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Transactions of the American Mathematical Society

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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
DOI: https://doi.org/10.1090/S0002-9947-04-03524-X
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.


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

  • [1999] E. Akin, Measures on Cantor space, Topology Proceedings (1999) 24: 1-34. MR 2002j:28002
  • [2003] E. Akin, M. Hurley and J. A. Kennedy, Dynamics of topologically generic homeomorphisms, Memoirs Amer. Math. Soc # 783 (2003).
  • [1988] D. Cooper and T. Pignataro, On the shape of Cantor sets, J. Diff. Geom. (1988) 28: 203-221. MR 89k:58160
  • [2002] E. Glasner, Topics in topological dynamics, 1991 to 2001. Recent progress in general topology, II, 153-175, North-Holland, Amsterdam, 2002.
  • [1995] E. Glasner and B. Weiss, Weak orbit equivalence of minimal Cantor systems, Internat. J. Math. (1995) 6: 559-579. MR 96g:46054
  • [2001] E. Glasner and B. Weiss, The topological Rohlin property and topological entropy, Amer. J. of Math (2001) 123: 1055-1070. MR 2002h:37025
  • [1995] T. Giordano, I. F. Putnam and C. F. Skau, Topological orbit equivalence and $C^{*}$-crossed products, J. Reine Angew. Math. (1995) 469: 51-111. MR 97g:46085

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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: https://doi.org/10.1090/S0002-9947-04-03524-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

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