A generalized palindromization map in free monoids

The palindromization map $\psi$ in a free monoid $A^$ was introduced in 1997 by the first author in the case of a binary alphabet $A$, and later extended by other authors to arbitrary alphabets. Acting on infinite words, $\psi$ generates the class of standard episturmian words, including standard Arnoux-Rauzy words. In this paper we generalize the palindromization map, starting with a given code $X$ over $A$. The new map $\psi_X$ maps $X^$ to the set $PAL$ of palindromes of $A^$. In this way some properties of $\psi$ are lost and some are saved in a weak form. When $X$ has a finite deciphering delay one can extend $\psi_X$ to $X^{\omega}$, generating a class of infinite words much wider than standard episturmian words. For a finite and maximal code $X$ over $A$, we give a suitable generalization of standard Arnoux-Rauzy words, called $X$-AR words. We prove that any $X$-AR word is a morphic image of a standard Arnoux-Rauzy word and we determine some suitable linear lower and upper bounds to its factor complexity. For any code $X$ we say that $\psi_X$ is conservative when $\psi_X(X^{})\subseteq X^{}$. We study conservative maps $\psi_X$ and conditions on $X$ assuring that $\psi_X$ is conservative. We also investigate the special case of morphic-conservative maps $\psi_{X}$, i.e., maps such that $\phi\circ \psi = \psi_X\circ \phi$ for an injective morphism $\phi$. Finally, we generalize $\psi_X$ by replacing palindromic closure with $\theta$-palindromic closure, where $\theta$ is any involutory antimorphism of $A^$. This yields an extension of the class of $\theta$-standard words introduced by the authors in 2006.

💡 Research Summary

The paper introduces a broad generalization of the classic palindromization map ψ, which originally maps any word in a free monoid A* to the shortest palindrome having that word as a prefix. By fixing a code X ⊆ A⁺ (a set of non‑empty words that admits unique factorization), the authors define a new operator ψ_X : X* → PAL, where PAL denotes the set of all palindromes over A. The definition is inductive: ψ_X(ε)=ε, ψ_X(x)=x^{(+)} for each code word x, and ψ_X(wx) = (ψ_X(w)·x)^{(+)} where (·)^{(+)} is the right‑palindromic closure. When X = A, ψ_X coincides with the original ψ.

The paper first studies elementary properties of ψ_X. Unlike ψ, ψ_X is not injective in general; however, if X is a prefix code, ψ_X becomes injective. Moreover, ψ_X(w) is always a prefix of ψ_X(wx), preserving the monotonicity that underlies the extension to infinite words.

If X possesses a finite deciphering delay (i.e., any infinite concatenation of elements of X can be uniquely factorized), ψ_X can be extended to a map ψ_X : X^ω → A^ω on infinite words. The image ψ_X(t) of an infinite word t is always closed under reversal: every factor’s reversal also occurs. When X is a prefix code, the extended map remains injective. The authors illustrate that standard Sturmian words arise from ψ_X when X = {ab, ba}, and that the Thue–Morse word can be generated using an appropriate infinite code.

A central contribution is the notion of X‑AR (generalized Arnoux‑Rauzy) words. Assuming X is a finite maximal code (hence a maximal prefix code with deciphering delay zero), an infinite word y = x₁x₂… ∈ X^ω is called X‑AR if each code word x ∈ X occurs infinitely often in the factorization of y. The associated infinite palindrome s = ψ_X(y) is then an X‑AR word. The authors prove several structural facts:

• s is ω‑power‑free: no non‑empty factor can be repeated arbitrarily many times.
• For sufficiently large n, the number S_r(n) of right‑special factors of length n satisfies a lower bound (|X|−1)/(d−1), where d = |A|.
• The factor complexity p_s(n) grows linearly, with a lower bound (|X|−1)n + c and an upper bound 2|X|n + b (c, b ∈ ℤ). The upper bound follows from a generalized Justin formula for right‑special factors.
• Every X‑AR word is a morphic image of a standard Arnoux‑Rauzy word over an alphabet of size |X|. Concretely, there exists an injective coding φ : A’⁎ → X⁎ (|A’| = |X|) such that ψ_X ∘ φ = φ ∘ ψ, where ψ is the classical palindromization map.

The paper then investigates “conservative” maps: ψ_X is called conservative if ψ_X(X*) ⊆ X*. A sufficient condition for conservativity is that X ⊆ PAL (all code words are palindromes) and X is a bifix code (both prefix and suffix). In the special case where an injective morphism φ satisfies φ(A) = X, ψ_X is called morphic‑conservative if φ∘ψ = ψ_X∘φ. The authors show that morphic‑conservativity forces X ⊆ PAL and X to be a prefix, bifix, and conservative code; consequently ψ_X is injective. Moreover, ψ_X is morphic‑conservative iff X ⊆ PAL, X is a prefix code, and ψ_X is conservative. They also define a weaker notion, “weakly conservative”: ψ_X(t) ∈ X^ω for every t ∈ X^ω. While every conservative map is weakly conservative, the converse fails in general; however, any ψ_X derived from a finite maximal code is weakly conservative.

Finally, the authors extend the framework by replacing ordinary palindromic closure with θ‑palindromic closure, where θ is any involutory antimorphism of A*. This yields a generalized map ψ_{θ,X} : X* → PAL_θ (θ‑palindromes). When X has finite deciphering delay, ψ_{θ,X} extends to X^ω, producing a class of infinite words strictly larger than the previously studied θ‑standard words. The key relationship ψ_{θ,X} = μ_θ ∘ ψ_X ∘ μ_θ^{-1} is established, where μ_θ maps each letter a to a if a = θ(a) and to aθ(a) otherwise. This shows that the θ‑generalization is essentially a conjugate of the original ψ_X.

Overall, the paper provides a comprehensive theory that unifies code theory, palindromic closure, and combinatorics on words. By allowing arbitrary codes as the directing alphabet, it vastly enlarges the family of words obtainable via iterated palindromic closure, offers precise complexity bounds, and connects these new families to classical episturmian and Arnoux‑Rauzy words through morphic images. The work opens avenues for further exploration of word families defined by other closure operators, as well as applications to symbolic dynamics and the study of low‑complexity sequences.