Dynamics and geometry of the Rauzy-Veech induction for quadratic differentials

TL;DR

This paper extends the Rauzy-Veech induction to quadratic differentials, linking geometric and combinatorial aspects of generalized permutations to classify connected components of moduli spaces.

Contribution

It introduces an analogue of the Rauzy-Veech induction for quadratic differentials and establishes a bijection with connected components of strata, advancing the understanding of their geometry and dynamics.

Findings

Characterization of attractors of the generalized Rauzy-Veech induction.

Bijection between Rauzy classes and strata components.

Classification of all exceptional strata components.

Abstract

Interval exchange maps are related to geodesic flows on translation surfaces; they correspond to the first return maps of the vertical flow on a transverse segment. The Rauzy-Veech induction on the space of interval exchange maps provides a powerful tool to analyze the Teichmueller geodesic flow on the moduli space of Abelian differentials. Several major results have been proved using this renormalization. Danthony and Nogueira introduced in 1988 a natural generalization of interval exchange transformations, namely the linear involutions. These maps are related to general measured foliations on surfaces (orientable or not). In this paper we are interested by such maps related to geodesic flow on (orientable) flat surfaces with Z/2Z linear holonomy. We relate geometry and dynamics of such maps to the combinatorics of generalized permutations. We study an analogue of the Rauzy-Veech induction and give an efficient combinatorial characterization of its attractors. We establish a natural bijection between the extended Rauzy classes of generalized permutations and connected components of the strata of meromorphic quadratic differentials with at most simple poles, which allows, in particular, to classify the connected components of all exceptional strata.