Golden gaskets: variations on the Sierpinski sieve

Broomhead, Dave and Montaldi, James and Sidorov, Nikita (2004) Golden gaskets: variations on the Sierpinski sieve. Nonlinearity, 17. pp. 1455-1480. ISSN 0951-7715

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Abstract

We consider the iterated function systems (IFSs) that consist of three general similitudes in the plane with centres at three non-collinear points, with a common contraction factor λ in (0, 1). As is well known, for λ = 1/2 the attractor, S_λ, is a fractal called the Sierpinski sieve and for λ < 1/2 it is also a fractal. Our goal is to study S_λ for this IFS for 1/2 < λ < 2/3 , i.e. when there are ‘overlaps’ in S_λ as well as ‘holes’. In this introductory paper we show that despite the overlaps (i.e. the breaking down of the open set condition (OSC)), the attractor can still be a totally self-similar fractal, although this happens only for a very special family of algebraic λ (so-called multinacci numbers). We evaluate the ausdorff dimension of S_λ for these special values by showing that S_λ is essentially the attractor for an infinite IFS that does satisfy the OSC. 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 S_λ has a non-empty interior. Finally we discuss higher-dimensional analogues of the model in question.

Item Type:	Article
Uncontrolled Keywords:	Iterated function system, fractal, self-similar, multinacci,
Subjects:	MSC 2010, the AMS's Mathematics Subject Classification > 11 Number theory MSC 2010, the AMS's Mathematics Subject Classification > 28 Measure and integration MSC 2010, the AMS's Mathematics Subject Classification > 37 Dynamical systems and ergodic theory
Depositing User:	Dr James Montaldi
Date Deposited:	16 Nov 2005
Last Modified:	20 Oct 2017 14:12
URI:	https://eprints.maths.manchester.ac.uk/id/eprint/43

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