On homeomorphic Bernoulli measures on the Cantor space |
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Authors: | Randall Dougherty R Daniel Mauldin Andrew Yingst |
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Institution: | IDA Center for Communications Research, 4320 Westerra Ct., San Diego, California 92121 ; Department of Mathematics, P.O. Box 311430, University of North Texas, Denton, Texas 76203 ; Department of Mathematics, P.O. Box 311430, University of North Texas, Denton, Texas 76203 |
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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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Keywords: | Homeomorphic measures Cantor space binomially reducible |
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