Quentin De Mourgues | Searcharxiv

arXiv2018

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 first paper proposed an ad hoc combinatorial proof of this classification. In this paper we define 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 second combinatorial proof of the classification.

math.CO↗

arXiv2017

A combinatorial approach to Rauzy-type dynamics I: permutations and the Kontsevich--Zorich--Boissy classification theorem

Rauzy-type dynamics are group 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 induced by 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 and algebraic geometry. We provide here a purely combinatorial proof of both classification theorems, and in passing establish a few previously unnoticed features. As will be shown elsewhere, our methods extend also to other Rauzy-type dynamics, both on labeled and unlabeled structures. Some of these dynamics have a geometrical interpretation (e.g., matchings, related to IET on non-orientable surfaces), while some others do not have one so far.

math.CO↗