Puiseux series polynomial dynamics and iteration of complex cubic polynomials

Kiwi, Jan

Annales de l'Institut Fourier, Tome 56 (2006) no. 5, pp. 1337-1404.

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

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 $𝕃\left[\zeta \right]$. 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.

Classification : 37F45, 12J25, 32S99
Keywords: Puiseux series, Julia sets
Mot clés : Séries de Puiseux, ensemble des Julia

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