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

### Mrinal Kanti Roychowdhury

Colorado State University, Fort Collins, USA

## 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

(i) $ϕ∣_{M}$ is continuous in the relative topology on $M$ and $ϕ_{−1}∣_{ϕ(M)}$ is continuous in the relative topology on $ϕ(M)$,

(ii) $ϕ(Orb_{T}(x))=Orb_{S}ϕ(x)$ 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.

## Cite this article

Mrinal Kanti Roychowdhury, Irrational Rotation of the Circle and the Binary Odometer are Finitarily Orbit Equivalent. Publ. Res. Inst. Math. Sci. 43 (2007), no. 2, pp. 385–402

DOI 10.2977/PRIMS/1201011787