Advertisement

Mathematische Annalen

, Volume 333, Issue 4, pp 703–739 | Cite as

On the dynamics near infinity of some polynomial mappings in Open image in new window

  • Tien Cuong Dinh
  • Romain DujardinEmail author
  • Nessim Sibony
Article

Abstract

We construct the Green current for a random iteration of horizontal-like mappings in Open image in new window . This is applied to the study of a polynomial map Open image in new window with the following properties:

i. infinity is f-attracting;

ii. f contracts the line at infinity to a point not in the indeterminacy set.

We study for such mappings the escape rates near infinity, i.e. the set of possible values of the function Open image in new window We show in particular that the set of possible values can contain an interval.

On the other hand the Green current T of f can be decomposed into pieces associated to an itinerary defined by the indeterminacy points. This allows us to prove that Open image in new window exists ||T||-a.e. and we give its value in terms of explicit quantities depending on f.

Mathematics Subject Classification (2000)

37F10 32H50 32U40 

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. 1.
    Bedford, E., Jonsson, M.: Dynamics of regular polynomial endomorphisms of Open image in new window. Amer. J. Math. 122, 153–212 (2000)Google Scholar
  2. 2.
    Bedford, E., Smillie, J.: Real polynomial diffeomorphisms of maximal entropy: II. Small Jacobian. PreprintGoogle Scholar
  3. 3.
    Demailly, J.-P.: Monge-Ampère operators, Lelong numbers and intersection theory. Complex analysis and geometry, Univ. Ser. Math. Plenum, New York, 1993 pp 115–193Google Scholar
  4. 4.
    Dinh, T.C., Sibony, N.: Dynamique des applications d'allure polynomiale. J. Math. Pures Appl. 82(9), 367–423 (2003)Google Scholar
  5. 5.
    Dinh, T.C., Sibony, N.: Dynamique des applications polynomiales semi-régulières. Ark. mat. 42, 61–85 (2004)Google Scholar
  6. 6.
    Dujardin, R.: Hénon-like mappings in Open image in new window. Amer. J. Math. 126, 439–472 (2004)Google Scholar
  7. 7.
    Duval, J., Sibony, N.: Polynomial convexity, rational convexity, and currents. Duke Math. J. 79, 487–513 (1995)CrossRefGoogle Scholar
  8. 8.
    Fornæss, J.E., Sibony, N.: Hyperbolic maps on ℙ2. Math. Ann. 311, 305–333 (1998)CrossRefGoogle Scholar
  9. 9.
    Gamelin, T.W.: Uniform algebras. Prentice-Hall, Inc., Englewood Cliffs, N. J. 1969Google Scholar
  10. 10.
    Guedj, V.: Dynamics of polynomial mappings of Open image in new window. Amer. J. Math. 124, 75–106 (2002)Google Scholar
  11. 11.
    Hubbard, J.H., Oberste-Vorth, R.W.: Hénon mappings in the complex domain. II. Projective and inductive limits of polynomials. Real and complex dynamical systems, NATO Adv. Sci. Inst. Ser. C Math. Phys. Sci., 464, Kluwer Acad. Publ. Dordrecht, 1995 pp 89–132Google Scholar
  12. 12.
    Katok, Anatole, Hasselblatt, Boris,: Introduction to the modern theory of dynamical systems. Cambridge University Press, 1995Google Scholar
  13. 13.
    Levenberg, N., Słodkowski, Z.: Pseudoconcave pluripolar sets in Open image in new window. Math. Ann. 312, 429–443 (1998)CrossRefGoogle Scholar
  14. 14.
    Łojasiewicz, S.: Introduction to complex analytic geometry. Birkhäuser Verlag, Basel, 1991Google Scholar
  15. 15.
    Pugh, C., Shub, M.: Ergodic attractors. Trans. Amer. Math. Soc. 312, 1–54 (1989)Google Scholar
  16. 16.
    Sibony, N.: Dynamique des applications rationnelles de ℙk. Dynamique et géométrie complexes (Lyon, 1997), Panoramas et Synthèses, 8, 1999Google Scholar
  17. 17.
    Sibony, N.: Quelques problèmes de prolongements de courants en analyse complexe. Duke Math. J. 52, 157–197 (1985)CrossRefGoogle Scholar
  18. 18.
    Słodkowski, Z.: Uniqueness property for positive closed currents in Open image in new window. Indiana Univ. Math. J. 48, 635–652 (1999)Google Scholar
  19. 19.
    Yamagishi, Y.: On the local convergence of Newton's method to a multiple root. J. Math. Soc. Japan 55, 897–908 (2003)Google Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2005

Authors and Affiliations

  • Tien Cuong Dinh
    • 1
  • Romain Dujardin
    • 2
    Email author
  • Nessim Sibony
    • 1
  1. 1.MathématiqueUniversité Paris-SudOrsay cedexFrance
  2. 2.Institut de Mathématiques de JussieuUniversité Denis Diderot, Case 7012Paris cedex 05France

Personalised recommendations