Published online by Cambridge University Press: 20 November 2018
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 ${{f}_{1}}\,\text{and}\,{{f}_{2}}$. It is shown that $\alpha $ and $\beta $ are approximately conjugate in ameasure theoretical sense if (and only if) $\overline{{{\theta }_{1}}\,\pm \,{{\theta }_{2}}}\,=\,0\,\text{in}\,\mathbb{R}\text{/}\mathbb{Z}$. Closely related to the classification of simple amenable ${{C}^{*}}$-algebras, it is shown that $\alpha $ and $\beta $ are approximately $K$-conjugate if (and only if) $\overline{{{\theta }_{1}}\,\pm \,{{\theta }_{2}}}\,=\,0\,\text{in}\,\mathbb{R}\text{/}\mathbb{Z}$ and $|{{d}_{1}}|\,=\,|{{d}_{2}}|$. This is also shown to be equivalent to the condition that the associated crossed product ${{C}^{*}}$-algebras are isomorphic.