Anosov Endomorphisms on the 2-Torus: Regularity of Foliations and Rigidity

The paper proves the T^2 special endomorphism equivalence: UBD for both 1D foliations ⇔ C∞ conjugacy to the linearization (Theorem A), by combining (i) UBD ⇒ equality of Lyapunov exponents with the linear model (Theorem B) and (ii) topological conjugacy plus matching periodic Lyapunov data ⇒ smooth conjugacy (Theorem C). The candidate solution follows exactly this structure: Step 1 invokes special ⇒ topological conjugacy to A (Prop. 2.6), Step 2 uses UBD to get linear rates along leaves and matching exponents, and Step 3 applies the rigidity criterion to upgrade the conjugacy to C∞; conversely, a C∞ conjugacy transports leaf-length to give UBD. The only issue is a minor mislabeling (they cite Theorem C where Theorem B is used, and vice versa), but the logical content matches the paper’s argument in substance. See Theorem A, B, C, and the UBD definition and construction details in the paper for support .