Solution
= Solution
Concatenate every finite binary word, ordered first by length and then lexicographically:
$$
x=0.\,0\,1\,00\,01\,10\,11\,000\,001\,010\,011\cdots{}_2.
$$
Every finite binary block occurs in this expansion. Since iterating $F$ shifts digits to the left, its orbit enters every binary cylinder and is therefore dense. The digit string is not eventually periodic, so it gives a concrete nonperiodic chaotic orbit.