Infinite self-shuffling words

Abstract

In this paper we introduce and study a new property of infinite words: An infinite word x ∈ A N , with values in a finite set A, is said to be k-self-shuffling (k ≥ 2) if x admits factorizations:In other words, there exists a shuffle of k-copies of x which produces x. We are particularly interested in the case k = 2, in which case we say x is self-shuffling. This property of infinite words is shown to be independent of the complexity of the word as measured by the number of distinct factors of each length. Examples exist from bounded to full complexity. It is also an intrinsic property of the word and not of its language (set of factors). For instance, every aperiodic word contains a non self-shuffling word in its shift orbit closure. While the property of being self-shuffling is a relatively strong condition, many important words arising in the area of symbolic dynamics are verified to be self-shuffling. They include for instance the Thue-Morse word fixed by the morphism 0 → 01, 1 → 10. As another example we show that all Sturmian words of slope α ∈ R \ Q and intercept 0 < ρ < 1 are self-shuffling (while those of intercept ρ = 0 are not). Our characterization of self-shuffling Sturmian words can be interpreted arithmetically in terms of a dynamical embedding and defines an arithmetic process we call the stepping stone model. One important feature of self-shuffling words stems from its morphic invariance: The morphic image of a self-shuffling word is self-shuffling. This provides a useful tool for showing that one word is not the morphic image of another. In addition to its morphic invariance, this new notion has other unexpected applications particularly in the area of substitutive dynamical systems. For example, as a consequence of our characterization of selfshuffling Sturmian words, we recover a number theoretic result, originally due to Yasutomi, on a classification of pure morphic Sturmian words in the orbit of the characteristic.

On Shuffling of Infinite Square-Free Words

“…if k = 3 A word w ∈ Σ ω k without factors of exponent greater than RT (k) is called a Dejean word. Charlier et al showed that the Thue-Morse word, which is a binary Dejean word, is self-shuffling [2].…”

“…We remark that we can immediately produce a square-free self-shuffling word over Σ 3 from g ω (0): Charlier et al [2] noticed that the property of being self-shuffling is preserved by the application of a morphism. Furthermore, Brandenburg [1] showed that the morphism f :…”

“…In this section we provide a positive answer to the question from [2] whether there exists a word for which no element of its shift orbit closure is self-shuffling. The Hall word H = 012021012102 • • • is defined as the fixed point of the morphism h(0) = 012, h(1) = 02, h(2) = 1.…”

“…A self-shuffling word, a notion which was recently introduced by Charlier et al [2], is an infinite word that can be reproduced by shuffling it with itself. More formally, given two infinite words x, y ∈ Σ ω over a finite alphabet Σ, we define S (x, y) ⊆ Σ ω to be the collection of all infinite words z for which there exists a factorization…”

“…The shift orbit closure S w of an infinite word w can be defined, e.g., as the set of infinite words whose sets of factors are contained in the set of factors of w. In [2] it has been proved that each word has a non-self-shuffling word in its shift orbit closure, and the following question has been asked: Does there exist a word for which no element of its shift orbit closure is self-shuffling (Question 7.2)? In Section 4 we provide a positive answer to the question.…”