Periodicity of Rauzy scheme and substitutional systems

TL;DR

This paper introduces Rauzy schemes as a new tool to analyze Rauzy graphs and proves their periodicity for morphic superwords, generalizing the periodicity of quadratic irrationals' chain fractions.

Contribution

The paper defines Rauzy schemes and establishes their periodicity for morphic superwords, extending known results about quadratic irrationals.

Findings

Rauzy schemes can be derived from Rauzy graphs by uniting certain vertices.

Morphic superwords exhibit periodic Rauzy schemes.

Generalization of quadratic irrationals' periodic chain fractions.

Abstract

In the paper the notion of {\em Rauzy scheme} is introduced. From Rauzy graph Rauzy Scheme can be obtaining by uniting sequence of vertices of ingoing and outgoing degree 1 by arches. This notion is a tool to describe Rauzy graph behavior. For morphic superword we prove periodicity of Rauzy schemes. This is generalization of fact that quadratic irrationals have periodic chain fractions.