Golden 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 λ ∈ (0, 1). As is well known, for λ = 1/2 the invariant set, Sλ, is a fractal called the Sierpiński sieve, and for λ < 1/2 it is also a fractal. Our goal is to study Sλ for this IFS for 1/2 < λ < 2/3, i.e., when there are “overlaps” in Sλ as well as “holes”. In this introductory paper we show that despite the overlaps (i.e., the Open Set Condition breaking down completely), the attractor can still be a totally self-similar fractal, although this happens only for a very special family of algebraic λ’s (so-called “multinacci numbers”). We evaluate dimH(Sλ) for these special values by showing that Sλ 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 λ we show that if λ is close to 2/3, then Sλ has a nonempty interior and that if λ < 1/ √ 3 then Sλ has zero Lebesgue measure. Finally we discuss higher-dimensional analogues of the model in question.