Division Point Measures from Primitive Substitutions

TL;DR

This paper extends previous work on substitution-based point measures, demonstrating that for a broad class of primitive substitutions on polygons in the plane, the associated atomic measures converge to Lebesgue measure.

Contribution

It generalizes existing results by establishing convergence of atomic measures to Lebesgue measure for primitive substitutions on polygons in two dimensions.

Findings

Atomic measures from primitive substitutions converge to Lebesgue measure

Results apply to a general class of polygon substitutions in 

Extends prior work on substitution limits in measure theory

Abstract

In this note we extend results of Olli concerning limits of point measures arising from substitutions. We consider a general primitive substitution on a finite polygon set in $\mathbb{R}^2$ and show that limits of certain atomic measures each converge to Lebesgue measure.