Puiseux series polynomial dynamics and iteration of complex cubic polynomials

[Dynamique polynomiale des séries de Puiseux]

Kiwi, Jan

Annales de l'Institut Fourier, Tome 56 (2006), p. 1337-1404 / Harvested from Numdam

Résumé

Nous considérons la complétion $\mathbb{L}$ du corps des séries formelles de Puiseux et nous étudions les polynômes à coefficients dans $\mathbb{L}$ 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 $\mathbb{L}$. Nous démontrons que la dynamique cubique sur $\mathbb{L}$ et sur $\mathbb{C}$ sont intimement liées. Plus précisement, nous montrons que certains éléments de $\mathbb{L}$ 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.

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.

Détail

Publié le : 2006-01-01

MR 2273859 | Zbl 1110.37036 | 1 citation dans Numdam

DOI : https://doi.org/10.5802/aif.2215
Classification:  37F45,  12J25,  32S99
Mots clés: Séries de Puiseux, ensemble des Julia

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