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| In 1932 Eberhard Hopf introduced a notion of an equivalent relation (called a Hopf-equivalence) for measurable sets with positive measure and for a non-singular and bi-measurable transformation $T$ on a Lebesgue space. He showed that $T$ has an equivalent finite invariant measure iff there exists no measurable subset of the space which is Hopf-equivalent under $T$ to its proper subset. Topological versions of the Hopf-equivalences were considered in many authors, i.e. B. Weiss, E. Glasner, T. Giordano, I. Putnam, C. Skau, H. Yuasa.
Let $T$ be a homeomorphism of a Cantor space $X$ and let $C, D \subset X$ be nonempty clopen sets. We say that $C$ and $D$ are finitely Hopf-equivalent [countable Hopf-equivalent] if there are integers $(n_i, 1 \leq i \leq k)$ [$n_i, i \geq 1$] and decompositions $C= \bigcup C_i$, $D = \bigcup D_i$ into nonempty pairwise disjoint clopen sets [into nonempty pairwise disjoint closed sets] such that $T^{n_i} (C_i) = D_i,\ i=1, \ldots ,k\quad [i=1,2,\ldots]$.
The topological Hopf-equivalences completely determine orbit structure of a Cantor minimal system.
Let $(X,T)$ and $(Y,S)$ be Cantor minimal systems. Then
\begin{itemize}
\item[-] $(X,T)$ and $(Y,S)$ are strong orbit equivalent [orbit equivalent] iff there is a homeomorphism $F \colon X \to Y$ which respects the finite Hopf-equivalence [the countable Hopf-equivalence].
\end{itemize}
A criterion for two clopen sets are countable equivalent was found by Glasner and Weiss (1995) using the invariant measures and the topological full groups. We give some information on a structure of full groups in Cantor dynamics and also we present another criterion for two nonempty clopen sets $C$ and $D$ to be countable Hopf-equivalent without the use of the invariant measures. |
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