Exact Hausdorff dimension in random recursive constructions

Proc. Nati. Acad. Sci. USA Vol. 84, pp. 3959-3961, June 1987 Mathematics

Exact Hausdorff dimension in random recursive constructions (iteration/fractals/probabiity)

R. D. MAULDINt, S. GRAF*, AND S. C. WILLIAMS§ tDepartment of Mathematics, North Texas State University, Denton, TX 76203; Mathematics Institute, University of Erlangen-Nurnberg, D-8520 Erlangen, Federal Republic of Germany; and §Department of Mathematics, Utah State University, Logan, UT 84322

Communicated by Gian-Carlo Rota, March 9, 1987

ABSTRACT The exact Hausdorff dimension function is determined for sets in R"' constructed by using a recursion that is governed by some given law of randomness.

We present a method of determining the exact Hausdorff dimension function for a wide class of random recursive constructions. Let us recall the setting. Fix the compact subset J of Rm with J = cl(int(J)) and a positive integer n. An n-ary random recursion modeled on J is a probability space (Ql, :, P) and a family of random subsets of Rm; J=

Jio- E {1,

. ..

n}*= U {1, . .

n}v

satisfying three properties. (i) J4(w) = J for almost all w E Q1. For every o- E {1, ... n}* and, for almost all co, if J,(w) is nonempty, then J,(w) is geometrically similar to J and the map c -* J,(w) is measurable with respect to the Hausdorff metric on the space of compact subsets of R". (ii) For almost every wt E f and for every o E {1, ..., n}*, Ja*1, J-*2 ...*, Jo*,n are nonoverlapping subsets of J,((w). (iii) The random vectors T. = (T,19, .. . , To*,n), a- E {1, ... . n}* are independent and identically distributed, where T,*i(o) equals the ratio of the diameter of J,.i(w) to the diameter of J,(w) [for convenience TO,(w) = diam(J)]. The system J is called an n-ary random recursive construction. We now define the random set K by

K(w) =

n

k=1

U

1..,n

J"(w).

Our interest centers on the Hausdorff dimension function of K. We will consider those constructions such that E(T? + . . n. + TO) > 1. These are really the only interesting constructions as is well known from the theory of branching processes (for discussion, see ref. 1). The Hausdorff dimension was determined by Mauldin and Williams. THEOREM 1. Assume E(T? + . . . + TO) > 1. Let a be the unique number such that E(Tc'

+ ...+

TOt)

=

and noticing that {Sn1a},'L forms a martingale. Therefore, there is a finite random variable X such that for almost all w lim Sk,a(W). X(c)) = k-.o

Since, for almost all co, lim

sup diam(J,) = 0, .ne it follows that 7Ca(K) s X(w) < +0. The proof that a is the dimension of K(Z) was completed by constructing an appropriate family of random measures. The formula determining a generalizes the formula given by Moran (2) in the deterministic case and, in that case, reduces to his formula. Moran also showed, in the deterministic case, that X'(K) > 0, where WCais Hausdorff's measure in dimension a. Graf (3) showed that this remains true for the random case, provided P(;,=JTr = 1) = 1 and there is some 8 such that 0 < 8< minl-i-nTP-a.s. Actually, for most constructions P(In T,' # 1) > 0. Graf showed Ha(K) = 0 a.s. under this assumption. S. J. Taylor conjectured the general form of the exact dimension in a letter. Our main theorems give upper and lower estimates on the exact Hausdorff dimension function associated with constructions such that P(Y.5; l7i 7 1) > 0. These estimates depend on (i) ri, the radius of convergence of the moment generating function of Xt, (ii) the probabilistic behavior of the construction, and (iii) the geometric properties of the modeling set, J. UPPER-BOUND THEOREM. Assume that E(T? + . . . + TO) > 1. Suppose 3 > 0 is such that rp < a). Let hi(t) = ta(log (log(l/t)))W10. Then there is a constant c such that k ox

Xthp(K(wt))

cX(cw)

< + oo

1.

(

/30 = sup{ 13> 1

< 1, P-a.s.J.

Then, 1 < 83 < +0 and

a.

The fact that XCa(K(w)) 1 and P(Ixn1T9 :& 1) > 0. Set

Then P(K 7 4)) > 0 and for P-a.e. o, Xea(K(w)) < A. Moreover, if K(w) # 4, then the Hausdorff dimension of K(Z) is

Sk,.

lE,1

0:s3 0}] 0 ensures that 13 < ao. Any one of conditions i, ii, and iii of the Exact-Dimension Theorem ensure that the radius of convergence of the moment generating function of XO is positive and finite where 1/,8 = 1 - (a/m). We apply our results to two examples.

(1/vf A tf)(1/(vt(1 - v

-

1/2

{

t)3)112)dv dt < mo.

Since

m = 1 and lim(v,-(3/l2,ll2) p(v, t) = X condition i is satisfied at (x1, x2) = (1/2, 1/2). Thus it follows from the Exact-Dimension Theorem that

0 < ch (K(co)) 0. Second, for all > 0, 1/min{TiflTj > O} -< nf So, for all f > 0,

E[1/min{TfIT,> 0}] s n.

Proc. NatL Acad. Scd USA 84 (1987)

Finally, condition ii is satisfied. To see this set 3 (1/n2). Then

E[(>L Ti)

6

3961 =

1

-

1{Ti2S1_2i}] 2 p2 11/tq, _

for large t and any given q. Thus, if K(w) 7 4, then 0 < Weh(K(c))