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

Bartosz Frej; Agata Kwaśnicka — 2016

Colloquium Mathematicae

Giordano et al. (2010) showed that every minimal free $ℤ^{d}$ -action of a Cantor space X is orbit equivalent to some 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∖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 $ℤ^{d}$ -action.

Minimal models for $ℤ^{d}$ -actions

Bartosz Frej; Agata Kwaśnicka — 2008

Colloquium Mathematicae

We prove that on a metrizable, compact, zero-dimensional space every $ℤ^{d}$ -action with no periodic points is measurably isomorphic to a minimal $ℤ^{d}$ -action with the same, i.e. affinely homeomorphic, simplex of measures.

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A map maintaining the orbits of a given ℤ d -action

Minimal models for ℤ d -actions

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

Minimal models for $ℤ^{d}$ -actions