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 y est conjugué au décalage unilatéral sur symboles.
@article{JTNB_2007__19_2_325_0,
author = {Robert Benedetto and Jean-Yves Briend and Herv\'e Perdry},
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 = {https://jtnb.centre-mersenne.org/articles/10.5802/jtnb.589/}
}
TY - JOUR AU - Robert Benedetto AU - Jean-Yves Briend AU - Hervé Perdry 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 - https://jtnb.centre-mersenne.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 Robert Benedetto %A Jean-Yves Briend %A Hervé Perdry %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 https://jtnb.centre-mersenne.org/articles/10.5802/jtnb.589/ %R 10.5802/jtnb.589 %G fr %F JTNB_2007__19_2_325_0
Robert Benedetto; Jean-Yves Briend; Hervé Perdry. 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. 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 | 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 -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 -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, -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 -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 -adique. Comptes Rendus Acad. Sci. Paris 339 (2004), 271–276. | Zbl
[H1] L-C. Hsia, A weak Néron model with applications to -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 -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 -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, -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, -adic chaos and random number generation. Experiment. Math. 7 (1998), 333–342. | MR | Zbl
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