Remark 8.5.9.17. Let $f: [0,m] \xrightarrow {\sim } [0,n]$ be a dyadic contraction. The following conditions are equivalent:
The dyadic contraction $f$ is elementary, in the sense of Notation 8.5.9.16.
For every point $x \in [0,m]$ where $f$ is differentiable, the derivative $f'(x)$ is either $1$ or $1/2$.
The dyadic contraction $f$ can be written as a concatenation $f_{1} \circledast f_2 \circledast \cdots \circledast f_ n$, where each $f_{i}$ is either the identity function $\operatorname{id}: [0,1] \xrightarrow {\sim } [0,1]$ or the homeomorphism
\[ H: [0,2] \xrightarrow {\sim } [0,1] \quad \quad x \mapsto x/2. \]