Semiperiodic words and root-conjugacy

• Authors: Arturo Carpi;Aldo de Luca
• Venue: Theoretical Computer Science
• Year: 2003

A factor u of a word w is called right special if there exist two distinct letters a and b such that both ua and ub are factors of w. Left special factors are defined symmetrically. By Rw (resp. Lw) we denote the minimal natural number such that there is no right (resp. left) special factor of w of length Rw (resp. Lw). Moreover, Hw (resp. Kw) denotes the length of the shortest prefix (resp. suffix) which cannot be extended on the left (resp. right) in w. The parameters Rw, Lw, Hw, and Kw give interesting information on the structure of the word w. We consider the class of all finite words w such that Rw Hw. These words are called semiperiodic. Any periodic word is semiperiodic, whereas the converse is not generally true. Several characterizations of semiperiodic words can be given. In particular, a word w is semiperiodic if and only if it has a period p ≤ |w| - Rw. A further characterization of semiperiodic words relates with their infinite extensions. From this characterization one derives the following result, deeply related to the theorem of Fine and Wilf: if w is a (semiperiodic) word having two periods p,q ≤ |w| - Rw, then also d = gcd(p,q) is a period of w. The root rw of a word w is its prefix whose length is equal to the minimal period of w. Two words u and v are root-conjugate if their roots ru and rv are conjugate. One of the main results of the paper is the following. Let w be a semiperiodic word. A word v has the same set of factors of length 1 +Rw of w if and only if v is semiperiodic and root-conjugate with w. Some applications and extensions of this result are proved.