Let $\mu(n)$ the Möbius function, we define $F:[0,1]\to[0,1]$ as $$F(x)=\sum_{n=1}^\infty\frac{\mu(n)}{n}x^n.\tag{1}$$

For a function of this kind (I presume that this continuous function has image $[0,1]$) was defined, for example in last paragraph of page 986, what is a periodic point, and its corresponding order of periodicity.

Question.Can you find examples of periodic points of $(1)$? Is it possible or feasible to prove or refute that our function has periodic points of order $3$?Many thanks.

Since I think that these questions are very difficult feel free to add comments about your thoughts and reasonings, heuristics or conjectures. I believe that this question concerning the function $(1)$ isn't in the literature.

## References:

[1] Tien-Yien Li, James A. Yorke, *Period Three Implies Chaos*, The American Mathematical Monthly, Vol. 82, No. 10 (1975).