Palindromic sequences generated from marked morphisms

TL;DR

This paper proves that palindromic fixed points of marked and primitive morphisms have a power conjugate to a morphism in class P, generalizing previous results and emphasizing the importance of marked morphisms in palindromic word structure.

Contribution

It introduces more general definitions of marked morphisms and proves that palindromic fixed points imply conjugacy to class P morphisms, extending prior work.

Findings

Palindromic fixed points imply conjugacy to class P morphisms.

Marked morphisms are crucial for the structure of palindromic words.

Generalized definitions encompass previous results over binary and ternary alphabets.

Abstract

Fixed points ${\bf u}=\varphi({\bf u})$ of marked and primitive morphisms $\varphi$ over arbitrary alphabet are considered. We show that if ${\bf u}$ is palindromic, i.e., its language contains infinitely many palindromes, then some power of $\varphi$ has a conjugate in class ${\mathcal P}$. This class was introduced by Hof, Knill, Simon (1995) in order to study palindromic morphic words. Our definitions of marked and well-marked morphisms are more general than the ones previously used by Frid (1999) or Tan (2007). As any morphism with aperiodic fixed point over binary alphabet is marked, our result generalizes the result of Tan. Labb\'e (2014) demonstrated that already on a ternary alphabet the property of morphisms to be marked is important for the validity of our theorem. The main tool used in our proof is the description of bispecial factors in fixed points of morphisms provided by Klouda (2012).