ANOSOV DIFFEOMORPHISMS OF OPEN SURFACES

The paper proves Theorem 1.5—existence of a Margulis leafwise system {µu} with (1) full support, (2) invariance under stable holonomies, (3) conformal covariance µu_x ∘ f^{-1} = e^{-h} µu_{f(x)}, and (4) infinite mass on rays in periodic unstable leaves—by building a countable Markov partition, constructing harmonic functions on the one–sided shift, and pushing conformal leaf-measures from the symbolic space to M, including a delicate Section 3.5 establishing global holonomy invariance and a proof that h = h_G(σ) > 0 (via finite Gurevich entropy) . The candidate solution instead proposes a direct Carathéodory measure on unstable plaques. It contains a critical error in the extension from local plaques to global leaves: it sends A forward (uses f^k(A) ⊂ W^u_ρ) when contraction along unstable requires pulling A back (use f^{-k}(A) ⊂ W^u_ρ). This breaks the claimed well-definedness and the conformal rule on full leaves. It also does not establish global holonomy invariance across rectangles, a point the paper treats as the main technical hurdle . Further, it unjustifiably asserts that the minimal covering numbers N_x(n) are independent of x. Finally, while the paper proves infinite mass on periodic rays (Lemma 3.33) , the model’s derivation relies on its flawed global extension. Therefore the paper’s argument is correct and complete for its stated hypotheses, whereas the model’s proof is incorrect/incomplete.