TY - JOUR

T1 - A pair of non-homeomorphic product measures on the Cantor set

AU - Austin, Tim D.

N1 - Funding Information:
Acknowledgements. The above work was carried out under a summer research studentship funded by Trinity College, Cambridge over the Long Vacation period 2005. My thanks go to Dr I. Leader for his advice and the official supervision of the project.
Copyright:
Copyright 2007 Elsevier B.V., All rights reserved.

PY - 2007/1

Y1 - 2007/1

N2 - For r ∈ [0,1] let μr be the Bernoulli measure on the Cantor set given as the infinite power of the measure on {0,1} with weights r and 1 - r. For r, s ∈ [0,1] it is known that the measure μr is continuously reducible to μs (that is, there is a continuous map sending μr to μs) if and only if s can be written as a certain kind of polynomial in r; in this case s is said to be binomially reducible to r. In this paper we answer in the negative the following question posed by Mauldin: Is it true that the product measures μr and μs are homeomorphic if and only if each is a continuous image of the other, or, equivalently, each of the numbers r and s is binomially reducible to the other?

AB - For r ∈ [0,1] let μr be the Bernoulli measure on the Cantor set given as the infinite power of the measure on {0,1} with weights r and 1 - r. For r, s ∈ [0,1] it is known that the measure μr is continuously reducible to μs (that is, there is a continuous map sending μr to μs) if and only if s can be written as a certain kind of polynomial in r; in this case s is said to be binomially reducible to r. In this paper we answer in the negative the following question posed by Mauldin: Is it true that the product measures μr and μs are homeomorphic if and only if each is a continuous image of the other, or, equivalently, each of the numbers r and s is binomially reducible to the other?

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U2 - 10.1017/S0305004106009741

DO - 10.1017/S0305004106009741

M3 - Article

AN - SCOPUS:33846986475

SN - 0305-0041

VL - 142

SP - 103

EP - 110

JO - Mathematical Proceedings of the Cambridge Philosophical Society

JF - Mathematical Proceedings of the Cambridge Philosophical Society

IS - 1

ER -