Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

Local generation of tilings

Tom Favereau, Mathieu Hoyrup

TL;DR

The paper investigates when tilings and subshifts can be generated locally, introducing two frameworks, L^0 (structure-based) and L^1 (input-window/narrow-based). It develops obstructions and tools, proves one-dimensional SFTs lie in L^0, and shows ramified two-dimensional subshifts (e.g., triangles, domino tilings) are not in L^0, while L^1 strictly contains L^0. By exploring weak factors, higher-power presentations, and Wang tilesets, the work maps which subshifts admit locally generated representations and identifies large gaps between the two notions. The results have implications for understanding locality in tiling generation and suggest directions for broader locality notions and classifications of tiling systems.

Abstract

In this article, we investigate the possibility of generating all the configurations of a subshift in a local way. We propose two definitions of local generation, explore their properties and develop techniques to determine whether a subshift satisfies these definitions. We illustrate the results with several examples.

Local generation of tilings

TL;DR

The paper investigates when tilings and subshifts can be generated locally, introducing two frameworks, L^0 (structure-based) and L^1 (input-window/narrow-based). It develops obstructions and tools, proves one-dimensional SFTs lie in L^0, and shows ramified two-dimensional subshifts (e.g., triangles, domino tilings) are not in L^0, while L^1 strictly contains L^0. By exploring weak factors, higher-power presentations, and Wang tilesets, the work maps which subshifts admit locally generated representations and identifies large gaps between the two notions. The results have implications for understanding locality in tiling generation and suggest directions for broader locality notions and classifications of tiling systems.

Abstract

In this article, we investigate the possibility of generating all the configurations of a subshift in a local way. We propose two definitions of local generation, explore their properties and develop techniques to determine whether a subshift satisfies these definitions. We illustrate the results with several examples.

Paper Structure

This paper contains 45 sections, 50 theorems, 22 equations, 12 figures, 2 tables.

Table of Contents

  1. Introduction
  2. Local generation
  3. Preliminaries
  4. The class C0
  5. The class nar
  6. Relevance of the definition
  7. Symbolic dynamics
  8. Definitions
  9. Dynamical properties
  10. Weak factors and weak conjugacies
  11. Properties of the classes L0,L2
  12. Closure by weak factors
  13. Weak mixing and countable unions
  14. Compact classes of functions
  15. Applications to local and narrow functions
  16. ...and 30 more sections

Key Result

Proposition 2.1

One has $X\in\mathscr{L}^0$ if and only if $X$ is a factor of a finite product of basic subshifts. Moreover, we can assume that this finite product is where $K$ is countable, $A$ is finite and $H_1,\ldots,H_k$ are non-trivial subgroups of $\mathbb{Z}^d$ of ranks at most $d-1$ (and possibly $k=0$).

Figures (12)

  • Figure 1: A tileset admitting a local generation procedure
  • Figure 2: The triangle $T_4$, illustrating that $X$ is not strongly irreducible: the left column and the right cell are not independent.□
  • Figure 3: A ramification: the cells $\lambda v$ are in green, the cells $\lambda v+\mu u$, $\mu>0$, are in red. In a black triangle, the purple and green regions are correlated, and the content of a red cell cannot be grafted in a green cell without changing the purple region.
  • Figure 4: The domino tileset
  • Figure 5: The domino subshift is not strongly irreducible: the purple pattern determines the content of the green cell.
  • ...and 7 more figures

Theorems & Definitions (115)

  • Definition 2.1: The class $\mathscr{L}^0$
  • Proposition 2.1: Canonical form
  • proof
  • Proposition 2.2
  • proof
  • Example 2.1: Cellular automaton
  • Definition 2.2: Narrow function
  • Example 2.2: Non-uniform cellular automaton
  • Definition 2.3: The class $\mathscr{L}^1$
  • Proposition 2.3
  • ...and 105 more