O ~ c}. , we obtain an equality. <=: It suffices to show that, if (xn) is a sequence of D - E converging to x (which is necessarily in D), then relation (11) is verified.

Theorem A. [13] Let {81 , ... ,8m } be a family of similarity contractions on ]Rd. Then there exists a unique, non-empty compact set E C ]Rd such that m i=1 Moreover, for any A E K{JRd) such that 8 i (A) 00 k=O where 8k is the k-th iterate of 8. C A for all i, 44 Zhi-Ying Wen This set E is called the self-similar set (888), or attractor, or invariant set of the family of similarity contractions {81 , . . ,8m }. The sets 8 il 0 . . 0 8 ik (A), 1 :S iI, ... , ik :S m are called the basic elements of A of order k.

Also, N",(c, n) = 0, so that f9(a) = -00. e. the set of a for which the spectrum belongs to [0,1]) is included in D. 2. Inequalities Since E", C E",(c), we have fd(a) ::; f~im(a) for all a. Also, d(E",(c,N))::; d(suPNE",(c,N)) implies that f~imsuP(a) ::; f~im(a). (5) (6) We will see that there is no relationship in general between fd and f~imsup. If dis O"-stable, then d(suPNE",(c,N)) = suPNd(E",(c,N)), so that f~im is identical to f~im suP. In this particular case, inequalities (5) and (6) reduce to fd(a) ::; f~im(a) = f~imSUP(a).