Application of signal analysis to the embedding problem of $${\mathbb {Z}}^k$$Zk-actions
Application of signal analysis to the embedding problem of $${\mathbb {Z}}^k$$Zk-actions
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DOI：
10.1007/s00039-019-00499-z
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发表时间：
2017-09
影响因子：
2.2
通讯作者：
Yonatan Gutman;Yixiao Qiao;M. Tsukamoto
中科院分区：
文献类型：
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作者：
Yonatan Gutman;Yixiao Qiao;M. Tsukamoto
研究了在无限维立方体上的移位作用中嵌入任意动作的问题。我们证明了如果a-actionX满足标记性(特别是如果X是一个没有周期点的极小系统)，并且如果它的平均维度小于Thand2，那么我们可以把它嵌入到移位中。这里的VALUTED/UF2是最优的。证明是通过信号分析得到的。我们发展了将动作编码成带限信号的理论，并将其应用于证明上述说法。主要的技术困难来自信号分析中的高维现象。我们通过探索为我们的动态环境量身定做的分析技术来克服它们。最重要的新思想是将一块瓷砖的信息编码成由另一块瓷砖构造的带限函数。
We study the problem of embedding arbitrary-actions into the shift action on the infinite dimensional cube. We prove that if a-actionXsatisfies the marker property (in particular ifXis a minimal system without periodic points) and if its mean dimension is smaller thanD/ 2 then we can embed it in the shift on. The valueD/ 2 here is optimal. The proof goes through signal analysis. We develop the theory of encoding-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 ofinto a band-limited function which is constructed from another tiling.