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 between
Θ-palindromic complexity and factor complexity. We generalize the notion
of richness to Θ-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.Comment: 14 page