Skip to main content
SpringerLink
Log in

Classification of invariant Fatou components for dissipative Hénon maps

  • Published:
Geometric and Functional Analysis Aims and scope Submit manuscript

Abstract

Fatou components for rational endomorphisms of the Riemann sphere are fully classified and play an important role in our view of one-dimensional dynamics. In higher dimensions, the situation is less satisfactory. In this work we give a nearly complete classification of invariant Fatou components for moderately dissipative Hénon maps. Namely, we prove that any such a component is either an attracting or parabolic basin, or the basin of a rotation domain. More specifically, recurrent Fatou components were classified about 20 years ago (modulo the problem of existence of Herman ring basins), while in this paper we prove that non-recurrent invariant Fatou components are semi-parabolic basins. Most of our methods apply in a more general setting.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Similar content being viewed by others

References

  1. Abate M.: Iteration theory, compactly divergent sequences and commuting holomorphic maps. Annali della Scuola Normale Superiore di Pisa—Classe di Scienze (4) 18(2), 167–191 (1991)

    MATH  MathSciNet  Google Scholar 

  2. Bedford E., Smillie J.: Polynomial diffeomorphisms of \({\mathbb{C}^2}\) : currents, equilibrium measure and hyperbolicity. Inventiones Mathematicae 103(1), 69–99 (1991)

    Article  MATH  MathSciNet  Google Scholar 

  3. Bedford E., Smillie J.: Polynomial diffeomorphisms of \({\mathbb{C}^2}\) : II. Stable manifolds and recurrence. Journal of the American Mathematical Society 4(4), 657–679 (1991)

    MATH  MathSciNet  Google Scholar 

  4. E. Bedford and J. Smillie. External rays in the dynamics of polynomial automorphisms of C2. In: Contemporary Mathematics, Vol. 222. AMS, Providence (1999).

  5. Benedicks M., Carleson L.: The dynamics of the Hénon mapping. Annals of Mathematics (2) 133(1), 73–169 (1991)

    Article  MATH  MathSciNet  Google Scholar 

  6. Bracci F., Molino L.: The dynamics near quasi-parabolic fixed points of holomorphic diffeomorphisms in \({\mathbb{C}^2}\). American Journal of Mathematics 126(3), 671–686 (2004)

    Article  MATH  MathSciNet  Google Scholar 

  7. Bracci F., Raissy J., Zaitsev D.: Dynamics of multi- resonant biholomorphisms. International Mathematics Research Notices 20, 4772–4797 (2013)

    MathSciNet  Google Scholar 

  8. Bracci F., Zaitsev D.: Dynamics of one-resonant biholomorphisms. Journal of the European Mathematical Society 15, 179–200 (2013)

    Article  MATH  MathSciNet  Google Scholar 

  9. De Carvalho A., Lyubich M., Martens M.: Renormalization in the Hénon family, I: universality but non-rigidity. Journal of Statistical Physics, Special issue dedicated to Feigenbaum’s 60th birthday 121, 611–669 (2005)

    MATH  MathSciNet  Google Scholar 

  10. P. Di Giuseppe. Topological Classification of Holomorphic, Semi-Hyperbolic Germs. In: "Quasi-Absence" of Resonances. available on arXiv:math/0501397.

  11. R. Dujardin and M. Lyubich. Stability and bifurcations for polynomial automorphisms of \({\mathbb{C}^2}\). Manuscript in preparation.

  12. Eremenko A., Levin G.: On periodic points of a polynomial. Ukrainian Mathematical Journal 41(11), 1258–1262 (1989)

    Article  MATH  MathSciNet  Google Scholar 

  13. Fornæss J.-E., Sibony N.: Complex Hénon mappings in \({\mathbb{C}^2}\) and Fatou-Bieberbach domains. Duke Mathematical Journal 65(2), 345–380 (1992)

    Article  MATH  MathSciNet  Google Scholar 

  14. Fornæss J.E., Sibony N.: Complex dynamics in higher dimension. I. Asterisque 222, 201–231 (1991)

    Google Scholar 

  15. Fornæss J.-E., Sibony N.: Classification of recurrent domains for some holomorphic maps. Mathematische Annalen, 301 (1995), no. 4, 813–820.

    Article  MATH  MathSciNet  Google Scholar 

  16. Fornæss J.-E., Sibony N.: Complex dynamics in higher dimension II, modern methods in complex analysis. Annals of Studies 137, 135–182 (1995)

    Google Scholar 

  17. Friendland S., Milnor J.: Dynamical properties of plane polynomial automorphisms. Ergodic Theory and Dynamical Systems 9(1), 67–99 (1989)

    MathSciNet  Google Scholar 

  18. A.A. Goldberg and I.V. Ostrovskii. Value distribution of meromorphic functions. Translations of Mathematical Monographs, Vol. 236. AMS (2008).

  19. Hakim M.: Attracting domains for semi-attractive transformations of \({\mathbb{C}^p}\). Publicationes Mathematicae 38, 479–499 (1994)

    MATH  MathSciNet  Google Scholar 

  20. W.K. Hayman and P.B. Kennedy. Subharmonic functions, Vol. I. London Mathematical Society Monographs, No. 9. Academic Press (1976).

  21. Hirsch M.W., Pugh C.C., Shub M.: Invariant Manifolds. Bulletin of the American Mathematical Society 76(5), 1015–1019 (1970)

    Article  MATH  MathSciNet  Google Scholar 

  22. L. Hörmander Notions of convexity. Progress in Mathematics, 127. Birkhäuser (1994).

  23. J.H. Hubbard. The Hénon mapping in the complex domain. In: M.F. Barnsely and S.G. Demko (eds) Chaotic Dynamics and Fractals. Academic, Orlando (1986).

  24. Hubbard J.H., Oberste-Vorth R.W.: Hénon mappings in the complex domain. I. The global topology of dynamical space. Institut des Hautes Etudes Scientifiques—Publications Mathmatiques 79, 5–46 (1994)

    Article  MATH  MathSciNet  Google Scholar 

  25. Hubbard J.H., Oberste-Vorth R.W.: Hénon mappings in the complex domain. I. Projective and inductive limits of polynomials. Advanced Science Institutes Series C: Mathematical and Physical Sciences 464, 89–132 (1995)

    MathSciNet  Google Scholar 

  26. Hubbard J.H., Papadopol P.: Superattractive fixed points in \({\mathbb{C}^n}\). Indiana University Mathematics Journal 43(1), 321–365 (1994)

    Article  MATH  MathSciNet  Google Scholar 

  27. Jupiter D., Lilov K.: Invariant Fatou components of automorphisms of \({\mathbb{C}^2}\). Far East Journal of Dynamical Systems 6(1), 49–65 (2003)

    MathSciNet  Google Scholar 

  28. Lyubich M., Lyubich Yu.: Spectral theory for almost periodic representations of semigroups. Ukrainian Mathematical Journal 36, 474–478 (1985) (Translation from Russian)

    Article  Google Scholar 

  29. J. Milnor. Singular points of complex hypersurfaces. In: Annals of Mathematics Studies, Vol. 61. Princeton University Press, Princeton (1968).

  30. J. Milnor. Dynamics in one complex variable. In: Annals of Mathematical Studies, Vol. 160. Princeton University Press, Princeton (1999).

  31. Newhouse. Diffeomorphisms with infinitely many sinks. Topology, 13 (1974), 9–18.

  32. Papakyriakopoulos C.D.: On Dehn’s lemma and the asphericity of knots. Annals of Mathematics 66(2), 1–26 (1957)

    Article  MATH  MathSciNet  Google Scholar 

  33. Peters H., Zeager C.: Tautness and Fatou components in \({\mathbb{P}^2}\). Journal of Geometric Analysis 22(4), 934–941 (2012)

    Article  MATH  MathSciNet  Google Scholar 

  34. M. Sodin. Lars Ahlfors’ thesis. Israel mathematical Conference Proceedings, Vol. 14 (2000).

  35. Sullivan D.: Quasiconformal homeomorphisms and dynamics. I. Solution of the Fatou-Julia problem on wandering domains. Annals of Mathematics 122, 401–418 (1985)

    Article  MATH  MathSciNet  Google Scholar 

  36. Ueda T.: Local structure of analytic transformations of two complex variables I. Journal of Mathematics of Kyoto University 26(2), 233–261 (1986)

    MATH  MathSciNet  Google Scholar 

  37. Ueda T.: Fatou sets in complex dynamics on projective spaces. Journal Mathematical Society of Japan 46, 545–555 (1994)

    Article  MATH  MathSciNet  Google Scholar 

  38. Ueda T.: Holomorphic maps on projective spaces and continuations of Fatou maps. Michigan Mathematical Journal 56(1), 145–153 (2008)

    Article  MATH  MathSciNet  Google Scholar 

  39. Wiman A.: Sur une extention d’un théoréme de M. Hadamard. Arkiv för Matematik, Astronomi och Fysik 2(14), 1–5 (1905)

    Google Scholar 

  40. Weickert B.J.: Nonwandering, nonrecurrent Fatou components in \({ \mathbb{P}^2 }\). Pacific Journal of Mathematics, 211(2), 391–397 (2003)

    Article  MATH  MathSciNet  Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Institute for Mathematical Sciences, Stony Brook University, Stony Brook, NJ, 11794, USA

    Mikhail Lyubich

  2. Korteweg de Vries Institute for Mathematics, University of Amsterdam, P.O. Box 94248, 1090 GE, Amsterdam, Netherlands

    Han Peters

Authors
  1. Mikhail Lyubich
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Han Peters
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Mikhail Lyubich.

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Lyubich, M., Peters, H. Classification of invariant Fatou components for dissipative Hénon maps. Geom. Funct. Anal. 24, 887–915 (2014). https://doi.org/10.1007/s00039-014-0280-9

Download citation

  • Received:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s00039-014-0280-9

Keywords

Search

Navigation