Random recursive constructions: asymptotic geometric and topological properties

We study some notions of "random recursive constructions" in Euclidean m 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 α α such that almost surely the random object has Hausdorff dimension α α . This number is the expected value of the sum of some ratios which in the deterministic case yields Moran’s formula.