C1 PESIN (UN)STABLE MANIFOLD WITHOUT DOMINATION

Yongluo Cao, Zeya Mi, Rui Zou

correcthigh confidence

Category: Not specified
Journal tier: Strong Field
Processed: Sep 28, 2025, 12:56 AM
arXiv Links: Abstract ↗PDF ↗

Audit review

The paper proves, for C^1 diffeomorphisms with a continuous invariant splitting TM=E⊕F and a measure whose Lyapunov exponents are positive on E and negative on F, the existence (μ-a.e.) of global Pesin unstable/stable manifolds tangent to E/F, via averaged domination and hyperbolic-time blocks. The candidate solution invokes precisely this result and sketches essentially the same mechanism (average domination, Pliss/selection of hyperbolic times, contraction on admissible graphs/disks, and globalization). The only notable issue is a likely typographical swap in the introduction’s Theorem A stating tangencies “to E and F respectively,” which contradicts the sign convention; the interior of the paper clearly aligns with unstable tangent to E and stable tangent to F. Otherwise, arguments align and are correct.

Referee report (LaTeX)

\textbf{Recommendation:} minor revisions

\textbf{Journal Tier:} strong field

\textbf{Justification:}

The manuscript removes a long-standing regularity/dominance barrier for the existence of Pesin (un)stable manifolds in the C\^1 setting under a continuous invariant splitting and hyperbolic measures. The averaged-domination approach is clear, effective, and well integrated with hyperbolic-time selections and compactness arguments. Aside from a minor wording inconsistency in the statement of Theorem A in the introduction (swapped tangency), the paper is well crafted and correct. The contribution is significant for smooth ergodic theory and nonuniform hyperbolicity.