Publications Meetings The Profession Membership Programs Math Samplings Policy & Advocacy In the News About the AMS

   
Mobile Device Pairing
Green Open Access
Proceedings of the American Mathematical Society
Proceedings of the American Mathematical Society
ISSN 1088-6826(online) ISSN 0002-9939(print)

 

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
Published electronically: May 13, 2005
MathSciNet review: 2159774
Full-text PDF Free Access

Abstract | References | Similar Articles | Additional Information

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 [Enhancements On Off] (What's this?)


Similar Articles

Retrieve articles in Proceedings of the American Mathematical Society with MSC (2000): 37H15, 37B05, 54H20

Retrieve articles in all journals with MSC (2000): 37H15, 37B05, 54H20


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: http://dx.doi.org/10.1090/S0002-9939-05-07777-4
PII: S 0002-9939(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