On Theta-palindromic Richness

TL;DR

This paper extends the concept of palindromes to $ heta$-palindromes using involutory antimorphisms, exploring their complexity, richness, and characterizations in various word sets.

Contribution

It generalizes the notion of palindromic richness to $ heta$-palindromes and provides characterizations and criteria for $ heta$-rich words, including $ heta$-episturmian words.

Findings

Established relation between $ heta$-palindromic and factor complexity.

Provided criteria for $ heta$-richness in $ heta$-episturmian words.

Presented examples of $ heta$-rich words.

Abstract

In this paper we study generalization of the reversal mapping realized by an arbitrary involutory antimorphism $\Theta$. It generalizes the notion of a palindrome into a $\Theta$-palindrome -- a word invariant under $\Theta$. For languages closed under $\Theta$ we give the relation between $\Theta$-palindromic complexity and factor complexity. We generalize the notion of richness to $\Theta$-richness and we prove analogous characterizations of words that are $\Theta$-rich, especially in the case of set of factors invariant under $\Theta$. A criterion for $\Theta$-richness of $\Theta$-episturmian words is given together with other examples of $\Theta$-rich words.