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On Countable SFT Covers of Sparse Multidimensional Shift Spaces

Ilkka Törmä

TL;DR

The paper investigates when countable two-dimensional sofic shifts admit countable SFT covers, focusing on sparse and gap-width shifts. It develops a modular construction toolbox to implement computations and geometric constraints inside countable SFTs, enabling embeddings of lifts and the simulation of tile-sets within countable frameworks. The main contributions show that one-dimensional gap width shifts with computable gap functions have lift-countable covers characterized by countable-cover conditions, and that a large class of two-dimensional gap width shifts with upper semicomputable gap functions bounded by $2^{\sqrt{n}}$ are countably covered. The results reveal subtle differences between one- and two-dimensional settings and provide tools potentially applicable to broader countable-to-one cover problems in symbolic dynamics.

Abstract

A multidimensional sofic shift is called countably covered if it has an SFT cover containing only countably many configurations. In contrast to the one-dimensional setting, not all countable sofic shifts are countably covered. We investigate the existence of countable covers for gap width shifts, where the number of nonzero symbols in a configuration is bounded by a function of the minimum distance between two such symbols. As our main results, we characterize those one-dimensional gap width shifts whose two-dimensional lift is a countably covered sofic shift, and show that a large class of two-dimensional gap width shifts are countably covered.

On Countable SFT Covers of Sparse Multidimensional Shift Spaces

TL;DR

The paper investigates when countable two-dimensional sofic shifts admit countable SFT covers, focusing on sparse and gap-width shifts. It develops a modular construction toolbox to implement computations and geometric constraints inside countable SFTs, enabling embeddings of lifts and the simulation of tile-sets within countable frameworks. The main contributions show that one-dimensional gap width shifts with computable gap functions have lift-countable covers characterized by countable-cover conditions, and that a large class of two-dimensional gap width shifts with upper semicomputable gap functions bounded by are countably covered. The results reveal subtle differences between one- and two-dimensional settings and provide tools potentially applicable to broader countable-to-one cover problems in symbolic dynamics.

Abstract

A multidimensional sofic shift is called countably covered if it has an SFT cover containing only countably many configurations. In contrast to the one-dimensional setting, not all countable sofic shifts are countably covered. We investigate the existence of countable covers for gap width shifts, where the number of nonzero symbols in a configuration is bounded by a function of the minimum distance between two such symbols. As our main results, we characterize those one-dimensional gap width shifts whose two-dimensional lift is a countably covered sofic shift, and show that a large class of two-dimensional gap width shifts are countably covered.
Paper Structure (31 sections, 12 theorems, 2 equations, 9 figures)

This paper contains 31 sections, 12 theorems, 2 equations, 9 figures.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Definitions
  4. Previous work
  5. Gap width shifts
  6. Definition and main results
  7. Gap functions
  8. The two-dimensional case
  9. Construction toolbox
  10. Proof of Theorem \ref{['thm:main']}
  11. Embedding $X^\dag$ in a countably covered $G(g)^\dag$
  12. Correctness of the embedding
  13. The simulation layer
  14. The simulated tile sets $T_n$
  15. The layers $L_{n,a,b}$ of the simulated tilings
  16. ...and 16 more sections

Key Result

Theorem 1

The two-dimensional lift $X^\dag$ of a $\mathbb{Z}$-shift space $X$ is sofic if and only if $X$ is effectively closed.

Figures (9)

  • Figure 1: A sample configuration of the grid shift $X_{\mathrm{grid}}$. Dotted lines are tile borders, along which colors must match.
  • Figure 2: The two layers of the alphabet $A_{\mathrm{count}}$ of the SFT $X_{\mathrm{count}}$: (a) the counting layer and (b) the synchronization layer.
  • Figure 3: A sample configuration of $X_{\mathrm{count}}$. Depicted are the grid lines of the grid layer, the counting layer, and the synchronization layer. The two arrows mark nonzero columns of the $A$-layer.
  • Figure 4: The tile set $A_{\mathrm{probe}}$ of the probe implementation layer. Here $b$ ranges over $B$, $b_0$ ranges over $\{0,1,2\} \cup (A \times \{0\}) \subset B$, $a_*$ ranges over $A \times \{0,1\}$, and $a_i$ for $i \in \{0,1\}$ ranges over $A \times \{i\}$.
  • Figure 5: A configuration over $A_{\mathrm{probe}}$, superimposed over a configuration of $X_{\mathrm{grid}}$. Each vertical segment is labeled with the type of symbol it carries (each tile of the segment carries the same symbol); $a_i$ stands for $A \times \{i\}$.
  • ...and 4 more figures

Theorems & Definitions (28)

  • Theorem 1: DuRoSh12AuSa13
  • Definition 2
  • Theorem 3: To20
  • Definition 4
  • Proposition 5
  • proof
  • Theorem 6
  • Corollary 7
  • proof
  • Theorem 8
  • ...and 18 more