gaskets: variations on the Sierpiński sieve

ABSTRACT. We consider the iterated function systems (IFSs) that consist of three general similitudes in the plane with centres at three non-collinear points, and with a common contraction factor ¸ 2 (0; 1). As is well known, for ¸ = 1=2 the attractor, S¸, is a fractal called the Sierpiński sieve, and for ¸ < 1=2 it is also a fractal. Our goal is to study S ¸ for this IFS for 1=2 < ¸ < 2=3, i.e., when there are “overlaps ” in S ¸ as well as “holes”. In this introductory paper we show that despite the overlaps (i.e., the breaking down of the Open Set Condition), the attractor can still be a totally self-similar fractal, although this happens only for a very special family of algebraic ¸’s (so-called “multinacci numbers”). We evaluate dimH(S¸) for these special values by showing that S ¸ is essentially the attractor for an infinite IFS which does satisfy the Open Set Condition. We also show that the set of points in the attractor with a unique “address ” is self-similar, and compute its dimension. For “non-multinacci ” values of ¸ we show that if ¸ is close to 2=3, then S ¸ has a nonempty interior. Finally we discuss higher-dimensional analogues of the model in question. INTRODUCTION AND SUMMARY Iterated function systems are one of the most common tools for constructing fractals. Usually, however, a very special class of IFSs is considered for this purpose, namely, those which satisfy th