THE SYMMETRIES OF AFFINE K-SYSTEMS AND A PROGRAM FOR CENTRALIZER RIGIDITY

DANIJELA DAMJANOVIĆ, AMIE WILKINSON, CHENGYANG WU, DISHENG XU

correctmedium confidence

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

Audit review

The candidate solution reproduces, in condensed form, the main theorems and implications of the paper: ergodic ⇒ Lie centralizer; weakly mixing ⇒ smooth centralizer is affine; non‑K systems can be approximated by affine maps with non‑Lie smooth centralizer; and K(X) equals the interior of both A(X) and L(X). These are precisely Theorem A, Theorem B, and Theorem C in the paper, together with the cited classical equivalence between stable ergodicity in Aff(X) and the K-property. No substantive logical divergences or missing hypotheses were found.

Referee report (LaTeX)

\textbf{Recommendation:} minor revisions

\textbf{Journal Tier:} strong field

\textbf{Justification:}

The paper makes substantial progress on centralizer rigidity for affine diffeomorphisms, tying the Lie structure of smooth centralizers directly to ergodic-theoretic properties and identifying K-systems as the precise interior for two natural centralizer classes. The blend of spectral, cocycle, and Lie-theoretic arguments is compelling and appears sound. Some expository enhancements would improve accessibility and self-containment, but the core results are strong and of broad interest in smooth dynamics and homogeneous dynamics.