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 -