Lipschitz sub-actions for locally maximal hyperbolic sets of a C1 flow

Xifeng Su, Philippe Thieullen

correctmedium confidence

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

Audit review

The paper proves Theorem 1.3 by introducing a continuous positive Livšic criterion and a continuous Lax–Oleinik semigroup to build a Lipschitz integrated subaction, followed by a local regularization that yields a Lipschitz u with Lipschitz Lie derivative and LVu ≤ φ−φ̄_Λ on a neighborhood of Λ (see the statement and construction around Theorem 1.3 and Theorems 2.3 and 3.3) . The candidate solution incorrectly relies on Lopes–Thieullen on the suspension to obtain a Lipschitz subaction and assumes bi-Lipschitz coding; the paper explicitly notes that such prior approaches produce at best Hölder regularity and hinge on non-Lipschitz stable manifolds .

Referee report (LaTeX)

\textbf{Recommendation:} minor revisions

\textbf{Journal Tier:} strong field

\textbf{Justification:}

The paper presents a clear and well-motivated path to a Lipschitz positive Livšic theorem for locally maximal hyperbolic sets of C1 flows, improving classical Hölder outcomes and avoiding reliance on non-Lipschitz stable manifolds. The decomposition into a positive Livšic criterion, a continuous Lax–Oleinik semigroup yielding a Lipschitz integrated subaction, and a local regularization to enforce a pointwise inequality and Lipschitz L\_V u is technically clean and conceptually illuminating. The main results are significant for ergodic optimization and Livšic-type cohomology for flows; a few expository clarifications (constants and dependencies, a compact summary of the regularization step) would further improve readability.