Equation $x^iy^jx^k=u^iv^ju^k$ in words

We will prove that the word $a^ib^ja^k$ is periodicity forcing if $j \geq 3$ and $i+k \geq 3$, where $i$ and $k$ are positive integers. Also we will give examples showing that both bounds are optimal.

💡 Research Summary

The paper investigates the periodicity‑forcing property of binary words of the form $a^{i}b^{j}a^{k}$. A word $w$ is called periodicity‑forcing if, for any two morphisms $g,h\colon A^{}\to\Sigma^{}$, the equality $g(w)=h(w)$ forces either $g=h$ or both morphisms are periodic (i.e., each image is a power of a shorter word). Earlier work by Karhumäki and Culik II showed that the shortest such words have length five and that the equation $x^{2}y^{3}x^{2}=u^{2}v^{3}u^{2}$ admits only periodic solutions. The present work generalises this result: it proves that $a^{i}b^{j}a^{k}$ is periodicity‑forcing whenever $j\ge3$ and $i+k\ge3$, with $i,k>0$. Moreover, the authors exhibit examples showing that these bounds cannot be lowered.

The authors begin with a concise review of combinatorics on words: prefixes, suffixes, maximal common prefix/suffix, primitive root, conjugacy, and binary codes. The central technical tool is the Periodicity Lemma (also known as the Fine‑Wilf theorem): if two infinite repetitions $p^{\omega}$ and $q^{\omega}$ share a factor of length at least $|p|+|q|-1$, then $p$ and $q$ are conjugate, and if they are also prefix‑comparable they must be equal. Several auxiliary lemmas (Lemmas 1–14) are proved, dealing with maximal common prefixes in binary codes, properties of $X$‑primitive words (words that cannot be expressed as a non‑trivial power within the submonoid generated by a binary code $X={x,y}$), and the structure of $X$‑primitive but imprimitive words (they must belong to $x^{}y\cup y^{}x$).

The main theorem (Theorem 1) states: let $x,y,u,v\in A^{*}$ with $x\neq u$ and $x^{i}y^{j}x^{k}=u^{i}v^{j}u^{k}$, where $i,k>0$, $j\ge3$, and $i+k\ge3$. Then $x,y,u,v$ all commute pairwise. The proof proceeds by a detailed case analysis:

1. Case (A) – $p_{x}=p_{u}$ (the primitive roots coincide). Lemma 9 immediately forces commutation.
2. Case (B) – $p_{y}$ and $p_{v}$ are conjugate. By examining the structure of the equation, the authors show that a long common factor appears between $x^{i}y^{j}x^{k}$ and $u^{i}v^{j}u^{k}$; the Periodicity Lemma then yields $y$ and $v$ conjugate, and Lemma 9 forces them to be equal, leading to commutation.
3. Case (C) – $p_{x}$ and $p_{v}$ are conjugate. Using the inequality $(i+k-1)|u|<|p_{x}|$ (derived from the Periodicity Lemma), they prove that $u^{i}v$ is a prefix of $p_{x}^{2}$, which forces $u$ to be a power of $p_{x}$. Consequently $x$ and $u$ commute.

After handling these three principal configurations, the authors split the remaining analysis according to the relative lengths of $x$ and $v$. For each subcase (e.g., $|x|\ge|v|$, $|x|<|v|$ with various relations among $i|x|$, $k|x|$, and $i|u|+|v|$) they apply Lemmas 10–14 to deduce that certain intermediate words $\sigma,\tau$ must commute; otherwise a contradiction arises via the Periodicity Lemma or Lemma 9. The most intricate parts (labelled 3a, 3b, 4a, etc.) involve constructing auxiliary words $r$, $z$, $z’$ and showing that if $\sigma$ and $\tau$ did not commute, one would obtain a factor of $v^{j-2}v’$ that cannot be expressed as a $j$‑th power of a word from ${\sigma,\tau}^{*}$, contradicting Lemma 9.

Finally, the paper presents counter‑examples showing optimality of the bounds: when $j=2$ or $i+k=2$, explicit non‑periodic solutions to the equation exist, demonstrating that the conditions $j\ge3$ and $i+k\ge3$ cannot be weakened.

In conclusion, the authors have completely characterised the periodicity‑forcing behaviour of the family $a^{i}b^{j}a^{k}$, extending previous results from the specific case $(i,j,k)=(2,3,2)$ to all admissible triples with $j\ge3$ and $i+k\ge3$. The work combines classical combinatorial word theory with careful case analysis, and the optimality discussion underscores the sharpness of the obtained criteria. This contribution deepens our understanding of word equations, periodicity, and the structure of morphisms that preserve specific words, and it sets a solid foundation for future investigations into more complex alphabets or higher‑arity word equations.