Puiseux series polynomial dynamics and iteration of complex cubic polynomials

Jan Kiwi[1]

  • [1] Facultad de Matemáticas Pontificia Universidad Católica Casilla 306, Correo 22, Santiago (Chile)

Annales de l’institut Fourier (2006)

  • Volume: 56, Issue: 5, page 1337-1404
  • ISSN: 0373-0956

Abstract

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We let 𝕃 be the completion of the field of formal Puiseux series and study polynomials with coefficients in 𝕃 as dynamical systems. We give a complete description of the dynamical and parameter space of cubic polynomials in 𝕃 [ ζ ] . We show that cubic polynomial dynamics over 𝕃 and ℂ are intimately related. More precisely, we establish that some elements of 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.

How to cite

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Kiwi, Jan. "Puiseux series polynomial dynamics and iteration of complex cubic polynomials." Annales de l’institut Fourier 56.5 (2006): 1337-1404. <http://eudml.org/doc/10179>.

@article{Kiwi2006,
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.},
affiliation = {Facultad de Matemáticas Pontificia Universidad Católica Casilla 306, Correo 22, Santiago (Chile)},
author = {Kiwi, Jan},
journal = {Annales de l’institut Fourier},
keywords = {Puiseux series; Julia sets; cubic polynomials},
language = {eng},
number = {5},
pages = {1337-1404},
publisher = {Association des Annales de l’institut Fourier},
title = {Puiseux series polynomial dynamics and iteration of complex cubic polynomials},
url = {http://eudml.org/doc/10179},
volume = {56},
year = {2006},
}

TY - JOUR
AU - Kiwi, Jan
TI - Puiseux series polynomial dynamics and iteration of complex cubic polynomials
JO - Annales de l’institut Fourier
PY - 2006
PB - Association des Annales de l’institut Fourier
VL - 56
IS - 5
SP - 1337
EP - 1404
AB - 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.
LA - eng
KW - Puiseux series; Julia sets; cubic polynomials
UR - http://eudml.org/doc/10179
ER -

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