On the Garden of Eden theorem for endomorphisms of symbolic algebraic varieties

TL;DR

This paper extends the Garden of Eden theorem to algebraic cellular automata over amenable groups and algebraic varieties, establishing a surjectivity characterization via a new pre-injectivity notion.

Contribution

It introduces a weak pre-injectivity condition for algebraic cellular automata and proves its equivalence to surjectivity, answering a question posed by Gromov.

Findings

Proves that algebraic cellular automata are surjective iff they are (*)-pre-injective.

Establishes the Myhill property for algebraic cellular automata.

Provides an algebraic analogue of the classical Garden of Eden theorem.

Abstract

Let $G$ be an amenable group and let $X$ be an irreducible complete algebraic variety over an algebraically closed field $K$. Let $A$ denote the set of $K$-points of $X$ and let $\tau \colon A^G \to A^G$ be an algebraic cellular automaton over $(G,X,K)$, that is, a cellular automaton over the group $G$ and the alphabet $A$ whose local defining map is induced by a morphism of $K$-algebraic varieties. We introduce a weak notion of pre-injectivity for algebraic cellular automata, namely $(*)$-pre-injectivity, and prove that $\tau$ is surjective if and only if it is $(*)$-pre-injective. In particular, $\tau$ has the Myhill property, i.e., is surjective whenever it is pre-injective. Our result gives a positive answer to a question raised by Gromov in~\cite{gromov-esav} and yields an analogue of the classical Moore-Myhill Garden of Eden theorem.