On homeomorphic Bernoulli measures on the Cantor space

• Published in: Transactions of the American Mathematical Society 2007-12, Vol.359 (12), p.6155-6166
• Main Authors: Dougherty, Randall, Mauldin, R. Daniel, Yingst, Andrew
• Format: Article
• Language: English
• Subjects: Algebra; Algebraic topology; Cylinders; Equivalence relation; Exact sciences and technology; Global analysis, analysis on manifolds; Homeomorphism; Integers; Manifolds and cell complexes; Mathematical theorems; Mathematics; Polynomials; Property partitioning; Sciences and techniques of general use; Topology. Manifolds and cell complexes. Global analysis and analysis on manifolds
• ISSN: 0002-9947; 1088-6850
• DOI: 10.1090/S0002-9947-07-04352-8

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.