Gespeichert in:
| Hauptverfasser: | , |
|---|---|
| Format: | Preprint |
| Veröffentlicht: |
2025
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| Schlagworte: | |
| Online-Zugang: | https://arxiv.org/abs/2502.03345 |
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Inhaltsangabe:
- Let $D$ be an endomorphism on $\mathbb{Z}_m^n$ so that \[D(x_1, x_2, ..., x_n)=(x_1+x_2 \; \text{mod} \; m, x_2+x_3 \; \text{mod} \; m, ..., x_n+x_1 \; \text{mod} \; m).\] We call the sequence $\{D^α(\mathbf{u})\}_{α=0}^{\infty}$ the Ducci sequence of $\mathbf{u} \in \mathbb{Z}_m^n$, which always enters a cycle. Now let $H$ be an endomorphism on $\mathbb{Z}_m^n$ such that \[H(x_1, x_2, ..., x_n)=(x_2, x_3, ..., x_n, x_1).\] In this paper, we will talk about a few cases when $\mathbf{u}$ and $H^β(\mathbf{u})$ have the same Ducci cycle for $β> 0$, as well as prove a few cases of $n,m$ where this is guaranteed for every $\mathbf{u} \in \mathbb{Z}_m^n$.