Computability of the packing measure of totally disconnected self-similar sets

TL;DR

This paper introduces an algorithm to accurately compute the packing measure of self-similar sets satisfying the SSC, with applications to deriving formulas for specific fractals like the Sierpinski gasket.

Contribution

It presents a convergent algorithm for exact packing measure computation of self-similar sets and demonstrates its effectiveness through examples and formulas.

Findings

Algorithm converges to the true packing measure

Accurate computation for regular self-similar sets

Derived formulas for Sierpinski gasket with certain contraction factors

Abstract

We present an algorithm to compute the exact value of the packing measure of self-similar sets satisfying the so called SSC and prove its convergence to the value of the packing measure. We also test the algorithm with examples that show both, the accuracy of the algorithm for the most regular cases and the possibility of using the additional information provided by it to obtain formulas for the packing measure of certain self-similar sets. For example, we are able to obtain a formula for the packing measure of any Sierpinski gasket with contractio factor in the interval $(0,1/3]$ (Theorem 2).