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

TL;DR

This paper investigates the packing measure of self-similar sets, establishing inequalities and density theorems that enable precise calculation of the measure and proving the continuity of the packing measure function under certain conditions.

Contribution

It provides new inequalities and density results for the packing measure of self-similar sets, and proves the continuity of the packing measure function for attractors satisfying the strong separation condition.

Findings

Established a lower bound for packing measure within open balls in the set

Derived density theorems enabling exact computation of packing measure

Proved the continuity of the packing measure function for certain self-similar 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.