Almost Rich Words as Morphic Images of Rich Words

We focus on Θ-rich and almost Θ-rich words over a finite alphabet A, where Θ is an involutive antimorphism over A∗. We show that any recurrent almost Θ-rich word u is an image of a recurrent Θ′-rich word under a suitable morphism, where Θ′ is also an involutive antimorphism. Moreover, if the word u is uniformly recurrent, we show that Θ′ can be set to the reversal mapping. We also treat one special case of almost Θ-rich words: we show that every Θ-standard word with seed is an image of an Arnoux-Rauzy word.