Two invertible dynamical systems (X, U ,mu,T) and (Y, B ,nu,S) where X, Y are topological spaces and T, S are homeomorphisms, are said to be finitarily orbit equivalent if there exists an invertible measure-preserving mapping &phis; from a subset of X of measure one to a subset of Y of full measure such that (i) &phis;(Orb T(x)) = OrbS&phis;( x) for mu-a.e. x ∈ X, (ii) there exists a set M ⊂ X such that mu( M) = 1, &phis;|M is continuous in the relative topology on M and &phis;-1| &phis;(M) is continuous in the relative topology on &phis;(M).; This dissertation focuses on the finitary orbit equivalence relation induced by measure-preserving actions of groups. An {lcub}mn{rcub}-odometer is an odometer with n-th wheel of length mn ≥ 2 (the sequence {lcub}mn{rcub} need not be bounded). In this dissertation in chapter 2 we prove: Theorem 1: Any {lcub}mn{rcub}-odometer and the binary odometer are finitarily orbit equivalent.; In chapter 3 we prove: Theorem 2: An irrational rotation and the binary odometer are finitarily orbit equivalent.; Since finitary orbit equivalence is an equivalence relation, our result implies that any {lcub}mn{rcub}-odometer, the binary odometer and an irrational rotation are finitarily orbit equivalent to each other.
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