## 2005.16: Golden gaskets: variations on the Sierpinski sieve

2005.16:
Dave Broomhead, James Montaldi and Nikita Sidorov
(2004)
*Golden gaskets: variations on the Sierpinski sieve.*
Nonlinearity, 17.
pp. 1455-1480.
ISSN 0951-7715

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DOI: 10.1088/0951-7715/17/4/017

## 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 |
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Uncontrolled Keywords: | Iterated function system, fractal, self-similar, multinacci, |

Subjects: | MSC 2000 > 11 Number theory MSC 2000 > 28 Measure and integration MSC 2000 > 37 Dynamical systems and ergodic theory |

MIMS number: | 2005.16 |

Deposited By: | Dr James Montaldi |

Deposited On: | 16 November 2005 |

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