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Jan Kiwi
Puiseux series polynomial dynamics and iteration of complex cubic polynomials
(Dynamique polynomiale des séries de Puiseux)
Annales de l'institut Fourier, 56 no. 5 (2006), p. 1337-1404
Article: subscription required | Reviews MR 2273859 | Zbl 1110.37036
Class. Math.: 37F45, 12J25, 32S99
Keywords: Puiseux series, Julia sets
[1] L. Ahlfors, Lectures on quasiconformal mappings, Van Nostrand, 1966 MR 200442 | Zbl 0138.06002
[2] M. Baker & R. Rumely, “Equidistribution of Small Points, Rational Dynamics, and Potential Theory”, to appear in Ann. Inst. Fourier
arXiv
[3] R.-L. Benedetto, “Wandering Domains and Nontrivial Reduction in Non-Archimedean Dynamics”, to appear
arXiv | MR 2157374 | Zbl 02212233
[4] R.-L. Benedetto, “Hyperbolic maps in $p$-adic dynamics”, Ergodic Theory and Dynamical Systems 21 (2001), p. 1-11 MR 1826658 | Zbl 0972.37027
[5] R.-L. Benedetto, “Examples of wandering domains in $p$-adic polynomial dynamics”, C. R. Math. Acad. Sci. Paris 7 (2002), p. 615-620 MR 1941304 | Zbl 01850790
[6] A.-S. Besicovitch, Almost periodic functions, Dover, 1954 MR 68029 | Zbl 0065.07102
[7] J.-P. Bézivin, “Sur la compacité des ensembles de Julia des polynômes $p$–adiques”, Math. Z. (2004), p. 273-289 MR 2031456 | Zbl 1047.37031
[8] P. Blanchard, “Complex analytic dynamics on the Riemann sphere”, Bull. Amer. Math. Soc. 11 (1984), p. 85-141
Article | MR 741725 | Zbl 0558.58017
[9] B. Branner, Cubic polynomials: turning around the connectedness locus, Publish or Perish, 1993, p. 391-427 MR 1215972 | Zbl 0801.58024
[10] B. Branner & J. H. Hubbard, “The iteration of cubic polynomials. Part I: The global topology of parameter space”, Acta math. 160 (1988), p. 143-206 MR 945011 | Zbl 0668.30008
[11] B. Branner & J. H. Hubbard, “The iteration of cubic polynomials. Part II: Patterns and parapatterns”, Acta math. 169 (1992), p. 229-325 MR 1194004 | Zbl 0812.30008
[12] E. Brieskorn & H. Knörrer, Plane algebraic curves, Birkhäuser Verlag, 1986 MR 886476 | Zbl 0588.14019
[13] E. Casas-Alvero, Singularities of plane curves, LMS Lecture Notes Series, Cambridge University Press, 2000 MR 1782072 | Zbl 0967.14018
[14] J.W.S. Cassels, Local fields, LMS student texts, Cambridge University Press, 1986 MR 861410 | Zbl 0595.12006
[15] A. Escassut, Analytic elements in $p$-adic analysis, World Scientific, 1995 MR 1370442 | Zbl 0933.30030
[16] A. Escassut, Ultrametric Banach Algebras, World Scientific, 2003 MR 1978961 | Zbl 1026.46067
[17] C. Favre & J. Rivera Letelier, “Théorème d’équidistribution de Brolin en dynamique $p$-adique”, C. R. Math. Acad. Sci. Paris 4 (2004), p. 271-276 MR 2092012 | Zbl 1052.37039
[18] G. Fernandez, Componentes de Fatou errantes en dinámica $p$-ádica, PUC, 2004
[19] S. Gelfand, Generalized Functions 1, Academic Press, 1964 Zbl 0115.33101
[20] D. Harris, “Turning curves for critically recurrent cubic polynomials”, Nonlinearity 12 (1999), p. 411-418 MR 1677771 | Zbl 0963.37040
[21] Y. Katnelzon, An introduction to harmonic analysis, Dover, 1976 Zbl 0352.43001
[22] C.-T. McMullen, Complex Dynamics and Renormalization, Annals of Math. Studies, Princeton University Press, 1994 MR 1312365 | Zbl 0822.30002
[23] J. Milnor, “On cubic polynomials with periodic critical points”, Unpublished, 1991
[24] J. Milnor, Dynamics in one complex variable, Vieweg, 1999 MR 1721240 | Zbl 0946.30013
[25] J. Milnor, Local connectivity of Julia sets: expository lectures, The Mandelbrot set, theme and variations, Cambridge Univ. Press, 2000, p. 67-116 MR 1765085 | Zbl 02096494
[26] M. Rees, “Views of parameter space: topographer and resident”, Astérique (2003) MR 2033172 | Zbl 1054.37020
[27] J. Rivera Letelier, “Points périodiques des fonctions rationelles dans l’espace hyperbolique $p$-adique”, Preprint Zbl 02205608
[28] J. Rivera Letelier, “Wild recurrent critical points”, arxiv.org/math.DS/0406417
arXiv | Zbl 02228487
[29] J. Rivera Letelier, Dynamique de fractions rationnelles sur des corps locaux, thesis, U. de Paris-Sud, 2000
[30] J. Rivera Letelier, “Dynamique des fonctions rationnelles sur des corps locaux, Geometric methods in dynamics. II”, Astérisque (2003), p. 147-230 MR 2040006 | Zbl 02066305
[31] J. Rivera Letelier, “Espace hyperbolique $p$-adique et dynamique des fonctions rationnelles”, Compositio Math. 138 (2003), p. 199-231 MR 2018827 | Zbl 1041.37021
Jan Kiwi
Puiseux series polynomial dynamics and iteration of complex cubic polynomials
(Dynamique polynomiale des séries de Puiseux)
Annales de l'institut Fourier, 56 no. 5 (2006), p. 1337-1404
Article: subscription required | Reviews MR 2273859 | Zbl 1110.37036
Class. Math.: 37F45, 12J25, 32S99
Keywords: Puiseux series, Julia sets
Résumé - Abstract
We let $\mathbb{L}$ be the completion of the field of formal Puiseux series and study polynomials with coefficients in $\mathbb{L}$ as dynamical systems. We give a complete description of the dynamical and parameter space of cubic polynomials in $\mathbb{L} [\zeta ]$. We show that cubic polynomial dynamics over $\mathbb{L}$ and $\mathbb{C}$ are intimately related. More precisely, we establish that some elements of $\mathbb{L}$ 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.
Bibliography
[2] M. Baker & R. Rumely, “Equidistribution of Small Points, Rational Dynamics, and Potential Theory”, to appear in Ann. Inst. Fourier
arXiv
[3] R.-L. Benedetto, “Wandering Domains and Nontrivial Reduction in Non-Archimedean Dynamics”, to appear
arXiv | MR 2157374 | Zbl 02212233
[4] R.-L. Benedetto, “Hyperbolic maps in $p$-adic dynamics”, Ergodic Theory and Dynamical Systems 21 (2001), p. 1-11 MR 1826658 | Zbl 0972.37027
[5] R.-L. Benedetto, “Examples of wandering domains in $p$-adic polynomial dynamics”, C. R. Math. Acad. Sci. Paris 7 (2002), p. 615-620 MR 1941304 | Zbl 01850790
[6] A.-S. Besicovitch, Almost periodic functions, Dover, 1954 MR 68029 | Zbl 0065.07102
[7] J.-P. Bézivin, “Sur la compacité des ensembles de Julia des polynômes $p$–adiques”, Math. Z. (2004), p. 273-289 MR 2031456 | Zbl 1047.37031
[8] P. Blanchard, “Complex analytic dynamics on the Riemann sphere”, Bull. Amer. Math. Soc. 11 (1984), p. 85-141
Article | MR 741725 | Zbl 0558.58017
[9] B. Branner, Cubic polynomials: turning around the connectedness locus, Publish or Perish, 1993, p. 391-427 MR 1215972 | Zbl 0801.58024
[10] B. Branner & J. H. Hubbard, “The iteration of cubic polynomials. Part I: The global topology of parameter space”, Acta math. 160 (1988), p. 143-206 MR 945011 | Zbl 0668.30008
[11] B. Branner & J. H. Hubbard, “The iteration of cubic polynomials. Part II: Patterns and parapatterns”, Acta math. 169 (1992), p. 229-325 MR 1194004 | Zbl 0812.30008
[12] E. Brieskorn & H. Knörrer, Plane algebraic curves, Birkhäuser Verlag, 1986 MR 886476 | Zbl 0588.14019
[13] E. Casas-Alvero, Singularities of plane curves, LMS Lecture Notes Series, Cambridge University Press, 2000 MR 1782072 | Zbl 0967.14018
[14] J.W.S. Cassels, Local fields, LMS student texts, Cambridge University Press, 1986 MR 861410 | Zbl 0595.12006
[15] A. Escassut, Analytic elements in $p$-adic analysis, World Scientific, 1995 MR 1370442 | Zbl 0933.30030
[16] A. Escassut, Ultrametric Banach Algebras, World Scientific, 2003 MR 1978961 | Zbl 1026.46067
[17] C. Favre & J. Rivera Letelier, “Théorème d’équidistribution de Brolin en dynamique $p$-adique”, C. R. Math. Acad. Sci. Paris 4 (2004), p. 271-276 MR 2092012 | Zbl 1052.37039
[18] G. Fernandez, Componentes de Fatou errantes en dinámica $p$-ádica, PUC, 2004
[19] S. Gelfand, Generalized Functions 1, Academic Press, 1964 Zbl 0115.33101
[20] D. Harris, “Turning curves for critically recurrent cubic polynomials”, Nonlinearity 12 (1999), p. 411-418 MR 1677771 | Zbl 0963.37040
[21] Y. Katnelzon, An introduction to harmonic analysis, Dover, 1976 Zbl 0352.43001
[22] C.-T. McMullen, Complex Dynamics and Renormalization, Annals of Math. Studies, Princeton University Press, 1994 MR 1312365 | Zbl 0822.30002
[23] J. Milnor, “On cubic polynomials with periodic critical points”, Unpublished, 1991
[24] J. Milnor, Dynamics in one complex variable, Vieweg, 1999 MR 1721240 | Zbl 0946.30013
[25] J. Milnor, Local connectivity of Julia sets: expository lectures, The Mandelbrot set, theme and variations, Cambridge Univ. Press, 2000, p. 67-116 MR 1765085 | Zbl 02096494
[26] M. Rees, “Views of parameter space: topographer and resident”, Astérique (2003) MR 2033172 | Zbl 1054.37020
[27] J. Rivera Letelier, “Points périodiques des fonctions rationelles dans l’espace hyperbolique $p$-adique”, Preprint Zbl 02205608
[28] J. Rivera Letelier, “Wild recurrent critical points”, arxiv.org/math.DS/0406417
arXiv | Zbl 02228487
[29] J. Rivera Letelier, Dynamique de fractions rationnelles sur des corps locaux, thesis, U. de Paris-Sud, 2000
[30] J. Rivera Letelier, “Dynamique des fonctions rationnelles sur des corps locaux, Geometric methods in dynamics. II”, Astérisque (2003), p. 147-230 MR 2040006 | Zbl 02066305
[31] J. Rivera Letelier, “Espace hyperbolique $p$-adique et dynamique des fonctions rationnelles”, Compositio Math. 138 (2003), p. 199-231 MR 2018827 | Zbl 1041.37021
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