Lyapunov stable chain recurrence classes for singular flows

Shaobo Gan, Jiagang Yang, Rusong Zheng

correctmedium confidence

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

Audit review

The paper explicitly proves Theorem A: for a C1-generic vector field away from homoclinic tangencies, any nontrivial Lyapunov stable chain recurrence class is a homoclinic class (hence contains a periodic orbit) and presents a detailed reduction to Propositions 6.1 and 6.2, with a measure-theoretic and dominated-splitting framework handling the singular case . By contrast, the model’s Step 1 incorrectly elevates “accumulation by periodic orbits” to “contains a periodic orbit” for general chain-transitive sets; the generic fact is only that every nontrivial compact chain-transitive set is the limit of periodic orbits (not that it contains one) . The model also ignores singularities, which are the core difficulty here; the paper’s argument develops significant machinery precisely to show periodic orbits must exist in Lyapunov stable singular classes and that such classes equal homoclinic classes .

Referee report (LaTeX)

\textbf{Recommendation:} no revision

\textbf{Journal Tier:} strong field

\textbf{Justification:}

The manuscript resolves a notable question about Lyapunov stable chain recurrence classes away from tangencies, unifying the nonsingular and singular cases via a carefully architected proof. The reduction to a main proposition, the use of dominated splittings near Lorenz-like singularities, and the measure-theoretic construction with C2 approximations convincingly produce periodic orbits in the class and identify it as a homoclinic class. The exposition is substantial but well organized, with precise statements and clear dependence on known results.