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No weakly factor-universal cellular automaton

Maja Gwozdz

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.

No weakly factor-universal cellular automaton

Abstract

Hochman asked whether there exists a cellular automaton such that every cellular automaton is a factor of 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 weakly factors onto the radius-zero -clock automaton , then every periodic point of has period divisible by . For a cellular automaton , define by , and let be the greatest common divisor of the cycle lengths of . We prove that if is a weak factor of , then holds. It follows that the action of on constant configurations yields an explicit divisibility obstruction to clock weak factors.
Paper Structure (2 sections, 4 theorems, 21 equations)

This paper contains 2 sections, 4 theorems, 21 equations.

Table of Contents

  1. Introduction
  2. Obstructions

Key Result

Theorem 1.1

Let $q\ge2$, $k\ge1$, and let $F:A^{\mathbb Z^d}\to A^{\mathbb Z^d}$ be a cellular automaton. Define $\varphi_F:A\to A$ by We use $\underline a$ to denote the constant configuration with value $a$. Let $m_1,\dots,m_r$ be the cycle lengths of $\varphi_F$, and set If there exists a continuous map with then

Theorems & Definitions (9)

  • Theorem 1.1
  • Proposition 2.1
  • proof
  • Corollary 2.2
  • proof
  • proof : Proof of Theorem \ref{['thm:main']}
  • Corollary 2.3
  • proof
  • Remark 2.4