Open and closed factors of Arnoux-Rauzy 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 $f_x^c:\mathbb{N}\to \mathbb{N}$ 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 ${\mathrm{liminf}}_{n\to \infty }{f}_x^c(n)=+\infty .$