Occurrences of palindromes in characteristic Sturmian words

Sturmian words have exactly two factors of length 1, and thus are infinite sequences over a two-letter alphabet A = {a, b}, say. Here, an infinite word (or sequence) x over A is a map x : N → A. For any i ≥ 0, we set x i = x(i) and write x = x 0 x 1 x 2 • • • , each x i ∈ A. Central to our study is the following characterization of Sturmian words, which was originally proved by Morse and Hedlund [21]. An infinite word s over A = {a, b} is Sturmian if and only if there exists an irrational α ∈ (0, 1), and a real number ρ, such that s is equal to one of the following two infinite words: (n ≥ 0)

The irrational α is called the slope of s and ρ is the intercept. If ρ = 0, we have s α,0 = ac α and s ′ α,0 = bc α , where c α is called the characteristic Sturmian word of slope α (see [2]). Our focus will be on palindromic factors of c α . In general terms, a palindrome is a finite word that reads the same backwards as forwards. Palindromes are important tools used in the study of factors of Sturmian words (e.g., [6,8,9,10]), and they have also become objects of great interest in computer science. The aim of this current paper is to completely describe where palindromes occur in c α (and hence s α,0 , s ′ α,0 ). In order to do this, we shall make use of some previous results concerning factorizations of c α into singular words, which are particular palindromes. Singular words were first defined for the Fibonacci word f (a special example of a Sturmian word) by Wen and Wen [25], who established a decomposition of f with respect to such words. This result was later extended by Melançon [19] to characteristic Sturmian words. More recently, Levé and Séébold [17] have generalized Wen and Wen’s ‘singular’ decomposition of f , by establishing a similar decomposition for each conjugate of f into what they called generalized singular words. This last result has now been further extended by the present author [14] to c α (and c 1-α ), where α has continued fraction expansion [0; 2, r, r, r, . . .] for some r ≥ 1.

It is well-known that any Sturmian word s is uniformly recurrent, i.e., any factor of s occurs infinitely often in s with bounded gaps [5]. Accordingly, any palindromic factor u of c α has infinitely many occurrences in c α and, as we shall see later (Corollary 5.2), the distance between any two adjacent occurrences of u is bounded above by an integer depending on u. Given any palindromic factor u of c α , we shall establish a decomposition of c α with respect to the occurrences of u. Such a decomposition shows precisely at which positions u occurs in c α , and this is directly related to the continued fraction expansion of the irrational slope α.

This paper is organized as follows. In Section 2, after some preliminaries on words and morphisms, we will recall some facts about c α and consider some of its singular decompositions (Section 2.2). Then, in Section 3, we consider the structure of palindromic factors of c α with respect to its singular factors. We also recall the important notion of a return word and the concept of overlapping occurrences of a word in c α . Section 4 contains the lemmas we need in order to establish the main result of this paper, which appears in Section 5. Lastly, using results of Section 4, we obtain decompositions of c α that show precisely where a given factor of length q n occurs in c α (where q n is the denominator of the n-th

Any of the following terminology that is not further clarified can be found in either [18] or [2], which give more detailed presentations.

In what follows, let A denote the two-letter alphabet {a, b}. A (finite) word is an element of the free monoid A * generated by A, in the sense of concatenation. The identity ε of A * is called the empty word, and the free semigroup over A is defined by A + := A * \ {ε}. We denote by A ω the set of all infinite words over A, and define A ∞ := A * ∪ A ω . The length |w| of a finite word w is defined to be the number of letters it contains. (Note that |ε| = 0.)

A finite word z is a factor of a word w ∈ A ∞ if w = uzv for some u ∈ A * and v ∈ A ∞ . Furthermore, z is called a prefix (resp. suffix ) of w if u = ε (resp. v = ε), and we write z ⊆ p w (resp. z ⊆ s w). The word z is said to have an occurrence (or occur) at pos

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