Periodicity of Rauzy Scheme for substitution words

TL;DR

This paper proves the periodicity of Rauzy schemes for substitution words, extending the analogy of periodic continued fractions for quadratic irrationals, with implications for discrete dynamics and logic.

Contribution

It establishes the periodicity of Rauzy schemes for morphic superwords, generalizing known properties of quadratic irrationals.

Findings

Rauzy schemes for substitution words are periodic

Periodic Rauzy schemes relate to quadratic irrational properties

Implications for discrete dynamic systems and logic

Abstract

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 fact has consequence in discrete dynamic systems and logic. This fact is also generalization of fact that quadratic irrationals have periodic chain fractions.