Fixed points avoiding Abelian $k$-powers

We show that the problem of whether the fixed point of a morphism avoids Abelian $k$-powers is decidable under rather general conditions

💡 Research Summary

The paper addresses the decision problem of whether the infinite fixed point of a morphism contains an Abelian k‑power. An Abelian k‑power is a concatenation X₁X₂…X_k where each consecutive block X_i is an anagram of X_{i+1}. While the classical problem of avoiding ordinary k‑powers is known to be decidable for all alphabets and all k, the Abelian version had only partial results.

The authors consider morphisms μ : Σ* → Σ* over a finite alphabet Σ={1,…,m} that satisfy two natural conditions: (1) μ(1)=1x for some non‑empty word x and |μ(a)|>1 for every a∈Σ, guaranteeing that the fixed point w=μ^ω(1) is infinite and that each iteration at least doubles the length of any factor; (2) the frequency matrix M of μ (the m×m matrix whose i‑th row is the Parikh vector of μ(i)) is nonsingular and its inverse has induced norm |M⁻¹|<1. The second condition ensures that the Parikh vectors of images do not expand too quickly and that repeated inverse transformations contract geometrically.

To capture Abelian repetitions the paper introduces the notion of a k‑template. A k‑template t consists of a (k+1)-tuple of optional boundary letters a₁,…,a_{k+1} (each either empty or a symbol from Σ) together with (k−1) integer vectors d₁,…,d_{k−1}∈ℤ^m. A word w realizes t if it contains a factor of the form a₁X₁a₂X₂…a_kX_k a_{k+1} where the Parikh difference ψ(X_{i+1})−ψ(X_i) equals d_i for each i. The special template T_k with all a_i empty and all d_i equal to the zero vector corresponds exactly to an Abelian k‑power.

The central technical tool is a parent‑child relation between templates. If a word u realizes a template t₂ and we apply μ, the resulting word μ(u) realizes a template t₁ that can be computed from t₂ via a linear equation
d_i = ψ(a’’i a{i+1}) − ψ(a’’_{i+1} a_i) + D_i·M,
where D_i are the difference vectors of t₂ and a’’_i, a_i are the boundary pieces obtained from the decomposition μ(A_i)=a’_i a_i a’’_i. Lemma 3.2 (Parent Lemma) proves this implication, and Lemma 3.3 shows that for any given t₁ the set of its parents is finite and effectively computable because the alphabet is finite and M is invertible.

Because |M⁻¹|<1, repeated inversion yields a geometric series bound on the size of the difference vectors of ancestors. Let C be the finite set of possible correction vectors c = ψ(a’’i a{i+1})−ψ(a’’{i+1} a_i). Define c* = max{c∈C}‖c‖ and r = c*/(1−|M⁻¹|). Then any ancestor of T_k has all its d‑vectors lying inside the Euclidean ball of radius r. Since only integer vectors are allowed, there are finitely many possible d‑vectors, and consequently only finitely many ancestors (Lemma 3.5).

The decision algorithm proceeds as follows.

1. Compute the full ancestor set T of the template T_k. This set is finite.
2. Collect all difference vectors appearing in any ancestor; let Δ be the maximal absolute component among them.
3. Lemma 3.6 (Inverse Parent Lemma) shows that if the fixed point μ^ω(1) contains an instance of any ancestor, then there exists such an instance whose length does not exceed
   L = N + k − 2 + (k−2)(N−2 + m·k·Δ),
   where N = max_{a∈Σ}|μ(a)|.
4. Enumerate all factors of μ^ω(1) of length ≤ L (this can be done by iterating μ a bounded number of times) and test each factor for the presence of any ancestor template.

If none of the factors contain an ancestor, the fixed point is Abelian k‑power‑free; otherwise, a violating factor is found. The algorithm runs in time polynomial in the size of the description of μ and in k, because the number of ancestors and the bound L depend only on μ and k, not on the infinite word itself.

The paper illustrates the method with a concrete example: Σ={1,2,3}, k=3, and μ defined by μ(1)=1123, μ(2)=133, μ(3)=223. The frequency matrix M is nonsingular and satisfies |M⁻¹|≈0.8589<1. Computing the ancestor set yields 1 294 templates, Δ=2, and the bound L=25. Exhaustively checking all length‑25 factors of the fixed point shows no Abelian 3‑power, reproducing Dekking’s earlier result in a fully mechanical way. The authors note that the same conditions hold for several morphisms studied by Dekking, Carpi, and Keränen, so their approach can re‑prove those results as well.

In conclusion, the paper establishes that, under the mild hypotheses of a non‑trivial expanding morphism with a contractive frequency matrix, the problem “does the fixed point avoid Abelian k‑powers?” is decidable. The key insight is the reduction of the infinite‑word property to a finite search over template ancestors, leveraging linear algebraic properties of the morphism. This contributes a robust, uniform framework for tackling Abelian repetition avoidance and opens avenues for extending the technique to more general patterns, non‑uniform morphisms, or weaker matrix conditions.