The geometry of self-affine fractals

Abstract

In this thesis we study the dimension theory of self-affine sets. We begin by introducing a number of notions from fractal geometry, in particular, dimensions, measure properties and iterated functions systems. We give a review of existing work on self-affine sets. We then develop a variety of new results on self-affine sets and their dimensional properties. This work falls into three parts: Firstly, we look at the dimension formulae for a class of self-affine sets generated by upper triangular matrices. In this case, we simplify the affine dimension formula into equations only involving the diagonal elements of the matrices. Secondly, since the Hausdorff dimensions of self-affine sets depend not only on the linear parts of the contractions but also on the translation parameters, we obtain an upper bound for the dimensions of exceptional sets, that is, the set of parameters such that the Hausdorff dimension of the attractor is smaller than the affine dimension. Thirdly, we investigate dimensions of a class of random self-affine sets, aiming to extend the ‘almost sure’ formula for random self-similar sets to random self-affine sets.