no. 6
Sur les ensembles de Julia et Fatou des fonctions entières ultramétriques
Bézivin, Jean-Paul1
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, Tome 51 (2001) no. 6, pp. 1635-1661.

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.

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.

DOI : 10.5802/aif.1869
Classification : 37F99, 11S99
Mot clés : fonctions entières $p$-adiques, ensemble de Julia, ensemble de Fatou, dynamique $p$-adique
Keywords: entire $p$-adic functions, Julia set, Fatou set, ultrametric dynamics

Bézivin, Jean-Paul 1

1 Université de Caen, Département de Mathématiques et Mécanique, Campus II, Boulevard du Maréchal Juin, BP 5186, 14032 Caen Cedex (France) 
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Bézivin, Jean-Paul. Sur les ensembles de Julia et Fatou des fonctions entières ultramétriques. Annales de l'Institut Fourier, Tome 51 (2001) no. 6, pp. 1635-1661. doi : 10.5802/aif.1869. https://aif.centre-mersenne.org/articles/10.5802/aif.1869/

[AV] D.K. Arrowsmith; F. Vivaldi Geometry of p-adic Siegel discs, Physica, Volume D 71 (1994) no. 1-2, pp. 222-236 | MR | Zbl

[B] J.-P. Bézivin Sur les points périodiques des applications rationnelles en dynamique ultramétrique (à paraître dans Acta Arithmetica) | Zbl

[BA1] I.N. Baker Permutable entire functions, Math. Zeitschrift, Volume 79 (1962), pp. 243-249 | DOI | MR | Zbl

[BA2] I.N. Baker Repulsive fixpoints of entire functions, Math. Zeitschrift, Volume 104 (1968), pp. 252-256 | DOI | MR | Zbl

[BE1] R. Benedetto Reduction, dynamics and Julia sets and reduction of rational functions (To appear in Journal of Number theory) | Zbl

[BE2] R. Benedetto Hyperbolic maps and p-adic dynamics (To appear in Ergodic theory and Dynamical systems) | Zbl

[BE3] R. Benedetto p-adic dynamics and Sullivan's No Wandering Domains Theorem, Compositio Mathematica, Volume 122 (2000), pp. 281-298 | DOI | MR | Zbl

[BE4] R. Benedetto Components and periodic points in non archimedean dynamics (July 1999) (preprint) | MR | Zbl

[BEA] A.F. Beardon Iteration of rational functions, Springer-Verlag, New-York, 1991 | MR | Zbl

[ER] A.E. Eremenko On the iteration of entire functions, Dynamical system and ergodic theory, Volume vol. 23 (1989), pp. 339-345 | Zbl

[FA] P. Fatou Sur l'itération analytique et les substitutions permutables, Journal de Mathématique, Volume 2 (1923) no. 4, pp. 343-384 | JFM

[FVDP] J. Fresnel; M. Van der Put Géométrie rigide et applications, Progress in Math., Birkhäuser, 1981 | MR | Zbl

[HS1] L. Hsia A weak Néron model with application to p-adic dynamical systems, Compositio Math., Volume 100 (1996), pp. 277-304 | Numdam | MR | Zbl

[HS2] L. Hsia Closure of periodic points over a non archimedean field, J. London Math. Soc., Volume 62 (2000), pp. 685-700 | DOI | MR | Zbl

[IY] G. Iyer On permutable integral functions, J. London Math. Soc., Volume 34 (1959) | MR | Zbl

[JU] G. Julia Œuvres, Volume II, pp. 64-100

[LI1] H.C. Li p-adic periodic points and sen's theorem, J. of Number Th., Volume 56 (1996), pp. 309-318 | DOI | MR | Zbl

[LI2] H.C. Li Counting periodic points of p-adic power series, Compositio Math., Volume 100 (1996), pp. 351-364 | Numdam | MR | Zbl

[LI3] H.C. Li p-adic dynamical systems and formal groups, Compos. Math., Volume 104 (1996) no. 1, pp. 41-54 | Numdam | MR | Zbl

[LI4] H.C. Li When is a p-adic power series an endomorphism of a formal group? Proc. Am. Math. Soc, Proc. Am. Math. Soc., Volume 124 (1996) no. 8, pp. 2325-2329 | DOI | MR | Zbl

[LU] J. Lubin Nonarchimedean dynamical systems, Compositio Math., Volume 94 (1994), pp. 321-346 | Numdam | MR | Zbl

[MI] J. Milnor Dynamics in one complex variable, Introductory lectures, Vieweg, Wiesbaden, 1999 | MR | Zbl

[MS1] P. Morton; J. Silverman Rational periodic points of rational functions, Inter. Math. Res. Notices, Volume 2 (1994), pp. 97-110 | DOI | MR | Zbl

[MS2] P. Morton; J. Silverman Periodic points, multiplicities and dynamical units, J. reine und ang. Math, Volume 461 (1995), pp. 81-122 | DOI | MR | Zbl

[NG] T.W. NG Permutable entire functions and their Julia sets (To appear in Math Proc Cambr Phil Soc.) | MR | Zbl

[PY] K.K. Poon; C.C. Yang Dynamical behaviour of two permutable functions, Ann. Polon. Math, Volume 68 (1998), pp. 159-163 | MR | Zbl

[RI] J.-F. Ritt Permutable rational functions, Trans. Amer. Math. Soc., Volume 25 (1923), pp. 399-448 | DOI | JFM | MR

[SW] N. Smart; C. Woodcock p-adic Chaos and ramdom number generation, Experiment. Math., Volume 7 (1998) no. 3, pp. 765-788 | MR | Zbl

[TVW] E. Thiran; D. Verstegen; J. Weyers p-adic dynamics, J. Stat. Phys., Volume 54 (1989) no. 3/4, pp. 893-913 | DOI | MR | Zbl

[VE] D. Verstegen p-adic dynamical systems, Number theory and physics, Proc. Winter Sch, Les Houches, 1989, Volume 47 (1990), pp. 235-242 | Zbl

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