On Nilpotency and Asymptotic Nilpotency of Cellular Automata

TL;DR

This paper proves that cellular automata which eventually stabilize all cells to zero are nilpotent across all dimensions, confirming a conjecture and exploring related properties on various subshifts.

Contribution

It confirms a conjecture that eventually fixing all cells to zero implies nilpotency in cellular automata across all dimensions and subshifts.

Findings

Proves that eventual fixation to zero implies nilpotency in all dimensions.

Shows that weak nilpotency implies nilpotency in all subshifts.

Discusses nilpotency properties on different subshifts.

Abstract

We prove a conjecture of P. Guillon and G. Richard by showing that cellular automata that eventually fix all cells to a fixed symbol 0 are nilpotent on S^Z^d for all d. We also briefly discuss nilpotency on other subshifts, and show that weak nilpotency implies nilpotency in all subshifts and all dimensions, since we do not know a published reference for this.