| Abstrakt: |
AbstractLet Nbe an integer with N?2 and let Xbe a compact subset of ?d. If $\mathsf {S}=(S_{1},\ldots ,S_{N})$is a list of contracting similarities Si:X?X, then we will write $K_{\mathsf {S}}$for the self-similar set associated with $\mathsf {S}$, and we will write Mfor the family of all lists $\mathsf {S}$satisfying the strong separation condition. In this paper we show that the maps (1)\begin{equation}\begin {array}{rcl} M &\to & \mathbb {R}\cr \mathsf {S} &\mapsto &\mathscr {H}^{\dim _{\mathsf {H}}(K_{\mathsf {S}})} (K_{\mathsf {S}}) \end {array}\label {eq1} \end{equation}and (2)\begin{equation}\begin {array}{rcl} M &\to & \mathbb {R}\cr \mathsf {S} &\mapsto & \mathscr {S}^{\dim _{\mathsf {H}}(K_{\mathsf {S}})}(K_{\mathsf {S}}) \end {array} \label {eq2} \end{equation}are continuous; here $\dim _{\mathsf {H}}$denotes the Hausdorff dimension, ?sdenotes the s-dimensional Hausdorff measure and ?sdenotes 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. [ABSTRACT FROM AUTHOR] |
