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A combinatorial approach to Rauzy-type dynamics

Abstract : 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.
Keywords : Teichmüller flow
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Submitted on : Thursday, September 12, 2019 - 9:13:10 PM
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Quentin de Mourgues. A combinatorial approach to Rauzy-type dynamics. Computational Geometry [cs.CG]. Université Sorbonne Paris Cité, 2017. English. ⟨NNT : 2017USPCD056⟩. ⟨tel-02285651⟩



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