Fractal Properties of Generalized Sierpiński Triangles

Abstract

We calculate the box-counting dimension of a self-affine version of the Sierpiński triangle. This is done by investigating the singular values of the affine transformations. We also investigate multifractal features of self-affine measures supported by certain generalized Sierpiński triangles.(1.2)Here we will discuss self-affine fractals in R 2 . A self-affine fractal F is the attractor of an IFS consisting of a set of affine contractions S i :for i = 1, 2, . . . k, where a i ∈ R 2 is a translation vector, and T i ∈ L(R 2 , R 2 ) is the linear part of the 31 Fractals 1998.06:31-41. Downloaded from www.worldscientific.com by UNIVERSITY OF QUEENSLAND on 02/03/15. For personal use only.

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“…There have also been several extensions of these ideas to nonlinear systems, see Falconer [8] and Barreira [2]. Due to their focus on upper triangular systems, the papers of Falconer-Miao [10], Falconer-Lammering [9], Manning-Simon [15] and Bárány [1] are particularly relevant to our study.…”

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