HOW DO TOPOLOGICAL ENTROPY AND FACTOR COMPLEXITY BEHAVE UNDER MONOID MORPHISMS AND FREE GROUP BASIS CHANGES?

Martin Lustig

correctmedium confidenceCounterexample detected

Category: math.DS
Journal tier: Specialist/Solid
Processed: Sep 28, 2025, 12:56 AM
arXiv Links: Abstract ↗PDF ↗

Audit review

Items (1)–(3) of the candidate solution match the paper’s Theorem 1.2 and its proofs: the upper bound p_Y(n) ≤ ||σ||·p_X(n) (Lemma 2.1), the same-length lower bound in the letter-to-letter recognizable case (Proposition 3.1), and the general recognizable case with a linear rescaling via the subdivision/letter-to-letter decomposition (Proposition 2.9) . However, the counterexample in item (4) is flawed: the morphism σ(a)=1, σ(b)=00 is not recognizable in the full shift under Definition 2.2 (the all-b periodic point yields two distinct phase offsets k=0 and k=1 with the same image), so it fails shift-period preservation/recognizability . The paper’s correct counterexample uses the subdivision morphism σ_II(a)=a^-a^+ and establishes precise complexity formulas (Lemma 3.2), which do yield separation at the same length under recognizability .

Referee report (LaTeX)

\textbf{Recommendation:} minor revisions

\textbf{Journal Tier:} specialist/solid

\textbf{Justification:}

This short note clarifies a subtle but important point about how subword complexity behaves under recognizable morphisms. The upper bound, the same-length lower bound for letter-to-letter recognizable maps, and the rescaled lower bound in general are presented cleanly. The counterexample showing that a same-length lower bound cannot be expected in general is correct and instructive. I recommend minor revisions to polish exposition (e.g., recall the recognizability definition right where it is used and give a brief intuitive remark on the repetition bound), but the mathematical content appears sound.