A 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](http://oeis.org/A010060), the [Kolakoski word](http://oeis.org/A000002), [Stewart's choral sequence](http://oeis.org/A116178), 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?


It's possible that one way to classify cube-free infinite binary words is to group them according to their subwords. For example, the Kolakoski word has the subword $00100$ whereas the Thue-Morse word does not, so they belong in different classes. The words created in Tony's answer (see below) have the same subwords (the Thue-Morse word is recurrent), 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 $2n$ (where $n$ is the length of the subword), so we can group it with other cfib words with subword complexity $2n$. Is the subword complexity of the Kolakoski word known?



Cube-free infinite binary words
Here are some deep facts relating to binary cfw's:


1) The set of right infinite binary cube-free words is a perfect set in the topological sense: For any given such sequence, there is a distinct one which agrees with it to the nth letter. In particular, there are uncountably many binary cfw's. 


2) Given any finite binary sequence, it is decidable whether it extends to an infinite binary cube-free word.


3) The number of (finite) binary cfw's of length n grows exponentially with n.


These results (and analogous ones for k-power free words over various alphabets) are proved in


<http://dl.acm.org/citation.cfm?id=873885> and <http://www.sciencedirect.com/science/article/pii/0195669895900519>


