Note on BDL property of fixed points of primitive morphisms

TL;DR

This paper investigates the conditions under which infinite words fixed by primitive morphisms can be geometrically represented in a way that closely resembles a lattice, focusing on bounded distance equivalence.

Contribution

It provides a necessary condition for such infinite words to have a geometric representation that is bounded distance equivalent to a lattice.

Findings

Identifies a necessary condition for geometric representations of fixed points of primitive morphisms.

Connects the BDL property with the structure of fixed points of primitive morphisms.

Advances understanding of the geometric properties of infinite words generated by morphisms.

Abstract

We consider an infinite word $\boldsymbol{u}$ fixed by a primitive morphism. We show a necessary condition under which $\boldsymbol{u}$ has a non-trivial geometric representation which is bounded distance equivalent to a lattice.