A new class of exceptional self-affine fractals

Abstract: Let be an integral self-affine set (not necessarily a self-similar set) satisfying , where is an integer expanding matrix and is a finite set of integer vectors. For “totally disconnected ”, in 1992, Falconer obtained formulas for lower and upper bounds for the Hausdorff dimension of . In order to have such bounds for arbitrary , 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. [Copyright &y& Elsevier]