Advertisement

The Journal of Geometric Analysis

, Volume 29, Issue 2, pp 1744–1762 | Cite as

Chaotic Holomorphic Automorphisms of Stein Manifolds with the Volume Density Property

  • Leandro Arosio
  • Finnur LárussonEmail author
Article
  • 57 Downloads

Abstract

Let X be a Stein manifold of dimension \(n\ge 2\) satisfying the volume density property with respect to an exact holomorphic volume form. For example, X could be \(\mathbb {C}^n\), any connected linear algebraic group that is not reductive, the Koras–Russell cubic, or a product \(Y\times \mathbb {C}\), where Y is any Stein manifold with the volume density property. We prove that chaotic automorphisms are generic among volume-preserving holomorphic automorphisms of X. In particular, X has a chaotic holomorphic automorphism. A proof for \(X=\mathbb {C}^n\) may be found in work of Fornæss and Sibony. We follow their approach closely. Peters, Vivas, and Wold showed that a generic volume-preserving automorphism of \(\mathbb {C}^n\), \(n\ge 2\), has a hyperbolic fixed point whose stable manifold is dense in \(\mathbb {C}^n\). This property can be interpreted as a kind of chaos. We generalise their theorem to a Stein manifold as above.

Keywords

Stein manifold Linear algebraic group Homogeneous space Holomorphic automorphism Volume-preserving automorphism Chaotic automorphism Andersén–Lempert theory Volume density property Algebraic volume density property Stable manifold 

Mathematics Subject Classification

Primary 32M17 Secondary 14L17 14R10 14R20 32H50 32M05 37F99 

References

  1. 1.
    Alpern, S., Prasad, V.S.: Maximally chaotic homeomorphisms of sigma-compact manifolds. Topol. Appl. 105, 103–112 (2000)MathSciNetCrossRefzbMATHGoogle Scholar
  2. 2.
    Andersén, E.: Volume-preserving automorphisms of \({\mathbb{C}}^n\). Complex Var. Theory Appl. 14, 223–235 (1990)zbMATHGoogle Scholar
  3. 3.
    Banks, J., Brooks, J., Cairns, G., Davis, G., Stacey, P.: On Devaney’s definition of chaos. Am. Math. Mon. 99, 332–334 (1992)MathSciNetCrossRefzbMATHGoogle Scholar
  4. 4.
    Devaney, R.L.: An Introduction to Chaotic Dynamical Systems. Addison-Wesley Studies in Nonlinearity, 2nd edn. Addison-Wesley, Boston (1989)zbMATHGoogle Scholar
  5. 5.
    Dixon, P.G., Esterle, J.: Michael’s problem and the Poincaré-Fatou-Bieberbach phenomenon. Bull. Am. Math. Soc. 15, 127–187 (1986)CrossRefzbMATHGoogle Scholar
  6. 6.
    Fornæss, J.E., Sibony, N.: The closing lemma for holomorphic maps. Ergod. Theory Dyn. Syst. 17, 821–837 (1997)MathSciNetCrossRefzbMATHGoogle Scholar
  7. 7.
    Forstnerič, F.: Stein manifolds and holomorphic mappings. The homotopy principle in complex analysis. Second edition. Ergebnisse der Mathematik und ihrer Grenzgebiete, 3. Folge, 56. Springer, Berlin (2017)Google Scholar
  8. 8.
    Heinzner, P.: Geometric invariant theory on Stein spaces. Math. Ann. 289, 631–662 (1991)MathSciNetCrossRefzbMATHGoogle Scholar
  9. 9.
    Jurdjevic, V.: Geometric Control Theory. Cambridge Studies in Advanced Mathematics, vol. 52. Cambridge University Press, Cambridge (1997)zbMATHGoogle Scholar
  10. 10.
    Kaliman, S., Kutzschebauch, F.: Algebraic volume density property of affine algebraic manifolds. Invent. Math. 181, 605–647 (2010)MathSciNetCrossRefzbMATHGoogle Scholar
  11. 11.
    Kaliman, S., Kutzschebauch, F.: On the present state of the Andersén-Lempert theory. Affine algebraic geometry. In: CRM Proceedings and Lecture Notes, vol. 54, pp. 85–122. American Mathematical Society (2011)Google Scholar
  12. 12.
    Kaliman, S., Kutzschebauch, F.: On algebraic volume density property. Transform. Gr. 21, 451–478 (2016)MathSciNetCrossRefzbMATHGoogle Scholar
  13. 13.
    Kaliman, S., Kutzschebauch, F.: Algebraic (volume) density property for affine homogeneous spaces. Math. Ann. 367, 1311–1332 (2017)MathSciNetCrossRefzbMATHGoogle Scholar
  14. 14.
    Kaup, W.: Infinitesimale Transformationsgruppen komplexer Räume. Math. Ann. 160, 72–92 (1965)MathSciNetCrossRefzbMATHGoogle Scholar
  15. 15.
    Leuenberger, M.: (Volume) density property of a family of complex manifolds including the Koras-Russell cubic threefold. Proc. Am. Math. Soc. 144, 3887–3902 (2016)MathSciNetCrossRefzbMATHGoogle Scholar
  16. 16.
    Morimoto, A.: On the classification of noncompact complex abelian Lie groups. Trans. Am. Math. Soc. 123, 200–228 (1966)MathSciNetCrossRefzbMATHGoogle Scholar
  17. 17.
    Peters, H., Vivas, L.R., Wold, E.F.: Attracting basins of volume preserving automorphisms of \({\mathbb{C}}^k\). Int. J. Math. 19, 801–810 (2008)CrossRefzbMATHGoogle Scholar
  18. 18.
    Ramos-Peon, A.: Non-algebraic examples of manifolds with the volume density property. Proc. Am. Math. Soc. 145, 3899–3914 (2017)MathSciNetCrossRefzbMATHGoogle Scholar
  19. 19.
    Touhey, P.: Yet another definition of chaos. Am. Math. Mon. 104, 411–414 (1997)MathSciNetCrossRefzbMATHGoogle Scholar
  20. 20.
    Varolin, D.: The density property for complex manifolds and geometric structures II. Int. J. Math. 11, 837–847 (2000)MathSciNetzbMATHGoogle Scholar
  21. 21.
    Varolin, D.: The density property for complex manifolds and geometric structures. J. Geom. Anal. 11, 135–160 (2001)MathSciNetCrossRefzbMATHGoogle Scholar

Copyright information

© Mathematica Josephina, Inc. 2018

Authors and Affiliations

  1. 1.Dipartimento Di MatematicaUniversità di Roma “Tor Vergata”RomaItaly
  2. 2.School of Mathematical SciencesUniversity of AdelaideAdelaideAustralia

Personalised recommendations