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The Rokhlin lemma for homeomorphisms of a Cantor set


Authors: S. Bezuglyi, A. H. Dooley and K. Medynets
Journal: Proc. Amer. Math. Soc. 133 (2005), 2957-2964
MSC (2000): Primary 37H15, 37B05; Secondary 54H20
DOI: https://doi.org/10.1090/S0002-9939-05-07777-4
Published electronically: May 13, 2005
MathSciNet review: 2159774
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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 $\mu_1,\mu_2,\ldots,\mu_k$ 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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Additional Information

S. Bezuglyi
Affiliation: Institute for Low Temperature Physics, National Academy of Sciences of Ukraine, Kharkov, Ukraine
Email: bezuglyi@ilt.kharkov.ua

A. H. Dooley
Affiliation: School of Mathematics, University of New South Wales, Sydney, Australia
Email: a.dooley@unsw.edu.au

K. Medynets
Affiliation: Institute for Low Temperature Physics, National Academy of Sciences of Ukraine, Kharkov, Ukraine
Email: medynets@ilt.kharkov.ua

DOI: https://doi.org/10.1090/S0002-9939-05-07777-4
Received by editor(s): October 20, 2003
Published electronically: May 13, 2005
Communicated by: Michael Handel
Article copyright: © Copyright 2005 American Mathematical Society

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