Checkpoint 3.4.3
Let \(H=[0,1)\) denote the half-open/half-closed unit interval and let \(d:H\to H\) denote the doubling map
\begin{equation*}
d(x) = 2x \mod 1.
\end{equation*}
Show that \(\varpi(x) = e^{2\pi i x}\) defines a conjugacy between \(d\) and the complex function \(f_0(z)=z^2\) restricted to the unit circle.