• DOI:10.1090/S0002-9947-1986-0831202-5
• Corpus ID: 53656002

Random recursive constructions: asymptotic geometric and topological properties

@article{Mauldin1986RandomRC,
  title={Random recursive constructions: asymptotic geometric and topological properties},
  author={R. Daniel Mauldin and S. C. Williams},
  journal={Transactions of the American Mathematical Society},
  year={1986},
  volume={295},
  pages={325-346}
}

• R. Mauldin, S. C. Williams
• Published 1986
• Mathematics
• Transactions of the American Mathematical Society

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, Sierpiriski curve or Menger curve. We demonstrate that associated with each such construction is a "universal" number a such that almost surely the random object has Hausdorff dimension a. This number is the expected value of the sum of some ratios which in the deterministic case yields Moran's formula. We introduce the notion of a "random recursive construction" and prove several basic facts about such constructions. We give specific examples which lead to random Cantor sets, Sierpinski curves or Menger universal curves. It is perhaps best to begin with a specific example of such a construction. To this end, let us make some notation. Let N be the set of positive integers and R the real numbers. If S is a set, let 5* be the set of all finite sequences of elements of S including 0, the empty sequence. If a = (ai,..., an) and ß — (bi,..., bm) are elements of S, then |q| = n, the length of a, and a * ß = (ai,... ,an,bi,... ,bm). Now, consider the following construction of a Cantor subset of [0,1], the unit interval. (Of course, by a Cantor set we mean a compact, perfect, 0-dimensional metric space.) Set J® = [0,1] and, by recursion, if Ja = [a,b], for a G {0,1}*, then set Jo-,0 = [a,a + x(b — a)] and JCT*i = [o + y(b — a), b], where the point (x, y) is chosen from the triangular region A = {(s,t) | 0 < s < t < 1} according to the uniform distribution. It follows from the results given in this paper that with probability one, the set K = f] (J ̂ n [ct6{0,1}TM is a Cantor set and the Hausdorff dimension of K, dimH(K), is (\/Ï7 3)/2. The paper is organized into four sections. In §1, we define the notion of a random construction and prove a few basic facts concerning such a construction. We demonstrate that with each construction there is a number a such that with probability one the object constructed has Hausdorff dimension < a. In this section, we relate our results to some deterministic results of P. A. P. Moran [15]. In §2 (Theorem 2.1), we show that certain commonly occurring constructions have finite moments of all orders. This result is necessary for our proof that with probability one the Hausdorff dimension is a. Received by the editors December 27, 1984 and, in revised form, July 9, 1985. 1980 Mathematics Subject Classification. Primary 54H20, 60B05; Secondary 28C10.