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Encoding subshifts through sliding block codes

Sophie MacDonald

TL;DR

This work extends Krieger's zero-ambiguity embedding theory to the setting of a mixing SFT $X$, a mixing sofic shift $Y$, and a factor map $\pi:X\to Y$. The authors give necessary and sufficient periodic-point criteria: a subshift $Z$ with entropy strictly below $Y$ embeds into $X$ with $\pi\circ\psi$ injective if and only if, for every period $n$, the number of least-period-$n$ points in $Z$ does not exceed the number of $Y$-periodic points that have a $\\pi$-preimage of the same period, denoted $q_n(Z)\le r_n(\pi)$. The core methodology introduces a sophisticated coding framework using markers, blanks, and stamps to separate and encode data blocks, together with Markov-approximation arguments and a blowing-up lemma to handle obstructions, yielding a robust construction that preserves entropy gaps and allows near-injective embeddings. These contributions bridge deterministic channel coding with symbolic dynamics, offering a precise, quantitative embedding criterion that generalizes Krieger’s theorem and clarifies when entropy is the governing barrier to zero-error transmission through a sliding-block channel. The results have potential implications for dimension-reduction and structural understanding of factor maps between complex symbolic systems.

Abstract

We prove a generalization of Krieger's embedding theorem, in the spirit of zero-error information theory. Specifically, given a mixing shift of finite type $X$, a mixing sofic shift $Y$, and a surjective sliding block code $π: X \to Y$, we give necessary and sufficient conditions for a subshift $Z$ of topological entropy strictly lower than that of $Y$ to admit an embedding $ψ: Z \to X$ such that $π\circ ψ$ is injective.

Encoding subshifts through sliding block codes

TL;DR

This work extends Krieger's zero-ambiguity embedding theory to the setting of a mixing SFT , a mixing sofic shift , and a factor map . The authors give necessary and sufficient periodic-point criteria: a subshift with entropy strictly below embeds into with injective if and only if, for every period , the number of least-period- points in does not exceed the number of -periodic points that have a -preimage of the same period, denoted . The core methodology introduces a sophisticated coding framework using markers, blanks, and stamps to separate and encode data blocks, together with Markov-approximation arguments and a blowing-up lemma to handle obstructions, yielding a robust construction that preserves entropy gaps and allows near-injective embeddings. These contributions bridge deterministic channel coding with symbolic dynamics, offering a precise, quantitative embedding criterion that generalizes Krieger’s theorem and clarifies when entropy is the governing barrier to zero-error transmission through a sliding-block channel. The results have potential implications for dimension-reduction and structural understanding of factor maps between complex symbolic systems.

Abstract

We prove a generalization of Krieger's embedding theorem, in the spirit of zero-error information theory. Specifically, given a mixing shift of finite type , a mixing sofic shift , and a surjective sliding block code , we give necessary and sufficient conditions for a subshift of topological entropy strictly lower than that of to admit an embedding such that is injective.
Paper Structure (12 sections, 22 theorems, 25 equations)

This paper contains 12 sections, 22 theorems, 25 equations.

Table of Contents

  1. Introduction
  2. Conventions, definitions, and background on symbolic dynamics
  3. Subshifts and sliding block codes
  4. Markers and Markov approximations
  5. Coding
  6. Blanks and stamps
  7. Blanks and markers
  8. Stamps and SFTs
  9. Counting
  10. Self-overlap and stamps
  11. Entropy and periodic points
  12. Proofs of \ref{['wlog-sft']} and \ref{['mmb-cor']}

Key Result

Theorem 1.1

Let $X$ be a mixing SFT, $Y$ a mixing sofic shift, and $\pi: X \to Y$ a factor code. Let $Z$ be a subshift with topological entropy strictly less than that of $Y$. Then there exists a subshift $Z'$ of $X$ conjugate to $Z$ such that $\pi|_{Z'}$ is injective, if and only if for every $n \geq 1$, the n

Theorems & Definitions (44)

  • Theorem 1.1
  • Theorem 1.2: Theorem 2 in wk-82-etds
  • Corollary 1.3
  • Corollary 1.4
  • Definition 2.1
  • Lemma 2.1: Proposition 3 in bm-1985-encod
  • Lemma 2.2: Lemma 10.3.2, lm-95-intro
  • Remark 2.3
  • Lemma 2.4: Lemma 2.3 in mb-1984-etds
  • Lemma 2.5: Lemma 2 in wk-82-etds
  • ...and 34 more