Uniform Morphisms Without Powers

• Uniform Morphisms Without Powers are fixed-length mappings that produce infinite words free from repeated patterns like squares and cubes.
• Constructions utilize methods such as the Thue–Morse morphism and computer-assisted designs to ensure primitivity and power-freeness.
• These morphisms bridge combinatorics, automata theory, and coding applications by generating regular, automatic sequences with practical pattern-avoidance properties.

A uniform morphism without powers is a morphism in the combinatorics on words that maps each letter of the input alphabet to a word of the same fixed length (uniformity) and, crucially, guarantees the avoidance of nontrivial word powers—potentially fractional, but often integer such as squares or cubes—in the images of power-free words. Uniform morphisms without powers constitute a central methodology for generating infinite sequences that avoid given patterns, and they play a foundational role in the construction of lexicographically least infinite words that avoid specified powers over both finite and infinite alphabets. The study of such morphisms directly connects to the structure of automatic and regular sequences, word avoidability, and the broader theory of symbolic dynamics and coding.

1. Definitions and Theoretical Framework

Let $\Sigma$ and $\Delta$ be alphabets. A morphism $\varphi:\Sigma^*\to\Delta^*$ is called $k$-uniform if $|\varphi(a)|=k$ for all $a\in\Sigma$. A word $w$ is $k$-power-free if it contains no factor $x^k$, $x\neq\varepsilon$; more generally, for rationals $a/b>1$, a word is $(a/b)$-power-free if no factor is of the form $v^{a/b}$, as formally detailed in (Pudwell et al., 2015). A morphism is $k$-power-free if it maps $k$-power-free words to $k$-power-free words, and similarly for fractional exponents.

A foundational result asserts that any uniform morphism that is $k$-power-free for $k\ge2$ is primitive: it sends primitive words to primitive words (Wlazinski, 27 Jan 2026). Primitive morphisms are injective and ensure that no nonempty word is mapped to a nontrivial power, underpinning their relevance in avoidability and symbolic dynamics.

2. Exemplary Constructions and Main Theorems

Systematic constructions of uniform morphisms without powers have been established for various powers and alphabets. Currie and Rampersad demonstrate that for every $k\ge0$, there exists a $k$-uniform morphism over the binary alphabet $\{0,1\}$ that is cubefree (i.e., avoids cubes $xxx$) (0812.4470). Explicit forms for small $k$ include: the identity for $k=1$, the Thue–Morse morphism for $k=2$, and computer-generated examples for $k\in\{3,5\}$. For large odd $k$, the construction uses cubefree words of the form $00x11$ and combinatorial arguments concerning the Thue–Morse sequence.

For infinite alphabets such as $\mathbb{N}$, morphisms avoiding fractional powers have been constructed, notably for lex least $(a/b)$-power-free words. The mapping often takes the form $\varphi(n)=0^{a-1}(n+1)$ for integer powers, with more intricate templates for fractional exponents. For example, the lexicographically least $5/4$-power-free word on $\mathbb{Z}_{\ge0}$ is generated by a 6-uniform morphism over an "8-typed" alphabet, followed by a coding that forgets the types (Rowland et al., 2020). The iteration structure typically presents as: $$w_{5/4}=p\,\cdot\,\tau\bigl(\varphi(z)\,\varphi^2(z)\,\varphi^3(z)\cdots\bigr)$$ with explicit formulas for the morphism, seed $z$, finite prefix $p$, and coding $\tau$.

3. Power-Avoidance Analysis and Regularity

The main challenge in establishing that a uniform morphism avoids powers is to prove that the morphism preserves the required power-freeness when iteratively applied. For integer powers, this links to classic results in code theory and synchronizing morphisms. For fractional powers, a three-step automated procedure—locating property (unique position classes), bounding the exponent, and exhaustive finite checking—forms the core methodology for verification (Pudwell et al., 2015). If the avoidance holds, the resulting infinite sequences are recognizably $k$-regular or automatic in the sense of Allouche–Shallit, exhibiting self-similar and shift-regular recurrence relations. For example, the $5/4$-power-free word's sequence is 6-regular of rank 188, with one residue class satisfying a linear-modular recurrence, and all others eventually purely periodic (Rowland et al., 2020).

4. Optimality, Primitive Properties, and Limitations

The condition that $k\ge2$ for primitivity of uniform $k$-power-free morphisms is sharp; uniform 1-power-free morphisms yield no nontrivial restrictions (Wlazinski, 27 Jan 2026). For nonuniform morphisms, the situation is more restrictive: only for $k\ge5$ does power-freeness imply primitiveness in the general (possibly nonuniform) case. Uniformity, therefore, crucially sharpens the link between power-freeness and the preservation of primitivity. Examples include classical construction failures: the morphism $g(a)=x^2$ is uniform but not primitive, as it maps $a$ to a square; by contrast, uniform morphisms that are genuinely $k$-power-free for $k\ge2$ must be primitive.

A summary table of key distinctions is as follows:

Property	Uniform $k$-power-free ($k\ge2$)	Nonuniform $k$-power-free ($k\ge5$)
Implies primitivity	Yes	Yes
Applies for $k=1$	No	No
Structure of images	Pure code	Possibly impure code

5. Applications and Extensions

Uniform morphisms without powers are instrumental in constructing infinite words for both theoretical and practical applications in symbolic dynamics, coding theory, and automatic sequence generation. For finite alphabets, they provide infinite words avoiding prescribed patterns (e.g., cubefree words in binary, squarefree in ternary). For the infinite alphabet $\mathbb{N}$, they generate lex least $(a/b)$-power-free words with regular and often automatic structure. The explicit morphism constructions yield regular sequences and facilitate mechanical verification of power-freeness via finite checks and recurrence analysis (0812.4470, Pudwell et al., 2015, Rowland et al., 2020).

Extensions include pursuit of squarefree $k$-uniform morphisms over the ternary alphabet, avoidance for higher powers (quartic and beyond), and the design of morphisms with additional combinatorial or arithmetic constraints.

6. Context, Open Problems, and Significance

The intersection of uniform morphisms without powers and the theory of regular sequences, primitive codes, and avoidability illuminates several central open directions. For example, the full classification of squarefree $k$-uniform morphisms over ternary alphabets remains unresolved except for $k\ge11$ (0812.4470). The algorithmic nature of the proofs and the automated construction of such morphisms suggest scalable approaches for further families of fractional powers and larger alphabets (Pudwell et al., 2015, Rowland et al., 2020).

Uniform morphisms without powers serve as canonical bridges between combinatorics on words, automata theory, and algebraic dynamics, underscoring the deep rigidity and universal structure present in the avoidance of low-complexity subword patterns across regular and automatic infinite strings.

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References:

• "Avoiding 5/4-powers on the alphabet of nonnegative integers" (Rowland et al., 2020)
• "There are k-uniform cubefree binary morphisms for all k ≥ 0" (0812.4470)
• "Avoiding fractional powers over the natural numbers" (Pudwell et al., 2015)
• "Etude des morphismes préservant les mots primitifs" (Wlazinski, 27 Jan 2026)