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k-abelian complexities under morphic images of Pisot substitutions

Generalize the automaticity of abelian complexity to k-abelian complexity by proving that if x is the fixed point of a Pisot substitution τ and y = σ(x) for a (possibly erasing) substitution σ, then for every k ≥ 1 the k-abelian complexity of y is automatic; additionally, develop an effective method to compute the length-k sliding-block code of the composition σ∘τ.

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Background

The paper establishes that the abelian complexity of fixed points of ultimately Pisot substitutions is automatic (in appropriate numeration systems) and discusses applications to Parikh-collinear morphisms. This opens the question of stability of automaticity for higher-order abelian equivalences under morphic images.

The authors specifically ask whether analogous results hold for all k and how to effectively compute the sliding-block code of compositions involving possibly erasing morphisms.

References

Let ${x}$ be a fixed point of a substitution $\tau$. Consider a substitution $\sigma \colon A \to A*$ that might be erasing and let ${y} = \sigma ({x})$. If $\tau$ is Pisot, then both ${x}$ and ${y}$ have automatic abelian complexities. Can we generalize this result to all $k$-abelian complexities? How do we compute the length-$k$ sliding-block code of the composition $\sigma \circ \tau$?

— Effective Computation of Generalized Abelian Complexity for Pisot Type Substitutive Sequences (2504.13584 - Couvreur et al., 18 Apr 2025) in Section 7 (Open problems and questions)