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Transactions of the American Mathematical Society

Published by the American Mathematical Society, the Transactions of the American Mathematical Society (TRAN) is devoted to research articles of the highest quality in all areas of pure and applied mathematics.

ISSN 1088-6850 (online) ISSN 0002-9947 (print)

The 2020 MCQ for Transactions of the American Mathematical Society is 1.43.

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Good measures on Cantor space
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by Ethan Akin PDF
Trans. Amer. Math. Soc. 357 (2005), 2681-2722 Request permission

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
  • 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