Golden gaskets: variations on the Sierpi\'nski 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 $\la\in(0,1)$. As is well known, for $\la=1/2$ the invariant set, $\S_\la$, is a fractal called the Sierpi\'nski sieve, and for $\la<1/2$ it is also a fractal. Our goal is to study $\S_\la$ for this IFS for $1/2<\la<2/3$, i.e., when there are "overlaps" in $\S_\la$ 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 $\la$'s (so-called "multinacci numbers"). We evaluate $\dim_H(\S_\la)$ for these special values by showing that $\S_\la$ 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 $\la$ we show that if $\la$ is close to 2/3, then $\S_\la$ has a nonempty interior and that if $\la<1/\sqrt{3}$ then \S_\la$ has zero Lebesgue measure. Finally we discuss higher-dimensional analogues of the model in question.