Fixed points avoiding Abelian $k$-powers

Given a morphism h : Σ * → Σ * with an infinite fixed point w and an integer k ≥ 2, determine if w is k-power free.

This decidability of this problem has been well studied. For instance, Berstel [3] showed that over a ternary alphabet, there is an algorithm to determine if w is squarefree. Similarly, Karhumäki [9] showed that over a binary alphabet, there is an algorithm to determine if w is cubefree. The problem was solved in general by Mignosi and Séébold [14], who showed that there exists an algorithm for this problem for all alphabet sizes and all k. (See also the work of Krieger [12] for extensions to fractional repetitions.)

In this paper we consider the analogous question for Abelian k-power freeness. In particular, we show that for morphisms h satisfying certain rather general conditions, the following problem is decidable.

Given a morphism h : Σ * → Σ * with an infinite fixed point w and an integer k ≥ 2, determine if w is Abelian k-power free.

Dekking [6] provided sufficient conditions for a morphism h to be Abelian k-power free (i.e., h maps Abelian k-power free words to Abelian k-power free words). Carpi [4] showed the existence of an algorithm to decide if a morphism satisfying certain technical conditions is Abelian squarefree. Our decision procedure for the problem stated above is based on the idea of “templates” used in [1] and [2]. The same idea also appears in the recent work of [5].

We freely use the usual notations of combinatorics on words and formal language theory. (See for example [7,13].) Fix positive integer m and alphabet Σ = {1, 2, . . . , m}. We use Z to denote the set of integers, and Z n to denote the set of 1 × n matrices (i.e. row vectors) with integer entries. For u, v ∈ Σ * we write u ∼ v if u and v are anagrams of each other, that is, if |u| a = |v| a for all a ∈ Σ. We define the Parikh map ψ : Σ * → Z m by

In other words, ψ(w) is a row vector which counts the frequencies of 1, 2 . . . , m in w. For w, v ∈ Σ * we have w ∼ v exactly when ψ(w) = ψ(v). Let k be a positive integer. An Abelian k-power is a non-empty word of the form

Let a morphism µ : Σ * → Σ * be fixed. It will be convenient and natural for us to make some assumptions on µ:

We will need some matrix theory. A standard reference is [8,Chapter 5]. An induced norm on matrices of R m×m is given by

where |v| is the usual Euclidean length of vector v. We make the additional restriction on µ that M is non-singular and that Proof: The alternative is that w is an interior factor of some word of µ(Σ), forcing |w| ≤ N -2.✷

Let k be a positive integer. A k-template is a (2k)-tuple

where the a i ∈ {ǫ, 1, 2, . . . , m} and the d i ∈ Z m . We say that a word w realizes k-template t if a non-empty factor I of w has the form

Call I an instance of t.

Remark 3.1 The particular k-template

will be of interest. Word I is an instance of T k if and only if I has the form and

be k-templates. We say that t 2 is a parent of t 1 if

Lemma 3.2 (Parent Lemma) Suppose that w ∈ Σ * realizes t 2 . Then µ(w) realizes t 1 .

Proof: Let w contain the factor

and for each i,

3 Given a k-template t 1 , we may calculate all of its parents.

Proof: The set of candidates for the A i in a parent, and hence for the a ′ i , a ′′ i is finite, and may be searched exhaustively. Since M is non-singular, a choice of values for a ′ i , a ′′ i , together with given values d i , determines the D i by (3).✷ Remark 3.4 Note that not all computed values for D i may be in Z m ; some k-templates may have no parents.

Rewriting (3),

Since the a ′ i , a ′′ i are factors of words of µ(Σ), there are finitely many possibilities

). Let C be the (finite) set of possible values for c. Let ancestor be the transitive closure of the parent relation. The D i vectors in any ancestor of k-template T k will have the form

Let c * = max{|c| : c ∈ C} and let r = c * /(1 -|M -1 |). We have

Thus, the D i lie within a ball of radius r in R n . It follows that there are only finitely many D i ’s in Z n .

Proof: There are finitely many choices for the A i ∈ {ǫ, 1, 2, . . . , m} and the D i in any ancestor.✷ Suppose that in k-template t 1 we have ⌊max i |d i |⌋ = ∆. Let I be an instance of t 1 ,

where ψ(X i+1 ) -ψ(X i ) = d i , i = 1, 2, . . . k -1.

If i > j, we have

This can be argued with the opposite inequality, showing in total that

If for some i we have

The greatest possible length of I would then be

and repeatedly using Lemma 2.1 we can write 6 Future work Keränen [10,11] has constructed an Abelian 2-power free quaternary word which is the fixed point of a cyclic 85-uniform morphism. His exhaustive searches have shown that this is the shortest cyclic uniform morphism which works. One would hope that much shorter, if less symmetric, morphisms exist. The result contained here suggests a new exhaustive search, considering shorter, not ne

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