Infinite Self-Shuffling Words

TL;DR

This paper introduces the concept of self-shuffling infinite words, characterizes their properties, and explores their significance in symbolic dynamics, including applications to Sturmian words and morphic invariance.

Contribution

It defines and studies the new property of self-shuffling in infinite words, providing characterizations, invariance results, and applications to Sturmian words and dynamical systems.

Findings

Self-shuffling is an intrinsic property of infinite words.

Many important words like Thue-Morse and Sturmian are self-shuffling.

Self-shuffling words are morphic invariant and have applications in dynamical systems.

Abstract

In this paper we introduce and study a new property of infinite words: An infinite word $x\in A^\mathbb{N}$, with values in a finite set $A$, is said to be $k$-self-shuffling $(k\geq 2)$ if $x$ admits factorizations: $x=\prod_{i=0}^\infty U_i^{(1)}\cdots U_i^{(k)}=\prod_{i=0}^\infty U_i^{(1)}=\cdots =\prod_{i=0}^\infty U_i^{(k)}$. 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 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 and all Sturmian words of intercept $0<\rho <1$ (while those of intercept $\rho=0$ are not self-shuffling). 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 {\it stepping stone model}. One important feature of self-shuffling words stems from its morphic invariance, which provides a useful tool for showing that one word is not the morphic image of another. The notion of self-shuffling 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 theoretic result, originally due to Yasutomi, on a classification of pure morphic Sturmian words in the orbit of the characteristic.