Random recursive constructions: asymptotic geometric and topological properties

Mauldin, R. Daniel; Williams, S. C.

doi:10.1090/S0002-9947-1986-0831202-5

Random recursive constructions: asymptotic geometric and topological properties
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by R. Daniel Mauldin and S. C. Williams PDF

Trans. Amer. Math. Soc. 295 (1986), 325-346 Request permission

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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