Packing measure of super separated iterated function systems

Abstract

Let J be the limit set of an iterated function system in $\mathbb {R}^d$ satisfying the open set condition. It is well known that the h-dimensional packing measure of J is positive and finite when h is given by Hutchinson’s formula. However, it may be hard to find a formula for the h-dimensional packing measure of J. We introduce the super separation condition and use it to reduce the problem of computing the packing measure to checking densities of a finite number of balls around each point in the limit set. We then use this fact to find formulas for the packing measure of a class of Cantor sets in $\mathbb {R}$, a class of fractals based on regular convex polygons in $\mathbb {R}^2$, and a class of fractals based on regular simplexes in $\mathbb {R}^d$ for $d \ge 3$.