Construction 8.5.7.13 (Speeding Up). Let $f: [0,1] \rightarrow [0,1]$ be an orientation-preserving homeomorphism. We define $\alpha (f): [0,1] \rightarrow [0,1]$ by the formula

\[ \alpha (f)(x) = \begin{cases} f(2x) / 2 & \text{ if $0 \leq x \leq 1/2$} \\ x & \text{ if $1/2 \leq x \leq 1$. } \end{cases} \]

Then $\alpha (f)$ is also an orientation-preserving homeomorphism of $[0,1]$ with itself. Moreover, if $f$ is dyadic, then $\alpha (f)$ is also dyadic. It follows that the construction $f \mapsto \alpha (f)$ determines a group homomorphism $\alpha $ from the Thompson group $\operatorname{Aut}_{ \mathrm{Dy} }([0,1])$ to itself.