Irrational Rotation of the Circle and the Binary Odometer are Finitarily Orbit Equivalent

Abstract

Two invertible dynamical systems (X, A, µ, T ) and (Y, B, ν, S) where X, Y are metrizable spaces and T, S are homeomorphisms on X and Y, are said to be finitarily orbit equivalent if there exists an invertible measure preserving mapping φ from a subset M of X of measure one to a subset of Y of full measure such that

1. φ|M is continuous in the relative topology on M and _φ_−1 |φ(M) is continuous in the relative topology on φ(M),
2. φ(Orb_T_(x)) = Orb_S__φ_(x) for µ-a.e. x ∈ X. In this article a finitary orbit equivalence mapping has been constructed between an irrational rotation of the circle and the binary odometer.