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

Abstract

AbstractLetNbe an integer withN≥2 and letXbe a compact subset of ℝd. If$\mathsf {S}=(S_{1},\ldots ,S_{N})$is a list of contracting similaritiesSi:X→X, then we will write$K_{\mathsf {S}}$for the self-similar set associated with$\mathsf {S}$, and we will writeMfor the family of all lists$\mathsf {S}$satisfying the strong separation condition. In this paper we show that the maps(1)and(2)are continuous; here$\dim _{\mathsf {H}}$denotes the Hausdorff dimension, ℋsdenotes thes-dimensional Hausdorff measure and 𝒮sdenotes thes-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.