1 research outputs found
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 \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.Comment: 27 pages, 10 figure