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On homeomorphic Bernoulli measures on the Cantor space


Authors: Randall Dougherty, R. Daniel Mauldin and Andrew Yingst
Journal: Trans. Amer. Math. Soc. 359 (2007), 6155-6166
MSC (2000): Primary 37B05; Secondary 28D05, 28C15
DOI: https://doi.org/10.1090/S0002-9947-07-04352-8
Published electronically: July 23, 2007
MathSciNet review: 2336321
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Abstract: Let $ \mu(r)$ be the Bernoulli measure on the Cantor space given as the infinite product of two-point measures with weights $ r$ and $ 1-r$. It is a long-standing open problem to characterize those $ r$ and $ s$ such that $ \mu(r)$ and $ \mu(s)$ are topologically equivalent (i.e., there is a homeomorphism from the Cantor space to itself sending $ \mu(r)$ to $ \mu(s)$). The (possibly) weaker property of $ \mu(r)$ and $ \mu(s)$ being continuously reducible to each other is equivalent to a property of $ r$ and $ s$ called binomial equivalence. In this paper we define an algebraic property called ``refinability'' and show that, if $ r$ and $ s$ are refinable and binomially equivalent, then $ \mu(r)$ and $ \mu(s)$ are topologically equivalent. Next we show that refinability is equivalent to a fairly simple algebraic property. Finally, we give a class of examples of binomially equivalent and refinable numbers; in particular, the positive numbers $ r$ and $ s$ such that $ s = r^2$ and $ r = 1-s^2$ are refinable, so the corresponding measures are topologically equivalent.


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Additional Information

Randall Dougherty
Affiliation: IDA Center for Communications Research, 4320 Westerra Ct., San Diego, California 92121
Email: rdough@ccrwest.org

R. Daniel Mauldin
Affiliation: Department of Mathematics, P.O. Box 311430, University of North Texas, Denton, Texas 76203
Email: mauldin@unt.edu

Andrew Yingst
Affiliation: Department of Mathematics, P.O. Box 311430, University of North Texas, Denton, Texas 76203
Address at time of publication: Department of Mathematics, University of South Carolina, Columbia, South Carolina 29208
Email: andyq@unt.edu, yingst@math.sc.edu

DOI: https://doi.org/10.1090/S0002-9947-07-04352-8
Keywords: Homeomorphic measures, Cantor space, binomially reducible
Received by editor(s): January 18, 2006
Received by editor(s) in revised form: May 1, 2006
Published electronically: July 23, 2007
Additional Notes: The second author was supported in part by NSF grant DMS 0400481
Article copyright: © Copyright 2007 American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication.

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