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 u 1 u 2 ⋯ u n with u i ∈ 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 f x c : N → 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 lim inf n → ∞ f x c ( n ) = + ∞ .