By considering the binary representation of the sawtooth $\operatorname{map}, F(x)=2 x[\bmod 1]$ for $x \in[0,1)$, show that:

(i) $F$ has sensitive dependence on initial conditions on $[0,1)$.

(ii) $F$ has topological transitivity on $[0,1)$.

(iii) Periodic points are dense in $[0,1)$.

Find all the 4-cycles of $F$ and express them as fractions.