### Abstract

We study the problem of embedding arbitrary Z^{k}-actions into the shift action on the infinite dimensional cube ([0,1]D)Zk. We prove that if a 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 ([0,1]D)Zk. The value D / 2 here is optimal. The proof goes through signal analysis. We develop the theory of encoding 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 R^{k} into a band-limited function which is constructed from another tiling.

Original language | English |
---|---|

Pages (from-to) | 1440-1502 |

Number of pages | 63 |

Journal | Geometric and Functional Analysis |

Volume | 29 |

Issue number | 5 |

DOIs | |

Publication status | Published - Oct 1 2019 |

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### All Science Journal Classification (ASJC) codes

- Analysis
- Geometry and Topology

### Cite this

^{k}-actions.

*Geometric and Functional Analysis*,

*29*(5), 1440-1502. https://doi.org/10.1007/s00039-019-00499-z

**Application of signal analysis to the embedding problem of Z ^{k} -actions.** / Gutman, Yonatan; Qiao, Yixiao; Tsukamoto, Masaki.

Research output: Contribution to journal › Article

^{k}-actions',

*Geometric and Functional Analysis*, vol. 29, no. 5, pp. 1440-1502. https://doi.org/10.1007/s00039-019-00499-z

^{k}-actions. Geometric and Functional Analysis. 2019 Oct 1;29(5):1440-1502. https://doi.org/10.1007/s00039-019-00499-z

}

TY - JOUR

T1 - Application of signal analysis to the embedding problem of Zk -actions

AU - Gutman, Yonatan

AU - Qiao, Yixiao

AU - Tsukamoto, Masaki

PY - 2019/10/1

Y1 - 2019/10/1

N2 - We study the problem of embedding arbitrary Zk-actions into the shift action on the infinite dimensional cube ([0,1]D)Zk. We prove that if a 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 ([0,1]D)Zk. The value D / 2 here is optimal. The proof goes through signal analysis. We develop the theory of encoding 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 Rk into a band-limited function which is constructed from another tiling.

AB - We study the problem of embedding arbitrary Zk-actions into the shift action on the infinite dimensional cube ([0,1]D)Zk. We prove that if a 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 ([0,1]D)Zk. The value D / 2 here is optimal. The proof goes through signal analysis. We develop the theory of encoding 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 Rk into a band-limited function which is constructed from another tiling.

UR - http://www.scopus.com/inward/record.url?scp=85066481978&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=85066481978&partnerID=8YFLogxK

U2 - 10.1007/s00039-019-00499-z

DO - 10.1007/s00039-019-00499-z

M3 - Article

AN - SCOPUS:85066481978

VL - 29

SP - 1440

EP - 1502

JO - Geometric and Functional Analysis

JF - Geometric and Functional Analysis

SN - 1016-443X

IS - 5

ER -