Hausdorff and packing measure functions of self-similar sets: continuity and measurability

Let N be an integer with N >= 2 and let X be a compact subset of R-d. If S = (S-1, ..., S-N) is a list of contracting similarities S-i : X -> X, then we will write K-S for the self-similar set associated with S, and we will write M for the family of all lists S satisfying the strong separation condition. In this paper we show that the maps

M -> R

S -> H-dimH(KS)(KS) (1)

and

M -> R

S -> S-dimH(KS)(KS) (2)

are continuous; here dim(H) denotes the Hausdorff dimension, H-s denotes the s-dimensional Hausdorff measure and S-s denotes the s-dimensional spherical Hausdorff measure. In fact, we prove a more general continuity result which, amongst other things, implies that the maps in (1) and (2) are continuous.