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In 1932 Eberhard Hopf introduced a notion of an equivalent relation (called a Hopfequivalence) for measurable sets with positive measure and for a nonsingular and bimeasurable 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 Hopfequivalent under $T$ to its proper subset. Topological versions of the Hopfequivalences 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 Hopfequivalent [countable Hopfequivalent] 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 Hopfequivalences 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 Hopfequivalence [the countable Hopfequivalence].
\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 Hopfequivalent without the use of the invariant measures. 
