A combinatorial approach to Rauzy-type dynamics II: The labelling method and a second proof of the KZB classification theorem

Rauzy-type dynamics are group actions on a collection of combinatorial objects. The first and best known example (the Rauzy dynamics) 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 induced by the group action have been classified by Kontsevich and Zorich, and by Boissy through methods involving both combinatorics algebraic geometry, topology and dynamical systems. Our precedent paper [ 5 ] as well as the one of Fickenscher [ 8 ] proposed an ad hoc combinatorial proof of this classification. However, unlike those two previous combinatorial proofs, we develop in this paper a general method, called the labelling method, which allows one to classify Rauzy-type dynamics in a much more systematic way. We apply the method to the Rauzy dynamics and obtain a third combinatorial proof of the classification. The method is versatile and will be used to classify three other Rauzy-type dynamics in follow-up works. Another feature of this paper is to introduce an algorithmic method to work with the sign invariant of the Rauzy dynamics. With this method, we can prove most of the identities appearing in the literature so far ([ 10 ], [ 6 ], [ 2 ], [ 5 ]...) in an automatic way.