Puiseux series polynomial dynamics and iteration of complex cubic polynomials
Annales de l'Institut Fourier, Volume 56 (2006) no. 5, p. 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 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 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 : https://doi.org/10.5802/aif.2215
Classification:  37F45,  12J25,  32S99
Keywords: Puiseux series, Julia sets
@article{AIF_2006__56_5_1337_0,
     author = {Kiwi, Jan},
     title = {Puiseux series polynomial dynamics and iteration of complex cubic polynomials},
     journal = {Annales de l'Institut Fourier},
     publisher = {Association des Annales de l'institut Fourier},
     volume = {56},
     number = {5},
     year = {2006},
     pages = {1337-1404},
     doi = {10.5802/aif.2215},
     mrnumber = {2273859},
     zbl = {1110.37036},
     language = {en},
     url = {http://www.numdam.org/item/AIF_2006__56_5_1337_0}
}
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. http://www.numdam.org/item/AIF_2006__56_5_1337_0/

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