A new class of exceptional self-affine fractals

Let 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.