti, Ch., Li, M. and Yang, D.. On the existence of attractors. Preprint, 2009, ArXiv 0904.4393.Google Scholar
[BV]Bonatti, Ch. and Viana, M.. SRB measures for partially hyperbolic systems whose central direction is mostly contracting. Israel J. Math. 115 (2000), 157–193.Google Scholar
[BPV]Bonatti, Ch., Pumariño, A. and Viana, M.. Lorenz attractors with arbitrary expanding dimension. C. R. Acad. Sci. Paris 325 (1997), 883–888.Google Scholar
[Bo]Bowen, R.. Equilibrium States and the Ergodic Theory of Axiom A Diffeomorphisms (Lecture Notes in Mathematics, 470). Springer, Berlin, 1975.Google Scholar
[Ca]Carvalho, M.. Sinaï Ruelle Bowen measures for N-dimensional derived from Anosov diffeomorphisms. Ergod. Th. & Dynam. Sys. 13 (1993), 21–44.Google Scholar
[Co]Conley, C.. Isolated Invariant Sets and Morse Index (CBMS Regional Conference Series in Mathematics, 38). American Mathematical Society, Providence, RI, 1978.Google Scholar
[Cr]Crovisier, S.. Periodic orbits and chain recurrent sets of C 1-diffeomorphisms. Publ. Math. Inst. Hautes Études Sci. 104 (2006), 87–141.Google Scholar
[Cr2]Crovisier, S.. Partial hyperbolicity far from homoclinic bifurcations. Preprint, 2008, ArXiv 0809.4965.Google Scholar
[Cr3]Crovisier, S.. Birth of homoclinic bifurcations: a model for the central dynamics of partially hyperbolic systems. Ann. of Math. (2) 172(3) (2010), 1641–1677.Google Scholar
[Cr4]Crovisier, S.. Perturbations de la dynamique de difféomorphismes en topologie C 1. Preprint, ArXiv 0912.2896.Google Scholar
[CP]Crovisier, S. and Pujals, E.. Essential hyperbolicity versus homoclinic bifurcations. Work in progress.Google Scholar
[DPU]Díaz, L. J., Pujals, E. R. and Ures, R.. Partial hyperbolicity and robust transitivity. Acta Math. 183 (1999), 1–43.Google Scholar
[D]Doering, C. I.. Persistently transitive vector fields on three dimensional manifolds. Proceedings on Dynamical Systems and Bifurcation Theory (Pitman Research Notes in Mathematics Series, 160). Logman Scientific & Technical, Harlow, 1987, pp. 59–89.Google Scholar
[F]Franks, J.. Necessary conditions for stability of diffeomorphisms. Trans. Amer. Math. Soc. 158 (1971), 301–308.Google Scholar
[GW]Gan, G. and Wen, L.. Nonsingular star flows satisfy Axiom A and the no-cycle condition. Invent. Math. 164(2) (2006), 279–315.Google Scholar
[Go]Gourmelon, N.. Generation of homoclinic tangencies by C 1-perturbations. Discrete Contin. Dyn. Syst. 26(1) (2010), 1–42.Google Scholar
[Ha]Hayashi, S.. Connecting invariant manifolds and the solution of the C 1-stability and Ω-stability conjectures for flows. Ann. of Math. (2) 145 (1997), 81–137 and Ann. of Math. (2) 150 (1999),353–356.Google Scholar
[Ha2]Hayashi, S.. Diffeomorphisms in ℱ1(M) satisfy Axiom A. Ergod. Th. & Dynam. Sys. 12(2) (1992), 233–253.Google Scholar
[Hu]Hurley, M.. Attractors: persistence, and density of their basins. Trans. Amer. Math. Soc. 269 (1982), 247–271.Google Scholar
[KH]Katok, A. and Hasselblatt, B.. Introduction to the Modern Theory of Dynamical Systems. With a Supplementary Chapter by Katok and Leonardo Mendoza (Encyclopedia of Mathematics and its Applications, 54). Cambridge University Press, Cambridge, 1995, p. xviii+802.Google Scholar
[Ku]Kupka, Y.. Contribution à la théorie des champs génériques. Contrib. Differential Equations 2 (1963), 457–484.Google Scholar
[Li]Liao, S.. Qualitative Theory of Differentiable Dynamical Systems. Science Press, Beijing, 1996, p. x+383; distributed by American Mathematical Society, Providence, RI.Google Scholar
[Ma1]Mañé, R.. Contributions to the stability conjecture. Topology 17 (1978), 386–396.Google Scholar
[Ma]Mañé, R.. An ergodic closing lemma. Ann. of Math. (2) 116 (1982), 503–540.Google Scholar
[Ma2]Mañé, R.. A proof of the C 1 stability conjecture. Publ. Math. Inst. Hautes Études Sci. 66 (1988), 161–210.Google Scholar
[MP]Morales, C. and Pacífico, M. J.. Lyapunov stability of ω-limit sets. Discrete Contin. Dyn. Syst. 8 (2002), 671–674.CrossRefGoogle Scholar
[MP2]Morales, C. and Pacífico, M. J.. A spectral decomposition for singular-hyperbolic sets. Pacific J. Math. 229(1) (2007), 223–232.Google Scholar
[MPP]Morales, C., Pacífico, M. J. and Pujals, E.. Robust transitive singular sets for 3-flows are partially hyperbolic attractors or repellers. Ann. of Math. (2) 160(2) (2004), 375–432.CrossRefGoogle Scholar
[Mo]Moreira, C. G.. There are no C 1-stable intersections of regular Cantor sets. Acta Math. to appear.Google Scholar
[N]Newhouse, S.. Nondensity of axiom A(a) on S 2. Global Analysis (Berkeley, CA, 1968) (Proceedings of Symposia in Pure Mathematics, XIV). American Mathematical Society, Providence, RI, 1970, pp. 191–202.Google Scholar
[N1]Newhouse, S.. Diffeomorphisms with infinitely many sinks. Topology 13 (1974), 9–18.Google Scholar
[N3]Newhouse, S.. The abundance of wild hyperbolic sets and nonsmooth stable sets for diffeomorphisms. Publ. Math. Inst. Hautes Études Sci. 50 (1979), 101–151.Google Scholar
[Pa]Palis, J.. A global view of dynamics and a conjecture on the denseness and finitude of attractors. Astérisque 261 (2000), 335–347.Google Scholar
[PT]Palis, J. and Takens, F.. Hyperbolicity and Sensitive Chaotic Dynamics at Homoclinic Bifurcations (Cambridge Studies in Advanced Mathematics, 35). Cambridge University Press, Cambridge, 1993.Google Scholar
[PV]Palis, J. and Viana, M.. High dimension diffeomorphisms displaying infinitely many sinks. Ann. of Math. (2) 140 (1994), 207–250.Google Scholar
[Po]Poincaré, H.. Les méthodes Nouvelles de la Mécanique céleste. 1899, new edition by Les grands classiques Gauthier-Villars, librairie Blanchard, Paris, 1987.Google Scholar
[Pu]Pugh, C.. The closing lemma. Amer. J. Math. 89 (1967), 956–1009.Google Scholar
[PS]Pujals, E. and Sambarino, M.. Homoclinic tangencies and hyperbolicity for surface diffeomorphisms. Ann. of Math. (2) 151(3) (2000), 961–1023.Google Scholar
[HHTU]Rodriguez Hertz, F., Rodriguez Hertz, M. A., Tahzibi, A. and Ures, R.. A criterion for ergodicity of non-uniformly hyperbolic diffeomorphisms. Electron. Res. Announc. Math. Sci. 14 (2007), 74–81.Google Scholar
[Sh]Shub, M.. Topological transitive diffeomorphism on T 4. Proceedings of the Symposium on Differential Equations and Dynamical Systems (Lecture Notes in Mathematics, 206). Springer, Berlin, 1971, p. 39.CrossRefGoogle Scholar
[Sh1]Shub, M.. Global Stability of Dynamical Systems. Springer, New York, 1987, p. xii+150.Google Scholar
[Si]Simon, R. C.. A three-dimensional Abraham–Smale example. Proc. Amer. Math. Soc. 34(2) (1972), 629–630.Google Scholar
[Sm]Smale, S.. Differentiable dynamical systems. Bull. Amer. Math. Soc. 73 (1967), 747–817.CrossRefGoogle Scholar
[Sm2]Smale, S.. Stable manifolds for differential equations and diffeomorphisms. Ann. Scuola Norm. Sup. Pisa (3) 17 (1963), 97–116.Google Scholar
[V]Vivier, T.. Flots robustement transitifs sur les variétés compactes. C. R. Acad. Sci. Paris 337 (2003), 791–796.Google Scholar
[W2]Wen, L.. Homoclinic tangencies and dominated splittings. Nonlinearity 15 (2002), 1445–1469.Google Scholar
[W3]Wen, L.. Generic diffeomorphisms away from homoclinic tangencies and heterodimensional cycles. Bull. Braz. Math. Soc. (N.S.) 35(3) (2004), 419–452.Google Scholar
[WX]Wen, L. and Xia, Z.. C 1-connecting lemmas. Trans. Amer. Math. Soc. 352 (2000), 5213–5230.Google Scholar
[Y]Yang, J.. Lyapunov stable chain recurrent class. Preprint, 2007.Google Scholar