Hostname: page-component-7b7567c8c-tnzcf Total loading time: 0 Render date: 2026-11-14T17:53:32.179Z Has data issue: false hasContentIssue false
  • English
  • Français

Hyperbolicity in the volume-preserving scenario

Published online by Cambridge University Press:  23 August 2012

ALEXANDER ARBIETO and
THIAGO CATALAN
ALEXANDER ARBIETO
Affiliation:
Instituto de Matemática, Universidade Federal do Rio de Janeiro, PO Box 68530, 21945-970 Rio de Janeiro, Brazil (email: arbieto@im.ufrj.br)
THIAGO CATALAN
Affiliation:
Faculdade de Matemática, Universidade Federal de Uberlândia, 34-32309442 Uberlândia-MG, Brazil (email: tcatalan@famat.ufu.br)
Rights & Permissions [Opens in a new window]

Abstract

Core share and HTML view are not available for this content. However, as you have access to this content, a full PDF is available via the 'Save PDF' action button.

Extending a result of Mañé, Hayashi proved that every diffeomorphism $f$ which has a $C^1$-neighborhood $\mathcal {U}$ where all periodic points of any $g\in \mathcal {U}$are hyperbolic is an Axiom A diffeomorphism. Here we prove the analogous result in the volume-preserving scenario.

Information

Type
Research Article
Information
Ergodic Theory and Dynamical Systems , Volume 33 , Issue 6 , December 2013, pp. 1644 - 1666
Copyright
Copyright © 2012 Cambridge University Press 

References

[1]Abdenur, F.. Generic robustness of spectral decompositions. Ann. Sci. Éc. Norm. Supér. (4) 36(2) (2003), 213–224.Google Scholar
[2]Abdenur, F., Bonatti, C., Crovisier, S., Diaz, L. and Wen, L.. Periodic points and homoclinic classes. Ergod. Th. & Dynam. Sys. 27 (2007), 1–22.CrossRefGoogle Scholar
[3]Araújo, V. and Pacífico, M. J.. Three-dimensional Flows (Results in Mathematics and Related Areas. 3rd Series, 53). Springer, Heidelberg, 2010.Google Scholar
[4]Arbieto, A. and Matheus, C.. A pasting lemma and some applications for conservative systems. Ergod. Th. Dynam. Sys. 27(5) (2007), 1399–1417.Google Scholar
[5]Arbieto, A., Senos, L. and Sodero, T.. The specification property for flows from the robust and generic viewpoint. Preprint, 2011, arXiv:1101.3445. J. Differential Equations to appear.CrossRefGoogle Scholar
[6]Arnaud, M.-C.. Le ‘Closing lemma’ en topologie $C^1$ (Mémoires de la Société Mathématique de France, 74). Société Mathématique de France, Paris, 1998.Google Scholar
[7]Avila, A.. On the regularization of conservative maps. Acta Math. 205(1) (2010), 5–18.Google Scholar
[8]Avila, A., Bochi, J. and Wilkinson, A.. Nonuniform center bunching and the genericity of ergodicity among $C^1$ partially hyperbolic symplectomorphisms. Ann. Sci. Éc. Norm. Supér. (4) 42(6) (2009), 931–979.Google Scholar
[9]Bessa, M.. Generic incompressible flows are topological mixing. C. R. Math. Acad. Sci. Paris 346 (2008), 1169–1174.Google Scholar
[10]Bessa, M. and Rocha, J.. On $C^1$-robust transitivity of volume-preserving flows. J. Differential Equations 245(11) (2008), 3127–3143.Google Scholar
[11]Bessa, M. and Rocha, J.. Three-dimensional conservative star flows are Anosov. Discrete Contin. Dyn. Syst. A 26 (2010), 839–846.Google Scholar
[12]Bessa, M., Ferreira, C. and Rocha, J.. On the stability of the set of hyperbolic closed orbits of a Hamiltonian. Math. Proc. Cambridge Philos. Soc. 149(2) (2010), 373–383.Google Scholar
[13]Bochi, J.. $C^1$-generic symplectic diffeomorphisms: partial hyperbolicity and zero centre Lyapunov exponents. J. Inst. Math. Jussieu 9(1) (2010), 49–93.Google Scholar
[14]Bonatti, C. and Crovisier, S.. Récurrence et générecité. Invent. Math. 158 (2004), 33–104.Google Scholar
[15]Bonatti, C., Diaz, L. and Pujals, H.. 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
[16]Carballo, C., Moralles, C. and Pacifico, M.. Homoclinic classes for generic $C^1$ vector fields. Ergod. Th. & Dynam. Sys. 23 (2003), 403–415.CrossRefGoogle Scholar
[17]Crovisier, S.. Perturbation de la dynamique de diffeomorphismes en topologie $C^1$. Preprint, 2009, arXiv:0912.2896.Google Scholar
[18]Crovisier, S.. Birth of homoclinic intersections: a model for the central dynamics of partially hyperbolic systems. Ann. of Math. (2) 172(3) (2010), 1641–1677.Google Scholar
[19]Ferreira, C.. Stability properties of divergence-free vector fields, Preprint, 2010, arXiv:1004.2893.Google Scholar
[20]Franks, J.. Necessary conditions for stability of diffeomorphisms. Trans. Amer. Math. Soc. 158 (1971), 301–308.Google Scholar
[21]Gan, S.. Another proof for $C^1$ stability conjecture for flows. Sci. China Ser. A 41(10) (1998), 1076–1082.Google Scholar
[22]Gan, S. and Wen, L.. Nonsingular star flows satisfy Axiom A and the no-cycle condition. Invent. Math. 164(2) (2006), 279–315.Google Scholar
[23]Hayashi, S.. Diffeomorphisms in $\mathcal {F}^1(M)$ satisfy Axiom A. Ergod. Th. & Dynam. Sys. 12 (1992), 233–253.CrossRefGoogle Scholar
[24]Hayashi, S.. Connecting invariant manifolds and the solution of the $C^1$ stability and $\Omega $-stability conjectures for flows. Ann. of Math. (2) 145(1) (1997), 81–137.Google Scholar
[25]Horita, V. and Tahzibi, A.. Partial hyperbolicity for symplectic diffeomorphisms. Ann. Inst. H. Poincaré Anal. Non Linéaire 23(5) (2006), 641–661.Google Scholar
[26]Li, M., Gan, S. and Wen, L.. Robustly transitive singular sets via approach of an extended linear Poincaré flow. Discrete Contin. Dyn. Syst. 13(2) (2005), 239–269.Google Scholar
[27]Liang, C., Liu, G. and Sun, W.. Equivalent conditions of dominated splittings for volume-preserving diffeomorphism. Acta Math. Sinica 23 (2007), 1563–1576.CrossRefGoogle Scholar
[28]Liao, S. T.. A basic property of a certain class of differential systems. Acta Math. Sinica 22 (1979), 316–343 (in Chinese).Google Scholar
[29]Mañé, R.. An ergodic closing lemma. Ann. of Math. (2) 116(3) (1982), 503–540.Google Scholar
[30]Mañé, R.. A proof of the $C^1$ stability conjecture. Publ. Math. Inst. Hautes Études Sci. 66 (1987), 161–210.Google Scholar
[31]Newhouse, S. E.. Quasi-elliptic periodic points in conservative dynamical systems. Amer. J. Math. 99(5) (1977), 1061–1087.CrossRefGoogle Scholar
[32]Palis, J.. Global perspective for non-conservative dynamics. Ann. Inst. H. Poincaré Anal. Non Linéaire 22 (2005), 485–507.Google Scholar
[33]Palis, J.. On the $C^1$$\Omega $-stability conjecture. Publ. Math. Inst. Hautes Études Sci. 66 (1988), 211–215.Google Scholar
[34]Pugh, C. and Robinson, C.. The $C^1$ closing lemma, including Hamiltonians. Ergod. Th. & Dynam. Sys. 3(2) (1983), 261–313.CrossRefGoogle Scholar
[35]Pujals, E. and Sambarino, M.. Homoclinic tangencies and hyperbolicity for surface diffeomorphisms. Ann. of Math. (2) 151 (2000), 961–1023.Google Scholar
[36]Robinson, C.. Generic properties of conservative systems. Amer. J. Math. 92 (1970), 562–603.Google Scholar
[37]Toyoshiba, H.. Vector fields in the interior of Kupka–Smale systems satisfy Axiom A. J. Differential Equations 177(1) (2001), 27–48.Google Scholar
[38]Vivier, T.. Projective hyperbolicity and fixed points. Ergod. Th. & Dynam. Sys. 26 (2006), 923–936.CrossRefGoogle Scholar
[39]Xia, Z.. Homoclinic points in symplectic and volume-preserving diffeomorphisms. Commun. Math. Phys. 177 (1996), 435–449.Google Scholar
[40]Xia, Z. and Wen, L.. $C^1$ connecting lemmas. Trans. Amer. Math. Soc. 352(11) (2000), 5213–5230.Google Scholar
[41]Zuppa, C.. Régularisation $C^{\infty }$ des champs vectoriels qui préservent l’élément de volume. Bol. Soc. Brasil. Mat. 10(2) (1979), 51–56.Google Scholar