ANOSOV ACTIONS: CLASSIFICATION AND THE ZIMMER PROGRAM

The paper’s Theorem 2.2 classifies totally Anosov C∞ actions of higher-rank semisimple G as smoothly conjugate to bi-homogeneous actions, via: (i) upgrading measurable structures to Hölder along coarse Lyapunov foliations using invariance principle and accessibility, (ii) classifying the induced A≅R^k action by bi-homogeneous models (Theorems 2.6/2.7), and (iii) extending from A to all of G using Zeghib-type centralizer arguments and root group generation to obtain a translation action on H/Γ . The candidate solution omits the needed accessibility/Oseledets-conformal steps before applying the R^k classification, and its Step 2 asserts incorrectly that every g∈G normalizes A, then deduces a homomorphism q: G→H purely from normalization of A—this contradicts the paper’s actual use of centralizer/affine-extension techniques to pass from A to G . It also misplaces the role of the invariance principle in the smoothing stage, which in the paper is handled via leafwise isometries/normal forms and then global regularity (Theorem 12.13) .