On the Julia and Fatou sets of ultrametric entire functions

Jean-Paul Bézivin[1]

  • [1] Université de Caen, Département de Mathématiques et Mécanique, Campus II, Boulevard du Maréchal Juin, BP 5186, 14032 Caen Cedex (France)

Annales de l’institut Fourier (2001)

  • Volume: 51, Issue: 6, page 1635-1661
  • ISSN: 0373-0956

Abstract

top
Let p a rational prime number. The paper is on the dynamics of p -adic entire functions. We prove results analogous to those known in complex dynamical system. In particular, for commuting entire transcendental functions, under the condition that they have a common periodical repulsive point, they have the same Julia and Fatou sets.

How to cite

top

Bézivin, Jean-Paul. "Sur les ensembles de Julia et Fatou des fonctions entières ultramétriques." Annales de l’institut Fourier 51.6 (2001): 1635-1661. <http://eudml.org/doc/115962>.

@article{Bézivin2001,
abstract = {Soit $p$ un nombre premier rationnel. Le sujet de l’article est l’étude de la dynamique des fonctions entières $p$-adiques. On démontre des résultats analogues à ceux connus dans le domaine complexe, en particulier si deux fonctions entières $p$-adiques qui ont un point répulsif commun commutent, alors leurs ensembles de Julia et de Fatou sont les mêmes.},
affiliation = {Université de Caen, Département de Mathématiques et Mécanique, Campus II, Boulevard du Maréchal Juin, BP 5186, 14032 Caen Cedex (France)},
author = {Bézivin, Jean-Paul},
journal = {Annales de l’institut Fourier},
keywords = {entire $p$-adic functions; Julia set; Fatou set; ultrametric dynamics},
language = {fre},
number = {6},
pages = {1635-1661},
publisher = {Association des Annales de l'Institut Fourier},
title = {Sur les ensembles de Julia et Fatou des fonctions entières ultramétriques},
url = {http://eudml.org/doc/115962},
volume = {51},
year = {2001},
}

TY - JOUR
AU - Bézivin, Jean-Paul
TI - Sur les ensembles de Julia et Fatou des fonctions entières ultramétriques
JO - Annales de l’institut Fourier
PY - 2001
PB - Association des Annales de l'Institut Fourier
VL - 51
IS - 6
SP - 1635
EP - 1661
AB - Soit $p$ un nombre premier rationnel. Le sujet de l’article est l’étude de la dynamique des fonctions entières $p$-adiques. On démontre des résultats analogues à ceux connus dans le domaine complexe, en particulier si deux fonctions entières $p$-adiques qui ont un point répulsif commun commutent, alors leurs ensembles de Julia et de Fatou sont les mêmes.
LA - fre
KW - entire $p$-adic functions; Julia set; Fatou set; ultrametric dynamics
UR - http://eudml.org/doc/115962
ER -

References

top
  1. D.K. Arrowsmith, F. Vivaldi, Geometry of p-adic Siegel discs, Physica D 71 (1994), 222-236 Zbl0822.11081MR1264116
  2. I.N. Baker, Permutable entire functions, Math. Zeitschrift 79 (1962), 243-249 Zbl0107.28301MR147650
  3. I.N. Baker, Repulsive fixpoints of entire functions, Math. Zeitschrift 104 (1968), 252-256 Zbl0172.09502MR226009
  4. J.-P. Bézivin, Sur les points périodiques des applications rationnelles en dynamique ultramétrique Zbl1025.11035
  5. A.F. Beardon, Iteration of rational functions, (1991), Springer-Verlag, New-York Zbl0742.30002MR1128089
  6. R. Benedetto, Reduction, dynamics and Julia sets and reduction of rational functions Zbl0978.37039
  7. R. Benedetto, Hyperbolic maps and p-adic dynamics Zbl0972.37027
  8. R. Benedetto, p-adic dynamics and Sullivan's No Wandering Domains Theorem, Compositio Mathematica 122 (2000), 281-298 Zbl0996.37055MR1781331
  9. R. Benedetto, Components and periodic points in non archimedean dynamics, (July 1999) Zbl1012.37005MR1863402
  10. P. Fatou, Sur l'itération analytique et les substitutions permutables, Journal de Mathématique 2 (1923), 343-384 Zbl50.0690.01
  11. J. Fresnel, M. Van der Put, Géométrie rigide et applications, (1981), Birkhäuser Zbl0479.14015MR644799
  12. A.E. Eremenko, On the iteration of entire functions, Dynamical system and ergodic theory vol. 23 (1989), 339-345, Banach center publication Zbl0692.30021
  13. G. Iyer, On permutable integral functions, J. London Math. Soc. 34 (1959) Zbl0085.29302MR105574
  14. L. Hsia, A weak Néron model with application to p-adic dynamical systems, Compositio Math. 100 (1996), 277-304 Zbl0851.14001MR1387667
  15. L. Hsia, Closure of periodic points over a non archimedean field, J. London Math. Soc. 62 (2000), 685-700 Zbl1022.11060MR1794277
  16. G. Julia, Œuvres Volume II, 64-100 
  17. H.C. Li, p-adic periodic points and sen's theorem, J. of Number Th. 56 (1996), 309-318 Zbl0859.11061MR1373554
  18. H.C. Li, Counting periodic points of p-adic power series, Compositio Math. 100 (1996), 351-364 Zbl0871.11084MR1387670
  19. H.C. Li, p-adic dynamical systems and formal groups, Compos. Math. 104 (1996), 41-54 Zbl0873.58060MR1420709
  20. H.C. Li, When is a p-adic power series an endomorphism of a formal group? Proc. Am. Math. Soc, Proc. Am. Math. Soc. 124 (1996), 2325-2329 Zbl0862.11067MR1322933
  21. J. Lubin, Nonarchimedean dynamical systems, Compositio Math. 94 (1994), 321-346 Zbl0843.58111MR1310863
  22. J. Milnor, Dynamics in one complex variable, (1999), Vieweg, Wiesbaden Zbl0946.30013MR1721240
  23. P. Morton, J. Silverman, Rational periodic points of rational functions, Inter. Math. Res. Notices 2 (1994), 97-110 Zbl0819.11045MR1264933
  24. P. Morton, J. Silverman, Periodic points, multiplicities and dynamical units, J. reine und ang. Math 461 (1995), 81-122 Zbl0813.11059MR1324210
  25. T.W. NG, Permutable entire functions and their Julia sets Zbl0979.30015MR1833078
  26. K.K. Poon, C.C. Yang, Dynamical behaviour of two permutable functions, Ann. Polon. Math 68 (1998), 159-163 Zbl0896.30021MR1610556
  27. J.-F. Ritt, Permutable rational functions, Trans. Amer. Math. Soc. 25 (1923), 399-448 Zbl49.0246.04MR1501252
  28. N. Smart, C. Woodcock, p -adic Chaos and ramdom number generation, Experiment. Math. 7 (1998), 765-788 Zbl0920.11052MR1678087
  29. E. Thiran, D. Verstegen, J. Weyers, p -adic dynamics, J. Stat. Phys. 54 (1989), 893-913 Zbl0672.58019MR988564
  30. D. Verstegen, p -adic dynamical systems, Number theory and physics, Proc. Winter Sch, Les Houches, 1989 47 (1990), 235-242, Springer Proc. Phys. Zbl0712.58021

NotesEmbed ?

top

You must be logged in to post comments.

To embed these notes on your page include the following JavaScript code on your page where you want the notes to appear.

Only the controls for the widget will be shown in your chosen language. Notes will be shown in their authored language.

Tells the widget how many notes to show per page. You can cycle through additional notes using the next and previous controls.

    
                

                
Note: Best practice suggests putting the JavaScript code just before the closing </body> tag.