For a Cantor set
X
X
, let
H
o
m
e
o
(
X
)
Homeo(X)
denote the group of all homeomorphisms of
X
X
. The main result of this note is the following theorem. Let
T
∈
H
o
m
e
o
(
X
)
T\in Homeo(X)
be an aperiodic homeomorphism, let
μ
1
,
μ
2
,
…
,
μ
k
\mu _1,\mu _2,\ldots ,\mu _k
be Borel probability measures on
X
X
, and let
ε
>
0
\varepsilon > 0
and
n
≥
2
n\ge 2
. Then there exists a clopen set
E
⊂
X
E\subset X
such that the sets
E
,
T
E
,
…
,
T
n
−
1
E
E,TE,\ldots , T^{n-1}E
are disjoint and
μ
i
(
E
∪
T
E
∪
…
∪
T
n
−
1
E
)
>
1
−
ε
,
i
=
1
,
…
,
k
\mu _i(E\cup TE\cup \ldots \cup T^{n-1}E) > 1 - \varepsilon ,\ i= 1,\ldots ,k
. Several corollaries of this result are given. In particular, it is proved that for any aperiodic
T
∈
H
o
m
e
o
(
X
)
T\in Homeo(X)
the set of all homeomorphisms conjugate to
T
T
is dense in the set of aperiodic homeomorphisms.