TITLE: Are there uncountably many cube-free infinite binary words?
QUESTION [21 upvotes]: In Cube-free infinite binary words it was established that there are infinitely many cube-free infinite binary words (see the earlier question for definitions of terms). The construction given in answer to that question yields a countable infinity of such words. In a comment on that answer, I raised the question of whether there is an uncountable infinity of such words. My comment has not generated any response; perhaps it will attract more interest as a question. 
I should admit that I ask out of idle curiosity, and have no research interest in the answer; it just seems like the logical question to ask once you know some set is infinite.

REPLY [7 votes]: I just saw and answered the earlier question, but perhaps I should repeat my post:
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