On homeomorphic Bernoulli measures on the Cantor space
Authors:
Randall Dougherty, R. Daniel Mauldin and Andrew Yingst
Journal:
Trans. Amer. Math. Soc. 359 (2007), 61556166
MSC (2000):
Primary 37B05; Secondary 28D05, 28C15
Published electronically:
July 23, 2007
MathSciNet review:
2336321
Fulltext PDF Free Access
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Abstract: Let be the Bernoulli measure on the Cantor space given as the infinite product of twopoint measures with weights and . It is a longstanding open problem to characterize those and such that and are topologically equivalent (i.e., there is a homeomorphism from the Cantor space to itself sending to ). The (possibly) weaker property of and being continuously reducible to each other is equivalent to a property of and called binomial equivalence. In this paper we define an algebraic property called ``refinability'' and show that, if and are refinable and binomially equivalent, then and 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 and such that and 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:
http://dx.doi.org/10.1090/S0002994707043528
PII:
S 00029947(07)043528
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.
