Conservative polynomials and yet another action of $Gal(\overline{\mathbb{Q}}/\mathbb{Q})$ on plane trees

Dans cet article nous étudions une action $D$ du groupe de Galois absolu $\Gamma =Gal(\overline{\mathbb{Q}}/\mathbb{Q})$ sur des arbres planaires bicolores. A l’encontre de l’action similaire fournie par la théorie des “dessins d’enfants” de Grothendieck, l’action $D$ est induite par l’action de $\Gamma$ sur des classes d’équivalence de polynômes conservateurs qui sont les exemples les plus simples de fonctions rationnelles finies postcritiques. Nous établissons les propriétés principales de l’action $D$ et la comparons avec l’action de Grothendieck.

In this paper we study an action $D$ of the absolute Galois group $\Gamma =Gal(\overline{\mathbb{Q}}/\mathbb{Q})$ on bicolored plane trees. In distinction with the similar action provided by the Grothendieck theory of “Dessins d’enfants” the action $D$ is induced by the action of $\Gamma$ on equivalence classes of conservative polynomials which are the simplest examples of postcritically finite rational functions. We establish some basic properties of the action $D$ and compare it with the Grothendieck action.

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