Introduction to Iterated Monodromy Groups

Sébastien Godillon (1)

1 AGM - UMR 8088 - Analyse, Géométrie et Modélisation

The theory of iterated monodromy groups was developed by Nekrashevych. It is a wonderful example of application of group theory in dynamical systems and, in particular, in holomorphic dynamics. Iterated monodromy groups encode in a computationally efficient way combinatorial information about any dynamical system induced by a post-critically finite branched covering. Their power was illustrated by a solution of the Hubbard Twisted Rabbit Problem given by Bartholdi and Nekrashevych. These notes attempt to introduce this theory for those who are familiar with holomorphic dynamics but not with group theory. The aims are to give all explanations needed to understand the main definition and to provide skills in computing any iterated monodromy group efficiently. Moreover some explicit links between iterated monodromy groups and holomorphic dynamics are detailed. In particular, some facts about combinatorial equivalence classes are provided, and matings of polynomials are discussed.

Mots clés: iterated monodromy group; holomorphic dynamics

Domaines: Systèmes dynamiques [math.DS]

Sébastien Godillon. Introduction to Iterated Monodromy Groups. Annales, 2012, 21 (S5), pp.1069-1118. ⟨hal-00796901⟩