Palindromic Recurrent Words Outside Morphic Class $P$

Definitions and mathematical statement

Let $A$ be a finite nonempty alphabet, let $A^*$ be the set of finite words over $A$, including the empty word $\varepsilon$, and write $u^{\mathrm{rev}}$ for the reversal of a finite word $u$. A palindrome is a word $u$ satisfying $u=u^{\mathrm{rev}}$. For a one-sided infinite word $w\in A^{\{0,1,2,\ldots\}}$, let $\operatorname{Fact}(w)$ denote its finite contiguous factors. The word $w$ is uniformly recurrent if, for every nonempty $u\in\operatorname{Fact}(w)$, there exists an integer $L\geq |u|$ such that every length-$L$ factor of $w$ contains $u$. It contains arbitrarily long palindromes if \[ \forall n\geq 1\ \exists u\in\operatorname{Fact}(w):\quad |u|\geq n\quad\text{and}\quad u=u^{\mathrm{rev}}. \] A morphism $\sigma:A^*\to A^*$ respects concatenation. It is primitive if there exists an integer $k\geq1$ such that, for every $a,b\in A$, the letter $b$ occurs in $\sigma^k(a)$. Such a morphism is nonerasing and acts on one-sided infinite words by concatenating letter images. Under the usual common-prefix convention, $\sigma$ belongs to class $P$ if there exist palindromes $p\in A^*$ and $q_a\in A^*$, one for each $a\in A$, such that \[ \sigma(a)=p q_a\qquad(a\in A). \] Here the empty word is allowed as a palindrome.

Conditional precise formulation. If "arise from primitive morphisms" means "are fixed points of primitive morphisms on their own alphabet," the supplied question asks whether there exist a finite nonempty alphabet $A$ and a one-sided infinite word $w$ whose occurring letters are exactly $A$, such that $w$ is uniformly recurrent, contains arbitrarily long palindromic factors, and satisfies \[ \begin{gathered} \exists\sigma:A^*\to A^*:\quad \sigma\text{ is primitive and }\sigma(w)=w,\\ \forall\tau:A^*\to A^*:\quad \left(\tau\text{ is primitive and }\tau(w)=w\right) \Longrightarrow \tau\notin P. \end{gathered} \] Unresolved qualifications. The supplied record does not define class $P$, specify whether infinite words are one-sided or two-sided, or define "arise from." The displayed formulation uses the stated one-sided fixed-point interpretation and the stated common-prefix definition of class $P$; these conventions require confirmation. If generation instead allows a letter-to-letter coding of a primitive fixed point on another alphabet, or refers to the factor language rather than an exact fixed point, both the generating objects and the universal quantifier must be specified anew. The supplied wording also requires confirmation that "none" excludes every primitive morphism generating the same word, as in the display. No unique unconditional formalization follows from the supplied excerpt alone.

Short English statement

Under the direct fixed-point interpretation, the question asks whether a uniformly recurrent infinite word can have arbitrarily long palindromic factors and be fixed by a primitive morphism, while no primitive morphism fixing that word belongs to class $P$. The intended generation relation and class convention remain unresolved. This is a statement-only draft requiring human mathematical review.

Sources

• Supplied local problem registry, data/unsolvedmath/problems.json, permanent UnsolvedMath ID $30001549$, exact source problem code catalogue entry; collection: Oberwolfach Reports Open Problems. No individual proposer is supplied. The proposed year is recorded as 2010.
• Supplied report citation: Mini-Workshop: Combinatorics on Words (2011), Oberwolfach Reports, pp. 2195--2244, DOI 10.4171/owr/2010/37. Supplied source URL and literature-source label "original OWR report": https://doi.org/10.4171/owr/2010/37. The citation's 2011 label and the record's 2010 year information are retained without reconciliation. The background describes Oberwolfach Reports as documenting mathematical workshops and their open-problem discussions.
• Supplied literature-source label "recent Fibonacci-word study": https://www.sciencedirect.com/science/article/pii/S0304397526001350. No authors, article title, or supporting passage are supplied.
• Literature assessment dated exactly 2026-08-21: the supplied curation labels the problem open and states that the extracted characterization problem for uniformly recurrent morphic words with arbitrarily long palindromes remains unresolved by the cited later work. This is supplied dated curation, not an independently verified literature assessment. The background explicitly requests independent verification of time-sensitive status and cited claims.
• The registry labels the statement corrected_verified; its supplied verification explanation says only that the named morphism class was placed in a mathematical delimiter without changing the question. This does not establish mathematical verification of the formulation or its open status. The report and linked study have not been independently consulted for this draft.