Infinite words rich and almost rich in generalized palindromes

Abstract

We focus on $\Theta$-rich and almost $\Theta$-rich words over a finite alphabet $\mathcal{A}$, where $\Theta$ is an involutive antimorphism over ${\mathcal{A}}^{\ast }$. We show that any recurrent almost $\Theta$-rich word $\uu$ is an image of a recurrent ${\Theta }^{\prime }$-rich word under a suitable morphism, where ${\Theta }^{\prime }$ is again an involutive antimorphism. Moreover, if the word $\uu$ is uniformly recurrent, we show that ${\Theta }^{\prime }$ can be set to the reversal mapping. We also treat one special case of almost $\Theta$-rich words. We show that every $\Theta$-standard words with seed is an image of an Arnoux-Rauzy word.

10 pages

Cited by in corpus (3)

• Languages invariant under more symmetries: overlapping factors versus palindromic richness
• Factor frequencies in languages invariant under more symmetries
• Factor frequencies in generalized Thue-Morse words