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

TL;DR

This paper investigates the behavior of Ducci sequences generated by a specific endomorphism on modular integer vectors, examining conditions under which these sequences share cycles when transformed by a cyclic permutation.

Contribution

It characterizes cases where vectors and their cyclic permutations have identical Ducci cycles, providing proofs for certain dimensions and moduli.

Findings

Identifies conditions for shared Ducci cycles under permutation.

Proves specific cases where cycle sharing is guaranteed.

Analyzes the structure of Ducci sequences on modular vector spaces.

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