On the Zero Defect Conjecture

TL;DR

This paper proves the Zero Defect Conjecture for binary alphabets and identifies classes of morphisms where it remains valid, advancing understanding of palindromic defects in fixed points of primitive morphisms.

Contribution

It confirms the conjecture for binary alphabets and describes conditions under which it holds for multiliteral alphabets, using properties of extension graphs.

Findings

The conjecture is true for binary alphabets.

Counterexamples exist over ternary alphabets.

Extension graphs are key to the proof.

Abstract

Brlek et al. conjectured in 2008 that any fixed point of a primitive morphism with finite palindromic defect is either periodic or its palindromic defect is zero. Bucci and Vaslet disproved this conjecture in 2012 by a counterexample over ternary alphabet. We prove that the conjecture is valid on binary alphabet. We also describe a class of morphisms over multiliteral alphabet for which the conjecture still holds. The proof is based on properties of extension graphs.