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Strong orbit realization for minimal homeomorphisms. (English) Zbl 0881.28013

From the author’s Introduction: …Let \((Y,T,\nu)\) be an ergodic automorphism of a non-atomic Lebesgue probability space, let \((X,S)\) be a minimal homeomorphism of the Cantor set, and let \(\mu\) be an ergodic \(S\)-invariant Borel probability. In one of our two main results, the Strong Orbit Realization Theorem (SORT, Theorem 2.5), we show that there is a topological realization \((S',\mu')\) of \((T,\nu)\), where \(S'\) is a minimal homeomorphism of \(X\), strongly orbit equivalent to \(S\) if and only if the finite rotations which are (topological) factors of \(S\) are (measurable) factors of \(T\).…For the second main result of this paper, the Orbit Realization Theorem (ORT, Theorem 7.2), we show that regardless of the factors of \(S\) and \(T\), there is a minimal homeomorphism \(S'\) which is orbit equivalent to \(S\) such that \((S',\mu')\) is a topological realization of \((T,\nu)\).
Our results, SORT and ORT, are preceded by Vershik’s realizations of ergodic transformations as homeomorphisms of ordered Bratteli diagrams…

MSC:

28D10 One-parameter continuous families of measure-preserving transformations
54H20 Topological dynamics (MSC2010)
37E99 Low-dimensional dynamical systems
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