Hausdorff measure of uniform self-similar fractals

Abstract

Let d ≥ 1 be an integer and E a self-similar fractal set, which is the attractor of a uniform contracting iterated function system (UIFS) on R d. Denote by D the Hausdorff dimension, by H D(E) the Hausdorff measure and by diam(E) the diameter of E. If the UIFS is parametrised by its contracting factor c, while the set ω of fixed points of the UIFS does not depend on c, we will show the existence of a positive constant depending only on ω, such that the Hausdorff dimension is smaller than one and H D(E) = diam(E)D if c is smaller than this constant. We apply our result to modified versions of various classical fractals. Moreover, we present a parametrised UIFS, where ω depends on c and show the inequatily H D(E) < diam(E)D, if c is small enough.