A curious dynamical system in the plane

Abstract

For any irrational $\alpha > 0$ and any initial value $z_{-1} \in \mathbb{C}$, we define a sequence of complex numbers $(z_n)_{n=0}^{\infty}$ as follows: $z_n$ is $z_{n-1} + e^{2 \pi i \alpha n}$ or $z_{n-1} - e^{2 \pi i \alpha n}$, whichever has the smaller absolute value. If both numbers have the same absolute value, the sequence terminates at $z_{n-1}$ but this happens rarely. This dynamical system has astonishingly intricate behavior: the choice of signs in $z_{n-1} \pm e^{2 \pi i \alpha n}$ appears to eventually become periodic (though the period can be large). We prove that if one observes periodic signs for a sufficiently long time (depending on $z_{-1}, \alpha$), the signs remain periodic for all time. The surprising complexity of the system is illustrated through examples.