Symmetric interval identification systems of order three

TL;DR

This paper investigates symmetric interval identification systems of order three, proving that Rauzy induction preserves symmetry and providing an example of a symmetric system of thin type.

Contribution

It demonstrates that Rauzy induction maintains symmetry in these systems and introduces an example of a symmetric system of thin type.

Findings

Rauzy induction preserves symmetry in order three systems

Symmetric systems of thin type exist

Finitely many Rauzy inductions lead to symmetric systems

Abstract

In the present paper we study interval identification systems of order three. We prove that the Rauzy induction preserves symmetry: for any symmetric interval identification system of order three after finitely many iterations of the Rauzy induction we always obtain a symmetric system. We also provide an example of symmetric interval identification system of thin type.