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

TL;DR

This paper introduces a systematic labelling method to classify Rauzy-type dynamics, providing a second combinatorial proof of the KZB classification theorem for these group actions on combinatorial objects.

Contribution

The paper develops a general labelling method for classifying Rauzy-type dynamics, enhancing the combinatorial approach and offering a new proof of the KZB classification theorem.

Findings

The labelling method effectively classifies Rauzy-type dynamics.

A second combinatorial proof of the KZB classification theorem is achieved.

The method simplifies understanding of the structure of Rauzy-type dynamics.

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.