2506.14027
Nowhere dense competing holes in open dynamical systems
F. Ciavattini, T.H. Steele
wrongmedium confidenceCounterexample detected
- Category
- math.DS
- Journal tier
- Specialist/Solid
- Processed
- Sep 28, 2025, 12:56 AM
- arXiv Links
- Abstract ↗PDF ↗
Audit review
The paper’s central nowhere-dense claim (Theorem 19) is false as stated: it implicitly uses “pi is not periodic,” yet this hypothesis is neither stated nor ensured. The identity homeomorphism on a compact perfect metric space yields a concrete counterexample where each T(i) = {p_i} and T is empty, contradicting complete indecisiveness, while the paper’s somewhere-dense negative result (Lemma 20) is sound and dynamics-free. See the theorem statement and its proof’s hidden nonperiodicity step, and the decisive-point definitions used to formalize T(i) and T .
Referee report (LaTeX)
\textbf{Recommendation:} major revisions
\textbf{Journal Tier:} specialist/solid
\textbf{Justification:}
The paper’s negative result for somewhere-dense closures is clean and correct, but the main positive result for nowhere-dense closures is stated without a necessary dynamical/nonperiodicity assumption that the proof tacitly uses. The identity-homeomorphism counterexample invalidates the theorem as written. With hypotheses explicitly added (e.g., topological transitivity and exclusion of periodic centers), the Baire-category argument can likely be repaired, preserving the interest of the contribution.