On Theta-palindromic Richness

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