On homeomorphic Bernoulli measures on the Cantor space

• Autores: Randall Dougherty, R. Daniel Mauldin, Andrew Yingst
• Localización: Transactions of the American Mathematical Society, ISSN 0002-9947, Vol. 359, Nº 12, 2007, págs. 6155-6166
• Idioma: inglés
• DOI: 10.1090/s0002-9947-07-04352-8
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• Resumen

  • 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.