MLD Relations of Pisot Substitution Tilings

TL;DR

This paper explores the relationships between 1D Pisot substitution tilings, focusing on conditions for local isomorphism and mutual local derivability using automorphisms of the free group, with examples illustrating these phenomena.

Contribution

It introduces a group-theoretic framework for analyzing tiling relations, showing how automorphisms determine LI and MLD properties, including cases where substitution matrices differ.

Findings

LI corresponds to conjugation by inner automorphisms

MLD can occur even with conjugate matrices in GL(n,Z)

Windows of MLD tilings can be reconstructed from each other

Abstract

We consider 1-dimensional, unimodular Pisot substitution tilings with three intervals, and discuss conditions under which pairs of such tilings are locally isomorhphic (LI), or mutually locally derivable (MDL). For this purpose, we regard the substitutions as homomorphisms of the underlying free group with three generators. Then, if two substitutions are conjugated by an inner automorphism of the free group, the two tilings are LI, and a conjugating outer automorphism between two substitutions can often be used to prove that the two tilings are MLD. We present several examples illustrating the different phenomena that can occur in this context. In particular, we show how two substitution tilings can be MLD even if their substitution matrices are not equal, but only conjugate in $GL(n,\mathbb{Z})$. We also illustrate how the (in our case fractal) windows of MLD tilings can be reconstructed from each other, and discuss how the conjugating group automorphism affects the substitution generating the window boundaries.