Good labeling property of simple nested fractals

TL;DR

This paper investigates criteria for nested fractals to possess the good labeling property (GLP), providing characterizations, reductions, and examples that clarify when GLP holds based on the number of essential fixed points.

Contribution

It introduces new criteria and characterizations for GLP in nested fractals, especially relating to the number of essential fixed points, including cases with prime, prime power, and composite counts.

Findings

Fractals with a number of fixed points as a power of two always have GLP.

Fractals with a prime number of fixed points also have GLP.

Examples show that for other fixed point counts, GLP may or may not hold.

Abstract

We show various criteria to verify if a given nested fractal has a good labeling property, inter alia we present a characterization of GLP for fractals with an odd number of essential fixed points. We show a convenient reduction of area to be investigated in verification of GLP and give examples that further reduction is impossible. We prove that if a number of essential fixed points is a power of two, then a fractal must have GLP and that there are no values other than primes or powers of two guaranteeing GLP. For all other numbers of essential fixed points we are able to construct examples having and other not having GLP.