No weakly factor-universal cellular automaton
This resolves a foundational problem in dynamical systems theory, specifically in cellular automata, by providing a negative answer to a key universality question.
The paper addresses Hochman's question on whether a cellular automaton exists that can factor onto every other cellular automaton, showing that no such automaton exists by proving a divisibility obstruction related to periodic points and clock automata.
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 $Ï_F:A\to A$ by $F(\underline a)=\underline{Ï_F(a)}$, and let $g_F$ be the greatest common divisor of the cycle lengths of $Ï_F$. 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.