# Morphisms generating antipalindromic words

## TL;DR

This paper introduces two classes of morphisms over binary alphabet whose fixed points contain infinitely many antipalindromic factors, conjectures their completeness, and proves the conjecture for specific subclasses.

## Contribution

It proposes a conjecture that these classes encompass all such morphisms and proves it for uniform morphisms and those with palindromic fixed points.

## Key findings

- Conjecture that these classes are complete for generating infinite words with antipalindromes.
- Proof of the conjecture for uniform morphisms.
- Proof of the conjecture for morphisms with palindromic fixed points.

## Abstract

We introduce two classes of morphisms over the alphabet $A=\{0,1\}$ whose fixed points contain infinitely many antipalindromic factors. An antipalindrome is a finite word invariant under the action of the antimorphism $\mathrm{E}:\{0,1\}^*\to\{0,1\}^*$, defined by $\mathrm{E}(w_1\cdots w_n)=(1-w_{n})\cdots(1-w_1)$. We conjecture that these two classes contain all morphisms (up to conjugation) which generate infinite words with infinitely many antipalindromes. This is an analogue to the famous HKS conjecture concerning infinite words containing infinitely many palindromes. We prove our conjecture for two special classes of morphisms, namely (i) uniform morphisms and (ii) morphisms with fixed points containing also infinitely many palindromes.