TY - JOUR

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

AU - Gutman, Yonatan

AU - Qiao, Yixiao

AU - Tsukamoto, Masaki

N1 - Funding Information:
This paper was written when the third named author stayed in the Einstein Institute of Mathematics in the Hebrew University of Jerusalem. He would like to thank the institute for its hospitality. The authors thank the referees for their many useful suggestions.
Publisher Copyright:
© 2019, Springer Nature Switzerland AG.

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.

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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 -