We study the problem of embedding arbitrary \({\mathbb {Z}}^k\)-actions into the shift action on the infinite dimensional cube \(\left( [0,1]^D\right) ^{{\mathbb {Z}}^k}\). We prove that if a \({\mathbb {Z}}^k\)-action X satisfies the marker property (in particular if X 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 \(\left( [0,1]^D\right) ^{{\mathbb {Z}}^k}\). 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 \({\mathbb {R}}^k\) into a band-limited function which is constructed from another tiling.

中文翻译：

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信号分析在$$ {\ mathbb {Z}} ^ k $$ Z k -actions嵌入问题中的应用

我们研究将任意\（{{mathbb {Z}} ^ k \）-动作嵌入到无限维立方体\（\ left（[0,1] ^ D \ right）^ {{\ mathbb {Z}} ^ k} \）。我们证明，如果\（{\ mathbb {Z}} ^ k \） -作用X满足标记属性（特别是如果X是没有周期点的最小系统），并且其平均尺寸小于D  / 2，那么我们可以将其嵌入到\（\ left（[0,1] ^ D \ right）^ {{\\ mathbb {Z}} ^ k} \）的移位中。 此处的值D / 2是最佳的。证明经过信号分析。我们开发了编码\（{\ mathbb {Z}} ^ k \）的理论-作用到带限信号中，并将其应用于证明以上陈述。主要的技术困难来自信号分析中的高维现象。我们通过探索适合我们动态设置的分析技术来克服它们。最重要的新想法是将\（{{mathbb {R}} ^ k \）的切片信息编码为由另一个切片构造的带限函数。