A curious dynamical system in the plane
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arXiv
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| Main Authors: | , |
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| Format: | Preprint |
| Published: |
2024
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| _version_ | 1866909386642817024 |
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| author | Steinerberger, Stefan Zeng, Tony |
| author_facet | Steinerberger, Stefan Zeng, Tony |
| contents | 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. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2409_08961 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | A curious dynamical system in the plane Steinerberger, Stefan Zeng, Tony Dynamical Systems 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. |
| title | A curious dynamical system in the plane |
| topic | Dynamical Systems |
| url | https://arxiv.org/abs/2409.08961 |