Dynamique des polynômes quadratiques sur les corps locaux
Journal de Théorie des Nombres de Bordeaux, Tome 19 (2007) no. 2, pp. 325-336.

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.

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.

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DOI : https://doi.org/10.5802/jtnb.589
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Robert Benedetto; Jean-Yves Briend; Hervé Perdry. Dynamique des polynômes quadratiques sur les corps locaux. Journal de Théorie des Nombres de Bordeaux, Tome 19 (2007) no. 2, pp. 325-336. doi : 10.5802/jtnb.589. https://jtnb.centre-mersenne.org/articles/10.5802/jtnb.589/

[A] E. Artin, Algebraic Numbers and Algebraic Functions. Gordon and Breach, New-York 1967. | MR 237460 | Zbl 0194.35301

[B1] J-P. Bézivin, Sur les points périodiques des applications rationnelles en dynamique ultramétrique. Acta Arithmetica 100 (2001), 63–74. | MR 1864626 | Zbl 1025.11035

[B2] J-P. Bézivin, Sur la compacité des ensembles de Julia des polynômes p-adiques. Math. Z. 246 (2004), 273–289. | MR 2031456 | Zbl 1047.37031

[BCSS] L. Blum, F. Cucker, M. Shub et S. Smale, Complexity And Real Computation. Spinger Verlag, Berlin 1998. | MR 1479636 | Zbl 0948.68068

[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 1813109 | Zbl 0978.37039

[BH] M. Baker et L-C. Hsia, Canonical heights, transfinite diameters, and polynomial dynamics. J. Reine Angew. Math. 585 (2005), 61–92. | MR 2164622 | Zbl 1071.11040

[BY1] M. Braverman et M. Yampolsky, Non-computable Julia sets. J. Amer. Math. Soc. 19 (2006), no. 3, 551–578. | MR 2220099 | Zbl 1099.37042

[BY2] M. Braverman et M. Yampolsky, On computability of Julia sets : answers to questions of Milnor and Shub. Preprint 2006. | MR 2220099

[D] V. A. Dremov, On a p-adic Julia set. Russian Math. Surveys 58 (2003), 1194–1195. | MR 2054097 | Zbl 1054.37027

[De] J. Denef, p-adic semi-algebraic sets and cell decomposition. J. Reine Angew. Math. 369 (1986), 154–166. | MR 850632 | Zbl 0584.12015

[DH] A. Douady et J. H. Hubbard, Itération des polynômes quadratiques complexes. Comptes Rendus Acad. Sci. Paris 294 (1982), 123–126. | MR 651802 | Zbl 0483.30014

[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 1052.37039

[H1] L-C. Hsia, A weak Néron model with applications to p-adic dynamical systems. Compositio Math. 100 (1996), 277–304. | Numdam | MR 1387667 | Zbl 0851.14001

[H2] L-C. Hsia, Closure of periodic points over a non archimedean field. J. London. Math. Soc. 62 (2000), 685-700. | MR 1794277 | Zbl 1022.11060

[J] M. Jakobson, Absolutely continuous invariant measures for one parameter families of one-dimensional maps. Comm. Math. Phys. 81 (1981), 161–185. | MR 630331 | Zbl 0497.58017

[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 1935840 | Zbl pre01850538

[M] J. Milnor, Dynamics in One Complex Variable. Friedr. Vieweg & Sohn, Braunschweig/Wiesbaden, 1999. | MR 1721240 | Zbl 0946.30013

[MS] P. Morton et J. Silverman, Periodic points, multiplicities, and dynamical units. J. Reine Angew. Math. 461 (1995), 81–122. | MR 1324210 | Zbl 0813.11059

[Na] W. Narkiewicz, Polynomial Mappings. Lecture Notes in Mathematics 1600 (1995), Springer Verlag, Berlin. | MR 1367962 | Zbl 0829.11002

[No] D. G. Northcott, Periodic points on an algebraic variety. Ann. of Math. 51 (1950), 167–177. | MR 34607 | Zbl 0036.30102

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

[P] T. Pezda, Polynomial cycles in certain local domains. Acta Arithmetica 66 (1994), 11–22. | MR 1262650 | Zbl 0803.11063

[R] J. Rivera–Letelier, Dynamique des fractions rationnelles sur les corps locaux, dans Geometric Methods in Dynamics, II. Astérisque 287 (2003), 199-231. | Zbl 1041.37021

[TVW] E. Thiran, D. Verstegen, J. Weyers, p-adic dynamics. J. Stat. Phys. 54 (1989), 893–913. | MR 988564 | Zbl 0672.58019

[W] A. Weil, Basic Number Theory. Die Grundlehren des mathematischen Wissenschaften 144 (1967), Springer Verlag, Berlin. | MR 234930 | Zbl 0176.33601

[WS] C. Woodcock et N. Smart, p-adic chaos and random number generation. Experiment. Math. 7 (1998), 333–342. | MR 1678087 | Zbl 0920.11052

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