Examining $H$-Closed Ducci Sequences on ${\mathbb{Z}}_m^n$

Abstract

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 $\{D^{\hat{I}}\pm (\mathbf{u}){\}}_{\hat{I}\pm =0}^{\infty }$ 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 $H^2(\mathbf{u})$ have the same Ducci cycle for ${\hat{I}}^2>0$, as well as prove a few cases of $n,m$ where this is guaranteed for every $\mathbf{u}\in {\mathbb{Z}}_m^n$.