Hausdorff dimension of boundaries of self-affine tiles in R^n

Abstract

We present a new method to calculate the Hausdorff dimension of a certain class of fractals: boundaries of self-affine tiles. Among the interesting aspects are that even if the affine contraction underlying the iterated function system is not conjugated to a similarity we obtain an upper- and lower-bounds for its Hausdorff dimension. In fact, we obtain the exact value for the dimension if the moduli of the eigenvalues of the underlying affine contraction are all equal (this includes Jordan blocks). The tiles we discuss play an important role in the theory of wavelets. We calculate the dimension for a number of examples.