In [Cube-free infinite binary words](https://mathoverflow.net/questions/61373) 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. 



Are there uncountably many cube-free infinite binary words?
Denote by $\mu$ the mapping from the Thue-Morse sequence, $\mu(0)=01$ and $\mu(1)=10$. Now define a sequence of maps from binary words to binary words, $g$, so that $g\_{\emptyset}(w)=w$, $g\_{0b}(w)=\mu^2(g\_{b}(w))$ and $g\_{1b}(w)=0\mu^2(g\_{b}(w))$. Now given an infinite binary sequence $B=b\_1b\_2\dots$ define $w\_{B}$ to be the limit of 
$$g\_{b\_1}(w),g\_{b\_1b\_2}(w),g\_{b\_1b\_2b\_3}(w),\dots$$
The $w\_B$ give you uncountably many $7/3$-power free words (so in particular, cube free) which moreover have infinitely many overlaps.


This stronger result is proved [here](http://arxiv.org/abs/math/0511425). I believe all known constructions of large families of such sequences are defined by iterative mappings.


