I'mA word $y$ is a subword of $w$ if there exist words $x$ and $z$ (possibly empty) such that $w=xyz$. Thus, $01$ is a subword of $0110$, but $00$ is not a subword of $0110$. I'm interested in right-infinite words over a two-letter alphabet that do not contain subwords of the form $xxx$, where $x$ is a word of one or more letters. (For example, the Thue-Morse word, the Kolakoski word, Stewart's choral sequence, and so on.) In particular, I would like to know if there are any general statements about all such words. For example, are there an infinite number of them? Is there any way to classify them?
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From the comments it seemsIt's possible that I needone way to clarify the terms I useclassify cube-free infinite binary words is to group them according to their subwords. A For example, the Kolakoski word $y$ is ahas the subword of $w$ if there exist$00100$ whereas the Thue-Morse word does not, so they belong in different classes. The words $x$ and $z$created in Tony's answer (possibly emptysee below) such thathave the same subwords $w=xyz$. Thus(the Thue-Morse word is recurrent), $01$so they belong to one class. I suppose there an infinite number of these classes.
Another possible way to classify cfib words is to group them according to their subword complexity. For example, Stewart's choral sequence has a subword complexity of $0110$, but$2n$ $00$(where $n$ is not a subwordthe length of the subword), so we can group it with other cfib words with subword complexity $0110$$2n$. A word is cube-free if it does not contain subwordsIs the subword complexity of the form $xxx$ where $x$ is aKolakoski word of one or more letters.known?