The Rokhlin lemma for homeomorphisms of a Cantor set

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