Infinite self-shuffling words

In this paper we introduce and study a new property of infinite words: An infinite word x ∈ A, with values in a finite set A, is said to be k-self-shuffling (k ≥ 2) if x admits factorizations: x = ∏∞ i=0 U (1) i · · ·U (k) i = ∏∞ i=0 U (1) i = · · · = ∏∞ i=0 U (k) i . 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 uniformly recurrent 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 t = t0t1t2 · · · {0, 1} N where tn is the sum modulo 2 of the digits in the binary expansion of n. 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 self-shuffling Sturmian words, we recover a number Email addresses: [email protected] (Émilie Charlier), [email protected] (Teturo Kamae), [email protected] (Svetlana Puzynina), [email protected] (Luca Q. Zamboni) Partially supported by the Academy of Finland under grant 251371, by Russian Foundation of Basic Research (grants 12-01-00448 and 12-01-00089). Partially supported by a FiDiPro grant (137991) from the Academy of Finland and by ANR grant SUBTILE. Preprint submitted to Elsevier November 26, 2013 theoretic result, originally due to Yasutomi, on a classification of pure morphic Sturmian words in the orbit of the characteristic.