  For a Cantor set $X$, let $Homeo(X)$ denote the group of all homeomorphisms
of $X$. The main result of this note is the following theorem. Let $T\in
Homeo(X)$ be an aperiodic homeomorphism, let $\mu_1,\mu_2,...,\mu_k$ be Borel
probability measures on $X$, $\e> 0$, and $n \ge 2$. Then there exists a clopen
set $E\subset X$ such that the sets $E,TE,..., T^{n-1}E$ are disjoint and
$\mu_i(E\cup TE\cup...\cup T^{n-1}E) > 1 - \e, i= 1,...,k$. Several corollaries
of this result are given. In particular, it is proved that for any aperiodic
$T\in Homeo(X)$ the set of all homeomorphisms conjugate to $T$ is dense in the
set of aperiodic homeomorphisms.
