no. 5
Puiseux series polynomial dynamics and iteration of complex cubic polynomials
Kiwi, Jan1
1Facultad de Matemáticas Pontificia Universidad Católica Casilla 306, Correo 22, Santiago (Chile)
Annales de l'Institut Fourier, Volume 56 (2006) no. 5, pp. 1337-1404

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.

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.

DOI: 10.5802/aif.2215
Classification: 37F45, 12J25, 32S99
Keywords: Puiseux series, Julia sets
Mots-clés : Séries de Puiseux, ensemble des Julia

Kiwi, Jan  1

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