SRB MEASURES FOR C∞ SURFACE DIFFEOMORPHISMS

The paper’s Main Theorem (countably many ergodic SRB measures covering the positive Lyapunov set and describing level sets) is explicitly stated and proved for C^\infty surface diffeomorphisms via an entropic construction using hyperbolic times, F\"olner sequences, entropy estimates, and absolute continuity of stable foliations. The manuscript also stresses that the C^\infty hypothesis is essential and formulates only a conjecture (not a theorem) for finite smoothness with the R(f)/r threshold. The candidate solution hinges on a nonexistent “Burguet’s C^r theorem” and builds the C^\infty result by taking r\to\infty, which contradicts the paper’s own statement that the C^r analog is conjectural (and that the C^\infty theorem is false in finite smoothness without an R(f)/r threshold). The model’s auxiliary appeal to Sarig’s symbolic coding and equilibrium states is not used in the paper’s proof and does not repair the central gap.