Acta Mathematica

On the regularization of conservative maps

Artur Avila

Full-text: Open access

Abstract

We show that smooth maps are C1-dense among C1 volume-preserving maps.

Article information

Source
Acta Math., Volume 205, Number 1 (2010), 5-18.

Dates
Received: 13 October 2008
First available in Project Euclid: 31 January 2017

Permanent link to this document
https://projecteuclid.org/euclid.acta/1485892482

Digital Object Identifier
doi:10.1007/s11511-010-0050-y

Mathematical Reviews number (MathSciNet)
MR2736152

Zentralblatt MATH identifier
1211.37029

Subjects
Primary: 37D20: Uniformly hyperbolic systems (expanding, Anosov, Axiom A, etc.)
Secondary: 37D25: Nonuniformly hyperbolic systems (Lyapunov exponents, Pesin theory, etc.) 37D30: Partially hyperbolic systems and dominated splittings 37J10: Symplectic mappings, fixed points 37C20C

Keywords
symplectic diffeomorphisms partial hyperbolicity Lyapunov exponents generic properties

Rights
2010 © Institut Mittag-Leffler

Citation

Avila, Artur. On the regularization of conservative maps. Acta Math. 205 (2010), no. 1, 5--18. doi:10.1007/s11511-010-0050-y. https://projecteuclid.org/euclid.acta/1485892482


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References

  • Arbieto, A. & Matheus, C., A pasting lemma and some applications for conservative systems. Ergodic Theory Dynam. Systems, 27 (2007), 1399–1417.
  • Avila, A., Bochi, J. & Wilkinson, A., Nonuniform center bunching and the genericity of ergodicity among C1 partially hyperbolic symplectomorphisms. Ann. Sci. Éc. Norm. Supér., 42 (2009), 931–979.
  • Bochi, J., Genericity of zero Lyapunov exponents. Ergodic Theory Dynam. Systems, 22 (2002), 1667–1696.
  • Bochi, J. & Viana, M., Lyapunov exponents: how frequently are dynamical systems hyperbolic?, in Modern Dynamical Systems and Applications, pp. 271–297. Cambridge Univ. Press, Cambridge, 2004.
  • — The Lyapunov exponents of generic volume-preserving and symplectic maps. Ann. of Math., 161 (2005), 1423–1485.
  • Bonatti, C., Díaz, L. J. & Pujals, E. R., A C1-generic dichotomy for diffeomorphisms: weak forms of hyperbolicity or infinitely many sinks or sources. Ann. of Math., 158 (2003), 355–418.
  • Bonatti, C., Díaz, L. J. & Viana, M., Dynamics Beyond Uniform Hyperbolicity. Encyclopaedia of Mathematical Sciences, 102. Springer, Berlin–Heidelberg, 2005.
  • Bonatti, C., Matheus, C., Viana, M. & Wilkinson, A., Abundance of stable ergodicity . Comment. Math. Helv., 79 (2004), 753–757.
  • Bonatti, C. & Viana, M., SRB measures for partially hyperbolic systems whose central direction is mostly contracting. Israel J. Math., 115 (2000), 157–193.
  • — Lyapunov exponents with multiplicity 1 for deterministic products of matrices . Ergodic Theory Dynam. Systems, 24 (2004), 1295–1330.
  • Burago, D. & Kleiner, B., Separated nets in Euclidean space and Jacobians of bi-Lipschitz maps. Geom. Funct. Anal., 8 (1998), 273–282.
  • Burns, K. & Wilkinson, A., On the ergodicity of partially hyperbolic systems . Ann. of Math, 171 (2010), 451–489.
  • Dacorogna, B. & Moser, J., On a partial differential equation involving the Jacobian determinant. Ann. Inst. H. Poincaré Anal. Non Linéaire, 7 (1990), 1–26.
  • Dolgopyat, D. & Pesin, Y., Every compact manifold carries a completely hyperbolic diffeomorphism. Ergodic Theory Dynam. Systems, 22 (2002), 409–435.
  • Franks, J., Necessary conditions for stability of diffeomorphisms. Trans. Amer. Math. Soc., 158 (1971), 301–308.
  • Hayashi, S., Connecting invariant manifolds and the solution of the C1 stability and -stability conjectures for flows. Ann. of Math., 145 (1997), 81–137.
  • Hirsch, M.W., Differential Topology. Graduate Texts in Mathematics, 33. Springer, New York, 1994.
  • McMullen, C. T., Lipschitz maps and nets in Euclidean space. Geom. Funct. Anal., 8 (1998), 304–314.
  • Moser, J., On the volume elements on a manifold. Trans. Amer. Math. Soc., 120 (1965), 286–294.
  • Palis, J. & Pugh, C. C. (eds.), Fifty problems in dynamical systems, in Dynamical Systems (Warwick, 1974), Lecture Notes in Math., 468, pp. 345–353. Springer, Berlin– Heidelberg, 1975.
  • Pugh, C.C., The closing lemma. Amer. J. Math., 89 (1967), 956–1009.
  • Rivière, T. & Ye, D., Resolutions of the prescribed volume form equation . NoDEA Nonlinear Differential Equations Appl., 3 (1996), 323–369.
  • Rodriguez Hertz, F., Stable ergodicity of certain linear automorphisms of the torus . Ann. of Math., 162 (2005), 65–107.
  • Rodriguez Hertz, F., Rodriguez Hertz, M. A., Tahzibi, A. & Ures, R., A criterion for ergodicity of non-uniformly hyperbolic diffeomorphisms . Electron. Res. Announc. Math. Sci., 14 (2007), 74–81.
  • Tahzibi, A., Stably ergodic diffeomorphisms which are not partially hyperbolic. Israel J. Math., 142 (2004), 315–344.
  • Thurston, W.P., Three-Dimensional Geometry and Topology. Vol. 1. Princeton Mathematical Series, 35. Princeton University Press, Princeton, NJ, 1997.
  • Ye, D., Prescribing the Jacobian determinant in Sobolev spaces. Ann. Inst. H. Poincaré Anal. Non Linéaire, 11 (1994), 275–296.
  • Zehnder, E., Note on smoothing symplectic and volume-preserving diffeomorphisms, in Geometry and Topology (Rio de Janeiro, 1976), Lecture Notes in Math., 597, pp. 828–854. Springer, Berlin–Heidelberg, 1977.
  • Zuppa, C., Régularisation C des champs vectoriels qui préservent l’élément de volume. Bol. Soc. Brasil. Mat., 10 (1979), 51–56.