A curious dynamical system in the plane

Abstract

For any irrational $\hat{I}\pm >0$ 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 $z_{n-1}+e^{2\ddot{I}€i\hat{I}\pm n}$ or $z_{n-1}-e^{2\ddot{I}€i\hat{I}\pm n}$, 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 $z_{n-1}\pm e^{2\ddot{I}€i\hat{I}\pm n}$ 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 $z_{-1},\hat{I}\pm$), the signs remain periodic for all time. The surprising complexity of the system is illustrated through examples.

19 pages, 9 figures