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The Rokhlin lemma for homeomorphisms of a Cantor set

Sergey Bezuglyi·Anthony H. Dooley·Konstantin Medynets
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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 $\mu_i(E\cup TE\cup...\cup T^{n-1}E) > 1 - \e, i= 1,...,k$. 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.

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Sergey Bezuglyi, Anthony H. Dooley, Konstantin Medynets. 2004-10-23. The Rokhlin lemma for homeomorphisms of a Cantor set. https://arxiv.org/abs/math/0410505

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