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

Full-text PDF

Abstract | References | Similar Articles | Additional Information

Abstract: Let be the Bernoulli measure on the Cantor space given as the infinite product of two-point measures with weights and . It is a long-standing 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:
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.