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Furstenberg Transformations and Approximate Conjugacy


Published:20080201
Printed: Feb 2008
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Abstract
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.