Dynamique des polynômes quadratiques sur les corps locaux

Benedetto, Robert; Briend, Jean-Yves; Perdry, Hervé

doi:10.5802/jtnb.589

Benedetto, Robert ¹; Briend, Jean-Yves ²; Perdry, Hervé ³

¹ Department of Mathematics and Computer Science Amherst College, P. O. Box 5000 Amherst, MA 01002-5000, USA
² Université de Provence Laboratoire Analyse, Topologie, Probabilités, UMR CNRS 6632 39 rue Joliot-Curie 13453 Marseille cedex 13, FRANCE
³ INSERM U535, Université Paris-Sud Pavillon Leriche Secteur Jaune - Porte 18 BP 1000, 94817 Villejuif Cedex, France

Journal de théorie des nombres de Bordeaux, Volume 19 (2007) no. 2, pp. 325-336.

Abstract
Résumé

We show that the dynamics of a quadratic polynomial over a local field can be completely decided in a finite amount of time, with the following two possibilities : either the Julia set is empty, or the polynomial is topologically conjugate on its Julia set to the one-sided shift on two symbols.

Dans cette note, nous montrons que la dynamique d’un polynôme quadratique sur un corps local peut être déterminée en temps fini, et que l’on a l’alternative suivante : soit l’ensemble de Julia est vide, soit $P$ y est conjugué au décalage unilatéral sur $2$ symboles.

DOI: 10.5802/jtnb.589

Author's affiliations:

Benedetto, Robert ¹; Briend, Jean-Yves ²; Perdry, Hervé ³

@article{JTNB_2007__19_2_325_0,
     author = {Benedetto, Robert and Briend, Jean-Yves and Perdry, Herv\'e},
     title = {Dynamique des polyn\^omes quadratiques sur les corps locaux},
     journal = {Journal de th\'eorie des nombres de Bordeaux},
     pages = {325--336},
     publisher = {Universit\'e Bordeaux 1},
     volume = {19},
     number = {2},
     year = {2007},
     doi = {10.5802/jtnb.589},
     mrnumber = {2394889},
     language = {fr},
     url = {http://archive.numdam.org/articles/10.5802/jtnb.589/}
}

TY  - JOUR
AU  - Benedetto, Robert
AU  - Briend, Jean-Yves
AU  - Perdry, Hervé
TI  - Dynamique des polynômes quadratiques sur les corps locaux
JO  - Journal de théorie des nombres de Bordeaux
PY  - 2007
SP  - 325
EP  - 336
VL  - 19
IS  - 2
PB  - Université Bordeaux 1
UR  - http://archive.numdam.org/articles/10.5802/jtnb.589/
DO  - 10.5802/jtnb.589
LA  - fr
ID  - JTNB_2007__19_2_325_0
ER  -

%0 Journal Article
%A Benedetto, Robert
%A Briend, Jean-Yves
%A Perdry, Hervé
%T Dynamique des polynômes quadratiques sur les corps locaux
%J Journal de théorie des nombres de Bordeaux
%D 2007
%P 325-336
%V 19
%N 2
%I Université Bordeaux 1
%U http://archive.numdam.org/articles/10.5802/jtnb.589/
%R 10.5802/jtnb.589
%G fr
%F JTNB_2007__19_2_325_0

Benedetto, Robert; Briend, Jean-Yves; Perdry, Hervé. Dynamique des polynômes quadratiques sur les corps locaux. Journal de théorie des nombres de Bordeaux, Volume 19 (2007) no. 2, pp. 325-336. doi : 10.5802/jtnb.589. http://archive.numdam.org/articles/10.5802/jtnb.589/

References
Cited by

[A] E. Artin, Algebraic Numbers and Algebraic Functions. Gordon and Breach, New-York 1967. | MR | Zbl

[BH] M. Baker et L-C. Hsia, Canonical heights, transfinite diameters, and polynomial dynamics. J. Reine Angew. Math. 585 (2005), 61–92. | MR | Zbl

[Be1] R. Benedetto, Fatou Components in $p$ -adic Dynamics. Thesis, Brown University, 1998.

[Be2] R. Benedetto, Reduction, dynamics, and Julia sets of rational functions. J. Number Theory 86 (2001), 175–195. | MR | Zbl

[B1] J-P. Bézivin, Sur les points périodiques des applications rationnelles en dynamique ultramétrique. Acta Arithmetica 100 (2001), 63–74. | MR | Zbl

[B2] J-P. Bézivin, Sur la compacité des ensembles de Julia des polynômes $p$ -adiques. Math. Z. 246 (2004), 273–289. | MR | Zbl

[BCSS] L. Blum, F. Cucker, M. Shub et S. Smale, Complexity And Real Computation. Spinger Verlag, Berlin 1998. | MR | Zbl

[BY1] M. Braverman et M. Yampolsky, Non-computable Julia sets. J. Amer. Math. Soc. 19 (2006), no. 3, 551–578. | MR | Zbl

[BY2] M. Braverman et M. Yampolsky, On computability of Julia sets : answers to questions of Milnor and Shub. Preprint 2006. | MR

[De] J. Denef, $p$ -adic semi-algebraic sets and cell decomposition. J. Reine Angew. Math. 369 (1986), 154–166. | MR | Zbl

[DH] A. Douady et J. H. Hubbard, Itération des polynômes quadratiques complexes. Comptes Rendus Acad. Sci. Paris 294 (1982), 123–126. | MR | Zbl

[D] V. A. Dremov, On a $p$ -adic Julia set. Russian Math. Surveys 58 (2003), 1194–1195. | MR | Zbl

[FR] C. Favre et J. Rivera–Letelier, Théorème d’équidistribution de Brolin en dynamique $p$ -adique. Comptes Rendus Acad. Sci. Paris 339 (2004), 271–276. | Zbl

[H1] L-C. Hsia, A weak Néron model with applications to $p$ -adic dynamical systems. Compositio Math. 100 (1996), 277–304. | Numdam | MR | Zbl

[H2] L-C. Hsia, Closure of periodic points over a non archimedean field. J. London. Math. Soc. 62 (2000), 685-700. | MR | Zbl

[J] M. Jakobson, Absolutely continuous invariant measures for one parameter families of one-dimensional maps. Comm. Math. Phys. 81 (1981), 161–185. | MR | Zbl

[Jo] R. Jones, Galois martingales and the hyperbolic subset of the $p$ -adic Mandelbrot set. PhD thesis (2005), Brown University.

[L] M. Lyubich, Almost every real quadratic map is either regular or stochastic. Ann. of Math. 156 (2002), 1–78. | MR

[M] J. Milnor, Dynamics in One Complex Variable. Friedr. Vieweg & Sohn, Braunschweig/Wiesbaden, 1999. | MR | Zbl

[MS] P. Morton et J. Silverman, Periodic points, multiplicities, and dynamical units. J. Reine Angew. Math. 461 (1995), 81–122. | MR | Zbl

[Na] W. Narkiewicz, Polynomial Mappings. Lecture Notes in Mathematics 1600 (1995), Springer Verlag, Berlin. | MR | Zbl

[NR] M. Nevins et T. Rogers, Quadratic maps as dynamical systems on the $p$ -adic numbers. Prépublication (2000).

[No] D. G. Northcott, Periodic points on an algebraic variety. Ann. of Math. 51 (1950), 167–177. | MR | Zbl

[P] T. Pezda, Polynomial cycles in certain local domains. Acta Arithmetica 66 (1994), 11–22. | MR | Zbl

[R] J. Rivera–Letelier, Dynamique des fractions rationnelles sur les corps locaux, dans Geometric Methods in Dynamics, II. Astérisque 287 (2003), 199-231. | Zbl

[TVW] E. Thiran, D. Verstegen, J. Weyers, $p$ -adic dynamics. J. Stat. Phys. 54 (1989), 893–913. | MR | Zbl

[W] A. Weil, Basic Number Theory. Die Grundlehren des mathematischen Wissenschaften 144 (1967), Springer Verlag, Berlin. | MR | Zbl

[WS] C. Woodcock et N. Smart, $p$ -adic chaos and random number generation. Experiment. Math. 7 (1998), 333–342. | MR | Zbl

Cited by Sources: