Year: 2024
Type: preprint
Abstract: For any irrational $α> 0$ 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 $z_{n-1} + e^{2 πi αn}$ or $z_{n-1} - e^{2 πi αn}$, 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 $z_{n-1} \pm e^{2 πi αn}$ 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 $z_{-1}, α$), the signs remain periodic for all time. The surprising complexity of the system is illustrated through examples.
Source: arXiv (Cornell University)
Authors: Steinerberger, Stefan, Zeng, Tony
Institution: University of Washington
Language: English
Subfield: Mathematical Physics
Field: Mathematics
Domain: Physical Sciences
Open Access status: green
OpenAlex ID: W4403662274