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A map maintaining the orbits of a given $\mathbb {Z}^d$-action

Volume 143 / 2016

Bartosz Frej, Agata Kwaśnicka Colloquium Mathematicum 143 (2016), 1-15 MSC: 37B05, 37B10, 37A20. DOI: 10.4064/cm6361-12-2015 Published online: 2 December 2015

Abstract

Giordano et al. (2010) showed that every minimal free $\mathbb {Z}^d$-action of a Cantor space $X$ is orbit equivalent to some $\mathbb {Z}$-action. Trying to avoid the K-theory used there and modifying Forrest's (2000) construction of a Bratteli diagram, we show how to define a (one-dimensional) continuous and injective map $F$ on $X\setminus \{\textrm {one point}\}$ such that for a residual subset of $X$ the orbits of $F$ are the same as the orbits of a given minimal free $\mathbb {Z}^d$-action.

Authors

  • Bartosz FrejFaculty of Pure and Applied Mathematics
    Wrocław University of Technology
    Wybrzeże Wyspiańskiego 27
    50-370 Wrocław, Poland
    e-mail
  • Agata KwaśnickaFaculty of Pure and Applied Mathematics
    Wrocław University of Technology
    Wybrzeże Wyspiańskiego 27
    50-370 Wrocław, Poland

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