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

Yonatan Gutman, Yixiao Qiao, Masaki Tsukamoto

Research output: Contribution to journalArticlepeer-review

5 Citations (Scopus)

Abstract

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.

Original languageEnglish
Pages (from-to)1440-1502
Number of pages63
JournalGeometric and Functional Analysis
Volume29
Issue number5
DOIs
Publication statusPublished - Oct 1 2019

All Science Journal Classification (ASJC) codes

  • Analysis
  • Geometry and Topology

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