[Paper Review] Application of signal analysis to the embedding problem of $\mathbb{Z}^k$-actions

Yonatan Gutman, Yixiao Qiao+1 more|arXiv (Cornell University)|Sep 1, 2017

Mathematical Dynamics and Fractals12 references3 citations

TL;DR

This paper establishes that any $Η^k$-action with the marker property and mean dimension less than $D/2$ can be embedded into the shift on $([0,1]^D)^{\mathbb{Z}^k}$, using a novel signal-theoretic approach that encodes tilings of $\mathbb{R}^k$ into band-limited functions. The bound $D/2$ is optimal, and the method overcomes higher-dimensional signal analysis challenges via analytic techniques tailored to dynamical systems.

ABSTRACT

We study the problem of embedding arbitrary $\mathbb{Z}^k$-actions into the shift action on the infinite dimensional cube $\left([0,1]^D ight)^{\mathbb{Z}^k}$. We prove that if a $\mathbb{Z}^k$-action satisfies the marker property (in particular if it is a minimal system without periodic points) and if its mean dimension is smaller than $D/2$ then we can embed it in the shift on $\left([0,1]^D ight)^{\mathbb{Z}^k}$. The value $D/2$ here is optimal. The proof goes through signal analysis. We develop the theory of encoding $\mathbb{Z}^k$-actions into band-limited signals and apply it to proving the above statement. Main technical difficulties come from higher dimensional phenomena in signal analysis. We overcome them by exploring analytic techniques tailored to our dynamical settings. The most important new idea is to encode the information of a tiling of the Euclidean space into a band-limited function which is constructed from another tiling.

Motivation & Objective

• To extend signal analysis techniques from one-dimensional to multi-dimensional dynamical systems.
• To solve the embedding problem for $\mathbb{Z}^k$-actions into shift systems on $([0,1]^D)^{\mathbb{Z}^k}$.
• To determine the sharp threshold for mean dimension in terms of $D$ under which such embeddings are possible.
• To develop a theory of encoding dynamical systems into band-limited signals in higher dimensions.
• To establish the optimality of the $D/2$ mean dimension bound in the embedding problem.

Proposed method

• The authors introduce a method to encode tilings of $\mathbb{R}^k$ into band-limited functions, using a secondary tiling to construct the signal.
• They apply the theory of band-limited signals to represent dynamical systems as discrete signals in $([0,1]^D)^{\mathbb{Z}^k}$.
• A key innovation is constructing a band-limited function from one tiling to encode information from another tiling.
• Analytic techniques are developed to handle higher-dimensional signal phenomena that obstruct standard approaches.
• The proof uses $\varepsilon$-embeddings and distance estimates via the map $\Pi_{-n+W(x,n)}$ to ensure injectivity.
• The construction relies on the marker property to ensure the existence of suitable partitions and windows $W(x,n)$.

Experimental results

Research questions

• RQ1 Can the embedding of $\mathbb{Z}^k$-actions into shift systems be achieved using signal analysis techniques in higher dimensions?
• RQ2 What is the sharp mean dimension threshold for such embeddings, and is $D/2$ optimal?
• RQ3 How can tilings of $\mathbb{R}^k$ be encoded into band-limited functions to represent dynamical systems?
• RQ4 Can the marker property be used to construct injective embeddings via signal encoding?
• RQ5 What are the limitations of this method when the underlying set $\Omega$ is not a rectangle?

Key findings

• Any $\mathbb{Z}^k$-action with the marker property and mean dimension less than $D/2$ can be embedded into the shift on $([0,1]^D)^{\mathbb{Z}^k}$.
• The bound $D/2$ is optimal, as shown by the existence of systems with mean dimension $\geq D/2$ that cannot be embedded.
• The embedding is constructed by encoding a tiling of $\mathbb{R}^k$ into a band-limited function derived from another tiling.
• The method overcomes higher-dimensional signal analysis challenges through tailored analytic techniques.
• The proof establishes injectivity via distance estimates using $\varepsilon$-embeddings and windowed projections $\Pi_{-n+W(x,n)}$.
• The result generalizes previous one-dimensional results and provides a foundation for embedding into more general signal spaces like $\mathcal{B}_1(\Omega)$.