Continuity of packing measure function of self-similar iterated function systems

Abstract

In this paper, we focus on the packing measure of self-similar sets. Let $K$ be a self-similar set whose Hausdorff dimension and packing dimension equal $s$, we state that if $K$ satisfies the strong open set condition with an open set $\mathcal{O}$, then $\mathcal{P}^s(K\cap B(x,r))\geq (2r)^s$ for each open ball $B(x,r)\subset \mathcal{O}$ centered in $K$, where $\mathcal{P}^s$ denotes the $s$-dimensional packing measure. We use this inequality to obtain some precise density theorems for packing measure of self-similar sets, which can be applied to compute the exact value of the $s$-dimensional packing measure $\mathcal{P}^s(K)$ of $K$. Moreover, by using the above results, we show the continuity of the packing measure function of the attractors on the space of self-similar iterated function systems satisfying the strong separation condition. This result gives a complete answer to a question posed by L. Olsen.