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Shifts of Finite Type Obtained by Forbidding a Single Pattern

Nishant Chandgotia, Brian Marcus, Jacob Richey, Chengyu Wu

TL;DR

This work studies shifts of finite type obtained by forbidding a single word, connecting combinatorial autocorrelation data with symbolic-dynamical invariants. It develops a finite-state, edge-labeled graph L_w that encodes the subshift X_{\{w\}} and proves that, up to alphabet permutation, w is recoverable from the unlabeled graph; PF theory and zeta/entropy data then compare different X_{\{w\}} via explicit comparators under a natural edge-ordering condition D(w,w'). The paper extends these ideas to the golden-mean ambient shift, provides detailed conjugacy results for one-dimensional shifts (full shift and golden mean) with precise chain-length bounds for swap-conjugacies, and begins a higher-dimensional program by introducing agreement sets and replacement principles. Overall, it links correlation polynomials, entropy, hitting times, and graph representations to illuminate when two single-pattern-forbidden SFTs are conjugate or ordered in complexity, while highlighting substantial open problems, especially in higher dimensions.

Abstract

Given a finite word $w$, Guibas and Odlyzko (J. Combin. Theory Ser. A, 30, 1981, 183-208) showed that the autocorrelation polynomial $φ_w(t)$ of $w$, which records the set of self-overlaps of $w$, explicitly determines for each $n$, the number $|B_n(w)|$ of words of length $n$ that avoid $w$. We consider this and related problems from the viewpoint of symbolic dynamics, focusing on the setting of $X_{\{w\}}$, the space of all bi-infinite sequences that avoid $w$. We first summarize and elaborate upon (J. Combin. Theory Ser. A, 30, 1981, 183-208) and other work to show that the sequence $|B_n(w)|$ is equivalent to several invariants of $X_{\{w\}}$. We then give a finite-state labeled graphical representation $L_w$ of $X_{\{w\}}$ and show that $w$ can be recovered from the graph isomorphism class of the unlabeled version of $L_w$. Using $L_w$, we apply ideas from probability and Perron-Frobenius theory to obtain results comparing features of $X_{\{w\}}$ for different $w$. Next, we give partial results on the problem of classifying the spaces $X_{\{w\}}$ up to conjugacy. Finally, we extend some of our results to spaces of multi-dimensional arrays that avoid a given finite pattern.

Shifts of Finite Type Obtained by Forbidding a Single Pattern

TL;DR

This work studies shifts of finite type obtained by forbidding a single word, connecting combinatorial autocorrelation data with symbolic-dynamical invariants. It develops a finite-state, edge-labeled graph L_w that encodes the subshift X_{\{w\}} and proves that, up to alphabet permutation, w is recoverable from the unlabeled graph; PF theory and zeta/entropy data then compare different X_{\{w\}} via explicit comparators under a natural edge-ordering condition D(w,w'). The paper extends these ideas to the golden-mean ambient shift, provides detailed conjugacy results for one-dimensional shifts (full shift and golden mean) with precise chain-length bounds for swap-conjugacies, and begins a higher-dimensional program by introducing agreement sets and replacement principles. Overall, it links correlation polynomials, entropy, hitting times, and graph representations to illuminate when two single-pattern-forbidden SFTs are conjugate or ordered in complexity, while highlighting substantial open problems, especially in higher dimensions.

Abstract

Given a finite word , Guibas and Odlyzko (J. Combin. Theory Ser. A, 30, 1981, 183-208) showed that the autocorrelation polynomial of , which records the set of self-overlaps of , explicitly determines for each , the number of words of length that avoid . We consider this and related problems from the viewpoint of symbolic dynamics, focusing on the setting of , the space of all bi-infinite sequences that avoid . We first summarize and elaborate upon (J. Combin. Theory Ser. A, 30, 1981, 183-208) and other work to show that the sequence is equivalent to several invariants of . We then give a finite-state labeled graphical representation of and show that can be recovered from the graph isomorphism class of the unlabeled version of . Using , we apply ideas from probability and Perron-Frobenius theory to obtain results comparing features of for different . Next, we give partial results on the problem of classifying the spaces up to conjugacy. Finally, we extend some of our results to spaces of multi-dimensional arrays that avoid a given finite pattern.
Paper Structure (15 sections, 27 theorems, 83 equations, 6 figures)

This paper contains 15 sections, 27 theorems, 83 equations, 6 figures.

Table of Contents

  1. Introduction
  2. Earlier results in one dimension
  3. Graphical representation for 1D SFTs with one forbidden word
  4. Ambient shift is the full shift
  5. The ambient shift is the golden mean shift
  6. Conjugacy of one-dimensional SFTs obtained by forbidding one word
  7. The ambient shift is the full shift
  8. The ambient shift is the golden mean shift
  9. Higher dimensional shift spaces
  10. Discussion
  11. The conjugacy problem for shift spaces with one forbidden word
  12. From follower set graph to correlation polynomial
  13. When the ambient shift space is an SFT with memory one
  14. Agreement sets in higher dimensions
  15. Higher dimensional SFTs

Key Result

Proposition 2.1

Let $u, v$ be two strings of length $k$ over the $q$-ary alphabet ($q\geq 2$). Then the following are equivalent:

Figures (6)

  • Figure 1: The graph $L_{21201;3}$ for the word $w = 21201$ over the alphabet $[3] = \{0,1,2\}$. Edges labeled $0, 1, 2$ are colored in red, blue, and green, respectively.
  • Figure 2: All six cases
  • Figure 3: "Base graphs" $S_1, S_2, S_3, S_4$
  • Figure 4: If $\mathcal{L}(I_1)=j+1, \mathcal{L}(I_2)=j, \mathcal{L}(I_3)=l$ for $S_1$, where $l<j$.
  • Figure 5: An illustration of why the in-degree of the vertex $0$ is $0$ for Case 5. Here, the red edge is invalid.
  • ...and 1 more figures

Theorems & Definitions (66)

  • Proposition 2.1
  • Proposition 2.2
  • Definition 3.1
  • Definition 3.2
  • proof
  • Proposition 3.4
  • proof
  • Proposition 3.5
  • proof
  • Remark 3.6
  • ...and 56 more