Labeled and Non-labeled Extended Rauzy Classes

TL;DR

This paper explores the relationship between labeled and non-labeled conventions in Interval Exchange Transformations by analyzing Extended Rauzy Classes, clarifying their connections for better understanding of IETs and Flat Surfaces.

Contribution

It establishes the relationship between Extended Rauzy Classes under labeled and non-labeled conventions, bridging a gap in the combinatorial understanding of IETs.

Findings

Unified understanding of Rauzy Classes across conventions

Clarified the combinatorial structure of IETs

Facilitated research on Flat Surfaces using different conventions

Abstract

Multiple conventions have been adopted for denoting Interval Exchange Transformations (IETs). The "non-labeled" convention was the original, while the "labeled" convention has proven convenient when investigating Flat Surfaces as described by IETs. We establish the relationship between Extended Rauzy Classes, an equivalence class of combinatorial data related to IETs, under each convention.