This is a “hint” for exercise 3.4. An infinite 2-free 3-sequence is sometimes called a Thue sequence, since the number theorist Axel Thue was the first to study them. There are several ways of generating Thue sequences. I mention three:
- One could define a map
as in the case of 3-free 2-sequences. Now set
and
and once again consider the iterates
- Thue’s original example was
and
- Another approach consists on taking the transformation
and
Now define
for
to be the string obtained from
by counting ones between consecutive zeros. For example,
so
while
so
Check that each
where
and
and conclude from this that the strings
If you need extra time, you have until Friday, September 11, to work on this question.
A kind of library 