A new class of exceptional self-affine fractals

Let FF be an integral self-affine set (not necessarily a self-similar set) satisfying F=T(F+A)F=T(F+A), where T−1T−1 is an integer expanding matrix and AA is a finite set of integer vectors. For “totally disconnected FF”, in 1992, Falconer obtained formulas for lower and upper bounds for the Hausdorff dimension of FF. In order to have such bounds for arbitrary FF, 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.