Entropy and ${\bi r}$ equivalence
1998 ◽
Vol 18
(5)
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pp. 1139-1157
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In this paper, the structure of $r$ equivalence, which was introduced by Vershik and which classifies group actions of the group $G=\sum_{n=1}^\infty{\Bbb Z}\slash r_n{\Bbb Z}$, $r_n\in{\Bbb N}\setminus\{1\}$, is examined. This is an equivalence relation that naturally arises from looking at certain sequences of $\sigma$-algebras. Vershik proved that if a sequence $r=(r_1,r_2,\ldots)$ does not satisfy a super-rapid growth rate, then entropy is an invariant for $r$ equivalence. In this paper, a strong converse of this is proven: for any $r$ which does satisfy this super-rapid growth rate, we can find a zero entropy action in every $r$ equivalence class.
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2016 ◽
Vol 74
(1)
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pp. 14-19
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1995 ◽
Vol 110
(3)
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pp. 531-537
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