• DOI:10.1007/s00039-019-00499-z
• Corpus ID: 685943

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}
}

• Yonatan Gutman, Yixiao Qiao, M. Tsukamoto
• Published 1 September 2017
• Mathematics, Computer Science
• Geometric and Functional Analysis

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. The proof goes through signal analysis. We develop the theory of encoding $${\mathbb {Z}}^k$$Zk-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 $${\mathbb {R}}^k$$Rk into a band-limited function which is constructed from another tiling.