Golden gaskets: variations on the Sierpi\'nski sieve

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