The classification of polynomial basins of infinity

DeMarco, Laura; Pilgrim, Kevin

Annales scientifiques de l'École Normale Supérieure, Série 4, Tome 50 (2017) no. 4, pp. 799-877.

Nous étudions la question de la classification de la dynamique des polynômes complexes $f:ℂ\to ℂ$ restreints à leur bassin de l'infini. Nous faisons la synthèse d'outils de combinatoire — tableaux, arbres, laminations — en un nouvel invariant du bassin dynamique que nous appelons pictogramme. Pour les polynômes dont tous les points critiques s'échappent vers l'infini, nous obtenons une description complète de l'ensemble des classes de conjugaison topologiques ayant un pictogramme donné. Plus généralement, pour tout polynôme, nous calculons le nombre de classes de conjugaison topologiques du bassin $(f,X(f\left)\right)$ à pictogramme donné. Nous définissons les pictogrammes de façon abstraite et prouvons que chacun d'eux est réalisable par un polynôme. Nous donnons plus de détails en degré 3 et donnons des exemples montrant que le pictogramme est un invariant plus fin que les tableaux de [5] et que les arbres de [10].

We consider the problem of classifying the dynamics of complex polynomials $f:ℂ\to ℂ$ restricted to the basins of infinity $X\left(f\right)$. We synthesize existing combinatorial tools—tableaux, trees, and laminations—into a new invariant of basin dynamics we call the pictograph. For polynomials with all critical points escaping to infinity, we obtain a complete description of the set of topological conjugacy classes with given pictograph. For arbitrary polynomials, we compute the total number of topological conjugacy classes of basins $(f,X(f\left)\right)$ with a given pictograph. We also define abstract pictographs and prove that every abstract pictograph is realized by a polynomial. Extra details are given in degree 3, and we provide examples that show the pictograph is a finer invariant than both the tableau of [5] and the tree of [10].

Classification : 37F10, 37F20.
Keywords: Complex dynamics, polynomial dynamics, basin of infinity, moduli space of polynomials, pictograph, tree.
Mot clés : Dynamique holomorphe, dynamiques polynômes, bassin d'infini, espace de modules de polynômes, pictogramme, arbre.

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