2026/07/29 by Akane Kawaharada
Mathematics · #math.DS #msc:11B50 #msc:11B85 #msc:28A80 #msc:37B15 #msc:68Q80
16 pages, 5 figures
arxiv created 2026/07/29 · arxiv updated 2026/07/30
This paper investigates the classification of spatio-temporal patterns generated by linear cellular automata with uniform weights (LCA-UW) over the ring \mathbb Z / n \mathbb Z. While these systems are governed by the state size n and a transition coefficient c, their combined influence produces a vast array of patterns that are difficult to organize through exhaustive observation. We introduce a binary projection operator B to focus on the fundamental structural evolution (infinite binary orbits) of these automata. Our main result demonstrates a fundamental reduction principle. For any coefficient c that shares prime factors with n, the generated infinite binary orbit eventually coincides with the orbit of an LCA-UW with some reduced state size and a unit coefficient c=1. We prove that for a fixed n, there exist exactly 2m - 1 distinct types of binary orbits, where m is the number of distinct prime factors of n. This theorem effectively collapses the two-dimensional parameter space (n, c) into a one-dimensional search over n, providing a streamlined framework for the topological and fractal classification of LCA-UW dynamics.