Skip to main content
SpringerLink
Log in

Dynamics of Transcendental Entire Maps on Berkovich Affine Line

  • Published:
Journal of Dynamics and Differential Equations Aims and scope Submit manuscript

Abstract

Dynamics of transcendental entire maps with coefficients in an algebraically closed and complete non-Archimedean field \(K\) is studied. It is shown, among other things, that periodic repelling points are dense in Berkovich Julia set, the forward union of any open set which intersects the Julia set is the whole Berkovich affine line, and a multi-connected Fatou component is wandering, in which all points go to infinity under iteration.

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

We’re sorry, something doesn't seem to be working properly.

Please try refreshing the page. If that doesn't work, please contact support so we can address the problem.

References

  1. Baker, M., Rumely, R.: Analysis and dynamics on the berkovich projective line (2004). Arxiv:math. NT/0407433

  2. Baker, M., Rumely, R.: Potential Theory and Dynamics on the Berkovich Projective Line, vol. 159. Mathematical Surveys and Monographs. American Mathematical Society, Providence (2010)

  3. Benedetto, R.: Non-Archimedean dynamics in dimension one: lecture note. http://math.arizona.edu/swc/aws/10/2010BenedettoNotes-09Mar.pdf

  4. Berkovich, V.: Spectral Theory and Analytic Geometry over Non-Archimedean Fields, vol. 33 of Mathematical Surveys and Monographs. American Mathematical Society, Providence (1990)

  5. Bézivin, J.: Sur les ensembles de Julia et Fatou des fonctions entières ultramétriques. Ann. Inst. Fourier (Grenoble) 51, 1635–1661 (2001)

    Article  MathSciNet  MATH  Google Scholar 

  6. Favre, C., Jonsson, M.: The Valuative Tree, vol. 1853 of Lecture Notes in Mathematics. Springer, Berlin (2004)

    Google Scholar 

  7. Favre C., Kiwi J., Trucco E., A non-Archimedean montel’s theorem. http://arxiv.org/abs/1105.0746

  8. Rivera-Letelier, J.: Dynamique des fonctions rationnelles sur des corps locaux. Geometric methods in dynamics, II. Astérisque 287, 147–230 (2003)

  9. Rivera-Letelier, J.: Points périodiques des fonctions rationnelles dans l’espace hyperbolique \(p\)-adique. Comment. Math. Helv. 80, 593–629 (2005)

    Article  MathSciNet  MATH  Google Scholar 

  10. Rivera-Letelier, J.: Théorie de Fatou et Julia dans la droite projective de Berkovich (in preparation)

  11. Silverman, J.: The Arithmetic of Dynamical Systems. Graduate Texts in Mathematics, vol. 241. Springer, New York (2007)

    Book  Google Scholar 

Download references

Acknowledgments

The research is supported in part by NSF of China (No. 10831004). The authors would like to thank Professor Charles Favre for helpful discussions and valuable suggestions. The authors would also like to thank the referee for useful suggestions and comments.

Author information

Authors and Affiliations

  1. Institute of Applied Mathematics, AMSS, Chinese Academy of Sciences, Beijing, 100190, China

    Shilei Fan

  2. Institute of Mathematics, AMSS, Chinese Academy of Sciences, Beijing, 100190, China

    Yuefei Wang

Authors
  1. Shilei Fan
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Yuefei Wang
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Yuefei Wang.

Rights and permissions

Reprints and permissions

About this article

Cite this article

Fan, S., Wang, Y. Dynamics of Transcendental Entire Maps on Berkovich Affine Line. J Dyn Diff Equat 25, 217–229 (2013). https://doi.org/10.1007/s10884-012-9281-2

Download citation

  • Received:

  • Revised:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s10884-012-9281-2

Keywords

Search

Navigation