On theta-palindromic richness

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