Hostname: page-component-78c5997874-xbtfd Total loading time: 0 Render date: 2024-11-09T15:56:50.512Z Has data issue: false hasContentIssue false
  • English
  • Français

Wild Milnor attractors accumulated by lower-dimensional dynamics

Published online by Cambridge University Press:  05 November 2012

RAFAEL POTRIE
RAFAEL POTRIE*
Affiliation:
CMAT, Facultad de Ciencias, Universidad de la República, Uruguay LAGA, Institute Galilée, Universit́e Paris 13, Villetaneuse, France (email: [email protected])
Get access Rights & Permissions [Opens in a new window]

Abstract

We present new examples of open sets of diffeomorphisms such that generic diffeomorphisms in those sets have no dynamically indecomposable attractors in the topological sense and have infinitely many chain-recurrence classes. We show that all other classes except one are contained in periodic surfaces. This study allows us to obtain the existence of Milnor attractors as well as study ergodic properties of the diffeomorphisms in those open sets by using ideas and results from Bonatti and Viana [SRB measures for partially hyperbolic diffeomorphisms whose central direction is mostly contracting. Israel J. Math.115 (2000), 157–193] and Buzzi and Fisher [Entropic stability beyond partial hyperbolicity. Preprint, 2011, arXiv:1103:2707].

Type
Research Article
Information
Ergodic Theory and Dynamical Systems , Volume 34 , Issue 1 , February 2014 , pp. 236 - 262
Copyright
Copyright © 2012 Cambridge University Press 

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

References

[A]Araujo, A.. Existência de atratores hiperbólicos para difeomorfismos de superfície. Thesis, IMPA, 1987.Google Scholar
[B]Bonatti, C.. Towards a global view of dynamical systems, for the $C^1$-topology. Ergod. Th. & Dynam. Sys. 31(4) (2011), 959–993.Google Scholar
[BC]Bonatti, C. and Crovisier, S.. Récurrence et généricité. Invent. Math. 158 (2004), 33–104.CrossRefGoogle Scholar
[BCDG]Bonatti, C., Crovisier, S., Díaz, L. and Gourmelon, N.. Internal perturbations of homoclinic classes: non-domination, cycles, and self-replication. Ergod. Th. & Dynam. Sys. to appear, Preprint, 2010, arXiv 1011.2935.Google Scholar
[BD1]Bonatti, C. and Díaz, L.. Persistent non-hyperbolic transitive diffeomorphisms. Ann. of Math. (2) 143 (1995), 357–396.Google Scholar
[BD2]Bonatti, C. and Díaz, L.. On maximal generic sets of generic diffeomorphisms. Publ. Math. Inst. Hautes Études Sci. 96 (2003), 171–197.Google Scholar
[BD3]Bonatti, C. and Díaz, L.. Abundance of $C^1$-robust homoclinic tangencies. Trans. Amer. Math. Soc. to appear, Preprint, 2009, arXiv:0909.4062.Google Scholar
[BDP]Bonatti, C., Díaz, L. and Pujals, E.. A $C^1$-generic dichotomy for diffeomorphisms: weak forms of hyperbolicity or infinitely many sinks or sources. Ann. of Math. (2) 158 (2003), 355–418.CrossRefGoogle Scholar
[BDV]Bonatti, C., Díaz, L. and Viana, M.. Dynamics beyond uniform hyperbolicity. A global geometric and probabilistic perspective. Mathematical Physics III (Encyclopaedia of Mathematical Sciences, 102). Springer, Berlin, 2005.Google Scholar
[BLY]Bonatti, C., Li, M. and Yang, D.. On the existence of attractors. Trans. Amer. Math. Soc. to appear, Preprint, 2009, arXiv:0904.4393.Google Scholar
[BV]Bonatti, C. and Viana, M.. SRB measures for partially hyperbolic diffeomorphisms whose central direction is mostly contracting. Israel J. Math 115 (2000), 157–193.Google Scholar
[BF]Buzzi, J. and Fisher, T.. Entropic stability beyond partial hyperbolicity, Preprint, 2011, arXiv:1103:2707.Google Scholar
[BFSV]Buzzi, J., Fisher, T., Sambarino, M. and Vasquez, C.. Maximal entropy measures for certain partially hyperbolic derived from Anosov systems. Ergod. Th. & Dynam. Sys. 32(1) (2012), 63–79.Google Scholar
[Car]Carvalho, M.. Sinai–Ruelle–Bowen measures for $N$-dimensional derived from Anosov diffeomorphisms. Ergod. Th. & Dynam. Sys. 13 (1993), 21–44.Google Scholar
[C1]Crovisier, S.. Partial hyperbolicity far from homoclinic bifurcations. Adv. Math. 226 (2011), 673–726.Google Scholar
[C2]Crovisier, S.. Perturbation de la dynamique de difféomorphismes en topologie $C^1$. Astérisque to appear, Preprint, 2009, arXiv:0912.2896.Google Scholar
[D]Daverman, R.. Decompositions of Manifolds. Academic Press, New York, 1986.Google Scholar
[F]Fathi, A.. Expansiveness hyperbolicity and Haussdorf dimension. Comm. Math. Phys. 126 (1989), 249–262.Google Scholar
[HPS]Hirsch, M., Pugh, C. and Shub, M.. Invariant Manifolds (Lecture Notes in Mathematics, 583). Springer, Heidelberg, 1977.Google Scholar
[Mi]Milnor, J.. On the concept of attractor. Comm. Math. Phys. 99(2) (1985), 177–195.Google Scholar
[PV]Palis, J. and Viana, M.. High dimensional diffeomorphisms displaying infinitely many periodic attractors. Ann. of Math. (2) 140 (1994), 207–250.Google Scholar
[Pot]Potrie, R.. A proof of the existence of attractors in dimension two. Unpublished note, not intended for publication. Available at http://www.cmat.edu.uy/∼rpotrie/documentos/pdfs/dimensiondos.pdf.Google Scholar
[PS]Pujals, E. and Sambarino, M.. Homoclinic tangencies and hyperbolicity for surface diffeomorphisms. Ann. of Math. (2) 151 (2000), 961–1023.Google Scholar
[R]Robinson, C.. Dynamical Systems, Stability, Symbolic Dynamics, and Chaos. CRC Press, Boca Raton, FL, 1994.Google Scholar
[RRTU]Rodriguez Hertz, F., Rodriguez Hertz, M. A., Tahzibi, A. and Ures, R.. Maximizing measures for partially hyperbolic systems with compact center leaves. Ergod. Th. & Dynam. Syst. 32(2) (2012), 825–839.Google Scholar
[VY]Viana, M. and Yang, J.. Physical measures and absolute continuity for one-dimensional center direction. Preprint, IMPA A683, 2010.Google Scholar
[W]Walters, P.. Anosov diffeomorphisms are topologically stable. Topology 9 (1970), 71–78.Google Scholar