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

DOI10.3934/DCDS.2022022arXiv1801.05973MaRDI QIDQ6296631

Publication date: 18 January 2018

Abstract: 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 Poincar'e 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.

Mathematics Subject Classification ID

Permutations, words, matrices (05A05) Dynamical aspects of measure-preserving transformations (37A05) Moduli of Riemann surfaces, Teichmüller theory (complex-analytic aspects in several variables) (32G15) Dynamical systems involving transformations and group actions with special properties (minimality, distality, proximality, expansivity, etc.) (37B05) Combinatorial dynamics (types of periodic orbits) (37E15) Group actions on combinatorial structures (05E18)

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