Open and closed factors in Arnoux-Rauzy words

Abstract

Given a finite non-empty set A, let A+ denote the free semigroup generated by A consisting of all finite words u1u2⋯un with ui∈A. A word u∈A+ is said to be closed if either u∈A or if u is a complete first return to some factor v∈A+, meaning u contains precisely two occurrences of v, one as a prefix and one as a suffix. We study the function fxc:N→N which counts the number of closed factors of each length in an infinite word x. We derive an explicit formula for fxc in case x is an Arnoux-Rauzy word. As a consequence we prove that lim infn→∞fxc(n)=+∞.