Mean dimension of $\mathbb{Z}^k$-actions

TL;DR

This paper investigates the properties of mean dimension in $bZ^k$-actions, addressing inverse limits, entropy detection at various resolutions, and embedding into infinite-dimensional shifts, especially when the system has the marker property.

Contribution

It extends the study of mean dimension from $bZ$-actions to $bZ^k$-actions, providing solutions for inverse limit characterization, entropy growth, and embedding problems under the marker property.

Findings

Complete solution for inverse limit characterization when the marker property holds.

Analysis of entropy detection growth as resolution becomes finer.

Conditions under which $X$ can be embedded into the infinite-dimensional cube shift.

Abstract

Mean dimension is a topological invariant for dynamical systems that is meaningful for systems with infinite dimension and infinite entropy. Given a $\mathbb{Z}^k$-action on a compact metric space $X$, we study the following three problems closely related to mean dimension. (1) When is $X$ isomorphic to the inverse limit of finite entropy systems? (2) Suppose the topological entropy $h_{\mathrm{top}}(X)$ is infinite. How much topological entropy can be detected if one considers $X$ only up to a given level of accuracy? How fast does this amount of entropy grow as the level of resolution becomes finer and finer? (3) When can we embed $X$ into the $\mathbb{Z}^k$-shift on the infinite dimensional cube $([0,1]^D)^{\mathbb{Z}^k}$? These were investigated for $\mathbb{Z}$-actions in [Lindenstrauss, Mean dimension, small entropy factors and an embedding theorem, Inst. Hautes \'Etudes Sci. Publ. Math. \textbf{89} (1999) 227-262], but the generalization to $\mathbb{Z}^k$ remained an open problem. When $X$ has the marker property, in particular when $X$ has a completely aperiodic minimal factor, we completely solve (1) and a natural interpretation of (2), and give a reasonably satisfactory answer to (3). A key ingredient is a new method to continuously partition every orbit into good pieces.