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# Furstenberg Transformations and Approximate Conjugacy

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Published:2008-02-01
Printed: Feb 2008
• Huaxin Lin
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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.
 Keywords: Furstenberg transformations, approximate conjugacy
 MSC Classifications: 37A55 - Relations with the theory of $C^*$-algebras [See mainly 46L55] 46L35 - Classifications of $C^*$-algebras

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