On the Dimension of Self-Affine Fractals

Abstract

An n ×n matrix M is called expanding if all its eigenvalues have moduli > 1. Let A be a nonempty finite set of vectors in the n-dimensional Euclidean space. Then there exists a unique nonempty compact set F satisfying \(MF\,=\,F + A\). F is called a self-affine set or a self-affine fractal. F can also be considered as the attractor of an affine iterated function system. Although such sets are basic structures in the theory of fractals, there are still many problems on them to be studied. Among those problems, the calculation or the estimation of fractal dimensions of F is of considerable interest. In this work, we discuss some problems about the singular value dimension of self-affine sets. We then generalize the singular value dimension to certain graph directed sets and give a result on the computation of it. On the other hand, for a very few classes of self-affine fractals, the Hausdorff dimension and the singular value dimension are known to be different. Such fractals are called exceptional self-affine fractals. Finally, we present a new class of exceptional self-affine fractals and show that the generalized singular value dimension of F in that class is the same as the box (counting) dimension.