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On injective endomorphisms of symbolic schemes

Tullio Ceccherini-Silberstein, Michel Coornaert, Xuan Kien Phung

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 $τ\colon A^{\mathbb Z} \to A^{\mathbb Z}$ whose inverse is not algebraic.

On injective endomorphisms of symbolic schemes

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 whose inverse is not algebraic.

Paper Structure

This paper contains 28 sections, 35 theorems, 66 equations.

Table of Contents

  1. Introduction
  2. Cellular automata
  3. The prodiscrete topology
  4. The closed image property for cellular automata
  5. Restriction of a cellular automaton to a subgroup
  6. Periodic configurations
  7. Cellular automata over schemes
  8. Change of memory set
  9. The monoid $\mathop{\mathrm{CA}}\nolimits(G,S,X,Y)$
  10. The closed image property
  11. Surjunctivity
  12. Reversibility
  13. Invertibility
  14. Counter-examples
  15. Schemes and algebraic varieties
  16. ...and 13 more sections

Key Result

Theorem 1.2

Let $G$ be a locally residually finite group and let $X$ be an algebraic variety over an algebraically closed field $K$. Let $A \coloneqq X(K)$ denote the set of $K$-points of $X$. Suppose that the variety $X$ is complete or that the field $K$ is uncountable. Then every injective cellular automaton

Theorems & Definitions (85)

  • Definition 1.1
  • Theorem 1.2
  • Theorem 1.3
  • Theorem 1.4
  • Corollary 1.5
  • Lemma 2.1
  • proof
  • Lemma 2.2
  • proof
  • Lemma 2.3
  • ...and 75 more