Self-similar sets with overlap

Mark McClure

University of North Carolina at Asheville

Abstract

Intuitively, a set is called self-similar if it is composed of scaled copies of the whole.  It is widely known that a set composed of n copies of itself each scaled by the factor r has fractal dimension log(n)/log(1/r), provided there is no significant overlap between the pieces.  The famous Sierpinski triangle, for example, which is composed of 3 copies each scaled by the factor 1/2, has fractal dimension log(3)/log(2).  Much less is known in the presence of significant overlap, although some important work has been done in recent years.  We present an exposition of the notion of finite-type iterated function system that can be used to generate self-similar sets with overlap and a method to compute the fractal dimension of such a set.  Unlike the non-overlapping case, there is no simple formula to compute the dimension.  Rather, there is an algorithm that can be used to produce a particular matrix; the fractal dimension is then computed in terms of the spectral radius of this matrix.  The algorithm has been implemented in Mathematica and we present examples of its use.