Good measures on Cantor space
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- by Ethan Akin
- Trans. Amer. Math. Soc. 357 (2005), 2681-2722
- DOI: https://doi.org/10.1090/S0002-9947-04-03524-X
- Published electronically: April 16, 2004
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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.References
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Bibliographic Information
- Ethan Akin
- Affiliation: Department of Mathematics, The City College (CUNY), 137 Street and Convent Avenue, New York City, New York 10031
- MR Author ID: 24025
- Email: ethanakin@earthlink.net
- Received by editor(s): April 9, 2002
- Received by editor(s) in revised form: July 24, 2003
- Published electronically: April 16, 2004
- © Copyright 2004 American Mathematical Society
- Journal: Trans. Amer. Math. Soc. 357 (2005), 2681-2722
- MSC (2000): Primary 37A05, 28D05; Secondary 37B10, 54H20
- DOI: https://doi.org/10.1090/S0002-9947-04-03524-X
- MathSciNet review: 2139523