We let be the completion of the field of formal Puiseux series and study polynomials with coefficients in as dynamical systems. We give a complete description of the dynamical and parameter space of cubic polynomials in . We show that cubic polynomial dynamics over and are intimately related. More precisely, we establish that some elements of naturally correspond to the Fourier series of analytic almost periodic functions (in the sense of Bohr) which parametrize (near infinity) the quasiconformal classes of non-renormalizable complex cubic polynomials. Our techniques are based on the ideas introduced by Branner and Hubbard to study complex cubic polynomials.
Nous considérons la complétion du corps des séries formelles de Puiseux et nous étudions les polynômes à coefficients dans en tant que systèmes dynamiques. Nous donnons une description complète de l’espace dynamique et l’espace des paramètres des polynômes cubiques à coefficients dans . Nous démontrons que la dynamique cubique sur et sur sont intimement liées. Plus précisement, nous montrons que certains éléments de correspondent de manière naturelle à des séries de Fourier de fonctions analytiques presque périodiques (au sens de Bohr) qui paramétrisent (à l’infini) les classes quasi-conformes des polynômes complexes cubiques non renormalisables. Nos techniques s’appuient sur des idées introduites par Branner et Hubbard pour l’étude des polynômes cubiques complexes.
Keywords: Puiseux series, Julia sets
Mot clés : Séries de Puiseux, ensemble des Julia
@article{AIF_2006__56_5_1337_0, author = {Kiwi, Jan}, title = {Puiseux series polynomial dynamics and iteration of complex cubic polynomials}, journal = {Annales de l'Institut Fourier}, pages = {1337--1404}, publisher = {Association des Annales de l{\textquoteright}institut Fourier}, volume = {56}, number = {5}, year = {2006}, doi = {10.5802/aif.2215}, zbl = {1110.37036}, mrnumber = {2273859}, language = {en}, url = {http://archive.numdam.org/articles/10.5802/aif.2215/} }
TY - JOUR AU - Kiwi, Jan TI - Puiseux series polynomial dynamics and iteration of complex cubic polynomials JO - Annales de l'Institut Fourier PY - 2006 SP - 1337 EP - 1404 VL - 56 IS - 5 PB - Association des Annales de l’institut Fourier UR - http://archive.numdam.org/articles/10.5802/aif.2215/ DO - 10.5802/aif.2215 LA - en ID - AIF_2006__56_5_1337_0 ER -
%0 Journal Article %A Kiwi, Jan %T Puiseux series polynomial dynamics and iteration of complex cubic polynomials %J Annales de l'Institut Fourier %D 2006 %P 1337-1404 %V 56 %N 5 %I Association des Annales de l’institut Fourier %U http://archive.numdam.org/articles/10.5802/aif.2215/ %R 10.5802/aif.2215 %G en %F AIF_2006__56_5_1337_0
Kiwi, Jan. Puiseux series polynomial dynamics and iteration of complex cubic polynomials. Annales de l'Institut Fourier, Volume 56 (2006) no. 5, pp. 1337-1404. doi : 10.5802/aif.2215. http://archive.numdam.org/articles/10.5802/aif.2215/
[1] Lectures on quasiconformal mappings, Van Nostrand, Princeton, 1966 | MR | Zbl
[2] Equidistribution of Small Points, Rational Dynamics, and Potential Theory (to appear in Ann. Inst. Fourier) | Numdam
[3] Wandering Domains and Nontrivial Reduction in Non-Archimedean Dynamics (to appear) | MR | Zbl
[4] Hyperbolic maps in -adic dynamics, Ergodic Theory and Dynamical Systems, Volume 21 (2001) no. 1, pp. 1-11 | DOI | MR | Zbl
[5] Examples of wandering domains in -adic polynomial dynamics, C. R. Math. Acad. Sci. Paris, Volume 7 (2002) no. 335, pp. 615-620 | MR | Zbl
[6] Almost periodic functions, Dover, 1954 | MR | Zbl
[7] Sur la compacité des ensembles de Julia des polynômes –adiques, Math. Z. (2004) no. 246, pp. 273-289 | DOI | MR | Zbl
[8] Complex analytic dynamics on the Riemann sphere, Bull. Amer. Math. Soc., Volume 11 (1984) no. 1, pp. 85-141 | DOI | MR | Zbl
[9] Cubic polynomials: turning around the connectedness locus, Publish or Perish (1993), pp. 391-427 | MR | Zbl
[10] The iteration of cubic polynomials. Part I: The global topology of parameter space, Acta math., Volume 160 (1988), pp. 143-206 | DOI | MR | Zbl
[11] The iteration of cubic polynomials. Part II: Patterns and parapatterns, Acta math., Volume 169 (1992) no. 3-4, pp. 229-325 | DOI | MR | Zbl
[12] Plane algebraic curves, Birkhäuser Verlag, Basel, 1986 | MR | Zbl
[13] Singularities of plane curves, LMS Lecture Notes Series, Volume 276, Cambridge University Press, 2000 | MR | Zbl
[14] Local fields, LMS student texts, Volume 3, Cambridge University Press, 1986 | MR | Zbl
[15] Analytic elements in -adic analysis, World Scientific, 1995 | MR | Zbl
[16] Ultrametric Banach Algebras, World Scientific, 2003 | MR | Zbl
[17] Théorème d’équidistribution de Brolin en dynamique -adique, C. R. Math. Acad. Sci. Paris, Volume 4 (2004) no. 339, pp. 271-276 | MR | Zbl
[18] Componentes de Fatou errantes en dinámica -ádica, PUC, Chile (2004) (Masters thesis)
[19] Generalized Functions, 1, Academic Press, 1964 | Zbl
[20] Turning curves for critically recurrent cubic polynomials, Nonlinearity, Volume 12 (1999) no. 2, pp. 411-418 | DOI | MR | Zbl
[21] An introduction to harmonic analysis, Dover, 1976 | Zbl
[22] Complex Dynamics and Renormalization, Annals of Math. Studies, Princeton University Press, 1994 no. 135 | MR | Zbl
[23] On cubic polynomials with periodic critical points (1991) (Unpublished)
[24] Dynamics in one complex variable, Vieweg, 1999 | MR | Zbl
[25] Local connectivity of Julia sets: expository lectures, The Mandelbrot set, theme and variations, Cambridge Univ. Press, 2000 no. 274, pp. 67-116 | MR | Zbl
[26] Views of parameter space: topographer and resident, Astérique (2003) no. 288, pp. vi+418 | Numdam | MR | Zbl
[27] Points périodiques des fonctions rationelles dans l’espace hyperbolique -adique (Preprint) | Zbl
[28] Wild recurrent critical points (arxiv.org/math.DS/0406417) | Zbl
[29] Dynamique de fractions rationnelles sur des corps locaux, U. de Paris-Sud, Orsay (2000) (Ph. D. Thesis)
[30] Dynamique des fonctions rationnelles sur des corps locaux, Geometric methods in dynamics. II, Astérisque (2003) no. 287, pp. 147-230 | Numdam | MR | Zbl
[31] Espace hyperbolique -adique et dynamique des fonctions rationnelles, Compositio Math., Volume 138 (2003) no. 2, pp. 199-231 | DOI | MR | Zbl
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