Notation 8.5.9.16. Let $n \geq 0$ be an integer and let $J$ be a subset of $\{ 1, 2, \cdots , n \} $. We let $b_{J}: [0,n] \xrightarrow {\sim } [0, n + |J| ]$ be the dyadic homeomorphism which is characterized by the following requirement: for every integer $1 \leq j \leq n$, the function $b_{I}$ is differentiable at every point $x \in (j-1, j)$, with derivative given by the formula
\[ b'_{J}(x) = \begin{cases} 1 & \text{if $j \notin J$} \\ 2 & \text{if $j \in J$.} \end{cases} \]
Note that, if this condition is satisfied, then the inverse homeomorphism $b_{J}^{-1}: [ 0, n + |J| ] \xrightarrow {\sim } [0,n]$ is a dyadic contraction. We say that a dyadic contraction is elementary if it has the form $b_{J}^{-1}$, for some integer $n \geq 0$ and some subset $J \subseteq \{ 1, 2, \cdots , n \} $.