Abstract
A narrow connection between infinite binary words rich in classical palindromes and infinite binary words rich simultaneously in palindromes and pseudopalindromes (the so-called $H$-rich words) is demonstrated. The correspondence between rich and $H$-rich words is based on the operation $S$ acting over words over the alphabet $\{0,1\}$ and defined by $S(u_0u_1u_2\ldots) = v_1v_2v_3\ldots$, where $v_i= u_{i-1} + u_i \mod 2$. The operation $S$ enables us to construct a new class of rich words and a new class of $H$-rich words. Finally, the operation $S$ is considered on the multiliteral alphabet $\mathbb{Z}_m$ as well and applied to the generalized Thue--Morse words. As a byproduct, new binary rich and $H$-rich words are obtained by application of $S$ on the generalized Thue--Morse words over the alphabet $\mathbb{Z}_4$.
Highlights
In the present paper we concentrate on construction of infinite words which are filled with palindromes or pseudopalindromes to the highest possible level, the so-called rich words
A finite word w = w0w1 · · · wn−1 is called palindrome if w coincides with its reversal R(w) = wn−1wn−2 · · · w1w0
Blondin Masse, Brlek, Garon and Labbeshowed in Blondin Masseet al. (2011) that rich words include complementary-symmetric Rote words. They can be defined as binary words with factor complexity Cu(n) = 2n for every nonzero integer n and with language closed under the exchange of letters, see Rote (1993). This implies that the language of a complementary-symmetric Rote word is closed under two mappings acting on the set {0, 1}∗ of all finite binary words: the first is R and the second is E defined by E(w0 · · · wn) = E(wn) · · · E(w0) for letters wi and E(0) = 1 and E(1) = 0
Summary
In the present paper we concentrate on construction of infinite words which are filled with palindromes or pseudopalindromes to the highest possible level, the so-called rich words. Namely k-ary Arnoux–Rauzy words and words coding k-interval exchange transformation with symmetric interval permutation, are rich as well Both mentioned classes have their language closed under reversal. (2011) that rich words include complementary-symmetric Rote words They can be defined as binary words with factor complexity Cu(n) = 2n for every nonzero integer n and with language closed under the exchange of letters, see Rote (1993). This implies that the language of a complementary-symmetric Rote word is closed under two mappings acting on the set {0, 1}∗ of all finite binary words: the first is R and the second is E defined by E(w0 · · · wn) = E(wn) · · · E(w0) for letters wi and E(0) = 1 and E(1) = 0.
Sign-in/Register to view highlights & summary Sign-in/Register to access full text options 
Free) 
Talk to us
Join us for a 30 min session where you can share your feedback and ask us any queries you have
Schedule a call
