On injective endomorphisms of symbolic schemes

TL;DR

This paper generalizes cellular automata over schemes, extending known results and establishing new properties like surjunctivity and reversibility, while also providing a negative answer to a previously open question.

Contribution

It introduces cellular automata over schemes, broadening the algebraic framework and extending results on properties like invertibility and surjunctivity.

Findings

Extended cellular automata to schemes setting.

Proved new results on closed image property and reversibility.

Provided a negative answer to an open question on algebraic bijective cellular automata.

Abstract

Building on the seminal work of Gromov on endomorphisms of symbolic algebraic varieties [10], we introduce a notion of cellular automata over schemes which generalize affine algebraic cellular automata in [7]. We extend known results to this more general setting. We also establish several new ones regarding the closed image property, surjunctivity, reversibility, and invertibility for cellular automata over algebraic varieties with coefficients in an algebraically closed field. As a byproduct, we obtain a negative answer to a question raised in [7] on the existence of a bijective complex affine algebraic cellular automaton $\tau \colon A^{\mathbb Z} \to A^{\mathbb Z}$ whose inverse is not algebraic.