Furstenberg Transformations and Approximate Conjugacy

Let $\alpha$ and $\beta$ be two Furstenberg transformations on 2-torus associated with irrational numbers ${\theta }_1,$ ${\theta }_2,$ integers $d_1,d_2$ and Lipschitz functions $f_1$ and $f_2\mathrm{.}$ We show that $\alpha$ and $\beta$ are approximately conjugate in a measure theoretical sense if (and only if) $\overline{{\theta }_1\pm {\theta }_2}=0$ in $\mathbb{R}\mathrm{/}\mathbb{Z}\mathrm{.}$ Closely related to the classification of simple amenable $C^{\ast }$-algebras, we show that $\alpha$ and $\beta$ are approximately $K$-conjugate if (and only if) $\bar{\theta_1\pm \theta_2}=0$in$\R/\Z$and$|d_1|=|d_2|.$ This is also shown to be equivalent to that the associated crossed product $C^{\ast }$-algebras are isomorphic.