ANOSOV ACTIONS: MINIMALITY OF FOLIATIONS OR SUSPENSION ACTION

R.R. Lopes, C. Maquera, R. Varão

correctmedium confidence

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

Audit review

The uploaded paper proves the dichotomy for R^k–Anosov actions transitive by regular subcones: either all strong leaves are dense or the action is (topologically) a suspension of a Z^k–Anosov action (Theorem A) . Its proof proceeds via: (i) density of the S-saturations of strong leaves implying density of weak leaves (Proposition 3.1) ; (ii) a reduction showing it suffices to check density on compact S-orbits (Proposition 3.2) ; (iii) a torus fibration and suspension structure when a strong leaf is not dense (Propositions 3.3–3.4) ; and (iv) non-minimality implies simultaneous integrability, which is equivalent to integrability of Ess⊕Euu (Propositions 3.6–3.7) . The candidate solution follows the same architecture and key lemmas (local product structure , density of compact S-orbits , compact-orbit reduction , transverse fibration and suspension , simultaneous integrability ⇒ integrability of Ess⊕Euu ). Minor presentational differences aside, the arguments agree step-by-step; no essential logical gaps beyond standard background are introduced by the model.

Referee report (LaTeX)

\textbf{Recommendation:} minor revisions

\textbf{Journal Tier:} strong field

\textbf{Justification:}

The paper presents a clear and effective extension of the classical Anosov alternative to higher-rank actions transitive by regular subcones. The proof is technically sound, building on standard tools (local product structure, density of compact S-orbits via closing lemma, and the simultaneous integrability criterion) and culminating in a neat suspension structure when non-minimality occurs. Minor editorial issues and opportunities for expanded exposition remain, but they do not impugn correctness.