Furstenberg Transformations and Approximate Conjugacy

http://dx.doi.org/10.4153/CJM-2008-008-2
Canad. J. Math. 60(2008), 189-207

  Published:2008-02-01
 Printed: Feb 2008

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 $f_2$. It is shown that $\alpha$ and $\beta$ are approximately conjugate in a measure theoretical sense if (and only if) $\overline{\theta_1\pm \theta_2}=0$ in $\R/\Z.$ Closely related to the classification of simple amenable \CAs, it is shown that $\af$ and $\bt$ are approximately $K$-conjugate if (and only if) $\overline{\theta_1\pm \theta_2}=0$ in $\R/\Z$ and $|d_1|=|d_2|.$ This is also shown to be equivalent to the condition that the associated crossed product \CAs are isomorphic.