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Transactions of the American Mathematical Society
Transactions of the American Mathematical Society
ISSN 1088-6850(online) ISSN 0002-9947(print)

 

Random recursive constructions: asymptotic geometric and topological properties


Authors: R. Daniel Mauldin and S. C. Williams
Journal: Trans. Amer. Math. Soc. 295 (1986), 325-346
MSC: Primary 60D05; Secondary 28A75, 54H20
MathSciNet review: 831202
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Abstract: We study some notions of "random recursive constructions" in Euclidean $ m$-space which lead almost surely to a particular type of topological object; e.g., Cantor set, Sierpiński curve or Menger curve. We demonstrate that associated with each such construction is a "universal" number $ \alpha $ such that almost surely the random object has Hausdorff dimension $ \alpha $. This number is the expected value of the sum of some ratios which in the deterministic case yields Moran's formula.


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Additional Information

DOI: http://dx.doi.org/10.1090/S0002-9947-1986-0831202-5
PII: S 0002-9947(1986)0831202-5
Keywords: Martingale, Hausdorff dimension, Cantor set, Sierpiński curve, Menger curve
Article copyright: © Copyright 1986 American Mathematical Society