Linear factor complexity of cyclic-shift-morphism words

Establish that, for the infinite word W_{A,\sigma,a_0} generated by the cyclic shift morphism \psi_{A,\sigma}(a)=a\sigma(a) over an alphabet A of size N\geq 3 with \sigma a cyclic permutation, the factor complexity p(k) is eventually p(k)=(N-1)k+C_N for some constant C_N depending on N.

Background

The paper defines the cyclic shift morphism \psi_{A,\sigma}:A^*\to A^* by \psi_{A,\sigma}(a)=a\sigma(a) and the associated infinite word W_{A,\sigma,a_0}=\lim_{n\to\infty}\psi_{A,\sigma}^n(a_0). For alphabets of size N\geq 3 and cyclic permutations \sigma, the paper studies the avoidance of strongly (3,\delta)-repetitions in these words, where \delta is a power of \sigma.

The unresolved issue concerns the factor complexity p(k), which counts the number of distinct factors of length k occurring in W_{A,\sigma,a_0}. The conjectured eventual formula predicts linear growth with slope N-1, analogous to the low complexity observed in structured families such as Sturmian and Arnoux–Rauzy words. The paper notes that proving this would require a detailed analysis of the language generated by the uniform morphism, potentially using return words, bispecial factors, or S-adic methods.