Application of signal analysis to the embedding problem of $${\mathbb {Z}}^k$$Zk-actions

@article{Gutman2017ApplicationOS,
  title={Application of signal analysis to the embedding problem of \$\$\{\mathbb \{Z\}\}^k\$\$Zk-actions},
  author={Yonatan Gutman and Yixiao Qiao and Masaki Tsukamoto},
  journal={Geometric and Functional Analysis},
  year={2017},
  pages={1-63}
}
We study the problem of embedding arbitrary $${\mathbb {Z}}^k$$Zk-actions into the shift action on the infinite dimensional cube $$\left( [0,1]^D\right) ^{{\mathbb {Z}}^k}$$[0,1]DZk. We prove that if a $${\mathbb {Z}}^k$$Zk-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 $$\left( [0,1]^D\right) ^{{\mathbb {Z}}^k}$$[0,1]DZk. The value D / 2 here is optimal… 
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