The Rokhlin lemma for homeomorphisms of a Cantor set

Abstract

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 $μ_i(E\cup TE\cup...\cup T^{n-1}E) > 1 - \e, i= 1,...,k$. 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.

9 pages. Proc. Ams, to appear