Non communiqué / Rauzy-type dynamics are group (or monoid) actions on a collection of combinatorial objects. The first and best known example concerns an action on permutations, associated to interval exchange transformations (IET) for the Poincaré map on compact orientable translation surfaces. The equivalence classes on the objects inducedby the group action are related to components of the moduli spaces of Abelian differentials with prescribed singularities, and, in two variants of the problem, have been classified by Kontsevich and Zorich, and by Boissy, through methods involving both combinatorics, algebraic geometry, topology and dynamical systems. In the first half of this thesis, we provide a purely combinatorial proof of both classification theorems. Our proof can be interpreted geometrically and the over archingidea is close to that of Kontsevich and Zorich, although the techniques arerather different. Not all Rauzy-type dynamics have a geometrical correspondence however, and some parts of this first proof do not seem to generalize well.In the second half of the thesis we develop a new method, that we call the labelling method. This second method is not completely disjoint from the first one, but it the new crucial ingredient of considering a sort of ‘monodromy’ for the dynamics, in away that we now sketch. Many statements in this thesis are proven by induction. It is conceivable to prove, by induction, a classification theorem for unlabelled objects. However, as the labelling method will show, it is easier to prove two statements in parallel within the same induction, the one on the unlabelled objects, and an apparently much harder one, on the monodromy of the labelled objects. Although the final result is stronger than the initial aim, by virtue of the stronger inductive hypothesis, the method may work more easily.This second approach extends to several other Rauzy-type dynamics. Our firststep is to apply the labelling method to derive a second proof of the classificationtheorem for the Rauzy dynamics. Then we apply it to the study of two other Rauzy-type dynamics (one of which is strictly related to the Rauzy dynamics on non-orientable surfaces), and finally we inventory a surprisingly high number of Rauzy-type dynamics for which the labelling.

Ce travail de thèse porte sur la dynamique du flot linéaire des surfaces de translation et de sa renormalisation par le flot de Teichmüller introduite par H. Masur et W. Veech en 1982. Une version combinatoire de cette renormalisation, l'induction de Rauzy sur les échanges d'intervalles, fût introduite auparavant par G. Rauzy en 1979. D'une part, nous faisons une étude combinatoire des classes de Rauzy qui forment une partition de l'ensemble des permutations irréductibles et interviennent dans l'algorithme d'induction de Rauzy. Nous donnons une formule pour la cardinalité de chaque classe. D'autre part, nous étudions un modèle de billard infini périodique dans le plan appelé le "vent dans les arbres" introduit dans une version stochastique par P. et T. Ehrenfest en 1912 et par J. Hardy et J. Weber en 1980 dans la version périodique. Nous construisons une famille de directions pour lesquelles le flot du billard est divergent donnant ainsi des exemples de Z^2-cocycles divergents au-dessus d'échanges d'intervalles. De plus, nous démontrons que le taux polynomial de diffusion générique est 2/3 autrement dit que la distance maximale atteinte par une particule au temps t est de l'ordre de t^2/3. / In this thesis, we study the dynamics of the linear flow of translation surfaces and its renormalization by the Teichmüller flow introduced by H. Masur and W. Veech in 1982. A combinatorial version of the renormalization, the Rauzy induction on interval exchange transformations, was introduced by G. Rauzy in 1979. First of all, we consider the combinatorics of Rauzy classes which form a partition of the set of irreducible permutations and are part of the Rauzy induction. In a second time, we consider an infinite Z^2-periodic billiard in the plane called the wind-tree model. It was introduced in a stochastic version by P. and T. Ehrenfest in 1912 and in the periodic version by J. Hardy and J. Weber in 1980. We construct a family of directions for which the flow of the billiard is divergent and hence give examples of divergent Z^2-cocycles over interval exchange transformations. Moreover, we prove that the polynomial rate of diffusion is generically 2/3. In other words, the maximal distance reached by a particule below time t has the order of t^2/3.