Global rigidity of smooth Z⋉λR-actions on T^2

The paper proves that any C^r (r ≥ 2) locally free Z ⋉_λ R-action on T^2 with f ∘ φ_t = φ_{λ t} ∘ f is C^{r−ε}-conjugate to an affine model generated by a hyperbolic toral automorphism A with unstable eigenvalue λ and a constant-speed linear flow along its unstable direction (Theorem A; see abstract and introduction ). The argument establishes: (i) uniform expansion along the φ_t-orbit foliation and its Denjoy/minimal/quasi-isometric structure (Lemma 2.1–2.2 ), (ii) hyperbolicity of the induced map on homology and a Franks semiconjugacy upgraded to a homeomorphism that maps F^u onto the linear unstable foliation L^u (Proposition 3.1 and Theorem 3.1 ), (iii) a dominated splitting TT^2 = E^{cs} ⊕ E^u and dynamic coherence (Proposition 4.1; cone criterion and coherence on T^2 ), (iv) Anosov property with Lyapunov exponents ±log λ on periodic points, leading to C^{r−ε} conjugacy to A via de la Llave’s periodic-data rigidity (Proposition 4.2 and its claims ), and (v) constant-speed linearity of the pushed-forward flow ψ_t = h φ_t h^{-1} along L^u (final step of the proof of Theorem A ). By contrast, the model’s solution hinges on an incorrect early claim that f is expansive simply from the relation f_* X = λ X and a local skew-product normal form; this is false in general (e.g., the paper constructs a C^{1+α} example not topologically conjugate to any affine model, hence not expansive on T^2; Example 1.1 ). The model also invokes Livšic/LMM before establishing that f is Anosov and improperly treats the topological conjugacy h_0 as differentiable along orbits. These gaps undermine the model’s proof, even though its final classification statement matches the paper.