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2209.13245

A mechanism for ejecting a horseshoe from a partially hyperbolic chain recurrence class

Christian Bonatti, Katsutoshi Shinohara

correcthigh confidence
Category
Not specified
Journal tier
Specialist/Solid
Processed
Sep 28, 2025, 12:56 AM

Audit review

The paper explicitly proves that “having a partially hyperbolic filtrating Markov partition containing a chain recurrence class satisfying condition (ℓ) is a C1-viral property,” see Definition 1.3 and Theorem 1.1, together with the proof sketch using Theorems 1.3, 1.4, 1.5 and Proposition 4.2 (flexibility ⇒ (ℓ)), and the final viralness argument in Section 4.3 . By contrast, the model’s sketch omits the indispensable “flexible periodic point” hypothesis needed for the expulsion step and for rebuilding (ℓ) in the ejected class; it also asserts preservation of a robust heterodimensional cycle inside the ejected class without supplying the paper’s mechanism (Proposition 4.2) that actually creates the cycle post‑expulsion. Hence the model’s argument is incomplete/incorrect on key technical points, even though its high‑level goal matches the paper.

Referee report (LaTeX)

\textbf{Recommendation:} minor revisions

\textbf{Journal Tier:} specialist/solid

\textbf{Justification:}

The manuscript develops a robust expulsion mechanism inside partially hyperbolic filtrating Markov partitions, proving that condition (ℓ) is C1-viral and yielding new aperiodic classes. The argument is carefully layered: preparation of an affine, generating refinement; an IFS-based perturbation that preserves designated orbits/intersections; and a flexibility-driven reconstruction of (ℓ) in the expelled class. The results consolidate and extend prior work on wild dynamics and flexible points. Minor editorial clarifications would further aid readability, but the core mathematics appears correct and significant.