The main topic of the book belongs to ergodic theory and measurable dynamics. It studies suitable notions of similarity of two dynamical systems using structure of orbits of corresponding systems. The authors discuss free and ergodic actions of countable discrete amenable groups. The key notions in the book are an orbit relation O={x,Tg(x)}gєG Ì X x X generated by the action, an orbit equivalence (a measure preserving map carrying one orbit relation to the other), and arrangements and rearrangements of orbits. Basic definitions and examples can be found in Chapter 2 (the vocabulary of arrangements and rearrangements, and m-equivalence classes of an arrangement). Fundamental results by Ornstein and Weiss on ergodic theory of actions of amenable groups are reviewed in Chapter 3. The next two chapters include key technical lemmas and entropy theory for restricted orbit equivalences. Chapter 6 contains a construction of a topological model for arrangements and rearrangements using a notion of Polish spaces and Polish actions. The last chapter contains a formulation and a proof of the equivalence theorem. Similar questions were studied carefully in the last decades for Z-action as well as for Zd -action (d≥ 1). Relations of the results described in the book to results obtained in these special cases can be found in the Appendix.

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