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An embedding theorem for multidimensional subshifts

Tom Meyerovitch

Abstract

Krieger's embedding theorem provides necessary and sufficient conditions for an arbitrary subshift to embed in a given topologically mixing $\mathbb{Z}$-subshift of finite type. For some $\mathbb{Z}^d$-subshifts of finite type, Lightwood characterized the \emph{aperiodic} subsystems. In the current paper we prove a new embedding theorem for a class of subshifts of finite type over any countable abelian group. Our main theorem provides necessary and sufficient conditions for an arbitrary subshift $X$ to embed inside a given subshift of finite type $Y$ that satisfies a certain condition. For the particular case of $\mathbb{Z}$-subshifts, our new theorem coincides with Krieger's theorem. In particular, our result gives the first complete characterization of the subsystems of the multidimensional full shift $Y= A^{\mathbb{Z}^d}$. The natural condition on the target subshift $Y$, introduced explicitly for the first time in the current paper, is called the map extension property. It was introduced implicitly by Mike Boyle in the early 1980's for $\mathbb{Z}$-subshifts, and is closely related to the notion of an absolute retract, introduced by Borsuk in the 1930's. A $\mathbb{Z}$-subshift has the map extension property if and only if it is a topologically mixing subshift of finite type. Over abelian groups, a subshift has the map extension property if and only if it is a contractible SFT as shown in work of Poirier and Salo. We also establish a new theorem regarding lower entropy factors of multidimensional subshifts, that extends Boyle's lower entropy factor theorem from the one-dimensional case.

An embedding theorem for multidimensional subshifts

Abstract

Krieger's embedding theorem provides necessary and sufficient conditions for an arbitrary subshift to embed in a given topologically mixing -subshift of finite type. For some -subshifts of finite type, Lightwood characterized the \emph{aperiodic} subsystems. In the current paper we prove a new embedding theorem for a class of subshifts of finite type over any countable abelian group. Our main theorem provides necessary and sufficient conditions for an arbitrary subshift to embed inside a given subshift of finite type that satisfies a certain condition. For the particular case of -subshifts, our new theorem coincides with Krieger's theorem. In particular, our result gives the first complete characterization of the subsystems of the multidimensional full shift . The natural condition on the target subshift , introduced explicitly for the first time in the current paper, is called the map extension property. It was introduced implicitly by Mike Boyle in the early 1980's for -subshifts, and is closely related to the notion of an absolute retract, introduced by Borsuk in the 1930's. A -subshift has the map extension property if and only if it is a topologically mixing subshift of finite type. Over abelian groups, a subshift has the map extension property if and only if it is a contractible SFT as shown in work of Poirier and Salo. We also establish a new theorem regarding lower entropy factors of multidimensional subshifts, that extends Boyle's lower entropy factor theorem from the one-dimensional case.
Paper Structure (12 sections, 48 theorems, 128 equations)

This paper contains 12 sections, 48 theorems, 128 equations.

Table of Contents

  1. Introduction and statement of the main results
  2. Notation and definitions
  3. Countable abelian groups
  4. Subshifts
  5. Markers in the domain: Krieger's Marker Lemma
  6. Retractions and the map extension property for subshifts
  7. Entropy minimal subshifts and necessity of the conditions for embedding
  8. Truncated Voronoi diagrams and Partial tilings
  9. Good marker patterns via the map extension property
  10. Proof of the embedding theorem for subshifts over a finitely generated abelian group
  11. Reduction to the finitely generated abelian case
  12. Lower entropy factors and the map extension property

Key Result

Theorem 1.1

Let $X$ be an arbitrary $\mathbb{Z}$-subshift and let $Y$ be a topologically mixing $\mathbb{Z}$-shift of finite type. Then $X$ embeds in $Y$ if and only if either $X$ is topologically conjugate to $Y$ or $h(X) < h(Y)$ and for every $n \in \mathbb{N}$ the number of points in $X$ that have least peri

Theorems & Definitions (127)

  • Theorem 1.1: Krieger's embedding theorem MR693975
  • Theorem 1.2: Lightwood's embedding theorem for square-filling mixing $\mathbb{Z}^2$ -SFTs MR1972240MR2085910
  • Theorem 1.3
  • Theorem 1.4
  • Corollary 1.5
  • Definition 2.1
  • Definition 2.2
  • Definition 2.3
  • Definition 2.4
  • Lemma 2.5
  • ...and 117 more