We now turn our attention to the dynamical behavior of \(f\) on the unit circle and we'll see that it's as complicated as it can be. Let's denote the unit circle by \(J\text{.}\) Our first order of business is to show that \(f\) displays sensitive dependence on \(J\text{,}\) so let \(z_0\in J\) and let \(D\) be an open disk centered at \(z_0\text{.}\) Then, no matter how small the radius of \(D\text{,}\) it will always contain points that are in the interior of \(J\) and points in the exterior of \(J\text{.}\) Thus, there are points arbitrarily close to \(z_0\) whose orbits converge to zero and other points arbitrarily close to \(z_0\) whose orbits diverge to \(\infty\text{.}\) That's pretty sensitive!

What about the behavior of \(f\) right on \(J\text{?}\) As it turns out, \(f\) is conjugate to the doubling map \(d\) and, therefore, chaotic on \(J\text{.}\) Recall that \(d\) is defined on \(H=\{x:0\leq x \lt 1\}\) by

In fact, it's trivial to show that a conjugacy from \(f\) to \(d\) is given by the natural parametrization of the unit circle \(x \to e^{2\pi ix}\text{.}\) This is one of those wonderful moments in mathematics where a lot of prior work comes together to yield an important result quite easily!