Skip to main content
SpringerLink
Log in

Puiseux series dynamics of quadratic rational maps

  • Published:
Israel Journal of Mathematics Aims and scope Submit manuscript

Abstract

We give a complete description for the dynamics of quadratic rational maps with coefficients in the completion of the field of formal Puiseux series.

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. P. Atela, Bifurcations of dynamic rays in complex polynomials of degree two, Ergodic Theory and Dynamical Systems 12 (1992), 401–423.

    Article  MathSciNet  MATH  Google Scholar 

  2. R. L. Benedetto, Fatou components in p-adic dynamics, Ph.D. thesis, Brown University, 1998.

  3. R. L. Benedetto, Examples of wandering domains in p-adic polynomial dynamics, Comptes Rendus Mathématique. Académie des Sciences. Paris 335 (2002), 615–620.

    Article  MathSciNet  MATH  Google Scholar 

  4. V. G. Berkovich, Spectral Theory and Analytic Geometry over Non-Archimedean Fields, Mathematical Surveys and Monographs, Vol. 33, American Mathematical Society, Providence, RI, 1990.

    MATH  Google Scholar 

  5. B. Bielefeld, Y. Fisher and J. H. Hubbard, The classification of critically preperiodic polynomials as dynamical systems, Journal of the American Mathematical Society 5 (1992), 721–762.

    Article  MathSciNet  MATH  Google Scholar 

  6. B. Branner and J. H. Hubbard, Iteration of cubic polynomials I: the global topology of parameter space, Acta Mathematica 160 (1988), 143–206.

    Article  MathSciNet  MATH  Google Scholar 

  7. M. Baker and R. Rumely, Potential Theory and Dynamics on the Berkovich Projective Line, Mathematical Surveys and Monographs, Vol. 159, American Mathematical Society, Providence, RI, 2010.

    MATH  Google Scholar 

  8. S. Bullett and P. Sentenac, Ordered orbits of the shift, square roots, and the devil’s staircase, Mathematical Proceedings of the Cambridge Philosophical Society 115 (1994), 451–481.

    Article  MathSciNet  MATH  Google Scholar 

  9. E. Casas-Alvero, Singularities of Plane Curves, London Mathematical Society Lecture Note Series, Vol. 276, Cambridge University Press, Cambridge, 2000.

    Book  MATH  Google Scholar 

  10. J. W. S. Cassels, Local fields, London Mathematical Society Student Texts, Vol. 3, Cambridge University Press, Cambridge, 1986.

    Book  MATH  Google Scholar 

  11. L. DeMarco, The moduli space of quadratic rational maps, Journal of the American Mathematical Society 20 (2007), 321–355.

    Article  MathSciNet  MATH  Google Scholar 

  12. L. De Marco and X. Faber, Degenerations of Complex Dynamical Systems, Available at arXiv:1302.4769

  13. A. Douady and J. H. Hubbard, Étude dynamique des polynômes complexes. I,II, With the collaboration of P. Lavaurs, Tan Lei and P. Sentenac, Publications Mathématiques d’Orsay [Mathematical Publications of Orsay], Vol. 85, Université de Paris-Sud, Département de Mathématiques, Orsay, 1985.

    Google Scholar 

  14. A. Douady, Systèmes dynamiques holomorphes, Bourbaki Seminar, Vol. 1982/83, Astérisque, Vol. 105, Soc. Math. France, Paris, 1983, pp. 39–63.

    Google Scholar 

  15. A. Ducros, Espaces analytiques p-adiques au sens de Berkovich, Séminaire Bourbaki, Vol. 2005/2006, Astérisque 311 (2007), Exp. No. 958, viii, 137–176

  16. A. Ducros, Géométries analytique p-adique: La théorie de Berkovich, Gazette des Mathématiciens 111 (2007), 12–27.

    MathSciNet  MATH  Google Scholar 

  17. A. L. Epstein, Bounded hyperbolic components of quadratic rational maps, Ergodic Theory and Dynamical Systems 20 (2000), 727–748.

    Article  MathSciNet  MATH  Google Scholar 

  18. X. Faber, Topology and geometry of the Berkovich ramification locus for rational functions, I, Manuscripta Mathematica 142 (2013), 439–474.

    Article  MathSciNet  MATH  Google Scholar 

  19. X. Faber, Topology and geometry of the Berkovich ramification locus for rational functions, II, Mathematische Annalen 356 (2013), 819–844.

    Article  MathSciNet  MATH  Google Scholar 

  20. C. Favre and M. Jonsson, The Valuative Tree, Lecture Notes in Mathematics, Vol. 1853, Springer-Verlag, Berlin, 2004.

    MATH  Google Scholar 

  21. L.-C. Hsia, Closure of periodic points over a non-Archimedean field, Journal of the London Mathematical Society 62 (2000), 685–700.

    Article  MathSciNet  MATH  Google Scholar 

  22. J. H. Hubbard, Local connectivity of Julia sets and bifurcation loci: three theorems of J.-C. Yoccoz, in Topological Methods in Modern Mathematics (Stony Brook, NY, 1991), Publish or Perish, Houston, TX, 1993, pp. 467–511.

    Google Scholar 

  23. J. Kiwi, Rational laminations of complex polynomials, in Laminations and Foliations in Dynamics, Geometry and Topology (Stony Brook, NY, 1998), Contemporary Mathematics, Vol. 269, American Mathematical Society, Providence, RI, 2001, pp. 111–154.

    Chapter  Google Scholar 

  24. J. Kiwi, Puiseux series polynomial dynamics and iteration of complex cubic polynomials, Universitde Grenoble. Annales de l’Institut Fourier 56 (2006), 1337–1404.

    Article  MathSciNet  MATH  Google Scholar 

  25. J. Kiwi, Rescaling limits of complex rational maps; available at ArXiv:1211.3397.

  26. C. T. McMullen, Complex Dynamics and Renormalization, Annals of Mathematics Studies, Vol. 135, Princeton University Press, Princeton, NJ, 1994.

    Google Scholar 

  27. J.W. Milnor, Dynamics in One Complex Variable, Introductory Lectures, Friedrich Vieweg & Sohn, Braunschweig, 1999.

    MATH  Google Scholar 

  28. J. W. Milnor, Local connectivity of Julia sets: Expository lectures, in Mandelbrot Set, Theme and Variations, London Mathematical Society Lecture Note Series, Vol. 274, Cambridge University Press, Cambridge, 2000, pp. 67–116.

    Chapter  Google Scholar 

  29. J.W. Milnor, Periodic orbits, externals rays and the Mandelbrot set: an expository account, Géométrie complexe et systèmes dynamiques (Orsay, 1995), Astérisque 261 (2000), xiii, 277–333

  30. A. Poirier, Critical portraits for postcritically finite polynomials, Fundamenta Mathematicae 203 (2009), 107–163.

    Article  MathSciNet  MATH  Google Scholar 

  31. M. Rees, Views of parameter space: Topographer and Resident, Astérisque 288 (2003).

  32. J. Rivera-Letelier, Dynamique des fonctions rationnelles sur des corps locaux, Geometric Methods in Dynamics. II, Astérisque 287 (2003), xv, 147–230.

    MathSciNet  Google Scholar 

  33. J. Rivera-Letelier, Espace hyperbolique p-adique et dynamique des fonctions rationnelles, Compositio Mathematica 138 (2003), 199–231.

    Article  MathSciNet  MATH  Google Scholar 

  34. J. Rivera-Letelier, Notes sur la droite projective de Berkovich, 2006.

  35. J. Rivera-Letelier, Personal communication, 2006.

  36. J. Stimson, Degree two rational maps with a periodic critical point, Ph.D. thesis, University of Liverpool, 1993.

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

    Article  MathSciNet  MATH  Google Scholar 

  38. L. Tan, Matings of quadratic polynomials, Ergodic Theory and Dynamical Systems 12 (1992), 589–620.

    MathSciNet  MATH  Google Scholar 

  39. W. P. Thurston, On the geometry and dynamics of iterated rational maps, in Complex Dynamics, A K Peters, Wellesley, MA, 2009, pp. 3–137.

    Chapter  Google Scholar 

  40. E. Trucco, Wandering Fatou components and algebraic Julia sets, Bulletin de la Société Mathématique de France, to appear; available at arXiv:0909.4528.

Download references

Author information

Authors and Affiliations

  1. Facultad de Matemáticas, Pontificia Universidad Católica, Casilla 306, Correo 22, Santiago, Chile

    Jan Kiwi

Authors
  1. Jan Kiwi
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Jan Kiwi.

Additional information

Partially supported by “Fondecyt #1110448”, Research Network on Low Dimensional Dynamics PBCT/CONICYT ACT17, Conicyt, Chile and ECOS C07E01.

Rights and permissions

Reprints and permissions

About this article

Cite this article

Kiwi, J. Puiseux series dynamics of quadratic rational maps. Isr. J. Math. 201, 631–700 (2014). https://doi.org/10.1007/s11856-014-1047-6

Download citation

  • Received:

  • Revised:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s11856-014-1047-6

Keywords

Search

Navigation