Golden gaskets: variations on the Sierpiński sieve
Golden gaskets: variations on the Sierpiński sieve
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DOI：
10.1088/0951-7715/17/4/017
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发表时间：
2003-09
期刊：
影响因子：
1.7
通讯作者：
D. Broomhead;J. Montaldi;N. Sidorov
中科院分区：
文献类型：
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作者：
D. Broomhead;J. Montaldi;N. Sidorov
本文研究了在平面上由三个中心在三个非共线点上的一般相似点组成的迭代函数系，其压缩因子λ在（0，1）中是公共的.众所周知，当λ = 1/2时，吸引子S_λ是一个分形，称为Sierpinski筛;当λ < 1/2时，吸引子S_λ也是一个分形。我们的目标是研究当1/2 < λ < 2/3，即S_λ中既有“重叠”又有“洞”时，该IFS的S_λ。在这篇介绍性的论文中，我们证明了尽管有重叠（即开集条件（OSC）的破坏），吸引子仍然可以是一个完全自相似的分形，尽管这只发生在一个非常特殊的代数λ族（所谓的多纳奇数）。我们通过证明S_λ本质上是满足OSC的无穷IFS的吸引子来计算S_λ在这些特殊值下的ausdorff维数。我们还表明，一组点的吸引子具有独特的“地址”是自相似的，并计算其尺寸。对于λ的非多项式值，我们证明了当λ接近2/3时，S_λ有非空内部.最后，我们讨论了更高维的类似物的模型的问题。
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.