No weakly factor-universal cellular automaton

TL;DR

This paper proves that no cellular automaton can weakly factor onto all others, establishing a divisibility obstruction related to periodic points and clock automata, thus answering Hochman's question negatively.

Contribution

It introduces a divisibility criterion for cellular automata to weakly factor onto clock automata, showing such universal factors do not exist.

Findings

No cellular automaton is a universal factor for all others.

Weak factors onto clock automata impose divisibility conditions on cycle lengths.

The action on constant configurations reveals explicit divisibility obstructions.

Abstract

Hochman asked whether there exists a cellular automaton $F$ such that every cellular automaton is a factor of $F$ in the dynamical sense. In particular, we do not require the factor map to commute with the spatial shifts. We show that no such cellular automaton exists. More generally, if $F$ weakly factors onto the radius-zero $q$-clock automaton $C_q^{(k)}$, then every periodic point of $F$ has period divisible by $q$. For a cellular automaton $F:A^{\mathbb Z^d}\to A^{\mathbb Z^d}$, define $\varphi_F:A\to A$ by $F(\underline a)=\underline{\varphi_F(a)}$, and let $g_F$ be the greatest common divisor of the cycle lengths of $\varphi_F$. We prove that if $C_q^{(k)}$ is a weak factor of $F$, then $q\mid g_F$ holds. It follows that the action of $F$ on constant configurations yields an explicit divisibility obstruction to clock weak factors.