Furstenberg Transformations and Approximate Conjugacy

Abstract

Let $α$ and $β$ be two Furstenberg transformations on 2-torus associated with irrational numbers $θ_1,$ $θ_2,$ integers $d_1, d_2$ and Lipschitz functions $f_1$ and $f_2.$ We show that $α$ and $β$ are approximately conjugate in a measure theoretical sense if (and only if) $\bar{θ_1\pm θ_2}=0$ in $\R/\Z.$ Closely related to the classification of simple amenable $C^*$-algebras, we show that $α$ and $β$ are approximately $K$-conjugate if (and only if) $\bar{θ_1\pm θ_2}=0$ in $\R/\Z$ and $|d_1|=|d_2|.$ This is also shown to be equivalent to that the associated crossed product $C^*$-algebras are isomorphic.