Sur les ensembles de Julia et Fatou des fonctions entières ultramétriques

Bézivin, Jean-Paul

Annales de l'Institut Fourier, Tome 51 (2001), p. 1635-1661 / Harvested from Numdam

Résumé

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.

Abstract

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.

Détail

Publié le : 2001-01-01

MR 1871284 | Zbl 01710113

DOI : https://doi.org/10.5802/aif.1869
Classification:  37F99,  11S99
Mots clés: fonctions entières $p$-adiques, ensemble de Julia, ensemble de Fatou, dynamique $p$-adique

Bibliographie

[AV] D.K. Arrowsmith; F. Vivaldi Geometry of p-adic Siegel discs, Physica, Tome D 71 (1994) no. 1-2, pp. 222-236 | MR 1264116 | Zbl 0822.11081

[B] J.-P. Bézivin Sur les points périodiques des applications rationnelles en dynamique ultramétrique (à paraître dans Acta Arithmetica) | Zbl 1025.11035

[BA1] I.N. Baker Permutable entire functions, Math. Zeitschrift, Tome 79 (1962), pp. 243-249 | Article | MR 147650 | Zbl 0107.28301

[BA2] I.N. Baker Repulsive fixpoints of entire functions, Math. Zeitschrift, Tome 104 (1968), pp. 252-256 | Article | MR 226009 | Zbl 0172.09502

[BE1] R. Benedetto Reduction, dynamics and Julia sets and reduction of rational functions (To appear in Journal of Number theory) | Zbl 0978.37039

[BE2] R. Benedetto Hyperbolic maps and p-adic dynamics (To appear in Ergodic theory and Dynamical systems) | Zbl 0972.37027

[BE3] R. Benedetto p-adic dynamics and Sullivan's No Wandering Domains Theorem, Compositio Mathematica, Tome 122 (2000), pp. 281-298 | Article | MR 1781331 | Zbl 0996.37055

[BE4] R. Benedetto Components and periodic points in non archimedean dynamics (July 1999) (preprint) | MR 1863402 | Zbl 1012.37005

[BEA] A.F. Beardon Iteration of rational functions, Springer-Verlag, New-York (1991) | MR 1128089 | Zbl 0742.30002

[ER] A.E. Eremenko On the iteration of entire functions, Dynamical system and ergodic theory, Banach center publication, Tome vol. 23 (1989), pp. 339-345 | Zbl 0692.30021

[FA] P. Fatou Sur l'itération analytique et les substitutions permutables, Journal de Mathématique, Tome 2 (1923) no. 4, pp. 343-384 | JFM 50.0637.02

[FVDP] J. Fresnel; M. Van Der Put Géométrie rigide et applications, Birkhäuser, Progress in Math. (1981) | MR 644799 | Zbl 0479.14015

[HS1] L. Hsia A weak Néron model with application to p-adic dynamical systems, Compositio Math., Tome 100 (1996), pp. 277-304 | Numdam | MR 1387667 | Zbl 0851.14001

[HS2] L. Hsia Closure of periodic points over a non archimedean field, J. London Math. Soc., Tome 62 (2000), pp. 685-700 | Article | MR 1794277 | Zbl 1022.11060

[IY] G. Iyer On permutable integral functions, J. London Math. Soc., Tome 34 (1959) | MR 105574 | Zbl 0085.29302

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

[LI1] H.C. Li p-adic periodic points and sen's theorem, J. of Number Th., Tome 56 (1996), pp. 309-318 | Article | MR 1373554 | Zbl 0859.11061

[LI2] H.C. Li Counting periodic points of p-adic power series, Compositio Math., Tome 100 (1996), pp. 351-364 | Numdam | MR 1387670 | Zbl 0871.11084

[LI3] H.C. Li p-adic dynamical systems and formal groups, Compos. Math., Tome 104 (1996) no. 1, pp. 41-54 | Numdam | MR 1420709 | Zbl 0873.58060

[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., Tome 124 (1996) no. 8, pp. 2325-2329 | Article | MR 1322933 | Zbl 0862.11067

[LU] J. Lubin Nonarchimedean dynamical systems, Compositio Math., Tome 94 (1994), pp. 321-346 | Numdam | MR 1310863 | Zbl 0843.58111

[MI] J. Milnor Dynamics in one complex variable, Vieweg, Wiesbaden, Introductory lectures (1999) | MR 1721240 | Zbl 0946.30013

[MS1] P. Morton; J. Silverman Rational periodic points of rational functions, Inter. Math. Res. Notices, Tome 2 (1994), pp. 97-110 | Article | MR 1264933 | Zbl 0819.11045

[MS2] P. Morton; J. Silverman Periodic points, multiplicities and dynamical units, J. reine und ang. Math, Tome 461 (1995), pp. 81-122 | Article | MR 1324210 | Zbl 0813.11059

[NG] T.W. Ng Permutable entire functions and their Julia sets (To appear in Math Proc Cambr Phil Soc.) | MR 1833078 | Zbl 0979.30015

[PY] K.K. Poon; C.C. Yang Dynamical behaviour of two permutable functions, Ann. Polon. Math, Tome 68 (1998), pp. 159-163 | MR 1610556 | Zbl 0896.30021

[RI] J.-F. Ritt Permutable rational functions, Trans. Amer. Math. Soc., Tome 25 (1923), pp. 399-448 | Article | JFM 49.0712.02 | MR 1501252

[SW] N. Smart; C. Woodcock $p$-adic Chaos and ramdom number generation, Experiment. Math., Tome 7 (1998) no. 3, pp. 765-788 | MR 1678087 | Zbl 0920.11052

[TVW] E. Thiran; D. Verstegen; J. Weyers $p$-adic dynamics, J. Stat. Phys., Tome 54 (1989) no. 3/4, pp. 893-913 | Article | MR 988564 | Zbl 0672.58019

[VE] D. Verstegen $p$-adic dynamical systems, Number theory and physics, Proc. Winter Sch, Les Houches, 1989, Springer Proc. Phys., Tome 47 (1990), pp. 235-242 | Zbl 0712.58021