A new class of exceptional self-affine fractals

Abstract

a b s t r a c tLet F be an integral self-affine set (not necessarily a self-similar set) satisfying F = T (F +A),where T −1 is an integer expanding matrix and A is a finite set of integer vectors. For ''totally disconnected F '', in 1992, Falconer obtained formulas for lower and upper bounds for the Hausdorff dimension of F . In order to have such bounds for arbitrary F , we consider an extension of Falconer's formulas to certain graph directed sets and define new bounds. For a very few classes of self-affine sets, the Hausdorff dimension and Falconer's upper bound are known to be different. In this paper, we present a new such class by using the new upper bound, and show that our upper bound is the box dimension for that class. We also study the computation of those bounds.