The Rokhlin lemma for homeomorphisms of a Cantor set

Bezuglyi, S.; Dooley, A.; Medynets, K.

doi:10.1090/S0002-9939-05-07777-4

The Rokhlin lemma for homeomorphisms of a Cantor set
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by S. Bezuglyi, A. H. Dooley and K. Medynets PDF

Proc. Amer. Math. Soc. 133 (2005), 2957-2964 Request permission

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 $\mu _1,\mu _2,\ldots ,\mu _k$ 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.

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