2011.05641
ON C0-GENERICITY OF DISTRIBUTIONAL CHAOS
Noriaki Kawaguchi
correctmedium confidence
- Category
- Not specified
- Journal tier
- Strong Field
- Processed
- Sep 28, 2025, 12:55 AM
- arXiv Links
- Abstract ↗PDF ↗
Audit review
The paper proves that strong distributional chaos (DC1*) is C0-generic for continuous maps on compact smooth manifolds and, in dimension >1, also generic for homeomorphisms; it does so by reducing zero-dimensional shadowing dynamics to inverse limits of SFTs under an MLC condition, constructing residual scrambled relations on appropriate components, and upgrading them to a single Mycielski set, then invoking known genericity facts to conclude residuality on manifolds. The candidate solution mirrors this strategy: Good–Meddaugh reduction to SFTs, residual DC1*-tuples via specification/markers, a Mycielski upgrade, and generic shadowing/entropy inputs to finish. Minor mismatches exist (e.g., it states generic chain-recurrence instead of the paper’s use of zero-dimensional chain-recurrent sets, and it cites a different source for shadowing genericity in H(M)), but these do not affect the main conclusion.
Referee report (LaTeX)
\textbf{Recommendation:} minor revisions
\textbf{Journal Tier:} strong field
\textbf{Justification:}
The paper establishes a sharp C0-generic DC1* result on manifolds by leveraging the Good–Meddaugh shadowing–SFT correspondence and a careful analysis of chain components under an MLC hypothesis. The core zero-dimensional lemma and its upgrade to a single Mycielski set across all n are technically solid and conceptually illuminating, answering prior questions in the literature. A few bibliographic clarifications (especially on residuality inputs in C(M)) would improve clarity, but the mathematical content appears correct and significant.