I have found plenty of questions regarding this section of the book but none regarding my specific query. This is what the book describes as a dyadic expansion:
For a mapping $F$ from $\Omega=(0,1]$ into itself given by $$F\omega=\begin{Bmatrix} 2\omega & \mbox{if } 0< \omega\leqslant\frac{1}{2} \\ 2\omega-1 & \mbox{if } \frac{1}{2}< \omega\leqslant1 \end{Bmatrix} $$ and $$d_1(\omega)=\begin{Bmatrix} 0 & \mbox{if } 0< \omega\leqslant\frac{1}{2} \\ 1 & \mbox{if } \frac{1}{2}< \omega\leqslant1 \end{Bmatrix} $$ and $d_i(\omega)=d_1(F^{i-1}\omega)$.
My question is, how is the exponent on $F$ to be interpreted? The way I read it, $d_i$ with $i = 1$ would mean that $d_1 = d_1(F^{0}\omega) = d_1(1)$ which obviously contradicts the definition just given above. What am I missing here?
(lifted the nice notation from this question)
It's a function composition.
$F^0(\omega)=\omega$ and $F^{i+1}(\omega) = F\circ F^i(\omega)$.