Open and closed factors of Arnoux-Rauzy words

TL;DR

This paper investigates the properties of closed factors within Arnoux-Rauzy words, deriving explicit formulas for their counts and demonstrating that the number of such factors grows unboundedly.

Contribution

It provides an explicit formula for the count of closed factors in Arnoux-Rauzy words and proves their unbounded growth.

Findings

Explicit formula for closed factors in Arnoux-Rauzy words

Unbounded growth of closed factors as length increases

Enhanced understanding of factor structure in complex words

Abstract

A finite word $u$ is called closed if its longest repeated prefix has exactly two occurrences in $u,$ once as a prefix and once as a suffix. We study the function $f_x^c:\mathbb N \rightarrow \mathbb N$ which counts the number of closed factors of each length in an infinite word $x.$ We derive an explicit formula for $f_x^c$ in case $x$ is an Arnoux-Rauzy word. As a consequence we prove that $\liminf _{n\rightarrow \infty}f_x^c(n)=+\infty.$