Let a rational prime number. The paper is on the dynamics of -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.
Soit un nombre premier rationnel. Le sujet de l’article est l’étude de la dynamique des fonctions entières -adiques. On démontre des résultats analogues à ceux connus dans le domaine complexe, en particulier si deux fonctions entières -adiques qui ont un point répulsif commun commutent, alors leurs ensembles de Julia et de Fatou sont les mêmes.
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
@article{AIF_2001__51_6_1635_0, author = {B\'ezivin, Jean-Paul}, title = {Sur les ensembles de {Julia} et {Fatou} des fonctions enti\`eres ultram\'etriques}, journal = {Annales de l'Institut Fourier}, pages = {1635--1661}, publisher = {Association des Annales de l{\textquoteright}institut Fourier}, volume = {51}, number = {6}, year = {2001}, doi = {10.5802/aif.1869}, zbl = {01710113}, mrnumber = {1871284}, language = {fr}, url = {https://aif.centre-mersenne.org/articles/10.5802/aif.1869/} }
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 SP - 1635 EP - 1661 VL - 51 IS - 6 PB - Association des Annales de l’institut Fourier UR - https://aif.centre-mersenne.org/articles/10.5802/aif.1869/ DO - 10.5802/aif.1869 LA - fr ID - AIF_2001__51_6_1635_0 ER -
%0 Journal Article %A Bézivin, Jean-Paul %T Sur les ensembles de Julia et Fatou des fonctions entières ultramétriques %J Annales de l'Institut Fourier %D 2001 %P 1635-1661 %V 51 %N 6 %I Association des Annales de l’institut Fourier %U https://aif.centre-mersenne.org/articles/10.5802/aif.1869/ %R 10.5802/aif.1869 %G fr %F AIF_2001__51_6_1635_0
Bézivin, Jean-Paul. Sur les ensembles de Julia et Fatou des fonctions entières ultramétriques. Annales de l'Institut Fourier, Volume 51 (2001) no. 6, pp. 1635-1661. doi : 10.5802/aif.1869. https://aif.centre-mersenne.org/articles/10.5802/aif.1869/
[AV] Geometry of p-adic Siegel discs, Physica, Volume D 71 (1994) no. 1-2, pp. 222-236 | MR | Zbl
[B] Sur les points périodiques des applications rationnelles en dynamique ultramétrique (à paraître dans Acta Arithmetica) | Zbl
[BA1] Permutable entire functions, Math. Zeitschrift, Volume 79 (1962), pp. 243-249 | DOI | MR | Zbl
[BA2] Repulsive fixpoints of entire functions, Math. Zeitschrift, Volume 104 (1968), pp. 252-256 | DOI | MR | Zbl
[BE1] Reduction, dynamics and Julia sets and reduction of rational functions (To appear in Journal of Number theory) | Zbl
[BE2] Hyperbolic maps and p-adic dynamics (To appear in Ergodic theory and Dynamical systems) | Zbl
[BE3] p-adic dynamics and Sullivan's No Wandering Domains Theorem, Compositio Mathematica, Volume 122 (2000), pp. 281-298 | DOI | MR | Zbl
[BE4] Components and periodic points in non archimedean dynamics (July 1999) (preprint) | MR | Zbl
[BEA] Iteration of rational functions, Springer-Verlag, New-York, 1991 | MR | Zbl
[ER] On the iteration of entire functions, Dynamical system and ergodic theory, Volume vol. 23 (1989), pp. 339-345 | Zbl
[FA] Sur l'itération analytique et les substitutions permutables, Journal de Mathématique, Volume 2 (1923) no. 4, pp. 343-384 | JFM
[FVDP] Géométrie rigide et applications, Progress in Math., Birkhäuser, 1981 | MR | Zbl
[HS1] A weak Néron model with application to p-adic dynamical systems, Compositio Math., Volume 100 (1996), pp. 277-304 | Numdam | MR | Zbl
[HS2] Closure of periodic points over a non archimedean field, J. London Math. Soc., Volume 62 (2000), pp. 685-700 | DOI | MR | Zbl
[IY] On permutable integral functions, J. London Math. Soc., Volume 34 (1959) | MR | Zbl
[JU] Œuvres, Volume II, pp. 64-100
[LI1] p-adic periodic points and sen's theorem, J. of Number Th., Volume 56 (1996), pp. 309-318 | DOI | MR | Zbl
[LI2] Counting periodic points of p-adic power series, Compositio Math., Volume 100 (1996), pp. 351-364 | Numdam | MR | Zbl
[LI3] p-adic dynamical systems and formal groups, Compos. Math., Volume 104 (1996) no. 1, pp. 41-54 | Numdam | MR | Zbl
[LI4] 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] Nonarchimedean dynamical systems, Compositio Math., Volume 94 (1994), pp. 321-346 | Numdam | MR | Zbl
[MI] Dynamics in one complex variable, Introductory lectures, Vieweg, Wiesbaden, 1999 | MR | Zbl
[MS1] Rational periodic points of rational functions, Inter. Math. Res. Notices, Volume 2 (1994), pp. 97-110 | DOI | MR | Zbl
[MS2] Periodic points, multiplicities and dynamical units, J. reine und ang. Math, Volume 461 (1995), pp. 81-122 | DOI | MR | Zbl
[NG] Permutable entire functions and their Julia sets (To appear in Math Proc Cambr Phil Soc.) | MR | Zbl
[PY] Dynamical behaviour of two permutable functions, Ann. Polon. Math, Volume 68 (1998), pp. 159-163 | MR | Zbl
[RI] Permutable rational functions, Trans. Amer. Math. Soc., Volume 25 (1923), pp. 399-448 | DOI | JFM | MR
[SW] -adic Chaos and ramdom number generation, Experiment. Math., Volume 7 (1998) no. 3, pp. 765-788 | MR | Zbl
[TVW] -adic dynamics, J. Stat. Phys., Volume 54 (1989) no. 3/4, pp. 893-913 | DOI | MR | Zbl
[VE] -adic dynamical systems, Number theory and physics, Proc. Winter Sch, Les Houches, 1989, Volume 47 (1990), pp. 235-242 | Zbl
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