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.References
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Additional Information
- S. Bezuglyi
- Affiliation: Institute for Low Temperature Physics, National Academy of Sciences of Ukraine, Kharkov, Ukraine
- MR Author ID: 215325
- 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
- MR Author ID: 752184
- Email: medynets@ilt.kharkov.ua
- Received by editor(s): October 20, 2003
- Published electronically: May 13, 2005
- Communicated by: Michael Handel
- © Copyright 2005 American Mathematical Society
- 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
- MathSciNet review: 2159774