On morphisms preserving palindromic richness
share
Droubay, Justin and Pirillo that each word of length n contains at most n+1 distinct palindromes. A finite "rich word" is a word with maximal number of palindromic factors. The definition of palindromic richness can be naturally extended to infinite words. Sturmian words and Rote complementary symmetric sequences form two classes of binary rich words, while episturmian words and words coding d-interval exchange transformations give us other examples on larger alphabets. In this paper we look for homomorphisms of the free monoid, which allow to construct new rich words from already known rich words. In particular we study two types of morphisms: Arnoux-Rauzy morphisms and morphisms from Class P_ret. These morphisms contain Sturmian morphisms as a subclass. We show that Arnoux-Rauzy morphisms preserve the set of all rich words. We also characterize P_ret morphisms which preserve richness on binary alphabet.
READ FULL TEXT