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Transactions of the American Mathematical Society
Transactions of the American Mathematical Society
ISSN 1088-6850(online) ISSN 0002-9947(print)

 

Good measures on Cantor space


Author: Ethan Akin
Journal: Trans. Amer. Math. Soc. 357 (2005), 2681-2722
MSC (2000): Primary 37A05, 28D05; Secondary 37B10, 54H20
Published electronically: April 16, 2004
MathSciNet review: 2139523
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Abstract: While there is, up to homeomorphism, only one Cantor space, i.e. one zero-dimensional, perfect, compact, nonempty metric space, there are many measures on Cantor space which are not topologically equivalent. The clopen values set for a full, nonatomic measure $\mu$ is the countable dense subset $\{ \mu(U) : U $ is clopen$\}$ of the unit interval. It is a topological invariant for the measure. For the class of good measures it is a complete invariant. A full, nonatomic measure $\mu$ is good if whenever $U, V$ are clopen sets with $\mu(U) < \mu(V)$, there exists $W$ a clopen subset of $V$ such that $\mu(W) = \mu(U) $. These measures have interesting dynamical properties. They are exactly the measures which arise from uniquely ergodic minimal systems on Cantor space. For some of them there is a unique generic measure-preserving homeomorphism. That is, within the Polish group of such homeomorphisms there is a dense, $G_{\delta}$ conjugacy class.


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Additional Information

Ethan Akin
Affiliation: Department of Mathematics, The City College (CUNY), 137 Street and Convent Avenue, New York City, New York 10031
Email: ethanakin@earthlink.net

DOI: http://dx.doi.org/10.1090/S0002-9947-04-03524-X
PII: S 0002-9947(04)03524-X
Keywords: Cantor set, measure on Cantor space, ordered measure spaces, unique ergodicity, generic conjugacy class, Rohlin property
Received by editor(s): April 9, 2002
Received by editor(s) in revised form: July 24, 2003
Published electronically: April 16, 2004
Article copyright: © Copyright 2004 American Mathematical Society