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Avoshifts, Unishifts and Nondeterministic Cellular Automata

Ville Salo

TL;DR

The paper investigates avoshifts and unishifts on $\mathbb{Z}^d$, defining a local-to-global validity property controlled by bounded subpatterns on convex sets. It develops an abstract construction framework of rules and manuals, then proves that every avoshift is, up to convex blocking and a linear change of basis, the two-sided spacetime subshift of a nondeterministic cellular automaton on a lower-dimensional avoshift. This central connection yields key corollaries: avoshifts contain periodic points, unishifts have dense periodic points and admit equal-entropy full-shift factors, and, through nondeterministic spacetime structure, concrete representations of classical examples like Ledrappier and the golden mean shift. The methods hinge on a novel notion of permissivity in manuals, enabling parallel handling of slanted halfspace extensions and a stepwise descent to lower dimensions. Overall, the results provide a structural bridge between multidimensional symbolic dynamics and nondeterministic CA, with implications for ergodicity, entropy, and dynamical representations of well-known subshifts.

Abstract

In this paper, we study avoshifts and unishifts on $\mathbb{Z}^d$. Avoshifts are subshifts where for each convex set $C$, and each vector $v$ such that $C \cup \{\vec v\}$ is also convex, the set of valid extensions of globally valid patterns on $C$ to ones on $C \cup \{v\}$ is determined by a bounded subpattern of $C$. Unishifts are the subshifts where for such $C, \vec v$, every $C$-pattern has the same number of $\vec v$-extensions. Cellwise quasigroup shifts (including group shifts) and TEP subshifts are examples of unishifts, while unishifts and subshifts with topological strong spatial mixing are examples of avoshifts. We prove that every avoshift is the spacetime subshift of a nondeterministic cellular automaton on an avoshift of lower dimension up to a linear transformation and a convex blocking. From this, we deduce that all avoshifts contain periodic points, and that unishifts have dense periodic points and admit equal entropy full shift factors.

Avoshifts, Unishifts and Nondeterministic Cellular Automata

TL;DR

The paper investigates avoshifts and unishifts on , defining a local-to-global validity property controlled by bounded subpatterns on convex sets. It develops an abstract construction framework of rules and manuals, then proves that every avoshift is, up to convex blocking and a linear change of basis, the two-sided spacetime subshift of a nondeterministic cellular automaton on a lower-dimensional avoshift. This central connection yields key corollaries: avoshifts contain periodic points, unishifts have dense periodic points and admit equal-entropy full-shift factors, and, through nondeterministic spacetime structure, concrete representations of classical examples like Ledrappier and the golden mean shift. The methods hinge on a novel notion of permissivity in manuals, enabling parallel handling of slanted halfspace extensions and a stepwise descent to lower dimensions. Overall, the results provide a structural bridge between multidimensional symbolic dynamics and nondeterministic CA, with implications for ergodicity, entropy, and dynamical representations of well-known subshifts.

Abstract

In this paper, we study avoshifts and unishifts on . Avoshifts are subshifts where for each convex set , and each vector such that is also convex, the set of valid extensions of globally valid patterns on to ones on is determined by a bounded subpattern of . Unishifts are the subshifts where for such , every -pattern has the same number of -extensions. Cellwise quasigroup shifts (including group shifts) and TEP subshifts are examples of unishifts, while unishifts and subshifts with topological strong spatial mixing are examples of avoshifts. We prove that every avoshift is the spacetime subshift of a nondeterministic cellular automaton on an avoshift of lower dimension up to a linear transformation and a convex blocking. From this, we deduce that all avoshifts contain periodic points, and that unishifts have dense periodic points and admit equal entropy full shift factors.

Paper Structure

This paper contains 23 sections, 38 theorems, 75 equations, 1 table.

Table of Contents

  1. Introduction
  2. Motivation and background
  3. New results
  4. Nondeterministic cellular automata representation for the examples
  5. The Ledrappier example
  6. The golden mean shift
  7. Definitions and basic facts
  8. Numbers
  9. Sets
  10. Symbolic dynamics
  11. Graphs
  12. Geometric conventions
  13. More geometry
  14. An abstract construction theory
  15. Rules and manuals
  16. ...and 8 more sections

Key Result

Theorem 1

Let $X \subset A^{\mathbb{Z}^d}$ be an avoshift in dimension $d \geq 2$. Then there is an avoshift $Z \subset B^{\mathbb{Z}^d}$ obtained from $X$ by a change of basis of $\mathbb{Z}^d$ and applying a convex block coding, such that $Z$ is the two-sided spacetime subshift of a nondeterministic cellula

Theorems & Definitions (80)

  • Example 1
  • Example 2
  • Theorem 1
  • Theorem 2
  • Theorem 3
  • Theorem 4
  • Theorem 5
  • Lemma 1
  • Lemma 2
  • Lemma 3
  • ...and 70 more