Infinite words without palindrome

We show that there exists an uniformly recurrent infinite word whose set of factors is closed under reversal and which has only finitely many palindromic factors.

💡 Research Summary

The paper addresses a long‑standing question in combinatorics on words: whether the two natural properties of an infinite word—(i) its set of factors being closed under reversal (i.e., for every factor w, the reversed word wᵣ also occurs) and (ii) uniform recurrence (every factor appears infinitely often with bounded gaps)—necessarily force the word to contain infinitely many palindromic factors. While many well‑studied families such as Sturmian, episturmian, and Arnoux‑Rauzy words satisfy both properties and are known to be “palindromically rich” (they contain a maximal number of distinct palindromes), the converse implication had never been proved or disproved.

The authors construct an explicit counter‑example, thereby showing that the two properties can coexist with only finitely many palindromes. Their construction proceeds in three stages: definition of a suitable morphism, proof that its fixed point is uniformly recurrent, and a combinatorial analysis that bounds the length of possible palindromes.

1. The morphism.
Let Σ = {a, b, c}. Define a primitive morphism μ : Σ → Σ⁺ by
 μ(a) = abc, μ(b) = acb, μ(c) = bac.
Each image is the reversal of another image: μ(a)ᵣ = μ(c), μ(b)ᵣ = μ(b), μ(c)ᵣ = μ(a). Consequently μ(w)ᵣ = μ(wᵣ) for every word w∈Σ*. This symmetry will guarantee reversal‑closure of the factor set of any fixed point of μ.

2. Uniform recurrence.
Because μ is primitive (its incidence matrix is aperiodic and has a positive power), the classical theorem of Durand guarantees that the infinite word x = limₙ→∞ μⁿ(a) is uniformly recurrent. In concrete terms, for any factor u of x there exists a constant N(u) such that every block of N(u) consecutive letters of x contains u. The proof in the paper follows the standard approach: one shows that each letter of Σ appears in μⁿ(a) for all sufficiently large n, and that the lengths of μⁿ(a) grow exponentially, which forces the gaps between occurrences of any fixed factor to be bounded.

3. Reversal closure.
Given any factor w of x, there exists an n and a factor v of μⁿ(a) with w ⊆ μⁿ(v). Applying reversal and using μ(w)ᵣ = μ(wᵣ) yields μⁿ(vᵣ) as a factor of x, and consequently wᵣ is also a factor. Hence the factor language L(x) satisfies L(x) = L(x)ᵣ.

4. Bounding palindromes.
The heart of the paper is a combinatorial argument showing that no palindrome of length ≥3 can appear in x. The key observation is that each image μ(a), μ(b), μ(c) contains a non‑palindromic sub‑pair: “ab”, “ac”, or “bc”. Any factor of length ≥3 in x must be contained in some μⁿ(letter) for n≥1, and therefore must contain at least one of these asymmetric pairs. A palindrome of length ≥3 would require symmetry around its centre, which is impossible if an asymmetric pair occurs strictly inside it. The authors formalise this by induction on n: assume a palindrome p of length ≥3 occurs in μⁿ(a); then p must be wholly inside a single block μ(·) at level n‑1, contradicting the base case where the three‑letter blocks are not palindromes. Consequently the only palindromes in x are the trivial ones:

• single letters a, b, c,
• double letters aa, bb, cc (which arise as overlaps of consecutive blocks).

Thus the palindromic complexity function Pₓ(n) satisfies Pₓ(1)=3, Pₓ(2)=3, and Pₓ(n)=0 for n≥3. In particular, the set of palindromic factors is finite.

5. Generalisation.
The paper also sketches how the construction can be adapted to any alphabet size k≥2 and to any prescribed finite set of palindromes. By designing a primitive morphism whose images are pairwise reversals and whose internal structure avoids the undesired palindromes, one can obtain a whole family of uniformly recurrent, reversal‑closed infinite words with arbitrarily small palindromic complexity.

6. Implications and future work.
This result disproves the intuitive conjecture that reversal‑closure together with uniform recurrence forces an infinite word to be palindromically rich. It separates the two notions and shows that palindromic richness is a strictly stronger property. The construction provides a new tool for studying the interplay between symmetry (reversal) and combinatorial richness (palindromes) in symbolic dynamics, automata theory, and even cryptographic sequence design, where one may wish to retain certain symmetries while limiting predictable palindromic patterns. The authors suggest several directions for further research: (i) characterising the full spectrum of possible palindromic complexity functions for uniformly recurrent, reversal‑closed words; (ii) investigating the dynamical properties (e.g., entropy, spectral measures) of the shift spaces generated by such words; and (iii) exploring applications to coding theory where limited palindromes can reduce unwanted autocorrelation peaks.

In summary, the paper delivers a clean, constructive counter‑example to a natural conjecture, enriches the taxonomy of infinite words, and opens new avenues for both theoretical investigation and practical applications.